advances in boundary element techniques xi

ECltd

Proceedings of the 11th International Conference Berlin, Germany 12-14 July 2010

Advances in B

oundary Elem

ent Techniques XI

Advances in Boundary Element Techniques XI

Edited by Ch Zhang MH Aliabadi M Schanz

ISBN 978-0-9547783-7-8 Publish by EC Ltd, United Kingdom

Advances In Boundary Element Techniques XI

Advances In Boundary Element Techniques XI Edited by Ch Zhang M H Aliabadi M Schanz Published by EC, Ltd, UK Copyright © 2010, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

ECltd

All Rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, except under terms of the Copyright, Designs and Patents Act 1988. Requests to the Publishers should be addressed to the Permission Department, EC, Ltd Publications, P.O.Box 425, Eastleigh, Hampshire, SO53 4YQ, England. ISBN: 978-0-9547783-7-8 The text of the papers in this book was set individually by the authors or under their supervision. No responsibility is assumed by the editors or the publishers for any injury and/or damage to person or property as a matter of product liability, negligence or other wise, or from any used or operation of any method, instructions or ideas contained in the material herein.

International Conference on Boundary Element Techniques XI 12-14 July 2010, Berlin, Germany

Organising Committee: Prof. Dr.-Ing. Chuanzeng Zhang, University of Siegen, Germany [email protected] Prof. Ferri M.H. Aliabadi Department of Aeronautics Imperial College London E-mail: [email protected] Prof. Martin Schanz Graz University of Technology Graz, Austria [email protected]

International Scientific Advisory Committee Abascal R (Spain) Abe K (Japan) Albuquerque EL (Brazil) Baiz P (UK) Baker G (USA) Beskos D (Greece) Blasquez A (Spain) Bonnet M (France) Chen JT (Taiwan) Chen Weiqiu (China) Chen Wen (China) Cheng A (USA) Cisilino A (Argentina) Davies A (UK) Denda M (USA) Dong C (China) Dumont N (Brazil) Estorff Ov (Germany) Gao XW (China) Garcia-Sanchez F (Spain) Gaul L (Germany) Gatmiri B (France) Gray L (USA) Gospodinov G (Bulgaria) Gumerov N (USA) Han X (China) Harbrecht H (Germany) Hartmann F (Germany) Hematiyan MR (Iran) Hirose S (Japan) Kinnas S (USA) Kuna M (Germany)

Langer S (Germany) Liu,G-R (Singapore) Mallardo V (Italy) Mansur WJ (Brazil) Mantic V (Spain) Marburg S (Germany) Marin L (Romania)) Matsumoto T (Japan) Mattheij RMM (The Netherlands) Mesquita E (Brazil) Millazo A (Italy) Minutolo V (Italy) Mohamad Ibrahim MN (Malaysia) Nishimura N (Japan) Niu Z (China) Ochiai Y (Japan) Pan E (USA) Panzeca T (Italy) Phan AV (USA) Partridge P (Brazil) Perez Gavilan JJ (Mexico) Pineda E (Mexico) Prochazka P (Czech Republic) Qin T (China) Qin Q (Australia) Rjasanow S (Germany) Saez A (Spain) Salvadori A (Italy) Sändig,A-M (Germany) Sapountzakis EJ (Greece) Sarler B (Slovenia) Schneider R (Germany) Sellier A (France) Seok Soon Lee (Korea) Shiah Y (Taiwan) Sladek J (Slovakia) Sollero P (Brazil) Stephan EP (Germany) Taigbenu A (South Africa) Tan CL (Canada) Tao W (China) Telles JCF (Brazil) Venturini WS (Brazil) Wang Y (China) Wen PH (UK) Wendland W (Germany) Wrobel LC (UK) Yao Z (China) Ye W (Hong Kong) Zhao MH (China)

PREFACE

The Conferences in Boundary Element Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques in both boundary element methods (BEM) and boundary integral equation methods (BIEM). Previous successful conferences devoted to Boundary Element Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008) and Athens, Greece (2009).

The present volume is a collection of edited papers that were accepted

for presentation at the Boundary Element Techniques Conference held at the Maritim Hotel Berlin, Germany, during 12th-14th July 2010. Research papers received from 18 counties formed the basis for the Technical Program. The themes considered for the technical program included solid mechanics, fluid mechanics, potential theory, composite materials, fracture mechanics, damage mechanics, contact and wear, optimization, heat transfer, dynamics and vibrations, acoustics and geomechanics.

A symposium “Recent Advances in Theory and Application of BEM”

was organized at the conference in honor of Professor Zhenhan Yao (Tsinghua University, Beijing. PR China), who is working on BEM for many years and has made many significant contributions to the Computational Mechanics especially to BEM. We would like thanks the organizers of the symposium (Prof. Ch. Zhang, Prof. C.Y.Dong and Prof. Y.H.Liu) for their effort.

The conference organizers would also like to express their appreciation to the International Scientific Advisory Board for their assistance in supporting and promoting the objectives of the meeting and for their assistance in the form of reviews of the submitted papers. Editors July 2010

Contents Study of contact stress evolution on fretting problems using a 3D boundary elements formulation L Rodriguez-Tembleque, R Abascal

1

Shape optimization with topological derivative and its application to noise barrier for railway viaducts K Abe, T Fujiu and K Koro

7

On the transient response of actively repaired damaged structures by the boundary element method A Alaimo, G Davì, A Milazzo

13

Computation of moments in thin plates of composite materials under dynamic load using the boundary element method K R Sousa, A P Santana, E L Albuquerque, and P Sollero

20

Drilling rotations in BEM P Baiz

26

Blob regularization of boundary integrals G Baker, H Zhang

32

On the accuracy of the fast hierarchical DBEM for the analysis of static and dynamic elastic crack problems I Benedetti, A Alaimo, M H Aliabadi

38

A boundary knot method for three-dimensional harmonic viscoelastic problems B Sensale, A Canelas

46

Non-Incremental boundary element discretization of non-linear heat equation based on the use of the proper generalized decompositions G Bonithon, P Joyot, F Chinesta and P. Villon

54

Three-dimensional boundary elements for the analysis of anisotropic solids F C Buroni, J E Ortiz, A Sáez

62

Sensitivity analysis of cracked structures with static and dynamic Green’s functions O Carl, Ch Zhang

69

A D-BEM approach with constant time weighting function applied to the solution of the scalar wave equation J A M Carrer and W J Mansur

77

A novel boundary meshless method for radiation and scattering problems Z Fu, W Chen

83

Anti-plane shear Green’s function for an isotropic elastic layer on a substrate with a material surface W. Q. Chen and Ch Zhang

91

Stress intensity factor formulas for a rectangular interfacial crack in three-dimensional bimaterials C-H Xu, T-Y Qin, Ch Zhang, N-A Noda

97

Iterative optimization methodology for sound scattering using the topological derivative approach and the boundary element method, A Sisamon, S C Beck, A P Cisilino, S Langer

104

A Laplace transform boundary element solution for the Cahn-Hilliard equation A J Davies and D Crann

110

Strategy for writing general scalable parallel boundary-element codes F C de Araújo, E F d'Azevedo, and L J Gray

118

Incomplete LU preconditioning of BEM systems of equations based upon the generic substructuring algorithm F C de Araújo, E F d'Azevedo, and L J Gray

124

Hypersingular BEM analysis of semipermeable cracks in magnetoelectroelastic solids R Rojas-Dıaz, M Denda, F Garcıa-Sanchez, A Saez

130

Boundary element analysis of cracked transversely isotropic and inhomogeneous materials C Y Dong, X Yang and E Pan

136

A family of 2D and 3D hybrid finite elements for strain gradient elasticity N A Dumont, D H Mosqueira

144

Transient thermoelastic crack analysis in functionally graded materials by a BDEM A Ekhlakov, O Khay, Ch Zhang

154

Time-Domain boundary element analysis of semicircular hill on viscoelastic media under vertically incident SV wave A Eslami Haghighat, S A Anvar, M Jahanandish, A Ghahramani

162

HEDD-FS method for numerical analysis of cracks in 2D finite smart materials C-Y Fan, G-Tao Xu and M-Hao Zhao

168

Recent developments of radial integration boundary element method in solving nonlinear and nonhomogeneous multi-size problems X W Gao, M Cui and Ch Zhang

174

A meshless boundary interpolation technique for solving the Stokes equations C Gáspá

184

A boundary element formulation based on the convolution quadrature method for the quasi-static behaviour analysis of the unsaturated soils P Maghoul, B Gatmiri, D Duhamel

190

Elastodynamic laminate element method for lengthy structures E V Glushkov, N V Glushkova and A A Eremin

196

Three-dimensional eigenstrain formulation of boundary integral equation method for spheroidal particle-reinforced materials H Ma, Q-H Qin

202

Green’s functions, boundary elements and finite elements F Hartmann

208

Crack identification in magneto-electro-elastic materials using neural networks and boundary element method G Hattori and A Saez

215

The singular nodal integration method for evaluation of domain integrals in the BEM M R Hematiyan, A Khosravifard, M Mohammadi

221

Application of convolution quadrature method to electromagnetic acoustic wave analysis S Hirose, Y Temma and T Saitoh

227

Boundary integral equations for unsymmetric laminated Composites C Hwu

231

BEM analysis of dynamic effects of microcracks and inclusions on a main crack J Lei, Ch Zhang, Q Yang, Y-S Wang

237

Nonlinear transient thermo-mechanical analysis of functionally graded materials by an improved meshless radial point interpolation method A Khosravifard, M R Hematiyan

245

Adaptive-hybrid meshfree method Leevan Ling

252

Analysis of acoustic wave propagation in a two-dimensional sonic crystal based on the boundary element method F Li, Y-S Wang, Ch Zhang

258

Analysis of two intersecting three-dimensional cracks L N Zhang, T Qin, Ch Zhang

266

Reconstruction of elasticity fields in isotropic materials via a relaxation of the alternating procedure L Marin and B T Johansson

272

Dual reciprocity boundary element formulation applied to the non-linear Darcian diffusive-advective problems C F Loeffler, F P Neves, P C Oliveira

280

Analysis of the dynamic response of deep foundations with inclined piles by a BEM-FEM model L A Padrón, J J Aznárez, O Maeso, A Santana

286

Fast Multipole Boundary Element Method (FMBEM) for acoustic scattering in coupled fluid-fluidlike problems V Mallardo, C Alessandri, M H Aliabadi

292

Galerkin projection for the potential gradient recovery on the boundary in 2D BEM V Mantic-Lugo, L J Gray, V Mantic, E Graciani, F Parıs

298

Shape sensitivity analysis of 3-D acoustic problems based on direct differentiation of hypersingular boundary integral formulation C J Zheng, T Matsumoto, T Takahashi and H B Chen

306

BEM and the Stoke system with a slip boundary condition D Medkova

312

The BEM on general purpose graphics processing units (GPGPU): a study on three distinct implementations J Labaki, E Mesquita, L O Saraiva Ferreira

316

Dynamic analysis of damaged magnetoelectroelastic laminated structures A Alaimo, A Milazzo, C Orlando

324

Seismic behaviour of structures on elastic footing, BEM-FEM analysis. S Ciaramella, V Minutolo, E Ruocco

330

Boundary element analysis of uncoupled transient thermo-elasticity involving non-uniform heat sources M Mohammadi, M R Hematiyan, L Marin

334

Three-dimensional thermo-elastoplastic analysis by triple-reciprocity boundary element method, Y Ochiai

340

Elastoplastic analysis for active macro-zones via multidomain symmetric BEM T Panzeca, E Parlavecchio, S Terravecchia, L Zito

346

Interaction problems between in-plane and out-plane loaded plates by SBEM. T Panzeca, F Cucco, A La Mantia, M Salerno

353

Genetic algorithm with boundary elements for simultaneous solution of minimum solution of minimum weight and shape optimization problems Li Chong Lee, Bacelar de Castro, P W Partridge

359

New boundary integral equations for evaluating the static and dynamic T-stresses, A.-V. Phan

365

The boundary element method applied to visco-plastic analysis E Pineda, M H Aliabadi, J Zapata

373

Optimal shape of fibers in composites with various ratios of phase stiffnesses P P Prochazka

381

Extended stress intensity factors of a three-dimensional crack in electromagnetothermoelastic solid T Y Qin, X J Li, L N Zhang

387

Adaptive cross approximation and its applications R Grzhibovskis and S Rjasanow

392

Nonlinear analysis of shear deformable beam-columns partially supported on tensionless Winkler foundation E J Sapountzakis and A E Kampitsis

398

Solution of hot shape rolling by the local radial basis function collocation method B Šarler, Siraj-ul-Islam, U Hanoglu

406

Regularization for a poroelastodynamic collocation BEM M Messner, M Schanz

412

A Fast BEM for the dynamic analysis of plates with bonded piezoelectric patches I Benedetti, Z S Khodaei, M H Aliabadi

418

On the displacement derivatives of the three-dimensional Green’s function for generally anisotropic bodies Y C Shiah, C L Tan, W X Sun, Y H Chen

426

Coupled thermoelastic analysis for interface crack problems J Sladek, V Sladek, P Stanak

433

Local integral equations combined with mesh free implementations and time stepping techniques for diffusion problems V Sladek, J Sladek, Ch Zhang

441

Computation of moments in thin plates of composite materials under dynamic load using the boundary element method K R Sousaa, A P Santanaa, E L Albuquerqueb, P Sollero

449

Meshless boundary element methods for exterior problems on spheroids E P Stephan, A Costea ,Q T Le Gia, T Tran

455

3-D Green element method for potential flows E Nyirenda, A Taigbenu

462

BEM Fracture Mechanics Analysis of 3D Generally Anisotropic Solids C L Tan, Y C Shiah, J R Armitage, W C Hsia

468

A BEM analysis of the fibre size effect on the debond growth along the fibre-matrix interface

474

L Tavara, V Mantic, E Graciani, F Parıs Nonlinear nonuniform torsional vibrations of shear deformable bars application to torsional postbuckling configurations E J Sapountzakis and V.J Tsipiras

482

Harmonic analysis of spatial assembled plate structures coupled with acoustic fluids using the boundary element method J Useche, E L Albuquerque, S Shoefel

490

On the numerical analysis of damage phenomena in saturated porous media E T Lima Junior, W S Venturini, A Benallal

498

Efficient solution of acoustic radiation problems by boundary elements and interpolated transfer functions O von Estorff, O Zalesk

508

A fast solver for boundary element elastostatic analysis J O Watson

514

Stress analysis of cracked structures considering crack surface contact by the boundary element method W Weber, K Willner, P Steinmann, G Kuhn

520

Fatigue crack growth in functional graded materials by meshless Method P H Wen, M H Aliabadi

526

An analysis of elastic plates under concentrated loads by non-singular boundary integral equations K-C Wu, Z-M Chang

534

A time-domain BEM for dynamic crack analysis in piezoelectric solids using non-linear crack-face boundary conditions M Wünsche, Ch Zhang, F García-Sánchez, A Sáez

541

Fast bEM for 3-D elastodynamics based on pFFT acceleration Technique Z Yan, J Zhang, W Ye

549

A new time domain boundary integral equation of elastodynamics Z H Yao

555

Regularization of the divergent integrals in boundary integral equations V.V. Zozulya

561

On Levi Functions W L Wendland

569

Domain integrals in a boundary element algorithm S Nintcheu Fata, L J Gray

570

Meshfree micro-scale modelling and stress analysis of 3D orthogonal woven composites L Li, P H Wen, M H Aliabadi

571

Study of contact stress evolution on fretting problems using a 3DBoundary Elements formulation

L. Rodrıguez-Tembleque1, R. Abascal2∗

Departamento de Mecanica de los Medios Continuos, Escuela Tecnica Superior de Ingenieros,Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN

[email protected], [email protected]

Keywords: Fretting Wear, Contact Mechanics, Boundary Elements Method.

Abstract. A Boundary Elements Method wear formulation is applied to simulate fretting wear ona cylinder-on-flat configuration for gross sliding and partial slip conditions. The present formulationapplies the Boundary Elements Method to approximate the solids elastic response, and an AugmentedLagrangian formulation to solve the contact problem. Wear on contact surfaces is computed usingthe Archard wear law. The numerical methodology is based on previous works [1, 2], and the worksof Stromberg [3], and Sfantos and Aliabadi [4, 5]. The evolution of the solids contact geometries,the contact pressure and the solids stresses can be predicted using this numerical tool. Also thisformulation allows to investigate the evolution of subsurface stress fields due to material removal byfretting wear.

Introduction

A Boundary Elements Method fretting wear formulation is applied to simulate contact tractionsevolution on a cylinder-on-flat configuration for gross sliding and partial slip conditions. The BoundaryElements Method is applied to approximate the solids elastic response, and an Augmented Lagrangianformulation to solve the contact problem. Wear on contact surfaces is computed using the Archardwear law. The numerical methodology is based on previous works [1, 2], and the works of Stromberg[3], and Sfantos and Aliabadi [4, 5].

The evolution of the solids contact geometries, the contact pressure and the solids stresses canbe predicted using this numerical tool. Also this formulation allows to investigate the evolution ofsubsurface stress fields due to material removal by fretting wear.

In the literature, some recent works have studied this kind of problem using a finite elements 2Dmodel: McColl et al.[6], Ding et al. [7] and Madge et al. [8]. This work uses the boundary elementsmethod, which allow to have analogous results a a very good accuracy using a very low number ofelements, compared with the finite elements 2D models, even using a general 3D fretting wear model.

Contact equations

The gap variable is defined for the pair I ≡ P 1, P 2 of points (Pα ∈ Ωα, α = 1, 2) at all times (τ), as[9]:

g = BT (X2 − X1) + BT (u2o − u1

o) + BT (u2 − u1) (1)

being BT (X2 − X1) the geometric gap between two solids in the reference configuration (gg), andBT (u2

o − u1o) the gap originated due to the rigid body movements (go). Therefore, the gap of the I

pair remains as follows:g = ggo + BT (u2 − u1) (2)

where ggo = gg + go. In this work, the reference configuration for each solid (Xα) that will beconsidered is the initial configuration (before applying load). Consequently, gg may also be termedinitial geometric gap. In the expression (2) two components can be identified: the normal gap,gn = ggo,n + u2

n − u1n, and the tangential gap or slip, gt = ggo,t +u2

t −u1t , being uα

n and uαt the normal

and tangential components of the displacements

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz 1

The unilateral contact condition and the law of friction defined for any pair I ≡ P 1, P 2 ∈ Γc

(Γc: Contact Zone) of points in contact can be compiled as follows, according to their contact status:

Contact-Adhesion: tn ≤ 0 ; gn = 0 ; gt = 0

Contact-Slip:

⎧⎨⎩

tn ≤ 0 ; gn = 0

‖tt‖ = µ |tn| ; gt · tt = −‖gt‖ ‖tt‖

No contact : tn = 0 ; gn ≥ 0 ; tt = 0

(3)

In the expression above gn is the pair I normal gap, and tn is the normal contact traction defined as:tn = BT

n t1 = −BTnt2, where tα is the traction of point P α ∈ Γα

c expressed in the global system ofreference, and Bn = [n] is the first column in the change of base matrix: B = [Bt|Bn] = [t1|t2|n]. Thenormal tractions acting upon the pair I points are of the same value and opposite signs, in accordancewith Newton’s third law.

Wear equations

The Holm-Archard’s wear law allow to compute the total volume of solid particles worn (W ) byadhesion wear, as

W = kadFn

HDs (4)

where Fn is the contact normal load, H is the surface hardness, Ds is the sliding distance, and kad

is the nondimensional wear coefficient, which represents the probability of forming a substantial wearparticle.

Expression (4) can be written locally as

gw = kw tnDs (5)

being gw the wear depth, tn the normal contact pressure, and kw = kad/H the dimensional wearcoefficient. The total volume worn (W ) can be computed integrating the state variable, gw, on thecontact zone:

W =∫

Γc

gw dΓ (6)

Wear process evolves over time, so equation (5) can be expressed in a differential for

gw = kw tnDs (7)

where Ds is the tangential slip velocity module: Ds = ‖gt‖.Considering wear on the contact surfaces, governed by the Holm-Archard’s law, the normal contact

gap (gn) is rewritten asgn = ggo,n + (u2

n − u1n) + gw (8)

for an instant τi. For quasi-static contact problems, wear depth defined on instant τi, is computed as

gw = gw(τk−1) + kw tn‖∆gt‖ (9)

being tn and ∆gt the normal contact pressure and the sliding distance (∆gt = gt(τk) − gt(τk−1)),respectively, calculated on the same instant, and gw(τk−1) the internal variable value on instant τk−1.

To obtain wear on each solid surface from the wear depth computed gw, we can apply two criteria:If the wear dimensional coefficient of each surface, kα

w (α = 1, 2), is known, wear depth on eachsurface is:

g1w =

gw

1 + (k2w/k1

w); g2

w =gw

1 + (k1w/k2

w)(10)

2 Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz

Boundary integral discrete equations

The boundary integral equations for a body Ω, can be written in a matrix form as:

Hu − Gp = F (11)

where the vector u represents the nodal displacements, and F contains the applied boundary condi-tions. These equations are well known and can be found in many books like [10] or [11].

For the two bodies Ωα (α = 1, 2) in contact, the Equations (11) have to be regrouped as:

Aαxxα + Aα

ppαc = Fα (12)

where xα is the nodal unknowns vector which contains the unknowns outside the potential contactzone (xα

p ), and the nodal displacements vector on the potential contact zone (uαc ). pα

c is the nodalcontact tractions vector, Aα

x are the columns of Hα and Gα matrices, and Aαp are the columns of Gα

matrix corresponding to the contact nodes.The equations (12), according to [9], can be rewritten for two solids contact, in the following way:

R1x1 + R2x2 + RλΛ + Rgk − F = 0, (13)

being,

R1 =

⎡⎣ A1

0(C1)T

⎤⎦ R2 =

⎡⎣ 0

A2

−(C2)T

⎤⎦ Rλ =

⎡⎣ A1

pC1

−A2pC2

0

⎤⎦

Rg =

⎡⎣ 0

0Cg

⎤⎦ F =

⎡⎣ b1

b2

Cgkgo

⎤⎦ +

⎡⎣ 0

0Cgnw

(k)

⎤⎦

(14)

Cα (α = 1, 2) is a boolean matrix which allows to extract the contact nodes displacements from xα,and Cg is the identity matrix (Cg = I).

Wear discrete equations

The discrete form of kinematic equation (8) for I pair, on instant k, is:

(k(k))I = (k(k)go )I + (d2(k))I − (d1(k))I + (Cgnw

(k))I (15)

where matrix Cgn is constituted of the Cg columns which affect the normal gap of contact pairs, andw(k) is a vector which contains the contact pairs wear depth.

According to the Holm-Archard’s law (7), wear is caused by the tangential slip ratio or the tan-gential slip velocity. In case of a contact problem, the discrete for of Expression (9) can be expressedfor I pair as

(w(k))I = (w(k−1))I + (∆w(k))I

(∆w(k))I = kw(Λ(k)n )I‖(kk

t )I − (k(k−1)t )I‖

(16)

where Λ(k)n is a vector which contains the normal traction components of contact pairs at instant k.

Contact discrete equations

The contact restrictions for every I pair, at instant k can be expressed as:

(Λ∗n

(k))I − PR−( (Λ∗n

(k))I) = 0 ; (Λ∗t(k))I − PCg( (Λ∗

t(k))I) = 0 (17)

The augmented contact variables are defines as: Λ∗n

(k) = Λ(k)n + rnk

(k)n and Λ∗

t(k) = Λ(k)

t − rt(k(k)t −

k(k−1)t ). The value of g for the tangential projection region, on I pair, is: g = µ|PR−( (Λ∗

n(k))I)| or

g = µ|PR−( (Λ(k)n )I)|.


(a) (b)

Figure 1: (a) Short cylinder over a flat. (b) Normal an tangential.

Results

The short cylinder-against-flat 3D fretting wear problem, see Fig.1(a), is considered in this work. Thecylinder of radii R = 6 mm and thickness e = 1 mm is subjected to fixed normal force per unit length,F , with superimposed cyclic tangential displacement, δ, as it is presented schematically in Fig.1(b),and collected on Table 1. The flat specimen geometrical dimensions are: e = 1 mm and L = 6 mm.Both solids have the same material properties, presented on Table 2.

Case 1: Gross slip fretting problem

Fig.2(a) shows the contact surface profiles versus number of fretting wear cycles for gross slip (Case 1).Note that horizontal and vertical positions refer, respectively, to the y and z coordinates. As fretting-wear proceeds, the contact surface profiles are modified. The contact tends towards conforming.Fig.2(b) shows the BE predicted contact pressure distribution versus the number of fretting cycles.

Case 2: Partial slip fretting problem

Using the same wear coefficient as in Case 1, the predicted development of the contact profiles forthe partial slip conditions of Case 2 are illustrated in Fig.3. There is no wear predicted in the stickregion due to the absence of slip. Slight wear occurs in the slip regions, resulting in an increasedgap for the initial configuration on these regions (Fig.3(a)). Fig.3(b) shows the BE predicted contactpressure distribution versus the number of fretting cycles. It can be observed that the initial pressuredistribution is consistent with the Hertz solution. However, after 10000 wear cycles, the normalpressure in the stick zone increases significantly, particulary at the stick-slip boundaries where sharppeaks develop. In contrast, in the slip zones is reduced to negligible values, due to the increased gapcaused by wear.

Case Normal load (F [N/mm]) Peak-to-Peak strokeamplitude (δ)

1 120 ±102 120 ±2.5

Table 1: Fretting cases parameters.


Material Nitrided CrMoVhigh strength steel

Young’s nodulus (Ec, Ef [N/mm2]) 200 · 103

Poisson’s ratio (ν) 0.3Friction coefficient (µ) 0.6Wear coefficient (kc

w = kfw [mm2/N ]) 1.0 · 10−7

Table 2: Material properties.

(a) (b)

Figure 2: (a) Surface profiles versus the number of fretting severe wear cycles. (b) Contact pressure.

(a) (b)

Figure 3: (a) Surface profiles versus the number of fretting slight wear cycles. (b) Contact pressure.


Summary and conclusions

This work applies the BEM wear methodology developed by authors [1, 2], and based on Stromberg [3],and Sfantos and Aliabadi [4, 5], for wear simulation wear on 3D contact problems to simulate frettingwear on a cylinder-on-flat configuration for gross sliding and partial slip conditions. The comparisonwith the previous models such as [6] and [7] has a very good agreement.

The evolution of the solids contact geometries and the contact pressures are predicted using thisnumerical tool. For the gross slip regime, the high wear leads to the contact edges moving rapidlyoutwards, leaving the material in a permanently compressive state, which prohibits fretting-wearinitiation. For the partial slip regime, wear increases the maximum contact pressure and shift itslocation to the boundaries between the stick-slip zones.

Acknowledgments

This work was co-funded by the DGICYT of Ministerio de Ciencia y Tecnologıa, Spain, researchproject DPI2006-04598, and by the Consegerıa de Innovacion Ciencia y Empresa de la Junta deAndalucıa, Spain, research projects P05-TEP-00882 and P08-TEP-03804.

References

[1] L. Rodrıguez-Tembleque, R. Abascal, and Aliabadi M.H. A boundary element formulation for 3dwear simulation in rolling-contact problems. In Abascal R. and Aliabadi M.H., editors, Advancesin Boundary Elements Techniques IX. EC, Ltd., UK, 2008.

[2] L. Rodrıguez-Tembleque, R. Abascal, and Aliabadi M.H. Wear prediction in tribometers using a3d boundary elements formulation. In Sapountzakis E.J. and Aliabadi M.H., editors, Advancesin Boundary Elements Techniques IX. EC, Ltd., UK, 2009.

[3] N. Stromberg. An augmented lagragian method for fretting problems. Eur. J. Mech. A/Solids,16(4):573–593, 1997.

[4] G.K. Sfantos and M.H. Aliabadi. Wear simulation using an incremental sliding boundary elementmethod. Wear, 260(9-10):1119–1128, 2006.

[5] G.K. Sfantos and M.H. Aliabadi. A boundary element formulation for three-dimensional slidingwear simulation. Wear, 5-6(262):672–683, 2007.

[6] I.R. McColl, J. Ding, and S.B. Leen. Finite element simulation and experimental validation offretting wear. Wear, 256:1114–1127, 2004.

[7] J. Ding, S.B. Leen, and I.R. McColl. The effect of slip regime on fretting wear-induced stressevolution. International Journal of Fatigue, 26:521–531, 2004.

[8] J.J. Madge, S.B. Leen, I.R. McColl, and P.H. Shipway. Contact-evolution based prediction offretting fatigue life: Effect of clip amplitude. Wear, 262:1159–1170, 2007.

[9] L. Rodrıguez-Tembleque and R. Abascal. A 3d fem-bem rolling contact formulation for unstruc-tured meshes. Int. J. Solids Struct., 47:330–353, 2010.

[10] C.A. Brebbia and J. Domınguez. Boundary Elements: An Introductory Course (second edition).Computational Mechanics Publications, 1992.

[11] M.H. Aliabadi. The Boundary Element Method Vol 2: Application in Solids and Structures.Wiley, London, 2002.


Shape Optimization with Topological Derivativeand Its Application to Noise Barrier for Railway Viaducts

Kazuhisa Abe1,a, Takasuke Fujiu2 and Kazuhiro Koro3,b

1 Department of Civil Engineering and Architecture, Niigata University8050 Igarashi 2-Nocho, Nishi-ku, Niigata 950-2181, JAPAN,

2 Graduate School of Science and Technology, Niigata University8050 Igarashi 2-Nocho, Nishi-ku, Niigata 950-2181, JAPAN

3 Department of Civil Engineering and Architecture, Niigata University8050 Igarashi 2-Nocho, Nishi-ku, Niigata 950-2181, JAPAN

[email protected], [email protected]

Keywords: BEM. Shape optimization. Topological derivative. Noise barrier.

Abstract. A shape optimization method is developed for sound insulating walls and applied to design of noisebarrier for railway viaducts. The optimization is achieved by the BE-based topology optimization method whichhas been proposed by the authors. To cope with local minima, the topological change which will improve theperformance is realized by the nucleation of small scatterers located around the main wall. The location andthe number of scatterers are determined based on the topological derivative. This value is formulated within theframework of the boundary element analysis. After the nucleation the shape is tuned by the shape optimizationprocess. The developed method is applied to the design of noise barrier installed in a railway viaduct. Throughnumerical results, capabilities of the method are demonstrated.

IntroductionNoise in urban area originating from vehicles is of serious issues which should be coped with. In general, toinsulate the traffic noise, highways and railways are equipped with walls. Since the capability of such a structurestrongly depends on its profile, it is worth to explore effective shapes. Therefore, the shape optimization methodcan be a practical tool for this task.

Abe et al. [1] have proposed the BE-based shape optimization method for sound scattering problems inwhich the coordinates of all boundary element junctions are used as design variables. This method enables theshape to be released from any restriction. While the complexity can thus be attained in the boundary shape, thereis a possibility of branch breaking during the optimization process. To cope with this anomaly, the topologicalchange was allowed in the context of shape optimization analysis. This was realized by the aid of the level setmethod [2].

Within the framework of the gradient-based shape optimization methods, the improvement of shape is to beachieved only in the neighborhood of the current form. Therefore, the shape may converge to a local optimal.Even if the above optimization method allows any topological change, still the obtained optimal shapes dependon the initial shapes.

In this paper, in order to cope with this problem, an optimization method is developed with the aid of thetopological derivative. Based on its value, small scattering bodies are allocated in the design domain so thatthe objective function will be decreased. The topological derivatives are given by the sensitivity to the growthof an infinitesimal void in accordance with the definition by Novotny et al. [3]. The BE-based formulationgiven in Ref.[4] for elastostatic analysis is applied to sound scattering problems. Distribution of the topologicalderivative enables us to determine the required number and allocation of small obstacles introduced in thecurrent shape. Therefore, it will be possible to relieve the optimal shape from local minima.

The developed method is applied to the design of noise barrier installed in a railway viaduct. Throughnumerical results, capabilities of the method are demonstrated.

Design Sensitivity Analysis with Boundary Element Method

Boundary Element Equation. Let us consider a section of a railway viaduct placed in a half-plane soundscattering field as illustrated in Fig.1. The noises are originating from the wheels and the pantograph. The


...

design domainsource pointsobservation points

20.0

7.01.5

y(m)

x(m)(0,0)

3.5

R.L.

5.0

40.02.0

0.5

0.50.5

Fig.1 Outline of the problem

noise barrier is to be arranged in the design domain which is set above the wall. Several observation points areallocated at a height of 0.5m from the ground.

The boundary element equation for this problem is given by

[H]P = P∗, (1)

where [H] is the coefficient matrix calculated with the fundamental solution for a half-plane, P is a vector ofnodal sound pressure and P∗ is a vector given by the point noises.

Notice that the Neumann condition is assumed on the boundary. Since in this study the Green’s functionof Helmholtz equation in a half-plane is used for the boundary integral equation, the element discretizationis needed only on the viaduct, the car body and the noise barriers. Moreover, in order to avoid the fictitiouseigenfrequencies, the Burton-Miller formulation [5] is employed.Design Sensitivity Analysis. The shape optimization problem for noise barrier is defined by

minXb

J(P; Xb) := F(P; Xb) + [λ]T HP − P∗ + λ+(V − Vmax),

subject to [λ]T HP − P∗ = 0 for ∀λ,λ+(V − Vmax) = 0, λ+ ≥ 0,

(2)

where J is the objective function, F is a cost function estimating the sound pressure. The second term of theright-hand side stands for the condition associated with the boundary element equation, and λ is a vector ofLagrange multiplier. The third term is the volume restriction. V is the volume (area) of the barrier, Vmax is anallowable limit and λ+ is a Lagrange multiplier. In eq(2) the all co-ordinates of element junctions Xb on thebarrier are used as the design variables.

The variation of J due to the geometrical change ∆Xb is given by

∆J =[∂F∂P

]T ∆P + [∂F∂Xb

]T ∆Xb

+ [λ]T

[∂H∂Xb· ∆Xb]P + [H]∆P − ∂P

∗

∂Xb· ∆Xb

+ λ+[

∂V∂Xb

]T ∆Xb,(3)

where ∆P is the change of pressure resulting from the shape change ∆Xb.As mentioned above, in this study every element junction on the barrier is used to represent the boundary

shape. In order to save the computational cost, the following adjoint equation is introduced,

[H]λ = −∂F∂P. (4)


contour−line segmentgrid point

boundary element

element junction

Fig.2 Boundary element discretization with level set function

Substituting the solution λ of eq(4) into eq(3), we can obtain the following expression

∆J = [Re(β) + λ+∂V∂Xb

]T ∆Xb, (5)

where β is a vector given by

β = ∂F∂Xb + [λ]T ∂H

∂XbP − ∂P

∗

∂Xb. (6)

The velocity which leads to an optimal shape is given by

V = −Re(β) + λ+∂V∂Xb. (7)

In this paper the nodal velocity V is directed to the outward normal at each boundary element junctionlocating on the barrier.

Shape Updating Process with Level Set Method

In this paper, to cope with the topological change which may happen during the shape optimization process,the shape is updated based on the Eulerian frame. The topological change is captured with the aid of the levelset method [2]. The level set function ψ is assigned at each fixed grid point (Fig.2). The boundary is implicitlydefined by the zero contour of the level set function. Once the contour line is drawn on the background grid,the element junctions are equidistantly distributed along the contour. The boundary element discretization isthen accomplished by connecting the element junctions with each other.

Shape change results from the advection of the level set function governed by the Hamilton-Jacobi equation,

∂ψ

∂t= −v · ∇ψ (8)

where v is the convective vector defined on the grid points. It is evaluated from the nodal velocity V definedon the boundary element junctions[6].

Topological Derivative for Sound Scattering Problems

Formulation of Topological Derivative. The topological derivative is defined by the rate of cost functionresulting from the growth of an infinitesimal hole [3]. To derive this within the framework of the BE analysis,let us consider a sound scattering field with a small obstacle of radius a centered at x. In this case the objectivefunction is modified as

J(P, P; a) := F + [λ]T HP − P∗ + P, (9)


where P is a vector given by the integration on the boundary of scattering body Γa, i.e.,

pi =

∫Γa

p(y)∂G(xi, y)∂ny

dΓy, (10)

here p is the sound pressure, G is the fundamental solution and ny stands for the outward normal. pi is evaluatedat a collocation point xi allocated on the boundary Γ except on Γa.

We assume that the topological derivative DT can be regularized for the objective function by

DT (x) = lima→ 0∆a→ 0

∆J∆Va, (11)

where Va is the volume of the circular obstacle, therefore, ∆Va = 2πa∆a.In this paper we consider a cost function given by

F =N∑i|pzi|, (12)

where pzi is sound pressure at the ith observation point zi and N is the number of observation points. pzi isevaluated by the integral representation,

pzi = −∫Γ

p(y)∂G∗(zi, y)∂ny

dΓy −∫Γa

p(y)∂G∗(zi, y)∂ny

dΓy + p∗zi, (13)

where G∗ is the half-plane fundamental solution with weak singularity.Variation ∆J due to the increment of radius ∆a is given by

∆J = Re

⎛⎜⎜⎜⎜⎜⎜⎝[N∑i

pzi

|pzi| ·∂pzi

∂P]T ∆P + (

N∑i

pzi

|pzi| ·∂pzi

∂a)∆a + [λ]T H∆P + ∆P

⎞⎟⎟⎟⎟⎟⎟⎠ , (14)

where ( ) denotes conjugate, and ∆P, ∆P are variations resulting from the increment ∆a. In the following,for the sake of brevity, Re( ) is omitted.

In order to eliminate the terms concerning ∆P in eq(14), the following adjoint equation is introduced,

[H]T λ = −N∑i

pzi

|pzi| ·∂pzi

∂P. (15)

Substituting the solution of eq(15) into eq(14), from eq(11) the topological derivative can be expressed by

DT (x) =N∑i

pzi

|pzi| pzi + [λ]T ˙P,

pzi := lima→0

12πa∂pzi

∂a, ˙P := lim

a→0

12πa

∂P∂a

.

(16)

Derivation of pzi, ˙P. From eq(16), pzi can be obtained by calculating ∂pzi/∂a. Since pzi is given by eq(13),we can evaluate ∂pzi/∂a from

∂pzi

∂a= − ∂∂a

(∫Γa

p∂G∗

∂nydΓy

). (17)

Let us consider a small scattering body of radius a embedded in waves propagating with wavenumber k inan infinite field. By letting a→ 0, the sound pressure on Γa can be given as,

p(y) = ps(x) + 2(y − x) · ∇ps y on Γa (18)


R.L.

600

3400

2890

1300

500 100

1000

718

150 300

3700

3500

2000

1500 5001000

200

600

[mm]

design domain

Fig.3 Car body, viaduct and initial shape of noisebarrier

0 200 400

50

60

optimization step

soun

d pr

essu

re le

vel[

dB]

Fig.4 Time history of sound pressure level

where ps is the sound pressure observed before the nucleation.Substituting eq(18) into eq(17), we can evaluate explicitly ∂pzi/∂a with the aid of far field approximation,

and then obtain pzi from eq(16) as

pzi = − ik2

4ps(x)H(1)

0 (krs) + H(1)0 (kr′s) −

ik2H(1)

1 (krs)∂ps

∂s+ H(1)

1 (kr′s)∂p′s∂s′, (19)

where H(m)n is mth Hankel function of order n, ∂ps/∂s is the directional derivative of ps in the direction of

rs = x− zi, rs = |rs|, and ( )′ stands for a quantity concerning the mirror image point with respect to the groundsurface.

We can derive ˙P in the similar manner.

Numerical Example

Analytical Conditions. As an example, a viaduct for the Sinkansen railway is considered. Outline of theanalysis region is illustrated in Fig.1. Detail around the design domain of the noise barrier is shown in Fig.3.The sources of noise originating from rolling wheels and pantograph of a train running at a speed of 200km/hare set to 9.5Pa and 2.0Pa [7], respectively, with a frequency at 500Hz. Five observation points are placed atdistances of 20, 25, 30, 35 and 40 (m) from the center of the track. A rectangular design domain of 2×1.5 (m)is set at the top of wall. Notice that only the upper part of the wall is optimized. The fixed grid of 0.02m sizeis embedded in the design domain for the level set analysis. The boundary of noise barrier is discretized withconstant elements of about 0.02m length. The volume of the barrier is allowed up to twice of the initial volumeV0, i.e., Vmax = 2V0.

Optimal Size of Scattering Body. A new scatterer will be nucleated at a portion having a negative DT .Therefore, the cost function will decrease with growing the hole to a certain extent. However excessive sizemay produce reverse effect. It implies the existence of optimal size for the nucleated hole. Based on numericalexperiments, we approximate the cost function F in terms of the volume of hole Va as

F(Va) ≈N∑i|Fi0 + Fi0Va +

12

Fi0V2a | (20)

where Fi0 is the sound pressure at zi, Fi0 and Fi0 are topological derivatives of first and second orders. Thesevalues are evaluated under the limit a→ 0.

Based on eq(20), the optimal size which will minimize the cost function F is determined.


9.25

9.50

8.00

8.25

8.50

8.75

9.00

2.40 4.402.65 2.90 3.15 3.40 3.65 3.90 4.15

9.25

9.50

8.00

8.25

8.50

8.75

9.00

2.40 4.402.65 2.90 3.15 3.40 3.65 3.90 4.15

9.25

9.50

8.00

8.25

8.50

8.75

9.00

2.40 4.402.65 2.90 3.15 3.40 3.65 3.90 4.15

9.25

9.50

8.00

8.25

8.50

8.75

9.00

2.40 4.402.65 2.90 3.15 3.40 3.65 3.90 4.15

0th step 85th step

225th step 500th step

(m)

Fig.5 Shape optimization of the noise barrier

Numerical Results. A small scattering body is located at a position where the minimum topological derivativetakes place. Time history of the sound pressure is shown in Fig.4. The nucleation is performed when thereduction of noise resulting from the shape optimization is relaxed. In this example new holes were introducedat the 85th and 225th steps. It can be found that the nucleation activates the optimization again, and noisereduction of about 15dB was achieved. After the nucleation the shape optimization with the level set method isresumed. The lower limit of the radius is determined by the resolution of the background grid. The nucleationis stopped when the optimal radius estimated from eq(20) is smaller than the grid size. Under this criterion,in this case two holes were created. Fig.5 shows the shapes at the 0th, 85th, 225th and 500th steps. From thefigure we can see that the second scatterer is smaller than the first one.

Conclusion

A shape optimization method has been developed for sound insulating wall installed in railway viaducts. Inorder to cope with the local minima, the nucleation of scattering bodies was attempted by the aid of the topo-logical derivative. This strategy enables us to discover an optimum topology. After the nucleation the profileis updated within the framework of the shape optimization. New holes are located based on the topologicalderivative. The optimal size of the nucleated obstacle is determined by eq(20). Through numerical example,the capabilities of the proposed method were proved.

References[1] K.Abe, S.Kazama and K.Koro Advances in Bound Elem Tech 2007, 8, 379-384 (2007).[2] G.Allaire, F.Jouve and A-M.Toader J Comput Phys, 194, 363-393 (2004).[3] A.A.Novotny, R.A.Feijoo, E.Taroco and C.Padra Comput Methods Appl Mech Engrg, 192, 803-829 (2003).[4] K.Abe, T.Fujiu and K.Koro Advances in Bound Elem Tech 2008, 9, 235-240 (2008).[5] AJ.Burton and GF.Miller Proc Roy Soc Lond A, 323, 201-210 (1971).[6] K.Abe, S.Kazama and K.Koro Commun Numer Meth Engng, 23, 405-416 (2007).[7] K.Nagakura QR of RTRI, 37, No.4, 210-215 (1996).


On the Transient Response of Actively Repaired Damaged Structures by the Boundary Element Method

A. Alaimo1, G. Davì2, A. Milazzo3

1 University of Palermo, Dipartimento di Ingegneria Strutturale Aerospaziale e Geotecnica,Viale delle Scienze Edificio 8, 90128 Palermo Italy, [email protected]



Keywords: Active repair, Piezoelectric patch, Fracture mechanics, Boundary Element Method, Transient analysis.

Abstract. The transient fracture mechanics behavior of damaged structures repaired through active piezoelectric patches is presented in this paper. The analyses have been performed through a boundary element code implemented in the framework of piezoelectricity to take account of the coupling between the elastic and the electric fields, which represents the peculiar feature of piezoelectric media. The multi-domain technique has been also involved to assemble the host structures and the active patches and to model the cracks. Moreover, the patches have been considered elastically bonded to the damaged structure by means of a zero thickness adhesive layer. This has been achieved through the implementation of an interface spring model which has allowed, coupled with an iterative procedure, to prevent overlapping at the interface between the host structure and the active patch as well as between the crack surfaces. The Dual Reciprocity Method (DRM) has been used in the present time dependent application for the approximation of the domain inertia terms. Numerical analyses have been carried out in order to characterize the dynamic repairing mechanism of the assembled structure by means of the computation of the dynamic Stress Intensity Factors.

Introduction. The unique feature of piezoelectric media to couple the elastic and the electric fields gives rise to the opportunity of employing this kind of material in the field of “Active Repair technology”. In fact, the actuating capability of piezoelectric materials, achieved through the converse piezoelectric effect [1], can be used to avoid the failure of a damaged structure by reducing the crack opening displacements [2, 3]. It follows that piezoelectric patches can effectively be used to repair flawed structures by replacing the most commonly used repair methods involving bonded or riveted metallic or composite patches, which act in a passive manner [4, 5]. The design of active patches, that can be arranged by bonding or embedding piezoelectric layers into the host damaged structures, needs the development of analysis tools in order to understand their repairing mechanisms and the overall fracture mechanics behavior of the repaired structures. Several analytical and numerical strategies have been developed to study the static electromechanical response of active repairs. Analytical models for cracked beam actively repaired through piezoelectric actuators have been developed by Wang et al. [6, 7]. The main idea proposed in the aforementioned works is to reduce the singularity at the crack tip by inducing, via the active repairs, a local moment. Finite element procedures have been developed by Duan et al. [8] and Liu et al. [3, 9] to analyze the active patch behavior applied on isotropic host damaged structures, while the boundary element method has been proposed by Alaimo et al. for the analysis of the piezoelectric patch activity in the repair of both isotropic and composite damaged structures [2, 10]. Alaimo et al. [11] have also analyzed the effect of the


Coulomb’s frictional contact on the fracture mechanics behavior of actively repaired delaminated composite structures. To the authors’ knowledge, although the static behavior of the active piezoelectric patch has been widely characterized by both numerical and analytical formulations, few works on the transient response of piezoelectric active patches have been presented yet. Among this Wang et al. [12] have proposed an analytical model to study the repair of cracked beam with piezoelectric patches under the effect of dynamical loadings. A repair criterion, based on the restoring of the natural frequency of the healthy beam through a suitable external voltage applied on the piezoelectric actuator, has been adopted in the aforementioned paper. A methodology for the optimal design of the voltage to be applied on the piezoelectric patch for the repair of vibrating delaminated beam has been also proposed by Wu et al. [13]. In the present work the boundary element method is used to model the dynamic fracture mechanics behavior of isotropic damaged structure actively repaired through piezoelectric patches. The BI formulation is developed by using the elastostatics fundamental solutions for two-dimensional anisotropic media, properly reformulated for the piezoelectric problem in terms of generalized variables. The inertia terms are considered as body forces and the Dual Reciprocity BEM is used to compute the mass matrix. The multi-domain technique [14], provided with an interface spring model, allows the assembling between the host structures and the patches as well as the modeling of the adhesive layer at the interface between contiguous domains. The transient behavior of the repaired structures is characterized in terms of the dynamic Stress Intensity Factors (dSIFs) directly computed from the crack tip opening displacements [14]. Numerical analyses are performed on a cracked isotropic beam under two different dynamic repairing voltage applied across the single layered piezoelectric repair.

Basic equations. The boundary integral procedure is formulated for the piezoelectricity problem under the assumption of plane strain conditions. According to Barnett and Lothe's generalized formalism for piezoelectricity [15], it is possible to write the governing equations of the piezoelectric problem as generalized Navier-like equation and, by applying the Betti's reciprocity theorem with the static electro-elastic fundamental solutions, the Somigliana identity for the electromechanical problem is obtained in terms of generalized variables [16],

* * * *0P d d

c U T U U T U F (1)

In Eq. (1) the generalized body force vector F is given by the inertial force components only, since the electric field is considered to be as quasi static. Then the generalized body force vector writes

F = U (2)

where denotes the 4 by 4 inertia matrix obtained from the scalar matrix by replacing the last diagonal term with zero.The piezoelectric dynamic problem is then solved numerically by means of the Boundary Element Method [17], in such a way the following equations of motion are obtained

M H = GP (3)

In Eq. (3), and P are the vectors of the generalized displacements and boundary tractions nodal values, respectively while H and G are influence matrices computed by integrating the kernel fundamental solutions weighted by linear shape functions, employed to express the generalized displacements and tractions on the boundary. The mass matrix M is instead approximated through the Dual Reciprocity Method [18,19], by transforming the domain inertia integral, represented by the right hand side of Eq. (1), into boundary integral.


The Houbolt method [20] is used to proceed the time integration of the equation of motion, Eq. (3). It

allows to approximate the acceleration at the instant t t as

22

1 2 5 4t t t t t t t t tt

= (4)

By substituting Eq. (4) into the equations of motion, Eq.(3), the following system of algebraic equations is obtained

H = GP h (5)

where and P are representative of the displacements and tractions at the instant t t , the term h take into account inertial effects related to the displacement history and is defined as

22 5 4t t t t tt Mh = (6)

while the influence matrix H depends on the integration time step t as

2

2t

H = H M (7)

Host Structure/Active Patch assembling and crack modeling strategy. The assembling between the host structure and the piezoelectric patch as well as the modeling of the crack is achieved through the multi-domain technique [16]. The equations of motion for each of the N homogeneous sub-region are then written as

1, 2..., k k k k k k k N M H = G P (8)

and the global system of equation pertaining the overall assembled structure is then obtained by applying the compatibility and equilibrium conditions along all the sub-region interfaces

; 1,..., 1; 1,...,ij ij ij ij

i j i j i N j i N P P (9)

In Eq. (9), the subscript ij indicates quantities associated with the nodes belonging to the interface

between the i-th and j-th sub-regions. The crack is modeled by providing the multi-domain technique with an interface spring model. By so doing, a zero thickness elastic layer, having vanishing stiffness, is considered between the crack surfaces and by means of an iterative procedure, deeply discussed in Alaimo et al. [2], the inconsistence of the overlap is also avoided. The spring model also allows the modeling of the adhesive layer among the host structure and the active repair. Since the elastic interface conditions [2] represent an uncoupled behavior between interlayer tractions and displacements jumps components, characterized by the compliance constants kN and kT, the modeling of the bonding layer through an equivalent zero-thickness elastic interface may be achieved by linking the mechanical properties of the adhesive to the compliance interface constants. For the interested reader the aforementioned procedure is fully described in [2]. Finally, the dynamic stress intensity factors, allowing the fracture mechanics characterization of the repaired structure, are directly computed from the crack opening displacements [14].


Numerical Results. The results obtained for a cantilever isotropic beam with a vertical crack actively repaired through a piezoelectric patch, shown in Fig. 1, are discussed in the following. The cracked beam is characterized by having length L = 0.08 m, height H = 0.02 m and a crack length a = 0.01 m. The beam is clamped on the left side while the crack is centered with respect to the span of the host structure. The material of the host damaged beam is PMMA plastics, having Young modulus E = 3.3 GPa, Poisson ratio = 0.35 and mass density = 1200 kg/m3. The active patch, made up by a single layer of PZT-4 piezoelectric ceramic whose material properties are those used by Liu [3], is bonded on the bottom side of the beam and across the crack, as depicted in Fig.1. The geometry of the patch is characterized by the following dimensions, LP = H / 0.75 and h = 0.25 H. Firstly, the natural frequencies of both the un-cracked and cracked beam are computed and the results compared with those obtained by the finite element computations performed through Comsol Multiphysics®.

Figure 1: Active repair configuration obtained with a single layered patch.

As shown in Table 1, the results obtained from the two different numerical analyses are in good agreement with the only exception for the fourth natural frequency which evidences a maximum percentage discrepancy of -12.9 % and -12.6 % for the healthy and cracked beam respectively. It is worth nothing that the presence of the crack leads to a reduction of the natural frequencies of the patched beam due to the less stiffness introduced by the damage.

Mode Healthy beam Cracked beam

fBEM [Hz] fFEM [Hz] % difference fBEM [Hz] fFEM [Hz] % difference

1 749 760 -1.47 % 727 735 -1.1 % 2 3953 3922 0.78 % 3212 3067 4.5 % 3 4631 4552 1.7 % 4400 4287 2.6 % 4 7503 8476 -12.9 % 7363 8295 -12.6 %

Table 1: Natural frequencies for both the healthy and the cracked beam.

The characterization of the free vibration behavior of the repaired beam has been followed by the analysis of its transient response for two different electric loads, applied on the piezoelectric patch in order to reduce the crack opening displacements corresponding to the static deformed configuration under the static transverse load F, see Fig. 1. Both the dynamical analyses are then performed by considering as initial

crack

P

x1

x2

L

V

F

H

h


conditions the static solution of the cracked repaired beam loaded by a transverse concentrated load F = 100 N/m, as shown in Fig. 1.

0 0.005 0.01 0.015 0.02 0.025Time [s]

0.15

0.2

0.25

0.3

0.35

0.4

KI/K

IU

0

0.2

0.4

0.6

0.8

1

V(t)

/V0

KI(t)/KIU

Applied Voltage

Figure 2: Dynamic Mode I SIF corresponding to quasi-static repairing voltage.

As previously highlighted by Liu [3] and by Alaimo et al. [2], for this particular mechanical load case, mode I is predominant. For this reason, the dynamic fracture mechanics behavior and consequently the repairing mechanism induced by the actuated patch is characterized in terms of KI only. The first electric load considered refers to a quasi-static voltage, see Fig. 2, whose time dependence is expressed as

t

eVtV 1)( 0 (10)

where V0 is set to 2000 V while the time constant is 8 ms. In Fig. 2 the dynamic KI/KIU behavior, being KIU the stress intensity factor characterizing the cracked beam without repair, is shown. It can be observed that, by increasing the voltage, the KI/KIU approaches quasi-statically its minimum value 0.173, which is very close to the KI/KIU value obtained through the static analysis corresponding to the repairing voltage VR = 1290 V, see Alaimo et al. [2]. Once reached its minimum value, the SIF turns to increase tending to KI/KIU = 0.2, corresponding to its static value at V=V0. The second analysis deals with the same repaired configuration under an Heaviside electrical loads having amplitude V = 1290 V, corresponding to the static repairing voltage VR [2]. Fig. 3 shows the time history of the dimensionless mode I stress intensity factor for both perfect and adhesive interface conditions between the host structure and the piezoelectric repair. The last condition is obtained by setting kN = 1.56*105 m/GPa and kT = 9.33*105 m/GPa in order to model an adhesive layer having Young modulus E = 3 GPa, Poisson ratio = 0.4 and thickness t = 0.1 mm. The results obtained point out that, after the transient behavior, the SIFs, characterizing both the perfect and imperfect bonding conditions, oscillates at the first natural frequency of the cracked structure. Moreover, the effect of the adhesive is to increase the amplitude of the dynamic SIF, while the effect of the inertia force is to drive the SIFs behind the limiting value, KI/KIU = 0.178 [2], characterizing the static repairing behavior of the piezoelectric patch.


0 0.002 0.004 0.006 0.008Time [s]

0.1

0.2

0.3

0.4

KI/K

IU

PerfectAdhesiveStatic SIF at V=VR

Figure 3: Dynamic Mode I SIF corresponding to the Heaviside electric loads.

Conclusion. The transient dynamic response of an actively repaired isotropic cracked beam has been analyzed in this paper. A boundary element code, implemented in the framework of piezoelectricity, has been used for the analyses and the multi-domain technique has been involved to interface the host damaged structure with the piezoelectric repair. The bonding layer has been modeled through the implementation of an interface spring model while the mass matrix has been approximated by means of the Dual Reciprocity Method. Numerical analyses have been performed on a cantilever cracked beam repaired with a single layered PZT-4 patch. The repairing mechanisms of the assembled structure under two different repairing voltage have been described in terms of the mode I dynamic stress intensity factor.

References

[1] I. Chopra AIAA Journal, 40 (10), 2145-87 (2002).

[2] A. Alaimo, A. Milazzo, C. Orlando Engineering Fracture Mechanics, 76, 500-511 (2009).

[3] T.J.C. Liu Theoretcial and Applied Fracture Mechanics, 47, 120-132 (2007).

[4] A.A. Baker, R. Jones Bonded Repair of Aircraft Structure, Dordrecht: Martinus Nijhoff (1988).

[5] C.H. Chue, L.A. Lin, S.C. Wang Engineering Fracture Mechanics, 48, 91-101 (1994).

[6] Q. Wang, S.T. Quek Smart Materials and Structures, 13, 1222-1229 (2004).

[7] Q. Wang, S.T. Quek, K.M. Liew Smart Materials and Structures, 11, 404-410 (2002).

[8] W.H. Duan, S.T. Quek, Q. Wang Smart Materials and Structures, 17, 0150017 (2008).

[9] T.J.C. Liu Engineering Fracture Mechanics, 75, 2566-2574 (2008).

[10] A. Alaimo, A. Milazzo, C. Orlando ICCES, 11 (1), 9-16 (2009).

[11] A. Alaimo, G. Davì, C. Orlando, Advances in Boundary Element Technique IX, 489-495, (2009).


[12] Q. Wang, W.H.Duan, S.T. Quek, International Journal of Mechanical Sciences, 46, 1517-1533 (2004).

[13] N. Wu, Q. Wang, Smart Materials and Structures, 19, 8pp. (2010).

[14] M.H. Aliabadi The boundary element method. Application in solids and structures. Vol.2, Chirchester: Wiley (2002).

[15] D.M. Barnett, J. Lothe, Physics State Solid (b), 67, 105-111 (1975).

[16] G. Davì, A. Milazzo, International Journal of solids and Structures, 38, 7065-7078 (2001).

[17] E.L. Albuquerque, P. Sollero, M.H. Aliabadi, International Journal of solids and Structures, 39, 1405-1422 (2002).

[18] P.W. Partridge, C.A. Brebbia, L.C. Wrobel, The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, and Elsevier, London, 1992.

[19] G. Dziatkiewich, P. Fedelinski, CMES, 17(1), 35-46 (2007).

[20] J.C. Houbolt, J. Aeronaut. Sci., 17, 540-550 (1950).


Computation of Moments in thin Plates of Composite Materials underDynamic Load using the Boundary Element Method

K. R. Sousaa, A. P. Santanaa, E. L. Albuquerqueb, and P. Solleroc

aFederal Institute of MaranhaoDepartment of Mechanical and Materials

65025-000, Sao Luis, MA, Brazilkerlles,[email protected]

bUniversity of Brazilia - UNBFaculty of Technology

70910-900, Brasilia, Bsb, [email protected]

cUniversity of Campinas - UNICAMPFaculty of Mechanical Engineering13083-970, Campinas, SP, Brazil

[email protected]

Keywords: Boundary element method, radial integration method, dual reciprocity boundary elementmethod, plates, composite materials, dynamic of plate, and stress analyses.

Abstract. This work presents a dynamic formulation of the boundary element method for moments ofanisotropic thin plates. The elastostatic fundamental solution for anisotropic thin plates is used and inertiaterms are treated as body forces. Domain integrals that come from body forces are transformed into bound-ary integrals using the radial integration method (RIM). In this method, the inertia term is approximatedas a sum of approximation functions times coefficients to be determined. In this work, the augmented thinplate spline is used as the approximation function. The time integration is carried out using the Houboltmethod. Only the boundary is discretized. Numerical results show good agreement with results available inliterature.

Introduction. Nowadays, BEM is a well-established numerical technique to deal with an enormous numberof engineering complex problems. Analysis of plate bending problems using the BEM has attracted theattention of many researchers during the past years, proving to be a particularly adequate field of applicationsfor that technique.

In recent years, the boundary element formulation for plate bending has included the analysis of anisotropicproblems. Shi and Bezine [10] presented a boundary element analysis of plate bending problems using fun-damental solutions proposed by [16] based on Kirchhoff plate bending assumptions. Rajamohan and Raa-machandran [6] proposed a formulation where singularities were avoided by placing source points outsidethe domain. Albuquerque et al [2] presented a method to transform domain integrals into boundary inte-grals in the classical plate theory for composite laminate materials. The transformation follows the radialintegration method, as proposed by Gao [3]. In [1], this formulation was extended for dynamic problems.Shear deformable shells have been analyzed using the boundary element method by [13] with the analyticalfundamental solution proposed by [14]. Wang and Huang [12] presented a boundary element formulation


for orthotropic shear deformable plates. Later, in Wang and Schweizerhof [15], the previous formulationwas extended to laminate composite plates.

Stress and moment computation by the BEM has been addressed by some works in literature. For ex-ample, Zao [18] and Zao and Lan [17] have discussed the computation of stresses in plane elastic problems,Knopke [4] presented and discussed the integral formulation for computation of stresses in isotropic thinplate, Rashed et al [7] presented an stress integral formulation in the BEM fo Reissner plate bending prob-lems. To the best of authors knowledge, the computation of moments by the BEM in anisotropic plates havestill not been addressed in literature.

This paper proposes a numerical procedure to compute moments at internal points and at the boundary ofcomposite laminated plates using a dynamic boundary element plate formulation that follows the Kirchhoffhypotheses.

Boundary integral equations. The boundary integral formulation for anisotropic thin plate problems usestwo integral equations, for displacement and rotation (see [2]). The transversal displacement equation isgiven by:

Kw(Q)+∫

Γ

[V ∗

n (Q,P)w(P)−m∗n(Q,P)

∂w(P)∂n

]dΓ(P)+

Nc

∑i=1

R∗ci(Q,P)wci(P) =

Nc

∑i=1

Rci(P)w∗ci(Q,P)+

∫Ωg

b(P)w∗(Q,P)dΩ+

∫Γ

[Vn(P)w∗(Q,P)−mn(P)

∂w∗

∂n(Q,P)

]dΓ(P), (1)

where P is the field point; Q is the source point; Γ is the boundary of the domain Ω of the plate; Ωg is thepart of the domain Ω where the body force b is applied; the constant K is introduced in order to considerthat the source point Q can be placed in the domain, on the boundary, or outside the domain (if the point Qis on a smooth boundary, then K = 1/2); ∂ ()

∂n is the derivative to the outward unity vector n that is normalto the boundary Γ at the field point P; mn and Vn are, respectively, the normal bending moment and theKirchhoff’s equivalent shear force on the boundary Γ; Rc is the thin plate reaction of corners; wc is thetransversal displacement of corners; Nc is the number of corners; and the symbol * stands for fundamentalsolutions.

The rotation equation is given by:

12

∂w(Q)∂n1

+∫

Γ

[∂V ∗

∂n1(Q,P)w(P)−

∂m∗n

∂n1(Q,P)

∂w∂n

(P)]

dΓ(P)+

Nc

∑i=1

∂R∗ci

∂n1(Q,P)wci(P) =

Nc

∑i=1

Rci(P)∂w∗

ci

∂n1(Q,P)+

∫Ωg

b(P)∂w∗

∂n1(Q,P)dΩ+

∫Γ

Vn(P)

∂w∗

∂n1(Q,P)−mn(P)

∂∂n1

[∂w∗

∂n(Q,P)

]dΓ(P), (2)

where ∂ ()∂n1

is the derivative to the outward unity vector n1 that is normal to the boundary Γ at the sourcepoint Q.


As can be seen, domain integrals arise in the formulation owing to the presence of the body force b.In order to transform these integrals into boundary integrals, consider, as in the DRM, that the body forceb is approximated over the domain Ωg as a sum of M products between approximation functions fm andunknown coefficients γm, that is:

b(P) =M

∑m=1

γm fm +ax+by+ c (3)

with

M

∑m=1

γmxm =M

∑m=1

γmym =M

∑m=1

γm = 0 (4)

The approximation functions used in this work is the well known thin plate spline given by:

fm3 = R2 log(R), (5)

used with the augmentation function given by equations (3) and (4) . It has been shown in some works fromliterature that this approximation function can give excellent results for many different formulations (seePartridge [5]).

Equations (3) and (4) can be written in a matrix form, considering all source points, as:

b = Fγ (6)

Thus, γ can be evaluated as:

γ = F−1b (7)

For transient analysis, the body force vector is given by:

b = ρhw. (8)

where ρ is the material density, h is the plate thickness, w is the acceleration (double dots stands for secondtime derivative).

To carry out the time integration during a interval T , this interval is divided into N equal intervals (timesteps) of size ∆τ (T = N∆τ). The acceleration for the time step τ +∆τ is given by:

wτ+∆τ =1

∆τ2 (2wτ+∆τ −5wτ +4wτ−∆τ −wτ−2∆τ) . (9)

Provided that wτ , wτ−∆τ , and wτ−2∆τ are known, we can compute wτ+∆τ by doing:

Axτ+∆τ = yτ+∆τ (10)

where xτ+∆τ is the vector of unknown variables and yτ+∆τ is the vector of known variables in which theelements are computed taking into account boundary conditions and computed values for prior time steps.

Moments are written in terms of transversal displacement as:


mx = −

(D11

∂ 2w∂x2 +D12

∂ 2w∂y2 +2D16

∂ 2w∂x∂y

),

my = −

(D12

∂ 2w∂x2 +D22

∂ 2w∂y2 +2D26

∂ 2w∂x∂y

), (11)

mxy = −

(D16

∂ 2w∂x2 +D26

∂ 2w∂y2 +2D66

∂ 2w∂x∂y

),

So, in order to compute moments, it is necessary to calculate second derivatives of transverse displace-ment w. These derivatives are given by (see [9]):

∂ 2w(Q)∂x2 =

∫Γ

[∂ 2V ∗

n∂x2 (Q,P)w(P)−

∂ 2m∗n

∂x2 (Q,P)∂w(P)

∂n

]dΓ(P)+

Nc

∑i=1

∂ 2R∗ci

∂x2 (Q,P)wci(P)−

∫Γ

[Vn(P)

∂ 2w∗

∂x2 (Q,P)−mn(P)∂ 3w∗

∂n∂x2 (Q,P)]

dΓ(P)+Nc

∑i=1

Rci(P)∂ 2w∗

ci

∂x2 (Q,P)+

∫Ω

b(P)∂ 2w∗

∂x2 (Q,P)dΩ (12)

∂ 2w(Q)∂y2 =

∫Γ

[∂ 2V ∗

n∂y2 (Q,P)w(P)−

∂ 2m∗n

∂y2 (Q,P)∂w(P)

∂n

]dΓ(P)+

Nc

∑i=1

∂ 2R∗ci

∂y2 (Q,P)wci(P)−

∫Γ

[Vn(P)

∂ 2w∗

∂y2 (Q,P)−mn(P)∂ 3w∗

∂n∂y2 (Q,P)]

dΓ(P)+Nc

∑i=1

Rci(P)∂ 2w∗

ci

∂y2 (Q,P)+

∫Ω

b(P)∂ 2w∗

∂y2 (Q,P)dΩ (13)

∂ 2w(Q)∂x∂y

=∫

Γ

[∂ 2V ∗

n∂x∂y

(Q,P)w(P)−∂ 2m∗

n∂x∂y

(Q,P)∂w(P)

∂n

]dΓ(P)+

Nc

∑i=1

∂ 2R∗ci

∂x∂y(Q,P)wci(P)−

∫Γ

[Vn(P)

∂ 2w∗

∂x∂y(Q,P)−mn(P)

∂ 3w∗

∂n∂x∂y(Q,P)

]dΓ(P)+

Nc

∑i=1

Rci(P)∂ 2w∗

ci

∂x∂y(Q,P)+

∫Ω

b(P)∂ 2w∗

∂x∂y(Q,P)dΩ (14)

Numerical results.Consider a square clamped-plate under a uniformely distributed step load applied at timeτ0 = 0 with amplitude q = 2,07.106 N/m2.The plate is orthotropic with the following material properties:E2 = 6895 MPa, E1 = 2E2, G12 = 2651.9 MPa, ν12 = 0.3, ρ = 7166 kg/m3. The edges of the plate isa = 254 mm and thickness h = 12.7 mm. This problem is equivalent to problem proposed by Sladek et al.(2007) which was analyzed using the MPLG. The static moment of the central node of the plate is givenby mstat

x = 9,54× 103 N.m and the normalization factor of time by to = a2/(4√

ρh/D). Twelve quadratic


discontinuous boundary elements (three per edge) with equal length and time steps ∆τ = 3.9447.10−5s areused in the discretization of space and time, respectively.

Results are obtained using 1, 9, and 25 internal points. They are shown in figurue 1. These moments arecompared with a meshless Petrov Galerkin formulation ([8]) and the finite element method ([11]). Figure 1shows moments mx at the central node of the plate as a function of time.

0 0.1 0.2 0.3 0.4 0.5−0.5

0

0.5

1

1.5

2

2.5

mx/m

est.

x

t/to

1 internal point9 internal points25 internal pointsFEMSladek et al. (2006)

Figure 1: Moment at the centre of the plate using different internal points.

Results obtained with 25 internal points were closer to the solution of the finite element method andmeshless than results with 1 and 9 internal points. However, results with 9 and 25 points are very close,indicating convergency. Results with one internal point are very smooth. So, the use of internal points areneeded to obtain better precision.

Conclusions. This paper analysed the use the radial integration method applied to transient analysis ofanisotropic plates. From results, we can conclude that

Acknowledgment. The authors would like to thank the State of Maranhao Research Foundation (FAPEMA)and the National Council for Scientific and Technological Development (CNPq) for the financial support ofthis work.

References

[1] E. L. Albuquerque, P. Sollero, and W. P. Paiva. The radial integration method applied to dynamicproblems of anisotropic plates. Communications in Numerical Methods in Engineering, 23:805–818,2007.

[2] E. L. Albuquerque, P. Sollero, W. Venturini and M. H. Aliabadi. Boundary element analysis ofanisotropic Kirchhoff plates. International Journal of Solids and Structures, 43:4029–4046, 2006.


[3] X.W.Gao. The radial integration method for evaluation of domain integrals with boundary only dis-cretization, Engineering Analysis with Boundary Elements, Vol. 26, pp. 905–916, (2002).

[4] B. Knopke. The hypersingular integral equation for the bending moments mxx, mxy, and myy ofKirchhoff plates, Computational Mechanics, Vol. 15, pp. 19-30, (1994).

[5] P. W. Partridge. Towards criteria for selection approximation functions in the dual reciprocity method.Engineering Analysis with Boundary Elements, 24:519–529, 2000.

[6] C. Rajamohan and J. Raamachandran. Bending of anisotropic plates charge simulation method, Ad-vances in Engineering Software, Vol. 30, pp. 369–373, (1999).

[7] Y. F. Rashed, M. H, Aliabadi, C. A. Brebbia. On the evaluation of stress in the BEM for Reissner platebending problems, Applied Mathematica Modeling, Vol. 21, pp. 155-163, (1997).

[8] J. Sladek, V. Sladek, Ch. Zhang, J. Krivacek and P.H. Wen. Analysis of orthotropic thick platesby meshless local PetrovGalerkin (MLPG) method. International Journal for Numerical Methods inEngineering, v. 67, p. 1830–1850, 2006.

[9] K. R. P. Sousa. Analysis of Stress in thin Plates of Composite Materials under Dynamic Load usingthe Boundary Element Method. Master Thesis, Faculty of Mechanical Engineering, University ofCampinas., 2009.

[10] G. Shi and G. Bezine. A general boundary integral formulation for the anisotropic plate bending prob-lems, Journal of Composite Material, Vol. 22, pp. 694–716, (1988).

[11] J. Useche. Shellcomp v3.4: Finite Element Analysis Program for Linear Static and Dynamic Analysisof Composite Shell Structures. Universidade Tecnolgica de Bolivar, Cartagena, Colmbia, 2008.

[12] J. Wang and M. Huang. Boundary element method for ortotropic thick plates, Acta Mechanica Sinica,Vol. 7 (3), pp. 258–266,(1991).

[13] J. Wang and K. Schweizerhof. Free vibration of laminated anisotropic shallow shells including trans-verse shear deformation by the boundary-domain element method, Computers and Structures, Vol. 62,pp. 151–156, (1997).

[14] J. Wang and K. Schweizerhof. The fundamental solution of moderately thick laminated anisotropicshallow shells, International Journal of Engineering Science, Vol. 33, pp. 995–1004, (1995).

[15] J. Wang and K. Schweizerhof. Fundamental solutions and boundary integral equations of moderatelythick symmetrically laminated anisotropic plates, Communications in Numerical Methods in Engineer-ing, Vol. 12, pp. 383–394, (1996).

[16] B.C. Wu and N.J. Altiero. A new numerical method for the analysis of anisotropic thin plate bendingproblems,Computer Methods in Applied Mechanics and Engineering, Vol. 25, pp. 343–353, (1981).

[17] Z. Zhao and S. Lan. Boundary stress calculation - a comparison study, Computers & Structures, Vol.71, pp. 77-85, (1999).

[18] Z. Zhao. On the calculation of boundary stress in boundary elements, Engineering Analysis withBoundary Elements, Vol. 16, pp. 317-322, (1995).


Drilling Rotations in BEM

P. M. Baiz

Department of Aeronautics, Imperial College London, London SW7 2AZ, [email protected]

Keywords: Drilling Rotation, Partition of Unity, 2D Elasticity.

Abstract. This paper presents an approach to include drilling rotations in the two-dimensionalboundary integral equation. The approach is based on a simple partition of unity strategy thatgives rise to a fictitious rotational degree of freedom. A functional based on the rotational residualties the average fictitious drilling rotation field to the true rotation field induced from the two-dimensional elasticity problem. The approach maintains the boundary only character of BEM andthe partition of unity enrichment makes it more general and efficient, as just certain areas of theboundary could be enriched. The accuracy of the proposed method is assessed with a well knownbenchmark problem (Cook’s membrane).

Introduction

When modelling thin walled structures (plate and shell assemblies) by BEM, similar to 2D and3D problems with domains of different material properties, the multi-domain BEM technique isrequired (see Figure 1 (a)). Now, contrary to 2D and 3D problems, when assembling plate segmentsas the ones shown in Figure 1 (a), a difficulty will arise on the assignment of the stiffness in theout-of-plane rotation of each plate. This is because classical plate bending formulations (see Figure1 (c)) do not produce equations associated with this rotational degree of freedom (d.o.f.), alsoknown as the ’drilling rotation’. Figure 1 (b), shows the classical d.o.f. in 2D elasticity (u1 and u2)and the out of plane rotational d.o.f. (drilling rotation).

Within the FEM community this has been a topic of intensive study [2]. The most commonapproaches employ various special devices to develop successful elements. The first of such tech-niques was proposed by Allman [3], who introduced a quadratic displacement approximation tosupplement drilling d.o.f. to nodes. More rigorous mathematical developments [4] have also beenproposed (based on variational principles employing independent rotation fields). Unfortunately,this has been an area totally neglected in the BEM community. The current approach [5] relieson the assumption that the plate flexural rigidity in its own plane is so large that it is possibleto ignore its associated deformation, in other words, there is no drilling rotation. This is a goodapproximation for wide plates, but not for narrow ones that behave like beams.

This paper aims to propose an approach to include drilling rotations in the classical BEM 2Delasticity formulation. The approach is based on a simple partition of unity strategy that gives riseto a fictitious rotational d.o.f. (quadratic displacement approximation to supplement drilling asproposed by [3]). Because of the local character of the approach (approximation on linear boundaryelements), the main characteristic of BEM will be maintained.

Classical BEM for 2D Elasticity

Lets consider flat isotropic sheets of thickness h, Young’s modulus E, Poisson’s ratio ν with aboundary Γ. The two-dimensional boundary integral equation for displacements at the boundarypoint x′ ∈ Γ in the absence of body forces can be written as [1],


Figure 1: Typical Thin Walled Assembly (a) and Degrees of Freedom in 2D (b) and Plate Bending(c).

cαβ(x′)uβ(x′) =∫Γ

U∗αβ(x′,x)tβ(x)dΓ −

∫Γ− T ∗

αβ(x′,x)uβ(x)dΓ (1)

where∫− denotes a Cauchy principal-value integral and cαβ(x′) is a function of the geometry at

the collocation points equal to 1/2δαβ for a smooth boundary. The boundary displacements andtractions are denoted by uα and tα, respectively. U∗

αβ(x′,x) and T ∗αβ(x′,x) are displacement and

traction fundamental solutions for 2D elasticity and can be found in [1].

Compatible quadratic displacements with vertex connectors

As shown by Allman [3], it is possible to write the normal and tangential components of displace-ment (un and ut) along a typical linear boundary element as,

un = a1 + a2s + a3s2, ut = a4 + a5s (2)

where the coordinate s is measured from one end of the side and where the coefficients a1,..,a5 are tobe evaluated in terms of the connectors at both ends of the side. Since there are five coefficients andsix d.o.f., this problem is mathematically overdetermined and the following strategy was adoptedin [3]: four boundary conditions are used to define the end-displacements and the fifth boundarycondition is chosen to define the difference of the derivatives of the end displacements:

un |s=0= un1, un |s=len= un2, ut |s=0= ut1, ut |s=len= ut2,

∂un

∂s|s=0 −∂un

∂s|s=len= −w2 + w1 (3)

Clearly, w1 and w2 are not true rotations in the context of plane elasticity analysis, but theyare closely related to the true rotations at the ends of the element. Following this approximationleads to the well known ’Allman triangle’ which rotational d.o.f.. have been employed by manyresearches to design quadrilateral and advanced solid elements.

This type of approximation was recently re-examined and reformulated in an extremely simplemanner. Tian and Yagawa [7] found that, although developed two decades ago, this type of elements(Allman) takes a typical form of the nowadays partition of unity (PU) based approximation, asfollows:


(ue

1

ue2

)=

n∑i=1

Ni

[1 0 −λyi

0 1 λxi

] ⎛⎜⎝ ui

1

ui2

wi

⎞⎟⎠ , λ = 0 (4)

PU [6] is a set of functions that sum to unit at an arbitrary point of the domain, such as usualFE and many meshfree shape functions. According to the property of a PU method, any PU can beused as the functions Ni. As pointed in [7], this theoretically explains why these rotation formulaecan be extended to different finite elements with translations only. This approximation has alsobeen extended to meshfree methods and, as shown in this work, to boundary element methods.

In equation (4), for the present work Ni represents the classical linear shape functions forboundary elements N1 = 0.5(1 − ξ) and N1 = 0.5(1 + ξ), ue

α represents the discretized in planedisplacements and ui

α, wi the nodal d.o.f.. Following exactly the same implementation in [3] λ istaken as 1/2 and the functions yi and xi are given as,

yi =2∑

j=1

Nj(yj − yi) xi =2∑

j=1

Nj(xj − xi) (5)

Enriched BEM formulation

To apply enrichment to BEM, equation (4) must be introduced in equation (1) as follows:

cαβ(x′)(

2∑i=1

Ni(ξp)uniβ +

2∑i=1

Ni(ξp) (−λyi(ξp)δβ1 + λxi(ξp)δβ2) wni

)=

Ne∑n=1

2∑i=1

Qniαβtni

β

−Ne∑n=1

2∑i=1

Pniαβuni

β −Ne∑n=1

2∑i=1

Pniαβ (−λyi(ξ)δβ1 + λxi(ξ)δβ2) wni (6)

where n is the number of the element containing x′ and ξp refers to the local coordinate of thesource point. The first and second term in the right hand side of equation (6) remain the sameas in standard BEM while the third term contains the enriched term (drilling rotation). A similarequation to (6) was recently presented by Simpson and Trevelyan [10] for mode I and II fractureanalysis.

Relation between true and assumed rotation fields

As mentioned in [10], implementing the above enrichment introduce an additional unknown to theclassical BEM. In their work, a technique based on additional collocations points was implemented.

In the present work a more physically based approach will be used to obtain the additionalequations necessary to have a well posed problem. Following a similar approach continuously usedin FEM, the following functional could be considered:

Π(u, w) =γ

2

∫Γ(Ψ − w)2dΓ (7)

The expression in (7) was recently used within the functional of the total potential energy byChoi et al. [9] for the hybrid Trefftz method (a type of boundary element method). This penaltyterm ties the average drilling rotation field w to the true rotation Ψ induced from uα, and is givenas Ψ = 1/2(u2,1 − u1,2) [2, 3, 9].

The penalty parameter γ in equation (7) was determined through some numerical tests in [9].In the present case such parameter (γ) can take any value without affecting the solution. This


parameter becomes only important when (7) is used within the functional of the total potentialenergy in order to balance its contribution to the classical two-dimensional problem.

Taking the first variation of equation (7) with respect to the drilling rotation field w andminimizing gives:

δΠ(u, w)δw

=∫Γ

δwT (Ψ − w)dΓ = 0 (8)

Introducing equation (4) in (8) gives the following expression,

[δwn1 δwn2

]1×2

∫ ξ=+1

ξ=−1

[N1(ξ)N2(ξ)

]2×1

12

[− 1

Jy(ξ)N1,ξ(ξ)un1

1 +1

Jx(ξ)N1,ξ(ξ)un1

2

+ (lenλ (N1,ξ(ξ)N2(ξ) + N2,ξ(ξ)N1(ξ)) − N1(ξ)) wn1 − 1Jy(ξ)

N2,ξ(ξ)un21

+1

Jx(ξ)N2,ξ(ξ)un2

2 − (lenλ (N1,ξ(ξ)N2(ξ) + N2,ξ(ξ)N1(ξ)) + N2(ξ)) wn2]J(ξ)dξ = 0 (9)

Equation 9 can be seen also as the final matrix representation of a weak form of the rotationresidual given in equation 7. In other words, equation 9 permits the element to satisfy localequilibrium of out of plane rotations in a weak sense.

Numerical Results (Cook’s Membrane)

The enriched BEM formulation proposed above was verified using the most common benchmarkexample in FEM for assessment of new elements (particularly those with drilling rotation capabili-ties). The reason for its popularity among FEM is because it shows the element’s ability to modelmembrane situations with distorted meshes.

This problem was first proposed by Cook [11] as a test case for non-rectangular quadrilateralelements. There is no known analytical solution but the most refined case of several references willbe used for comparison. The material properties, geometry and boundary conditions are shown inFigure 2. The tip displacements and rotations at point A (see Figure 2) for different meshes areshown in Table 1.

Table 1: Tip Displacements for Cook’s membrane problem.

u1 u2 w

16x16Q (AL) [9] — 23.86350 —32x32Q (Green strain-Allman) [8] -10.675 23.916 0.92553

32x32Q (Right stretch strain-Allman) [8] -10.663 23.876 0.8904632x32Q (FEAP shell 6 dof) in [8] based on [12] -10.678 23.922 0.85404

64x64x2T (OPT) [13] — 23.95 —Present 16 Linear (4 per side) -8.05093 23.20510 1.61773Present 64 Linear (16 per side) -10.85637 23.99848 0.77387Present 128 Linear (32 per side) -10.75679 24.14329 0.81364

Present Uniform 172 Linear -10.74402 24.07600 0.81050

Figures 3 presents the deformed shapes for the 16 Linear Boundary Elements (4 per side)and for the more refined uniform 172 Linear Boundary Elements. As shown in Table 1, excellentagreement is obtained with the reported literature with less than 1% difference for deflection and5% for rotation.


Figure 2: Cook’s membrane problem.

Figure 3: Deformed Shape for more refined and less refined meshes.


Conclusions

This paper presented the first attempt to develop a BEM formulation that includes drilling rotationsbased on Allman’s triangular finite element. This rotations are necessary for accurate simulationsof complex multi-domain plate assemblies (formulation that include 6 degrees of freedom per node:3 displacements and 3 rotations). Some of the most important advantages of the approach are: itsboundary only character and the PUM approximation that could lead to great efficiency by justadding drilling rotations along the boundary that needs them (junction boundary).

References

[1] M.H. Aliabadi,The Boundary Element Method, vol II: Application to Solids and Structures,Chichester, Wiley (2002).

[2] 0.C. Zienkiewicz, L.R. Taylor, The Finite Element Method (vol. 2, Solid Mechanics),Butterworth-Heinemann, 5 edition (2000).

[3] D.J. Allman, A compatible triangular element including vertex rotations for plane elasticityanalysis, Computers and Structures, v57, pp. 1-8 (1984).

[4] T.R.J. Hughes, F.Brezzi, On drilling degrees of freedom, Comput. Methods Appl. Mech. Engrg.,v72, pp. 105-121 (1989).

[5] P.M. Baiz, M.H. Aliabadi, Local buckling of thin-walled structures by the boundary elementmethod, Eng. Anal. Bound. Elem., v33, pp. 302-313 (2009).

[6] I. Babuska, G. Caloz, J. Osborn, Special finite element methods for a class of second orderelliptic problems with rough coefficients, SIAM Journal of Numerical Analysis, v31, pp. 945-98(1994).

[7] R. Tian, G. Yagawa, Allmans triangle, rotational DOF and partition of unity, Int. J. Numer.Meth. Engng, v69, pp. 837 - 858 (2007).

[8] K. Wisniewski, E. Turska, Enhanced Allman quadrilateral for finite drilling rotations, Comput.Methods Appl. Mech. Engrg., v195, pp. 6086 - 6109 (2006).

[9] N. Choi, Y. S. Choo, B. C. Lee, A hybrid Trefftz plane elasticity element with drilling degreesof freedom, Comput. Methods Appl. Mech. Engrg., v195, pp. 4095 - 4105 (2006).

[10] R. Simpson, J. Trevelyan, Enrichment of the Boundary Element Method through the partitionof unity method for mode I and II fracture analysis, Advances in Boundary Element TechniquesX, Athens, Greece (2009).

[11] R.D. Cook, Improved two-dimensional finite element, J. Struct. Div., ASCE 100 (ST6), pp.1851 - 1863 (1974).

[12] O.C. Zienkiewicz, R.L. Taylor, The finite element method, 4th ed. Basic Formulation andLinear Problems, vol. 1, McGraw-Hill, 1989.

[13] C.A. Felippa, A study of optimal membrane triangles with drilling freedoms, Comput. MethodsAppl. Mech. Engrg., v192, pp. 21252168 (2003).


Blob regularization of boundary integrals

Gregory Baker, Huaijian ZhangDepartment of Mathematics, The Ohio State University

231 W. 18th Ave, Columbus, OH43210,[email protected]

Abstract: Boundary integral methods have proved very useful in the simulation of free surface motion, in part,because only information at the surface is necessary to track its motion. However, the velocity of the surface mustbe calculated quite accurately, and the error must be reasonably smooth, otherwise the surface buckles as numericalinaccuracies grow, leading to a failure in the simulation. For two-dimensional motion, the surface is just a curve andthe boundary integrals are simple poles that may be removed, allowing spectrally accurate numerical integration.For three-dimensional motion, the singularity in the integrand, although weak, prevents highly accurate numericalmethods, and the intuitive requirement that the errors be smooth can be difficult to achieve. By smoothing theintegrand, or equivalently the Green’s function, by multiplication with an appropriate function, the singularitymay be removed at a cost in accuracy. This loss of accuracy can be balanced with the accuracy of the numericalintegration to produce an overall third-order accuracy. We demonstrate the results with some test cases.

Key–Words: Free Surface Flows, Singular Integrals, Vortex Methods, Blob Regulariztion.

1 IntroductionBlob methods have been developed primarily for tracking vorticity in incompressible, inviscid flows numerically[1]. The Euler equations of motion in vorticity form are:

∂ω

∂t+ u · ∇ω = ω · ∇u , (1)

∇ · u = 0 , (2)

where the velocity is u and the vorticity ω = ∇ × u. The velocity may be determined from the vorticity by theBiot-Savart integral [2],

u(x, t) = −∫

ω(x′, t) ×∇G(x− x′) dx′ , (3)

where G is the free-surface Green’s function for Laplace’s equation. The result is valid in the absence of solidboundaries, but additional contribution to the velocity can be added to account for them. If ω is known at sometime t, then (3) can be integrated to determine u and then (1) can be advanced in time to update ω. Clearly, themethod is well-suited for representing the vorticity at a collection of Lagrangian markers that then track with thefluid velocity. Only points where the vorticity is non-zero need be tracked.

An important special case is when the vorticity is distributed as a delta function along a surface

ω = γ δ(n) , subject to n · γ = 0 , (4)

where n is the distance along the normal n to the surface and γ is called the vortex sheet strength. Since thevorticity must point along a tangent to the surface, we may write

γ = γ t , (5)

where t is a unit tangent vector. As a consequence, the fluid velocity can be expressed as

u(x, t) = −∫

γ(p, q) t(p, q) ×∇G(x− x(p, q)

)dS(p, q) , (6)


where the surface location is written in parametric form x(p, q).There are several important properties of vortex sheets that are obtained by taking the limit as x → x(η, ζ), a

point on the surface, along the normal there. The result is

u±(η, ζ) = uP (η, ζ) ± γ(η, ζ)

2n(η, ζ) × t(η, ζ) , (7)

where the negative subscript is from the side into which the normal points and

uP (η, ζ) =1

4π

∫γ(p, q) t(p, q) ×

(x(η, ζ) − x(p, q)

)∣∣x(η, ζ) − x(p, q)

∣∣3 dS(p, q) (8)

is a principal-valued integral. Thus the vortex sheet strength measures the jump in tangential components of thevelocity across the surface,

γ(p, q)n(p, q) × t(p, q) = u+(p, q) − u−(p, q) , (9)

while the normal components are continuous,

n(p, q) · u+ = n(p, q) · u− = n(p, q) · uP . (10)

The nature of vortex sheets lends itself to a representation for free surfaces between immiscible fluids becausethe normal component of the fluid velocity is automatically continuous and the jump in tangential componentcaptures the generation of vorticity in the presence of a jump in densities. Indeed, a vortex sheet can be viewed asa free surface between two immiscible fluids of equal densities where there is no generation of vorticity and thevortex sheet merely advects with the average fluid velocity at the surface. The extension to fluids with differentdensities on either side of the surface has been derived in two-dimensional motion [3] and in three-dimensionalmotion [4]; a comprehensive treatment is also available [5]. What these studies also show is that spectrally accuratemethods are easily obtainable in two-dimensional motion because the pole singularity in boundary integrals canbe removed leaving the integrands analytic. The 3/2-power singularity in (8) cannot be removed completely byanalytic techniques, although it can be weaken further [4]. The challenge is to find accurate numerical methods forthe integration (8) that give errors that are smooth, avoiding artificial buckling of the surface as it moves with thecalculated velocity.

One way forward has been suggested by an appropriate regularization of the Green’s function and testedthoroughly for two-dimensional motion with good success [6]. A version has been proposed for three-dimensionalmotion [7] with a specific application to deep water waves. Here, we are interested only in testing the success ofregularization of the Green’s function in three-dimensional boundary integral methods.

2 Blob RegularizationBlob regularization is based on modifying the Green’s with a smoothing function. The approach adopted here isgiven in [7]. A smoothed Green’s function takes the form

Gδ(x) = − 1

4πrS

(r

δ

), with r = |x| . (11)

The smoothing function S(r) is chosen so that S(r) → 1 rapidly as r → ∞. Since

∇2Gδ(r) = − 1

4πrδ2d2S

dr2

(r

δ

)≡ 1

δ3ψ

(r

δ

),

the result may be interpreted as an approximation to the delta function as δ → 0.The smoothing function S(r) must satisfy additional conditions to ensure accuracy in using Gδ in place of G.

The error is dominated by the integral,∫ (

Gδ(x) −G(x))pn(x) dS(x) , (12)


where pn is a homogeneous polynomial of degree n. Expressed in polar coordinates, (12) takes the form,

Cnδn+1

∫ (S(r) − 1

)rn dr . (13)

The accuracy is determined by how many moment conditions (13) are satisfied. The choice,

S(r) = erf(r) +2r√π

e−r2 , (14)

ensures (13) is satisfied for both n = 0, 1. The error in using the blob regularization is then O(δ3).

Since the integration is now smooth with the regularized Green’s function, a standard integration such as thetrapezoidal rule can by applied. The error can be shown to be O

(h3

)provided δ/h is kept fixed [6, 7]. Here h is a

measure of the spacing between the Lagrangian points. Thus altogether, the error is O(h3

)with δ/h kept fixed. A

simple case will be used to confirm these estimates.

3 Test Case: A Cylindrical Vortex SheetThe surface is given by

x(p, q) = R cos(p) , y(p, q) = R sin(p) , z(p, q) = q , (15)

with corresponding surface vectors,

t1 = (0, 0, 1) , t2 = (− sin(p), cos(p), 0) , n = (− cos(p),− sin(p), 0) . (16)

The easiest way to construct velocity components (uP , vP , wP ) that correspond to a vortex sheet,

γ = γ1(p, q) t1 + γ2(p, q) t2 , (17)

is to introduce the velocity potential,u = ∇φ , (18)

since ∇× u = 0 away from the surface. By invoking (2), φ must satisfy Laplace’s equation inside and outside thevortex sheet. Now simply make a choice, for example,

φ = f(r) cos(nθ) cos(αz) , (19)

where f(r) must satisfy

rd

dr

(r

df

dr

)− (n2 + α2r2) f = 0 , (20)

the modified Bessel equation of integer order n. The appropriate choice of solutions, suitably normalized, are

f(r) = AIn(αr)

αI ′n(αR), r < R , (21)

f(r) = BKn(αr)

αK ′n(αR)

, r > R . (22)

The requirement that the normal component of the velocity is continuous on the surface (10) makes A = B. Thejump in velocity across the surface is

u+ − u− =AΓn

αR

(−n sin(p) sin(np) cos(αq), n cos(p) sin(np) cos(αq), αR cos(np) sin(αq)), (23)

which must match vortex sheet strength through (9). Here,

Γn =In(αR)

I ′n(αR)− Kn(αR)

K ′n(αR)

. (24)


Finally,

γ1 =nAΓn

αRsin(np) cos(αq) , (25)

γ2 = −AΓn cos(np) sin(αq) . (26)

The vortex sheet distribution generates the surface velocity up by the surface integral (8). In this specificexample,

up(η, ζ) =R2

4π

∫ ∞

−∞

∫2π

0

γ1(p, q)

(sin(p) − sin(η), cos(η) − cos(p), 0

)[R2 (cos(η) − cos(p))2 + R2 (sin(η) − sin(p))2 + (ζ − q)2

]3/2 dp dq

+R

4π

∫ ∞

−∞

∫2π

0

γ2(p, q)

(cos(p) (ζ − q), sin(p) (ζ − q), R (1 − cos(η − p)

)[R2 (cos(η) − cos(p))2 + R2 (sin(η) − sin(p))2 + (ζ − q)2

]3/2 dp dq . (27)

On the other hand, (7) means that 2up = u+ + u−, or up = (uP , vP , wP ) where

2uP (η, ζ) = 2A cos(η) cos(nη) cos(αζ) +nAVn

αRsin(η) sin(nη) cos(αζ) , (28)

2vP (η, ζ) = 2A sin(η) cos(nη) cos(αζ) − nAVn

αRcos(η) sin(nη) cos(αζ) , (29)

2wp(η, ζ) = −AVn cos(nη) sin(αζ) , (30)

whereVn =

In(αR)

I ′n(αR)+

Kn(αR)

K ′n(αR)

(31)

The choice for γ1 (25) and γ2 (26) will produce the velocity components given above. This, then, provides a testcase for the numerical integration of the boundary integrals (27) using the regularized Greens function (11).

4 Numerical ResultsA specific test case is chosen with A = R = n = 1. The surface variables p and q are divided into N and M evenlyspaced intervals respectively. This means the spacing can have different values, h1 = 2π/N and h2 = 2π/(αM)depending the integration variable. By making different choices of N,M , we can assess the behavior of the erroron either δ/h1 or δ/h2.

The numerical integration is performed by applying the standard trapezoidal rule in both integrations. Theinfinite range of integration in q may be replaced by a finite range of integration [0, 2π/α] since the vortex sheetstrength is periodic and the method of images may be applied. Specifically,

∫ ∞

−∞

F (q) dq =

∫2π/α

0

∞∑k=−∞

F (q + 2kπ/α) dq . (32)

Bearing in mind that the integrands in (27) must be multiplied by H(r/δ) = S(r/δ) − (r/δ)S′(r/δ) to accountfor the influence of the smoothing function, the parts of the integrands that must be summed are

∞∑k=−∞

H(rk/δ)

r3kand

∞∑−∞

(ζ − q − 2kπ/α)H(rk/δ)

r3k, (33)

where r2k = 2R2 (1 − cos(η − p)) + (ζ − q− 2kπ/α)2 is the denominator. The convergence of these sums can beimproved by using the symmetries in the sums for large |k| and subtracting a common part whose sum is known.For example, the first sum can be written as (without the contribution from k = 0)

1

2

∞∑k=−∞k =0

[H(rk/δ)

r3k+

H(r−k/δ)

r3−k

− α3

4π3|k|3]

+α3

4π3

∞∑k=1

1

k3. (34)


0 0.5 1 1.5 20

1

2

3

4

5

6

δ/h1

−log

10(e

rror

)

0 0.5 1 1.5 20

1

2

3

4

5

6

δ/h1

−log

10(e

rror

)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

δ/h1

−log

10(e

rror

)

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

δ/h1

−log

10(e

rror

)

Figure 1: The absolute error in the velocity for different ratios h1/h2: top left figure h1/h2 = 1; top right figureh1/h2 = 2; bottom left figure h1/h2 = 4; bottom right corner h1/h2 = 0.5. Each curve corresponds to a differentresolution, starting at N = 16 and increasing up to N = 512 in powers of 2.

The first sum in (34) converges as 1/K4 if the sum is truncated (−K,K) and the second sum has a known analyticresult. The second sum in (33) can be treated similarly.

First, let’s consider the relative role of h1 and h2 on the absolute error in the velocity at the point θ = 0and z = 0 for the case α = 1. The ratio h1/h2 is kept fixed while δ/h1 is varied. In this way we can assessthe impact of the smoothed Green’s function for various choices of resolution. The results are shown in Fig. 1for the ratios h1/h2 = 1, 2, 4, 0.5. Each curve corresponds to a different resolution, starting at N = 16 andincreasing in powers of 2 up to N = 512. All the results show a peak in accuracy although this peak depends onthe resolution. For h1 = h2, this peak is around δ = h1. To the left of the peak, the accuracy drops to first-orderas the effective range of the smoothing on the Green’s function δ falls below the grid spacing. To the right of thepeak, the accuracy downgrades slowly but retains third-order convergence. A curious feature appears in the figurefor h1/h2 = 4 where an improvement in accuracy spikes around the value δ/h1 = 0.25. At this value, δ/h2 = 1.The interpretation is clear; there is improvement when δ exceeds the spacing h2, but full accuracy is only reachedwhen δ also exceeds h1. When h1 and h2 are close, as in the results for h1 = 2h2, the different in scales is lessnoticeable. The results for h1 = 0.5h2 confirm the interpretation; here δ/h1 = 2 or δ/h2 = 1 must be exceededfor full accuracy. The conclusion is that min(δ/h1, δ/h2) should exceed 1 to obtain the best accuracy. A goodchoice seems to be h1 = h2 with δ/h1 between 1.0 and 1.5.

Besides creating accurate results for boundary integrals associated with free surface motion in incompressible,inviscid fluids, the expectation is that the errors will prove to be smooth functions of the surface variables. Thisexpectation is confirmed in Fig. 2, which shows the absolute error in the velocity as a function of p = θ and q = zfor the case h1 = h2, δ = 1.5h1 and N=256.

Results have been presented for the choice α = 1. We have confirmed that the results are similar for otherchoices of α, specifically α = 2, 0.5.

5 ConclusionBlob regularization of boundary integrals can produce very accurate results provided the blob size δ exceedsthe largest grid spacing. The numerical evidence suggests that the errors are smooth functions of the surface


0

24

6

0

2

4

60

0.5

1

1.5

x 10−5

θz

erro

r

Figure 2: The absolute error in the velocity as a function of the surface coordinatesθ and z.

parameters.

Acknowledgements: The research was supported by NSF (grant OCE-0620885).

References:

[1] J.T. Beale and A.J. Majda, High order accurate vortex methods with explicit velocity kernels, J. Com-put. Phys., 58, 1985, p. 188.

[2] P.G. Saffman, Vortex Dynamics, Cambridge University Press, 1992.[3] G.R. Baker, D.I. Meiron and S.A. Orszag, Generalized vortex methods for free-surface flow problems,

J. Fluid Mech., 123, 1982, pp. 477–501.[4] G. R. Baker, D. I. Meiron and S. A. Orszag, Boundary integral methods for axi-symmetric and three-

dimensional Rayleigh-Taylor instability problems, Physica 12D, 1984.[5] G.R. Baker, Boundary Element Methods in Engineering and Sciences, Chapter 8, Imperial College Press,

2010.[6] G.R. Baker and J.T. Beale, Vortex blob methods applied to interfacial motion, J. Comput. Phys., 196, 2004,

pp. 233–258.[7] J.T. Beale, A convergent boundary integral method for three-dimensional water waves, Math. Comput., 70,

2001, p. 977.


On the accuracy of the fast hierarchical DBEM for the analysis of static and dynamic elastic crack problems

I. Benedetti1, A. Alaimo1, M.H. Aliabadi21 Dipartimento di Ingegneria Strutturale Aerospaziale e Geotecnica,

Università di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy, [email protected], [email protected]

2 Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW72AZ, London, UK,

[email protected]

Keywords: DBEM, Adaptive Cross Approximation, Hierarchical Matrices, Fast BEM solvers, Elastodynamics, Laplace Transform Method, Stress Intensity Factors.

Abstract. In this paper the main features of a fast dual boundary element method based on the use of hierarchical matrices and iterative solvers are described and its effectiveness for fracture mechanics problems, both in the static and dynamic case, is demonstrated. The fast solver is built by representing the collocation matrix in hierarchical format and by using a preconditioned GMRES for the solution of the algebraic system. The preconditioner is computed in hierarchical format by LU decomposition of a coarse hierarchical representation of the collocation matrix. The method is applied to elastostatic problems and to elastodynamic cases represented in the Laplace transform domain. The application of the hierarchical format in the Laplace domain is straightforward and offers some interesting advantages related to the use of some local preconditioners. The accuracy in the determination of both static and dynamic stress intensity factors is assessed and the effectiveness of the technique is successfully demonstrated.

Introduction

The Boundary Element Method (BEM) is nowadays a powerful numerical tool for the analysis and solution of many physical and engineering problems and represents a sensible alternative to other numerical approaches, such as the Finite Element Method (FEM), especially in some fields such as Fracture Mechanics [1, 2]. In general the main advantage offered by the BEM is related to the reduction in the degrees of freedom needed to model a given physical system. Such reduction relies on the underlying boundary integral formulation which requires, for its numerical solution, only the discretization of the boundary of the analyzed domain. This results not only in the reduced size of the solving systems, but also in faster data preparation.

Although these appealing advantages, as the size of the analyzed problems increases BEM techniques lose part of their attractiveness, mainly due to the time required to solve the final system of equations. The solution matrix produced by BEM is in fact generally fully populated and neither symmetric nor definite. This circumstance results in the main drawbacks of the method, that is increased memory requirements as well as increased solution time with respect to other numerical approaches for problems of comparable numerical size. Such considerations have limited the size of the problems that could be effectively tackled on common computers using the standard BEM and have hindered for many years the industrial development of the method and confining its use to the analysis of small or medium size problems.

However, in the recent years, considerable efforts have been devoted to the development of strategies aimed at reducing the computational complexities of the BEM, reducing both memory requirements and time consumption. Many investigations have been carried out and different techniques have been developed such as the fast multipole method (FMM) [3, 4], the panel clustering method [5], the mosaic-skeleton approximation [6] and the methods based on the use of hierarchical matrices [7]. The general aim of such techniques is to reduce the computational complexity of the matrix-vector multiplication which is the core operation in iterative solvers for linear systems. However, while FMMs and panel clustering tackle the problem from an analytical point of view and require the knowledge of some kernel expansion in advance to carry out the integration, mosaic-skeleton approximations and hierarchical matrices provide purely


algebraic tools for the approximation of the boundary element matrices, thus proving particularly suitable for problems where analytic closed form expressions of the kernels are not available or difficult to expand.

In the present work the use of hierarchical matrices and iterative solvers for the rapid solution of three-dimensional elastostatic and elastodynamic dual boundary element crack problems is described. The use of hierarchical matrices allows a noticeable reduction of the storage memory requirements for a given problem as well as a solution time reduction. In the following sections the basic equations of the dual boundary element method (DBEM) are briefly reviewed for both the static case and the dynamic formulation in the Laplace transform domain. The main features of the fast hierarchical solution strategy for both cases are then described. Finally, some problems are analyzed with a focus on the accuracy of the hierarchical solution in the determination of the crack parameters.

The DBEM for the static and dynamic analysis of elastic crack problems

The boundary integral equations governing the static behavior of an elastic body can be written as

0 0 0 0, ,i j j i j j i j jc u T u d U t d

x x x x x x x x (1)

where i jU and i jT are the fundamental solution of elastostatic problems. If an elastic dynamic problem is tackled by using the BEM in the Laplace transform domain, the following boundary integral equations can be written

0 0 0 0, , , ,i j j i j j i j jc u T s u d U s t d

x x x x x x x x (2)

where the tilde indicates transformed quantities and s is the Laplace parameter. The boundary integral representation of the elastodynamic problem in the Laplace domain has the same form as that of the elastostatic problem. Eq.(2) is to be used in conjunction with the transformed boundary conditions to solve any specific problem. If cracks are present in the analyzed domain, also the following equations have to be used to close the problem

0 0 0 0 0 0

0 0 0 0 0 0 0 0

, ,

= , ,

ij j ij j ij j ij j

ij j ij j j ijk k j ijk j

c u c u T u d U t d

c t c t n T u d n U t d

x x x x x x x x x x

x x x x x x x x x x x x (3)

The previous equations are the displacement and traction boundary integral equations collocated at opposite crack surfaces [8]. Their form is the same for both static and dynamic problems in the Laplace transform domain, with the obvious difference that, in the last case, they involve the transformed kernels and transformed quantities depending on the Laplace parameter. After discretization and application of suitable boundary conditions, Eqs.(1-3) lead to a linear system of equations of the form

A x y (4)

in the static case and of the form

( )s A x y (5)

in the dynamic case in the Laplace transform domain, where the dependence on the complex parameter has been highlighted. To analyze a general elastodynamic problem by using the Laplace transform technique, one has generally to compute the solution of the system (5) for a set of Laplace parameters ks , with

1,...,k L , in order to calculate the time-dependent values of any relevant variable by means of some Laplace inverse transformation technique. Wen et al. [9] obtained for example accurate results for long durations in the time domain by using


2 0,...,25kkis k

T (6)

with 5T and 0 20T t , where 0t is the unit time and by using the Durbin’s inversion technique to get time domain quantities. The static and dynamic stress intensity factors are computed by means of crack-front interpolation functions, which are used on the crack surface elements adjacent to the crack front in order to catch the rbehavior of the displacements field. Discontinuous eight node quadrilateral elements are then used to represent the crack surface and the corresponding special shape functions, derived for eight-noded and nine-noded discontinuous Lagrangian elements by Mi et al. [19] and Cisilino et al.[18], are considered to compute the crack tip opening displacements. Once the crack displacement field is computed, the SIFs at a point P of the crack front can be directly obtained from the crack tip opening displacements

2

2

2

4(1 ) 2

4(1 ) 2

4(1 ) 2

P Pu PlI b b

P Pu PlII n n

P Pu PlIII t t

EK u ur

EK u ur

EK u ur

(7)

where Pu and Pl are the nodes of the upper and lower crack surfaces behind the crack front while , , and b n tu u u are the displacement components with respect to a local crack coordinate system [1]. The

dynamic SIFs are computed directly from the transformed displacements of the crack surfaces and the corresponding time dependent values are obtained by the Durbin's Laplace transform inversion [9].

Hierarchical matrices for static and dynamic DBEM crack problems

To reduce both storage memory and solution time required by the elastostatic and elastodynamic BEM analysis in the Laplace domain, systems (4) and (5) are represented in hierarchical format. It is worth noting that, in the Laplace domain, system (5) has to be set for each value of the Laplace parameter.

The hierarchical or low rank representation of a BEM matrix is built by generating the matrix itself as a collection of sub blocks, some of which admit a special approximated and compressed format. Such blocks, referred to as low rank blocks, can be stored in the form

1

kT T

k i ii

B B u v U V (8)

The block B of order m n is approximately generated through the product of U, of order m k , and TV ,of order k n . If k, i.e. the rank of the block, is low, then the representation (8) allows to reduce both memory storage and the computational cost of the matrix-vector multiplication, which is the bottleneck of any iterative solver. The approximation of the low rank blocks (8) is built by computing only some of the entries of the original blocks through adaptive algorithms known as Adaptive Cross Approximation (ACA) [10, 11], that allow to reach an initially selected accuracy . Low rank blocks represent the numerical interaction, through asymptotic smooth kernels, between sets of collocation points and clusters of integration elements which are sufficiently far apart from each other. The distance between clusters of elements enters a certain admissibility condition of the form

, ( , ) coll int coll intmin diam diam dist (9)

where coll int and are clusters of elements and 0 is a parameter influencing the number of admissible blocks on one hand and the convergence speed of the adaptive approximation of the low rank blocks on the other hand [12]. The blocks that do not satisfy such condition are called full rank blocks and they need to be computed and stored entirely, without approximation. Once low and full rank blocks have been generated, some recompression techniques can be used to further reduce the storage memory and


computational complexity of the single blocks and of the overall hierarchical matrix (reduced SVD [13] and coarsening [14]).

As an almost optimal representation is obtained, the solution of the system can be tackled either directly, through hierarchical matrix inversion [15], or indirectly, through iterative methods [16]. In both cases, the efficiency of the solution relies on the use of a special arithmetic, i.e. a set of algorithms that implement the operations on matrices represented in hierarchical format, such as addition, matrix-vector multiplication, matrix-matrix multiplication, inversion and hierarchical LU decomposition. A collection of algorithms that implement many of such operations is given in [12] while the hierarchical LU decomposition is discussed in [16].

The use of iterative methods takes full advantages of the hierarchical representation, exploiting the efficiency of the low-rank matrix-vector multiplication. The convergence of iterative solvers can be improved by using suitable preconditioners. In this work a hierarchical LU preconditioner is built starting from a coarse approximation of accuracy p of the collocation matrix.

In this work, an iterative GMRES algorithm is used in conjunction with the hierarchical preconditioner for solving the static system and the system set up for each value of the Laplace parameter of interest. Even using the hierarchical format, the setup of a preconditioner is an expensive procedure. In the dynamic analysis a new preconditioner should be built for each value ks of the Laplace parameter. In order to further speed up the overall dynamic analysis, local preconditioners are then used. For the interested reader, the full description of the use of local preconditioners can be found in the work of Benedetti et al.[17]. Here is only mentioned that a preconditioner computed for a certain Laplace parameter ks can be successfully used to precondition the system set up for other close Laplace parameters k js , thus eliminating the need to compute a new preconditioner for each new Laplace parameter

, , ,k p k j c k j k p k js s s s s P A Px y (10)

Numerical experiments

To assess the accuracy of the hierarchical DBEM in the determination of the relevant crack parameters, the stress intensity factors for a static problem and two dynamic cases are computed.

Static analysis. An inclined embedded penny crack in a cylindrical bar subjected to static load acting over the bases is first considered, Fig.1. The crack is inclined of an angle 45 with respect to the bases of the bar and has radius a. The bar has radius 5R a and height 6H R . The analyzed mesh is comprised of 800 elements and 3652 nodes. The analytical solution of this problem is known in the case of infinite domain, that can be simulated with the abovementioned dimensions, and the expression for the stress intensity factors IK , IIK and IIIK is reported in the literature [1].

Figure 1 Penny crack in a cylindrical bar and non-dimensional stress intensity factor for mode I.


Figure 2 Non-dimensional stress intensity factors for mode II and mode III.

The hierarchical solution is computed by setting: min 36n , 3 , 210p and 810GMRES . Three

different accuracies are evaluated for the collocation matrix, 410c , 310c

and 210c , and the

effect on the accuracy of the SIFs is checked. In Fig.1 and Fig.2 the non-dimensional stress intensity factors for mode I, II and III, as computed by the hierarchical scheme for the different accuracies, are compared to the exact values and to the values obtained by the standard DBEM. All the SIFs are made dimensionless by dividing by 0 , ,iK i I II III , 0iK being the maximum value of the analytical SIFs. As it can be noted, the hierarchical analysis gives very accurate results in all the considered cases, and the error of the hierarchical solution with respect to the analytical values is in the range of 2-4%. The achieved speed-up ratios (hierarchical over standard time) are 0.34, 0.18 and 0.16 going from the highest to the lowest accuracy, while the required storage memory for the collocation matrix is 18.09%, 13.20% and 8.84% of the original allocation.

Dynamic analysis. The dynamic stress intensity factor (DSIF) for two crack problems has also been computed. The transformed DSIFs are computed directly from the transformed displacements on the crack surface and the time-dependent values are then obtained by the Durbin's Laplace transform inversion, following the procedure used in [9], to which the reader is referred for further details.

An embedded crack case is considered first. A bar of size 1 22 2 2w w h , containing a central penny-shaped crack, is subjected to the Heaviside traction load 0( ) ( )t H t acting on the ends, as shown in Fig.3. The radius of the crack is a and 1 2w w , 0.5a w , 2h w . The Poisson's ratio is 0.2 .

This problem has been considered by Wen et al. [9] and it has been dealt with by using the standard dynamic DBEM on which the present work is based. The standard DBEM has been validated in [9] by comparison with the displacement discontinuity method (DDM) [20] and it has been shown to produce highly accurate results. Here the results obtained by using the original code developed by Wen et al. are compared with the results given by the hierarchical solver. Fig.3 shows the normalized DSIF for mode I, i.e.

0IK t K with 0 02 /K a , at the crack front point shown in Fig.3 with the thick marker. The DSIF given by the standard DBEM is compared with that computed by prescribing for the collocation matrix, in the hierarchical scheme, the accuracies 310c

and 210c and setting the other hierarchical parameters

to: min 36n , 3 , 110p , max 200N . In both cases the approximated solution, computed by using the

fast solver, is not distinguishable from that computed by using the standard direct solver. The present results have been obtained from the mesh shown in Fig.3, which is comprised of 530 elements and 1792 nodes. Each crack surface is modeled with 20 eight-node discontinuous elements. It is worth noting that, on the external boundary, this mesh is finer than that used by Wen et al. to capture the DSIF. However, it has been used here to force the computation of large matrix blocks through ACA and to investigate the effects of the hierarchical approximation on the quality of the computed crack parameters. It was observed that also the DSIF satisfies, in terms of 2L norm, the accuracy required to the hierarchical scheme.


Figure 3 Embedded crack in a prismatic bar and dynamic SIF for mode I.

It may be interesting to know that, for 310c , the overall assembly, solution and total speed up ratios

were 0.81, 0.16 and 0.26 respectively. For 210c , the same ratios were 0.73, 0.15 and 0.24. Moreover, in

both cases, only around 75% of the original memory allocation was used for storing both the collocation matrix and the preconditioner.

As second case, a surface breaking semi-circular crack of radius a is contained in a plate of size 2 2w h b subjected to the Heaviside load 0( ) ( )t H t acting as shown in Fig.4. The dimensions of the plate are: 1 a cm , 3 w cm , 3 h cm , 2.5 b cm . The Young’s modulus is 5 210 /E N m , the Poisson’s ratio 0.3 and the mass density 3 210 /kg m . The load amplitude is 2

0 100 /N m . A non-uniform mesh with 462 elements and 1533 nodes is used for the computation.

The normalized DSIF for mode I at the thick point shown in Fig.4 is computed. This case has already been considered by Zhang and Shi [21] and Zhong and Zhang [22] and the results obtained in the present work by using the standard DBEM are in good agreement with their results, although some differences in detail may exist.

Fig.4 shows the comparison of the DSIFs computed by the standard DBEM, using the code developed by Wen et al. [9], and those obtained by using the hierarchical solver. Two hierarchical solutions have been considered, setting the prescribed accuracy to 310c

and 210c and the other hierarchical parameters

to: min 36n , 3 , 110p , max 200N . In both cases the hierarchical solution is not graphically

distinguishable from the direct solution. The existing differences can be seen in terms of 2L norm for the DSIF values, and they are within the accuracy set for the hierarchical scheme.

Overall assembly, solution and total speed up ratios were: 1.01, 0.20 and 0.39 for 310c ; 0.92, 0.18

and 0.36 for 210c . Between 75% and 85% of the original memory allocation was used for storing the

collocation matrix. The hierarchical technique proves then to be a powerful, accurate and reliable tool for the dynamic

analysis of crack problems, either in the case of embedded or surface breaking cracks, with uniform and non-uniform meshes. For further details about the structure of the hierarchical matrix with respect to the distance of the crack from the external surface the reader is referred to the previous works of the authors [23, 24].


Figure 4 Surface breaking crack in a prismatic bar and dynamic SIF for mode I.

Summary

In this work the use of a fast DBEM solver for the analysis of static and dynamic elastic crack problems has been described and its effectiveness in the determination of the static and dynamic stress intensity factors has been demonstrated. In particular, it has been shown that accurate SIFs can be obtained also setting a moderate accuracy for the hierarchical representation of the collocation matrix, obtaining then relevant savings in terms of storage memory and solution time.

References

[1] M.H. Aliabadi, The Boundary Element Method: Applications in Solids and Structures, vol. 2. John Wiley & Sons Ltd, 2002.

[2] M.H. Aliabadi, Applied Mechanics Reviews, 50, 83–96, 1997.

[3] H. Rokhlin, J Comp Phys, 60, 187-207, 1985.

[4] V. Popov, H. Power, Eng. An. Bound. Elem., 25, 7–18, 2001.

[5] W. Hackbusch and Z.P. Nowak, Numerische Mathematik, 73, 207-243, 1989.

[6] E. E. Tyrtyshnikov, Calcolo, 33, 47-57, 1996.

[7] W. Hackbusch, Computing, 62, 89-108, 1999.

[8] M.H. Aliabadi, International Journal of Fracture, 86, 91-125, 1997.

[9] P.H. Wen, M.H. Aliabadi and D.P. Rooke, Comp Meth App Mech Eng, 167, 139-151, 1998.

[10] M. Bebendorf, Numerische Mathematik, 86, 565-589, 2000.

[11] M. Bebendorf, S. Rjasanow, Computing, 70, 1-24, 2003.

[12] S. Börm, L. Grasedyck and W. Hackbusch, Eng. An. Bound. Elem., 27, 405–422, 2003.

[13] M. Bebendorf, Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von Niedrigrang-Matrizen, Ph.D. Thesis, Universität Saarbrücken, 2000. dissertation.de, Verlag im Internet, ISBN 3-89825-183-7, 2001.


[14] L. Grasedyck, Computing, 74, 205-223, 2005.

[15] L. Grasedyck, W. Hackbush, Computing, 70, 295-334, 2003.

[16] M. Bebendorf, Computing, 74, 225-247, 2005.

[17] I. Benedetti, M.H. Aliabadi, Int J Num Meth Eng, In Press (ref. NME-Aug-09-0529.R1), 2010. [18] A.P. Cisilino, M.H. Alaibadi, International Journal of Pressure Vessel and Piping, 70, 135-144, 1997.

[19] Y. Mi, M.H. Alaibadi, International Journal of Fracture, 67, R67-R71, 1994.

[20] P.H. Wen, Dynamic Fracture Mechanics: Displacement Discontinuity Method, Computational Mechanics Publications, Southampton and Boston, 1996.

[21] Y.Y. Zhang, W. Shi, Engineering Fracture Mechanics, 47(5), 715-722, 1994.

[22] M. Zhong, Y.Y. Zhang, Applied Mathematics and Mechanics, 22(11), 1344-1351, 2001.

[23] I. Benedetti, M.H. Aliabadi, G.Davì, Int. J. Solids Structures, 45, 2355-2376, 2008.

[24] I. Benedetti, A. Milazzo, M.H. Aliabadi, Int J Num Meth Eng, 80(10), 1356-1378, 2009.


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Non-Incremental Boundary Element Discretization of non-linear heatequation based on the use of the Proper Generalized Decompositions

G. Bonithon1,4, P. Joyot1, F. Chinesta2 and P. Villon3

1 ESTIA-Recherche, technopole izarbel, 64210 Bidart, France, [email protected] EADS Corporate Foundation International Chair, GEM CNRS-ECN, 1 rue de la Noë BP 92101,

44321 Nantes cedex 3, France, [email protected] UTC-Roberval UMR 6253, 60200 Compiègne, France, [email protected]

4 EPSILON Ingénierie, 10 rue Jean Bart, BP 97431, France, 31674 Labège Cedex

Keywords: Boundary element method, Separated representations, Proper Generalized Decomposition

Abstract. In this work, we propose a new approach for solving the heat equation within the BoundaryElements method framework. This technique lies in the use of a separated representation of theunknown field that allows decoupling the space problem (that results steady state) from the temporalone (one dimensional that only involves the time coordinate).

IntroductionThe Boundary Elements Method (BEM) allows efficient solution of partial differential equationswhose kernel functions are known. The heat equation is one of these candidates when the thermalparameters are assumed constant (linear model). When the model involves large physical domainsand time simulation intervals the amount of information that must be stored increases significantly.We propose here an alternative strategy able to change the nature of the problem. Thus, the temper-ature field involved by the so called heat equation is approximated using a separated representationinvolving products of space and time functions. This kind of approximation is not new, in fact, properorthogonal decomposition [1] allows such one decomposition, but in this case this decompositionmust be applied a posteriori, i.e. on the transient solution of the considered model.

The technique that we propose in this paper allows to transform the transient model in a sequence ofspace problems (all of them steady state) and time problems (that only involve the time coordinate).This iteration procedure leads to a proper space-time generalized decomposition of the model solu-tion. The efficiency of such one approach was proven in [2, 3, 4]. In our knowledge, this techniquehas never been coupled with a BEM for solving the resulting steady problem defined in the physicaldomain.

We start summarizing the main ideas of the Proper Generalized Decomposition and we will focus onthe application of such technique in the context of the BEM. Finally, numerical example, with a nonlinear source term, will be presented and discussed.

A Proper Generalized Decomposition Boundary Element MethodLet us consider the heat equation

∂u∂ t

−a∆u = f (u) in Ω× (0,Tmax] (1)

with homogeneous initial and boundary conditions, where a is the diffusion coefficient, Ω ⊂ Rd,d ≥

1, Tmax > 0. The aim of the separated representation method is to compute N couples of functions


(Xi,Ti)i=1,...,N such that Xii=1,...,N and Tii=1,...,N are defined respectively in Ω and (0,Tmax] andthe solution u of this problem can be written in the separate form

u(x, t) ≈N

∑i=1

Ti(t) ·Xi(x) (2)

The weak formulation yields: Find u(x, t) such that

∫ Tmax

0

∫Ω

u(

∂u∂ t

−a∆u− f (x, t))

dxdt = 0 (3)

for all the functions u(x, t) in an appropriate functional space.

We compute now the functions involved in the sum (eq (2)). We suppose that the set of functionalcouples (Xi,Ti)i=1,...,n with 0 ≤ n < N are already known (they have been previously computed)and that at the present iteration we search the enrichment couple (R(t),S(x)) by applying an alternat-ing directions fixed point algorithm that after convergence will constitute the next functional couple(Xn+1,Tn+1). Hence, at the present iteration, n, we assume the separated representation

u(x, t) ≈n

∑i=1

Ti(t) ·Xi(x) + R(t) ·S(x) (4)

The weighting function u is then assumed as

u = S ·R + R ·S (5)

Introducing (eq (4)) and (eq (5)) into (eq (3)) it results

∫ Tmax

0

∫Ω

(S ·R + R ·S) ·(

S · ∂R∂ t

−a∆S ·R)

dxdt =

=∫ Tmax

0

∫Ω

(S ·R + R ·S) ·(

f (x, t)−n

∑i=1

Xi ·∂Ti

∂ t+ a

n

∑i=1

∆Xi ·Ti

)dxdt (6)

We apply an alternating directions fixed point algorithm to compute the couple of functions (R,S):

• Computing the function S(x).First, we suppose that R is known, implying that R vanishes in (eq (5)). Thus, eq (6) writes

∫Ω

S · (αtS−aβt∆S) dx =∫

ΩS ·

(γt(x)−

n

∑i=1

α it Xi + a

n

∑i=1

β it ∆Xi

)dx (7)


where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

αt =∫ Tmax

0R(t) · ∂R

∂ t(t)dt

α it =

∫ Tmax

0R(t) · ∂Ti

∂ t(t)dt

βt =∫ Tmax

0R2(t)dt

β it =

∫ Tmax

0R(t) ·Ti(t)dt

γt(x) =∫ Tmax

0R(t) · f (x, t) dt; ∀x ∈ Ω

(8)

The weak formulation (eq (7)) is satisfied for all S, therefore we could come back to the asso-ciated strong formulation

αtS−aβt∆S = γt −n

∑i=1

α it Xi + a

n

∑i=1

β it ∆Xi (9)

that one could solve by using any appropriate discretization technique for computing the spacefunction S(x).

• Computing the function R(t).From the function S(x) just computed, we search R(t). In this case S vanishes in (eq (5)) andeq (6) reduces to ∫ Tmax

0

∫Ω

(S ·R) ·(

S · ∂R∂ t

−a∆S ·R)

dxdt =

=∫ Tmax

0

∫Ω

(S ·R) ·(

f (x, t)−n

∑i=1

Xi ·∂Ti

∂ t+ a

n

∑i=1

∆Xi ·Ti

)dxdt (10)

where all the spatial functions can be integrated in Ω. Thus, by using the following notations⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

αx =∫

ΩS(x) ·∆S(x)dx

α ix =

∫Ω

S(x) ·∆Xi(x)dx

βx =∫

ΩS2(x)dx

β ix =

∫Ω

S(x) ·Xi(x)dx

γx(t) =∫

ΩS(x) · f (x, t) dx; ∀t

(11)

equation (eq (10)) reads

∫ Tmax

0R ·

(βx

∂R∂ t

−aαxR− γx(t) +n

∑i=1

β ix∂Ti

∂ t−

n

∑i=1

aα ix ·Ti

)dt = 0 (12)

As eq (12) holds for all S, we could come back to the strong formulation


βx∂R∂ t

= a ·αx ·R + γx(t)−n

∑i=1

β ix ·

∂Ti

∂ t+

n

∑i=1

a ·α ix ·Ti (13)

which is a first order ordinary differential equation that can be solved easily (even for extremelysmall times steps) from its initial condition.

These two steps must be repeated until convergence, that is, until verifying that both functions reacha fixed point.

The BEM is used to solve eq (9). We can notice that this equation defines a steady state ellipticequation with constant coefficients.

Numerical exampleWe considered a simple rectangular domain Ω = (0,1)× (0,1) and a time interval I = (0,1]. Thesource term is set to f (u) = u2 (1−u), the boundary conditions and the initial condition is set to anexact solution of this problem given by :

ure f (x, t) =eη(x,t)

2 + eη(x,t)

with η (x, t) = 1√2

(x + t√

2

)The domain boundary Γ consists of nΓ ×nΓ segments Γi. The time interval I is discretized by usingnτ nodes, uniformly distributed.

First we are analyzing the convergence rates as a function of the space discretization (i.e. nΓ). For allthe space meshes the time discretization (i.e. nτ ) is adapted in order to reach the maximum precision.

Figure 1 show the evolution of the L2 error in time and space as a function of the level of approxima-tion, that is, as a function of the number of functional couples Xi(x) ·Ti(t) involved in the approxima-tion of u(x, t) for different meshes. This error is defined by:

en =

∥∥∥∥∥n

∑i=1

Xi(x) ·Ti(t)−ure f (x, t)

∥∥∥∥∥L2

Ω×I

‖ure f (x, t)‖L2Ω×I

We can notice that for a given number of functional couples the error en decreases when nΓ increases,reaching an asymptotic value. For reducing the value of the error we must increase nΓ as well as thenumber of functional couples Xi(x) ·Ti(t) involved in the functional approximation. In the case of theexample here addressed we must consider 4 functional couples for reaching a quatratic convergencerate for 4 nΓ 16.

Figure 3 depicts functions X1 (x) ,T1 (t), X2 (x) ,T2 (t), X3 (x) ,T3 (t) for nΓ = 16 and nτ = 256.Finally, figure 2 depicts the unknown field u(x, t).

ConclusionThe proposed approach transforms an incremental BEM procedure into a decoupled one that needsthe solution of some steady problems defined in space (Poisson equation in the case here addressed),


1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6

log10(1/nΓ )

5.5

5.0

4.5

4.0

3.5

3.0

log 1

0(e

n)

2

nRS=1

nRS=2

nRS=3

nRS=4

Fig. 1: Evolution of the error en versus the space discretization for different levels of approximationn.

and some ordinary differential equations that only involve the time coordinate. Significant reductionof CPU time is expected due to the non-incremental nature of the proposed technique, as well as asignificant reduction of the amount of information to be stored. As shown by the numerical exemple,this technique seems specially well adapted for the treatment of non-linear transient BEM models.


[min=0.68, max=1.00] × 5.3e−01

0.700

0.750

0.800

0.850

0.900

0.950

1000

1.000

t=2.4e−01

[min=0.69, max=1.00] × 5.7e−01

0.700

0.750

0.800

0.850

0.900

0.950

1000

1.000

t=5.0e−01

[min=0.71, max=1.00] × 6.0e−01

0.750

0.800

0.850

0.900

0.950

1000

1.000

t=7.6e−01

[min=0.72, max=1.00] × 6.3e−01

0.760

0.800

0.840

0.880

0.920

0.960

1000

1.000

t=1.0e+00

Fig. 2: u(x, t) and ure f (x, t) (dashed line) for t = 0.24s,0.5s,0.76s,1s, nΓ = 4 and nτ = 256


-0.900

-0.750

-0.600

-0.450

-0.300

-0.150

X1(x)

0 1.00.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2×1.7e−03 T1(t)

-0.200

0.000

0.2000.400

0.600

0.800

X2(x)

0 1.00.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2×6.7e−04 T2(t)

-0.900

-0.750

-0.600

-0.450

-0.300

-0.150

0.000

0.000

X3(x)

0 1.01.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6×5.5e−05 T3(t)

Fig. 3: Functional couples Xi (x) ,Ti (t) for nΓ = 16 and nτ = 256


References[1] F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, and F. Boust. Alleviating mesh constraints:

Model reduction, parallel time integration and high resolution homogenization. Computer Meth-ods in Applied Mechanics and Engineering, 197(5):400 – 413, 2008.

[2] A. Ammar, F. Chinesta, and P. Joyot. The nanometric and micrometric scales of the structure andmechanics of materials revisited: An introduction to the challenges of fully deterministic numeri-cal descriptions. International Journal for Multiscale Computational Engineering, 6(3):191–213,2008.

[3] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes ofmultidimensional partial differential equations encountered in kinetic theory modeling of complexfluids. Journal of Non-Newtonian Fluid Mechanics, 139(3):153 – 176, 2006.

[4] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes ofmultidimensional partial differential equations encountered in kinetic theory modelling of com-plex fluids: Part ii: Transient simulation using space-time separated representations. Journal ofNon-Newtonian Fluid Mechanics, 144(2-3):98 – 121, 2007.


Three-dimensional Boundary Elements for the Analysis of AnisotropicSolids

Federico C. Buroni†,1, Jhonny E. Ortiz2, Andrés Sáez3

School of Engineering, University of Seville, Camino de los descubrimientos s/n, E41092 Sevilla, [email protected], [email protected], [email protected]

Keywords:Green’s function; Fundamental solutions; Cauchy’s residue theory; Degenerate materials; Anisotropy;Boundary element method (BEM)

Abstract.An alternative boundary element formulation for the analysis of anisotropic three-dimensional

(3D) elastic solids is presented. This numerical procedure uses explicit expressions for the fundamentalsolution displacements and tractions developed by others authors by means of Stroh formalism andCauchy’s residue theory, plus it implements a multiple pole residue approach to additionally account forthe case of mathematically degenerate materials when Stroh’s eigenvalues are coincident. Meanwhile,the numerical instabilities that may be observed in quasi-degenerate cases when Stroh’s eigenvaluesare nearly equal are addresed as well. Thus, an explicit Boundary Element Method (BEM) approachfor the numerical solution of 3D linear elastic problems for solids with general anisotropic behavior isdeveloped and validated with some numerical examples.

Introduction

Meanwhile for the two-dimensional anisotropic case many works have been reported in the literature,the three-dimensional (3D) modeling of general anisotropic elastic solids with BEM has not beenstudied so profoundly. The main difficulties deal with the numerical treatment of rather complexfundamental solutions. Ting & Lee [1] deduced a unique generic explicit expression in terms ofthe Stroh’s eigenvalues which is valid for mathematical degenerate cases, but the computation of theGreen’s functions derivatives was not addressed in their work. Sales & Gray [2] and Phan et al. [3]later presented a multiple pole residue approach for the numerical evaluation of the Green’s functionand its derivative which covers all the mathematical degenerate cases. However, this set of solutionshas never been implemented into a BEM code to the best of the authors’ knowledge.

By considering the integral expressions developed by Barnett [4] and applying the Cauchy’s residuetheory, Lee [5] deduced explicit expressions for the first- and second-order derivatives of the Green’sfunction in terms of the Stroh’s eigenvalues for non-degenerate materials. The resulting expressions forthe first-order derivative have been recently implemented and validated in the work of Shiah et al. [6]and even more recently implemented in a BEM code by Tan et al. [7]. In these works the degenerateproblem is overcome by introducing a small perturbation to the repeated Stroh’s eigenvalues.

The aim of the present work is to develop an alternative approach to numerically evaluate explicit3D anisotropic displacement and traction fundamental solutions, both for mathematical degenerateand non-degenerate materials, and validate their implementation into a general BEM code. In thiswork, a multiple pole residue approach is proposed to treat mathematical degenerate cases based inthe Ting & Lee’s [1] and Lee’s solution [5]. First the integral-form of the fundamental solutions arepresented. Then, and starting from these integral-form solutions, expressions for the explicit evaluationof the displacement and traction fundamental solutions in mathematical degenerate and non-degeneratematerials are developed. Validation of the proposed fundamental solutions in degenerate cases arepresented. The boundary element formulation is illustrated by solving a benchmark example andfinally the conclusions of the work are presented.

Integral-form of displacement fundamental solution

Let U∗jk(x,x

′) the displacement fundamental solution (free-space Green’s function) at a point x due toa unit force applied at point x′. The tensor U∗

jk depends on the relative vector x − x′ so, henceforth


it is considered that the Cartesian coordinate system xi has the origin at the point x′ for simplicity.Following Ting & Lee’s approach [1], the Green’s function can be expressed as a singular term by amodulation function H as

U∗jk(x) =

1

4πrHjk(x) (1)

where x = re with r = |x|. The modulation function Hjk(x) depends on the direction of x but not on itsmodulus, so Hjk(x) = Hjk(e) and that is one of the three Barnett-Lothe tensors which is symmetric,positive-definite and H(e) = H(−e). Hence, U∗(x) is also symmetric and U∗(x) = U∗(−x). Thetensor Hjk can be evaluated as [1]

Hjk(e) =1

π

∫ +∞

−∞Γ−1jk (p)dp (2)

withΓjk(p) = Qjk + (Rjk +Rkj)p+ Tjkp

2

expressed in terms of the parameter p, and being

Qjk = Cijklninl; Rjk = Cijklniml; Tjk = Cijklmiml (3)

where ni and mi are the components of any two mutually orthogonal unit vectors such that n,m, eis a right-handed triad. Note that Qjk and Tjk are symmetric and positive-definite.

It is well-known that the kernel in (2) is a single-valued analytic function except for three complexpoles with positive imaginary part and their conjugates. These poles correspond to the roots of thesixth-order polynomial equation

|Γ(p)| = 0 (4)

and in the Stroh’s formalism context are the Stroh’s eigenvalues pα [8]. The determinant in (4) canbe factorized as [1] |Γ(p)| = |T|∏3

ξ=1(p− pξ)(p − pξ) with T as defined in (3), the overbar · denotingcomplex conjugate and pξ = αξ + iβξ, βξ > 0 ∧ αξ, βξ ∈ R with i =

√−1.Let Γjk be the adjoint of Γjk, defined as Γpj(p)Γjk = |Γ(p)|δpk where δpk is the Kronecker delta,

then Hjk can be expressed as [1]

Hjk(e) =1

π|T|∫ +∞

−∞

Γjk(p)3∏

ξ=1

(p− pξ)(p− pξ)

dp (5)

In Ting & Lee’s work [1] the above integral for the Green’s function is further reduced to a generalexplicit expression in terms of the Stroh’s eigenvalues. However, direct derivation of the resultingexpression is somehow cumbersome, so that an alternative approach is needed for the numerical im-plementation of the derivatives, as next shown [5].

Integral-form of traction fundamental solution

The traction fundamental solution follows easily from the derivative of the free-space Green’s functionas

T ∗ik = CijlmU∗

lk,mηj (6)

where ηj are the components of the external unit normal vector to the boundary ∂Ω at point xand Cijkm denote the components of the fourth-rank elasticity tensor. The first-order derivative of theGreen’s function may be expressed in a similar way to equation (1), as a singular term by a modulationfunction which only depends on e as

U∗pj,q(x) =

1

4πr2U∗pj,q(e) (7)


where, according to Lee’s approach [5], the modulation function is given by

U∗ij,l(e) = −elHij +

Cpqrs

π(Mlqiprj es +Msliprj eq) (8)

where the Msliprj integrals have the following representation in terms of the parameter p [5]

Mijklmn =1

|T|2∫ +∞

−∞

Φijklmn(p)

(p− p1)2(p− p2)2(p− p3)2dp (9)

where T has been previously defined in (3), pα are the Stroh’s eigenvalues and the function

Φijklmn(p) :=Bij(p)Γkl(p)Γmn(p)

(p− p1)2(p− p2)2(p − p3)2(10)

has been introduced together with the definition of

Bij := ninj + (nimj +minj)p+mimjp2 (11)

The Φijklmn(p) function is analytic everywhere in the upper half-plane ((p) > 0) and the kernel in theintegral (9) has three complex-double poles with positive imaginary part corresponding to the roots of|Γ(p)|2 = 0.

Because of the symmetry of the adjoint matrix Γ, the components Mijklmn satisfy the followingsymmetry conditions Mijklmn = Mijmnkl = Mijlkmn = Mijklnm. The matrix Bij is also symmetric,resulting in an additional symmetry Mijklmn = Mjiklmn. This leads to a considerable reduction inthe number of components Mijklmn to be calculated, and it must be considered in the numericalimplementation. Mijklmn represents 729 components, but only 126 of them are different.

It is necessary to remark that in the Lee’s work [5] explicit expressions in terms of Stroh’s eigenvaluesare further obtained, but only for mathematical non-degenerate cases (p1 = p2 = p3).

Multiple pole residue approach

Both the integrals in the Barnett-Lothe tensor Hij (e), Equation (5), and the integrals Mijklmn (e) inthe modulation function of the derivative of the Green’s function, Equation (9), can be evaluated byCauchy’s residue theory. This approach leads to explicit expressions in terms of the Stroh’s eigenvalues.

From a numerical point of view, it is actually necessary to provide a scheme for Green’s functionevaluation of general validity. Such scheme should be able to deal with both mathematical degenerateand non-degenerate materials. The degenerate case occurs when repeated Stroh’s eigenvalues areobtained, and this may happen due both to the material properties Cijkm and the direction of thevector x. Furthermore, numerical instabilities should be avoided in quasi-degenerate cases as well, whenStroh’s eigenvalues are sufficiently close. In this work, a multiple pole residue approach is proposedin order to overcome such degeneracies and obtain accurate results, leading to new expressions for theexplicit evaluation of the derivative of the Green’s function.

In order to compute an integral of a rational function of complex variable of the form∫ +∞−∞ f(p)dp,

calculus of residues yields ∫ +∞

−∞f(p)dp = 2πiσ (12)

where σ is the sum of the residues Res(po) of f(p) at the poles po which lie in the upper half plane.Let the point po be a pole of m-order of f(p). A formula for evaluating the residue at this pole is givenas (see, e.g., [3, 10])

Res(po) =1

(m − 1)!limp→po

dm−1

dpm−1[(p− po)

mf(p)] (13)

Equation (13) allows to write explicit solutions in terms of the Stroh’s eigenvalues for degenerate andnon-degenerate cases.


Table 1: Material A. Transversely isotropic symmetryElastic constants

[109 N

m2

]C1111 C1122 C3333 C2323 C1133

49.4 34.6 30(C1133C1122

)2√C1111C3333−C1133

2 9.7

Although in Ting & Lee’s work [1] a solution valid for degenerate cases has been derived for thedisplacement fundamental solution, in the present work a multiple pole residue approach is proposedalso for the displacement solution for the sake of completeness, in a similar fashion to the works byWang [9] and Phan et al. [3].

At most, there are N (1 N 3) distinct Stroh’s eigenvalues pα of mα-multiplicity. Hence, ageneral expression, valid for degenerate and non-degenerate materials, of the Barnett-Lothe tensor Hjk

may be obtained as

Hjk(e) =2i

|T|N∑

α=1

1

(mα − 1)!

⎡⎢⎢⎢⎢⎢⎢⎣

dmα−1

dpmα−1 Γjk(p)

(p − pα)mα

N∏ξ=1ξ =α

[(p − pξ)(p − pξ)]mξ

⎤⎥⎥⎥⎥⎥⎥⎦at p=pα

(14)

Similarly, general expressions both for degenerate and non-degenerate materials may be derived forthe Mijpkmn components to yield

Mijpkmn(e) =2πi

|T|2N∑

α=1

1

(2mα − 1)!

⎡⎢⎢⎢⎢⎢⎢⎣

d2mα−1

dp2mα−1 Φijpkmn(p)

N∏ξ=1ξ =α

[(p − pξ)]2mξ

⎤⎥⎥⎥⎥⎥⎥⎦at p=pα

(15)

Validation and a BEM example

In order to assess and validate the proposed scheme, a material with transverserly isotropic symmetryis next considered, since it exhibit degeneracy [11] and analytic closed-form solutions are available [12].An artificial Material A is considered which has two repeated Stroh’s eigenvalues for any directionof the evaluation vector x, except for evaluation points x along the isotropy axis (x3-axis) where thedegeneracy (p1 = p2 = p3) is observed. The elastic constants for Material A are listed in Table1. Moreover, for transversely isotropic materials it is satisfied C1212 = C1111−C1122

2 , C1313 = C2323,C2222 = C1111, C2233 = C1133.

Observation points x = 0, sin φ, cos φ are considered. In Figure 1(a) the modulus of the differencebetween the Stroh’s eigenvalues |pα−pβ| (α = β), in function of the φ-angle with respect to the x3-axis,are shown for the Material A. The degeneracy behavior previously mentioned is clearly observed. Hence,this material is degenerate anywhere and non-degenerate solutions are not defined. Arbitrarily, U∗

22

and T ∗22 components are considered. Figures 1(b) and 1(c) show these components of the fundamental

solutions, respectively, for r = 1 and an outward unit normal defined by η = 0, sin φ, cos φ.The displacement fundamental solution component U22 is evaluated by using Equations (14) (as-

suming that p1 = p2 = p3) and (assuming that p1 = p2 = p3). In the same way, the tractionfundamental solution component T ∗

22 is evaluated assuming the p1 = p2 = p3 degeneracy and assumingthat p1 = p2 = p3 by using Equations (14) and (15). The analytic solution by Pan & Chou [12] isconsidered for comparison and plotted together with the present solution. The proposed degeneratesolutions provide a stable and accurate solution in function of the kind of the mathematical degeneracy.

The proposed multiple pole residue approach is implemented in a three-dimensional collocation-BEM code. Nine-node quadratic boundary elements are used. A simple example is presented in order


! "#

"$%&

"$%&

&

! "#

"$%&

"$%&

&

'(

')(

'*(

Figure 1: (a) Absolute values of difference between eigenvalues, (b) U∗22 and (c) T ∗

22 components vs.φ-angle for Material A.

to futher validate the correctness of the expressions for the displacement fundamental solutions U∗ij and

the traction fundamental solutions T ∗ij developed in the present work. A cube of length a subjected to

uniaxial tension σ0 is considered, as sketched in Figure 2. Symmetry boundary conditions are imposedon the coordinate planes, i.e., u1|x1=0 = 0, u2|x2=0 = 0 and u3|x3=0 = 0. The cube is discretized usingone element per face, totalizing 6 elements and 26 nodes. Two set of materials are considered. First,an isotropic material (Material B) with the elastic constants listed in Table 2. In such a case, a tripleeigenvalue p0 = i, ∀x exists. The previously defined Material A is considered as well. The presentformulation is robust enough to model both degeneracies.

The displacements values obtained at points A, B, C, D and E shown in Figure 2 are listed inTables 3 and 4 for the two materials B and A, respectively. The values are normalized and comparedwith the corresponding analytic solutions in both Tables. Excellent accuracy is observed for all thecases.

Figure 2: Cube under tension stress.


Table 2: Material B. Isotropic materialElastic constants

[109 N

m2

]C1111 C1122

15.7264 6.4957

Table 3: Results of BEM example. Cube undersimple tension with isotropic material properties (Material B).

Point Coordinates Result u1/ua3 u2/u

a3 u3/u

a3

A (a, 0.5a, 0.5a) Present work -0.14615 -0.14615 1.00001Analytic solution -0.14615 -0.14615 1.00000

B (a, a, 0.5a) Present work -0.29231 -0.14615 1.00001Analytic solution -0.29231 -0.14616 1.00000

C (0.5a, a, a) Present work -0.58463 -0.58463 1.00002Analytic solution -0.58463 -0.58463 1.00000

D (a, 0.5a, a) Present work -0.14615 -0.29231 1.00001Analytic solution -0.14615 -0.29231 1.00000

E (a, a, a) Present work -0.29231 -0.29231 1.00001Analytic solution -0.29231 -0.29231 1.00000


A multiple pole residue approach has been proposed to derive explicit expressions for the numericalevaluation of the 3D anisotropic fundamental solutions. These expressions are based on Ting & Lee’s[1] and Lee’s [5] solutions for the Green’s function and its derivative, and result in terms of the Stroh’seigenvalues, so there is no integration needed in the proposed approach. The solution covers all thepossible mathematically degenerate and non-degenerate materials, while being numerically stable forquasi-degenerate cases when Stroh’s eigenvalues are nearly equal. Subsequently, an explicit BEMimplementation for modeling three-dimensional materials with general anisotropic behavior has beenpresented. In contrast with previous works, the media is allowed to be both mathematically degenerateor non-degenerate, and the formulation is independent of the approximation order of the physicalvariables and geometry.

Table 4: Results of BEM example. Cube under simple tensionwith degenerate transversely isotropic material properties (Material A).

Point Coordinates Result u1/ua3 u2/u

a3 u3/u

a3

A (a, 0.5a, 0.5a) Present work -0.23061 -0.11535 0.99929Analytic solution -0.23095 -0.11547 1.00000

B (a, a, 0.5a) Present work -0.23074 -0.23079 0.99929Analytic solution -0.23095 -0.23095 1.00000

C (0.5a, a, a) Present work -0.05772 -0.11546 0.99936Analytic solution -0.05774 -0.11547 1.00000

D (a, 0.5a, a) Present work -0.11539 -0.05774 0.99924Analytic solution -0.11547 -0.05774 1.00000

E (a, a, a) Present work -0.11553 -0.11551 0.99995Analytic solution -0.11547 -0.11548 1.00000


Acknowledgments

This work was supported by the Ministerio de Ciencia e Innovación of Spain and the Consejería deInnovación, Ciencia y Empresa of Andaluca (Spain) under projects DPI2007- 66792-C02-02 and P06-TEP-02355. J.E. Ortiz was supported by the Ramón y Cajal Program of the Ministerio de Ciencia eInnovación of Spain. F.C. Buroni is gratefully acknowledged to the Junta de Andalucía of Spain forfinancial support throughout the Excellence Scholarship Program.

References

[1] Ting TCT, Lee VG. The three-dimensional elastostatic Green’s function for general anisotropiclinear elastic solids Q. J. Mech. Appl. Math.1997; 50:407-426.

[2] Sales MA, Gray LJ. Evaluation of the anisotropic Green’s function and its derivatives Computersand Structures 1998; 69:247-254.

[3] Phan AV, Gray LJ, Kaplan T. Residue approach for evaluating the 3D anisotropic elastic Green’sfunction: multiple roots. Engineering Analysis with Boundary Elements 2005; 29:570-576.

[4] Barnett DM. The precise evaluation of derivatives of the anisotropic elastic Green’s functions.Phys. Stat. Sol. (b) 1972; 49, 741-748.

[5] Lee VG. Explicit expressions of derivatives of elastic Greens functions for general anisotropicmaterials. Mech. Res. Comm. 2003; 30, 241-249.

[6] Shiah YC, Tan, CL, Lee VG. Evaluation of Explicit-form Fundamental Solutions for Displacementsand Stresses in 3D Anisotropic Elastic Solids. Computer Modelling in Engineering and Sciences2008; 34:205-226.

[7] Tan CL, Shiah YC, Lin CW. Stress analysis of 3D generally anisotropic elastic solids using theboundary element method Computer Modelling in Engineering and Sciences 2009; 41:195-214.

[8] Ting TCT. 1996 Anisotropic Elasticity, Oxford University Press, Oxford.

[9] Wang CY. Elastic fields produced by a point source in solids of general anisotropy. Journal ofEngineering Mathematical 1997; 32:41-52.

[10] Sveshnikov A, Tikhonov A. 1978 The theory of functions of a complex variable, 2nd edn. Moscow:Mir Publishers

[11] Tanuma K. Surface-impedance tensors of transversely isotropic elastic materials.Quarterly Journalof Mechanics and Applied Mathematics1996;49:29-48.

[12] Pan Y, Chou T. Point force solution for an infinite transversely isotropic solid. Journal of AppliedMechanics and Transactions ASME1976; 98:608-612.


Sensitivity analysis of cracked structures with static and dynamic Green’s functions

Oliver Carl, Chuanzeng ZhangDepartment of Civil Engineering, University of Siegen,

Paul-Bonatz-Str. 9-11, 57076 Siegen, Germany,

E-mails: [email protected], [email protected]

Keywords: Green’s functions, sensitivity analysis, linear elastic fracture mechanics, spring model

Abstract. In this paper, a simplified analytical method for sensitivity analysis of cracked or damaged structures is presented. The method enables the prediction of the difference between the solutions for an uncracked and a cracked structure by considering only the cracked or damaged region of a structure, which results in a local analysis and consequently lower computational effort. The method is based on the comparison between the strain energies of an uncracked and a cracked structure as well as the exact or approximate Green's functions of the problem.

Introduction From the point of view of structural mechanics, a static or dynamic analysis of a structure is nothing else than the establishment of a correlation between the external loading (action or input) and the structural response to the loading (reaction or output). For linear problems, the correlation can be described by the so-called Green's functions. If a structural or material change appears in a part of a structure, such as stiffness changes and cracking of the structural components during the manufacturing process or under in-service loading conditions, the static and dynamic responses or the problem solutions of the structures will be also altered. In the linear structural analysis, the corresponding new solutions can be found by using modified Green's functions. However, such Green’s functions have to be formulated and derived for the cracked or modified structures, which requires a new global analysis of the structures and will be hence computationally expensive. In this paper, sensitivity analysis by the Green’s function (SAGF) approach [4, 12], derived for the local analysis of stiffness modifications and loss of supports, is applied to static and dynamic problems. Furthermore stiffness modifications in structures due to damage and cracking are modeled by the concept of linear elastic fracture mechanics and applied to beam-like structures. For analytical solutions, cracked regions of beam or truss structures are approximated by spring models based on linear elastic fracture mechanics. The analytical solutions are suitable for sensitivity analysis of weakened or cracked beam and truss structures consisting of homogeneous materials or fiber-reinforced composite like reinforced concrete beams and bars.For more complicated problems, sensitivity analysis is applied to derive cracked finite elements based on cracked or weakened Green's functions. Then, the cracked finite elements can be utilized for the numerical solutions of the more complicated problems in engineering applications.

Global analysis by Betti’s theorem and Green’s functions A two-dimensional linear elastic problem namely the plane stress state in a bounded body with a continuous boundary D N is considered. The displacement vector iuu satisfies the following partial differential equations

, , , , 1, 2 ,ij j i i jj j ji i iL u u u u u p i j (1)

with Lamé operator ijL L , inertia force iu according to d’Alembert and body force vector ip .

Additionally we consider the following boundary conditions * *( , ) on , on i i D ij j i Nu t u n t x , (2)


and to complete the problem formulation the following initial conditions at time 0t 0 0( ,0) and ( ,0)i i i iu u u v x x . (3)

The stress tensor ij is defined by Hooke’s law

2ij kk ij ij , (4)

and the linearized strain tensor ij by

, ,12ij i j j iu u (5)

with the isotropic elastic constants and . An equivalent formulation of the initial boundary value problem is the variational equation of motion [2] with the test function ˆ ˆiuu for finding a function

Vu for all t such that *ˆ ˆ ˆ ˆ

N

ij ij i i i i i i Nd u u d p u d t u d

. (6)

Considering a structure under time-harmonic loading with an excitation frequency , i tt e q x q x , (7)

the displacements are also time-harmonic or in a steady state , i tt e u x u x . (8)

From eq (6) we obtain Green’s first identity or the principle of virtual displacements allowing for homogeneous initial conditions

2 *ˆ ˆ ˆ ˆN

ij ij i i i i i i Nd u u d p u d t u d

. (9)

In this case, Betti’s reciprocal theorem for the Lamé operator ijL L leads to a global analysis, in which

we have to consider the complete structure regardless whether the domain is uncracked or cracked 2 2ˆˆ ˆ ˆ ˆ ˆ, ( ) ( ) ( ) 0B d d d

u u Lu u u tu ut u Lu u . (10)

To obtain an integral representation for the displacements u x at a point 1 2,x xx , we substitute for uthe associated Green’s functions ,j

iG y x into the reciprocal theorem. The Green’s functions are the solution of the dual problem

2, , on j j ji i i yL G y x G y x y x I (11)

and satisfy homogeneous boundary conditions on D and N . To represent a point load ( 0i ) in

jx direction acting at point x , we introduce the Dirac delta function ji y x and the identity matrix I .

Substituting for u the solution of eq (1) with 0u on D , we find from Betti’s reciprocal theorem

*0 0, , ,

N

j i t j i tj y yu t e d e d

x G y x q y G y x t y (12)

To obtain an integral expression for the stresses we find analogously *

1 1, , ,N

jk i t jk i tjk y yt e d e d

x G y x q y G y x t y (13)

where the lower subscript 1i denotes that 1 ,jkG y x are the Green’s functions based on first-order derivatives of the Dirac delta function [12]. Exact closed-form Green’s functions from eq (11) can be obtained only for some special problems using for example Laplace- or Fourier transform. For more complicated problems we have to use approximate Green’s functions , ,j

i hG y x . A tool for this purpose is the boundary element method [11] or the finite element method. Working with the finite element method the displacement fields representing the Green’s functions are computed by nodal loads based on Dirac delta function or their derivatives [12]. To improve the approximated Green’s functions from FE analysis the Green’s function decomposition approach is suggested [9]. The method is originally developed for point-wise error estimation and adaptivity


in finite element method. The Green’s functions are decomposed into the known fundamental solutions and the unknown regular solutions, which will be found by a FE calculation

, ,, , ,j j ji h i R h G y x g y x u y x . (14)

Sensitivity analysis by Green’s functions for the local analysis of damaged regions In this section, we derive an analytical method for sensitivity analysis of cracked or weakened structures, in which we only need to consider the damaged regions of a structure to predict changes in the deformations, stresses, eigenfrequencies or mode shapes. This method leads to a lower computational effort in contrast to the classical procedure in structural mechanics. The basic idea of sensitivity analysis by the Green’s functions (SAGF) approach is based on the weak-form eq (9) for an uncracked system

2 *ˆ ˆ ˆ ˆN

ij ij i i i i i i Nd u u d q u d t u d

. (15)

Taking the stiffness modifications due to damage into account, the weak-form for a cracked system with additional symmetric terms of the damaged region c can be written as

2 2ˆ ˆ ˆ ˆ2 2

c c

c c c c c ckk ij ij ij kk ij ij ij c i i i i cd d u u d u u d

*ˆ ˆ .

N

i i i i Nq u d t u d

(16)

The displacement vector of the cracked system is denoted by c u u u , and the mass and stiffness modifications are represented by and the modified elastic constants and . After some mathematical manipulations we obtain the difference of the strain energies and inertia terms of the original (uncracked) system and the modified (cracked) system

2 2ˆ ˆˆ ˆc c

c cij ij i i ij ij c i i cd u u d d u u d

, (17)

with 2ij kk ij ij and 2c c cij kk ij ij . Substituting the virtual displacements u with

the associated Green’s functions jiG we obtain the central equation of the approach

2 2j ji i

c c

G Gj js k i c c c i cJ u E E d d d d

u G u G , (18)

where sE and kE represent the changes in strain and kinetic energy. Moreover we know from [12], that the changes in deformations or internal forces at a position x can be expressed by a linear functional cJ u J u J u . Equation (18) is superior to Betti’s reciprocal theorems eq (12) and (13).

It should be noted that eq (18) can also be rewritten as 2 2

,

j ji i

c c c c

G Gj jc c c i c c c i c cd d d d

u G u G , (19)

in which a dot denotes the scalar product of two vectors or matrices and 2ij kk ij ij . To

apply the derived method we must know the solution cu of a cracked problem and the Green’s functions for the uncracked system or vice versa. But this is not very practical, because if we solve the boundary value problems for both systems, we are also able to compare the two solutions u and cu directly. Using the following approximation, we only consider terms of the original (uncracked) system to predict response changes due cracks or damages

2ji

c c

G jc i cJ u d d

u G . (20)

If we apply the SAGF approach to crack problems, then the mass density will not change in the damaged region and with 0 we have

ji

c

GcJ u d

. (21)


Beam problems and approximation of cracks with spring models In beam problems the cracks are approximated by spring models. For mode-I cracks we use rotational springs and for mode-II cracks translational springs as depicted in Fig. 1. The spring stiffnesses c and wc are obtained by using the concept of linear elastic fracture mechanics and derived from the energy release rate [10]. The compliances or inverse spring stiffnesses are

1 2

2 2

21 I

a

bc K daE M

and

1 22 2

21w II

a

bc K daE Q

, (22)

with elastic modulus ,E Poisson’s ratio , beam width ,b bending moment ,M transverse shear force Q ,and the stress intensity factors depending on the crack configuration [14].

M M

ah

(mode I)

ah

Q

Q

(mode II)

Q

QMM

cwc

Fig. 1 Approximation of cracks for different kinds of loading and crack configurations by spring models.

Cracking in fiber-reinforced composites like reinforced concrete beams can also be approximated by spring models, Fig. 2. The spring stiffness of a rotational spring used for bending cracks is split into a concrete and a reinforcement term

, ,concrete ,reinforcementj j jc c c , (23)in which the stiffness of the cracked concrete is modeled by eq (22) also taking the interaction of multiple cracks into account [6, 15]. The spring stiffness of the reinforcement depends on some geometrical considerations in combination with the steel’s strain, the cross-sectional area, the details of the reinforcement, and the crack width [5].

MM

,2c

M M

,1c ,3c

Fig. 2 Reinforced concrete beam with bending cracks approximated by rotational springs.

Applying eq (18) to cracked beam-like structures with spring models, we obtain the change in deflection, slope, bending moment or transverse shear force ( 0,1, 2, 3i ) due to cracking in an initially uncracked system as

, ,, , , ,

, ,

1 1i c i c

ii ic

c j G c j c j G c ji i ij j w j

ww w M x M x Q x Q xx x x c c

!! !

! ! ! " (24)

with the bending moment and the shear force resulting from the load case of the uncracked system, and the bending moment and the shear force from the associated Green’s functions ,i cG for the cracked system, which are, similar to eq (21), approximated by the Green’s functions iG of the uncracked system at the position cx of the crack number j .

Closed-form solutions. For Euler-Bernoulli beams, closed-form analytical solutions are obtained by using static or dynamic Green's functions for uncracked beams, which are modified and include terms taking


cracks into account. Applying the SAGF approach in combination with the superposition principle it is also possible to take stiffness jumps into account and add new supports if they are elastic and modeled by springs. Furthermore for computations with the theory of plastic limit analysis of continuous beams in which plastic hinges are modeled by rotational springs with spring stiffness 0.c # This means an extension of the closed-form solutions for continuous beams under static loadings from [8] and for dynamic problems from [1].

Numerical solutions. For numerical solutions of more complicated beam-like structures, modified beam elements based on Green’s functions of a cracked clamped-clamped beam as shown in Fig. 3 are derived and can be easily integrated into standard FE codes. Considering a mode-I crack, e.g. the entry 33

ck of the modified element stiffness matrix is given by

2 3

33 33 33 2 2 3

3 44 3 3

cc

c c

EI x c l EIk klEI l x l x c l

$

. (25)

Comparing the modified element stiffness matrix with that from [3], in which cracks are also approximated by spring models and their degrees of freedom are eliminated by Guyan’s reduction, we find that the two solutions are identical.

1 1w 1 23cM k

2 26cM k

2 25cQ k

1 22cQ k

1 1

1 33cM k 2 36

cM k

1 32cQ k 2 35

cQ k

cx cl x

c

c

cracked Green's functions

11 14

22 23 25 26

33 35 36

44

55 56

66

0 0 0 00 0

00 0

.

c c

c c c c

c c c

c c

c c

c

k kk k k k

k k kk

sym k kk

% &% &% & % &% &% &% &% &

K

modified stiffness matrix

Fig. 3 Cracked Green’s functions to obtain the modified stiffness matrix of a cracked beam element.

Numerical Examples Example 1: Cracked dynamic Green’s functions of a two-span beam for closed-form analytical solutions

The cracked static or dynamic Green’s functions of a two-span beam can be derived from the Green’s functions ,iG y x of an uncracked single-span beam extended by terms taking cracks into account as shown in eq (24)

, , , , ,, ,

1 1( , ) ( , )i ii c i c j G c j c j G c j

j j w j

G y x G y x M x M x Q x Q xc c

" , (26)

where y denotes the field point and x indicates the source point of the load. By applying the geometrical compatibility condition 0, 0,0 ( , ) ( , )B c B c B Bw x G x x G x x B at the position

Bx of the support B, we obtain the support reaction force as

0,

0,

( , )( , )

c B

c B B

G x xB

G x x . (27)

Now the cracked Green’s functions of a two-span beam (TSB) using the superposition principle are obtained as

, , ,( , ) ( , ) ( , )TSBi c i c i c BG y x G y x B G y x . (28)


To verify the presented method we consider a two-span beam with three equal parallel edge cracks under a time-harmonic loading with the frequency 2 2 3f' ' Hz, 0 50F kN, rectangular cross-section and span length 10L m.

3 equal parallel edge cracks

0( ) i tF t F e =

.EI const=1c 2c 3cL2L 2L

crack distance = crack depth s a( )F t

Fig. 4 A two-span beam with three equal parallel edge cracks under time-harmonic loading.

Fig. 4 shows the first and second eigenfrequency for a ratio of the crack-depth to the beam size / 0,2a h from closed-form analytical solutions in comparison to a two-dimensional FE calculation with ANSYS using crack-tip elements. Furthermore the results of deflection, bending moment and transverse shear force over the beam length for / 0,5a h obtained by using closed-form cracked Green’s functions in eq (28) (exact) and approximate solutions in eq (21) (approximate) are presented in Fig. 4. In this example the translational springs are neglected, because their influences on deformations and internal forces are very small (< 1%).


The improvement of the approximate solutions in Fig. 4 (dashed lines) for more complicated problems is in progress. To achieve a higher accuracy of the approximate solutions a recurrence relation based on the SAGF approach could be applied, which is similar to applications for linear discrete systems, where the inverse of a modified stiffness matrix is approximated by a Taylor expansion [7].

Example 2: Two-dimensional cracked cantilever plate under a self-weight load

x

y

11

10,0ml =

5,0mh =

a

crack

Fig. 5 Cracked cantilever plate under a self-weight load and stress changes xx and yy in section 1-1.

To show the sensitivity of the stresses xx and yy in section 1-1 to a crack of length a as shown in Fig. 5, we model the cracked region as a loss of support stiffness. From the SAGF approach eq (18) we obtain


1xx c G a

a

d x u y A y , (29)

with the support reactions 1GA from the uncracked approximate Green’s functions calculated by a two-

dimensional FE analysis, and the displacement vector cu of the cracked system approximated by beam models with springs ( c – only mode-I, [ wc c ]– mode-I and mode-II) to approximate the vertical and the horizontal nodal displacements.

Summary In this paper, a simplified analytical method for sensitivity analysis of cracked or weakened structures subjected to static or dynamic loading is presented. The basic idea of the method is to predict changes in deformations, stresses, eigenfrequencies and mode shapes from stiffness modifications by using the differences of strain energies of an uncracked and a cracked structure in combination with static or dynamic Green’s functions. For a given structure, the intensity of theses changes in deformations or internal forces depends on the number, size and location of cracks, and the external loads. If the Green’s functions are obtained exactly or approximately, then they can be used to obtain solutions for different kinds of load cases. The present approach has the advantage that we only need to consider the cracked regions of a structure to calculate its response changes, which leads to a lower computational effort. The extension of the approach to more complicated problems is in progress. In principle, the method can also be extended to nonlinear problems as presented in references [12, 13]. Potential applications of structural sensitivity analysis by the Green’s functions approach could be found for example in damage monitoring, crack identification and predictive maintenance of structures or structural elements.

References

[1] M. Abu-Hilal Journal of Sound and Vibration, 267, 191-207 (2003).

[2] J.D. Achenbach Wave Propagation in Elastic Solids, North-Holland (1973).

[3] A.S. Bouboulas and N.K. Anifantis Engineering Structures, 30, 894-901 (2008).

[4] O. Carl and F. Hartmann 3rd MIT Conference on Computational Fluid and Solid Mechanics, Boston, USA (2005).

[5] O. Carl and Ch. Zhang 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2010, Karlsruhe, Germany (2010).

[6] M.B. Civelek and F. Erdogan International Journal of Fracture, 19, 139-159 (1982).

[7] A. Deif Sensitivity Analysis in Linear Systems, Springer (1986).

[8] G. Failla and A. Santini International Journal of Solids and Structures, 44, 7666-7687 (2007).

[9] T. Grätsch and F. Hartmann Computational Mechanics, 37, 394-407 (2006).

[10] D. Gross and T. Seelig Fracture Mechanics, Springer (2006).

[11] F. Hartmann Introduction to Boundary Elements, Springer (1989).

[12] F. Hartmann and C. Katz Structural Analysis with Finite Elements, Springer (2007).

[13] F. Hartmann and T. Kunow Proceedings of the Ninth International Conference on Computational Structures Technology, Civil-Comp Press (2008).

[14] Y. Murakami Stress Intensity Factors Handbook, Volume 1 and 2, Pergamon (1987).

[15] L. Rohde, R. Kienzler and G. Herrmann Philosophical Magazine, 85, 4231-4244 (2005).


A D-BEM Approach with Constant Time Weighting Function Applied to the Solution of the Scalar Wave Equation

J. A. M. Carrer1 and W. J. Mansur2

1PPGMNE: Programa de Pós-Graduação em Métodos Numéricos em Engenharia, Universidade Federal do Paraná,

Caixa Postal 19011, CEP 81531-990, Curitiba, PR, Brasil email: [email protected]

2Programa de Engenharia Civil, COPPE/UFRJ,

Universidade Federal do Rio de Janeiro, Caixa Postal 68506, CEP 21945-970, Rio de Janeiro, Brasil

email: [email protected]

Keywords: scalar wave equation, DS-BEM, time weighting D-BEM

Abstract. A D-BEM approach, based on time weighting residuals, is developed for the solution of 2D scalar wave propagation problems. The basic equation of the proposed formulation is generated by weighting, with respect to time, the basic D-BEM equation, under the assumption of linear and cubic time variation for the potential and for the flux. A constant time weighting function is adopted. As the time integration reduces the order of the time derivative that appears in the domain integral, the initial conditions are directly taken into account. The potentialities of the proposed formulation are verified by the examples included at the end of the work. Introduction

The solution of time dependent problems by the Boundary Element Method (BEM) offers to the researcher a vast range of possibilities, according to the way the problem will be solved. In this work the scalar wave equation in 2D is solved by employing the so-called D-BEM formulation, D meaning domain. This formulation employs the static fundamental solution, instead of the time dependent fundamental solution. As a consequence, it is characterized by the presence of a domain integral whose kernel is constituted by the static fundamental solution multiplied by the second order time derivative of the potential (acceleration), e.g. Carrer et al. [1], Hatzigeorgiou and Beskos [2]. The selection of an adequate approximation to the acceleration, in order to perform the march in time, is a task that deserves attention. Although the Houbolt method [3] has been widely used and seems to be a good choice, alternative time marching schemes can be found in Carrer and Mansur [4], Souza et al. [5], Chien et al. [6]. Here, another approach is developed, in which the basic D-BEM equation is integrated from the initial time, say t0, to a specified final time, say tF. Over each time interval tn t tn+1 a constant time weighting function is adopted in a feature that can be identified as the subdomain collocation or the first approximation of the method of moments, e. g. Zienkiewicz and Morgan [7], Finlayson [8]. Initially, linear time approximation was assumed to both the potential and to the flux in the interval tn t tn+1. This approach, called DSL-BEM (D for the D-BEM formulation, S for the subdomain method and L for the linear time variation), did not produce reliable results, see Carrer and Mansur [9]. For this reason, another attempt to improve the results was carried out, in which a cubic time variation was assumed, in the interval tn2 t tn+1, for the potential and for the flux, thus generating the DSC-BEM approach (C meaning cubic). In order to avoid handling with values related to times previous to t0, this approach is not employed at the beginning of the analysis; in this way, the time marching process begins with the DSL-BEM approach and, after some time has elapsed, employs the DSC-BEM approach. The use of the DSL-BEM and DSC-BEM approaches constitutes the DS-BEM formulation, which produced the expected reliable results. It is important to mention that the contribution of the non-homogeneous initial conditions is directly taken into account, as the domain integral now contains only the first order time derivative of the potential. At the end of the article two examples are presented and discussed. In the examples, the DS-BEM results are compared with the corresponding analytical solutions, with the aim of verifying the potentialities of the proposed formulation.


Constant Time Weighting of the D-BEM Equation The basic integral equation of the D-BEM formulation is written as follows:

c() u(,t) =

u*(,X)p(X,t) d(X)

p*(,X)u(X,t) d(X)

1c2

u*(,X)u

..(X,t) d(X) (1)

where u*(,X) is the fundamental solution, p*(,X) is the normal derivative of the fundamental solution and the coefficient c() is the same of the static problem, see Carrer et al. [1]. In order to solve eq (1), initially it is assumed a linear time variation for both the potential and the flux at each time interval tn t tn+1. Subsequently, integration on time, interpreted as a time weighting statement, is carried out from the initial time t0 = 0 to the final time of analysis, say tF. With these comments in mind, noting that W(t) is the time weighting function, eq (1) is rewritten as:

c()

t0

tF W(t)u(,t)dt =

u*(,X)

t0

tF W(t)p(X,t)dt d(X)

p*(,X)

t0

tF W(t)u(X,t)dt d(X)

1c2

u*(,X)

t0

tF W(t)u

..(X,t)dt d(X) (2)

In the subdomain collocation or the first approximation of the method of moments, e. g. Zienkiewicz and Morgan [7], Finlayson [8], the time weighting function W(t) is chosen to satisfy:

W(t) = 1 if tn t tn+1

0 if t tn or t tn+1 (3)

With the use of the time weighting function defined by eq (3), the time integration is restricted to the interval tn t tn+1. In this interval one has: u(X,t) = n(t)un(X) + n+1(t)un+1(X) and p(X,t) = n(t)pn(X) + n+1(t)pn+1(X) (4) where n(t) and n+1(t) are the linear interpolation functions:

n(t) = tn+1 tt and n+1(t) =

t tn

t (5) In this work, only time intervals with constant length, t = tn+1 tn, were used. The substitution of eqs (3, 4) in eq (2), followed by the time integration, gives:

c() t2

un+1()

+ un() =

u*(,X)

t2

pn+1(X)

+ pn(X) d(X)

p*(,X)

t2

un+1(X)

+ un(X) d(X)

1c2

u*(,X)

u.

n+1(X) u.

n(X) d(X) (6)

In eq (6), the subscripts (n + 1) and n represent the time tn+1 = (n + 1)t and tn = nt. Note that the initial conditions can be directly applied in eq (6) at the beginning of the analysis, i.e. when the subscript n = 0. In this work, the boundary is approximated by linear elements and the domain, by triangular linear cells. After performing the boundary and the domain integrations, the matrix form of eq (6) is:

Hbb 0

Hdb I

ub

n+1 + ubn

udn+1 + ud

n

=

Gbb

Gdb

pbn+1 + pb

n 2

c2t

Mbb Mbd

Mdb Mdd

u

. bn+1 u

. bn

u. d

n+1 u. d

n

(7)


in which the superscripts b and d correspond to the boundary and to the domain, respectively. In the sub-matrices, the first superscript corresponds to the position of the source point and the second superscript, to the position of the field point. The identity matrix is related to the coefficients c() = 1 of the internal points.

In eq (7), the derivatives u. b

n+1 and u. d

n+1 are approximated by employing the backward finite difference formula below, see Zienkiewicz and Morgan [7]:

u.

n+1 = u

n+1 u n

t (8)

After substituting eq (8) in eq (7) and rearranging, yields:

(ct)2Hbb

+ 2Mbb 2Mbd

(ct)2Hdb

+ 2Mdb

(ct)2I

+ 2Mdd

ub

n+1

udn+1

=

(ct)2Gbb

(ct)2Gdb

pbn+1

(ct)2Hbb

0

(ct)2Hdb (ct)2I

ub

n

udn

+

(ct)2Gbb

(ct)2Gdb

pbn +

2Mbb 2Mbd

2Mdb 2Mdd

ub

n + t u. b

n

udn + t u

. dn

(9)

As usual in D-BEM formulations, the unknowns in eq (9) are the potential and the flux at the boundary and the potential at the domain . All the development described so far can be identified as the DSL-BEM formulation. The results provided by this formulation are not reliable, due to a significant presence of numerical damping, see Carrer and Mansur [9]. In this way, the search for another formulation became imperative. With the subdomain collocation, a natural choice was to assume a quadratic behaviour for u(X,t) and p(X,t) in an interval tn1 t tn+1; this assumption, however, was not useful, being characterized by instability in the results (not presented here). The adoption of the cubic Lagrange interpolation for u(X,t) and p(X,t), in an interval tn2 t tn+1, produced accurate results. This formulation, named DSC-BEM, is described in what follows. The time weighting function W(t), now, is chosen to satisfy:

W(t) = 1 if tn2 t tn+1

0 if t tn2 or t tn+1 (10)

and the time integration is restricted to the interval tn2 t tn+1. In this interval one has: u(X,t) = n2(t)un2(X) + n1(t)un1(X) + n(t)un(X) + n+1(t)un+1(X) (11) and p(X,t) = n2(t)pn2(X) + n1(t)pn1(X) + n(t)pn(X) + n+1(t)pn+1(X) (12) where the i(t) functions, i = (n 2), (n 1), n, (n + 1), are the Lagrange interpolation functions, see [9]. After substitution of eqs (10, 11, 12) in eq (2), and bearing in mind that the time integration is carried out from time tn2 to time tn+1, one has:

c() t

3

8 un2() +

98 un1() +

98 un()+

38 un+1() =

u*(,X) t

3

8 pn2(X) +

98 pn1(X) +

98 pn(X)+

38 pn+1(X) d(X)

p*(,X) t

3

8 un2(X) +

98 un1(X) +

98 un(X)+

38 un+1(X) d(X)

1c2

u*(,X)

u.

n+1(X) u.

n2(X) d(X) (13)


The first order time derivatives u.

n+1 and u.

n2 that appear in the domain integral in eq (13) are given by:

u.

n+1 = 1

6t 11un+1

18un + 9un1 2un2 and u.

n2 = 1

6t 2un+1

9un + 18un1 11un2 (14) One must note that the first expression in eq (14) is the well-known Houbolt approximation for the first order time derivative, see Houbolt [3]. This result was already expected, as the Houbolt method is obtained by cubic Lagrange interpolation of u = u(t) in the interval tn2 t tn+1. After substituting eq (14) in eq (13) and integrating over the boundary and the domain, the matrix form for the DSC-BEM formulation is written as:

(ct)2Hbb

+ 4Mbb 4Mbd

(ct)2Hdb

+ 4Mdb

(ct)2I

+ 4Mdd

ub

n+1

udn+1

=

(ct)2Gbb

(ct)2Gdb

pbn+1

(ct)2Hbb

0

(ct)2Hdb (ct)2I

3ub

n + 3ubn1 + ub

n2

3udn + 3ud

n1 + udn2

+

(ct)2Gbb

(ct)2Gdb

3pb

n + 3pbn1 + pb

n2 +

4Mbb 4Mbd

4Mdb 4Mdd

ub

n + ubn1 ub

n2

udn + ud

n1 udn2

(15)

The DSL-BEM, eq (9), is employed initially, as it takes into account the initial conditions directly and avoids handling with values related to times previous to t0; then, after a few time steps, the DSC-BEM, eq (15), is used. As mentioned at the Introduction to this work, the use of both formulations is referred to as DS-BEM approach. A measure of the time step is provided by the dimensionless variable t defined as follows:

t = ct

(16)

in which c is the wave propagation velocity, t is the time interval and is the length of the smallest element used in the boundary discretization. It is important to mention that the best values for t are determined empirically and differs according to the formulation employed.

Examples In what follows, the DS-BEM results are compared with the corresponding analytical solutions, see Stephenson [10], Kreyszig [11]. One-dimensional bar. This example consists of a one-dimensional bar defined in the domain 0 x a, 0 y a/2, fixed at one side (x = 0) and free at the other (x = a), subjected to the initial conditions given by:

u0(x) = Ux and u.

0(x) = 0 (17) The mesh, constituted of 48 boundary elements and 256 cells, is depicted in Fig. 1.

Figure 1. One-dimensional bar: boundary and domain discretization.


Results corresponding to the potential at boundary node A(a,a/4) and to the flux at boundary node B(0,a/4) are depicted, respectively, in Fig. 2 and Fig. 3. The time interval was selected by taking t = 2/3. The analytical solution is:

u(x,t) = 8Ua2

n = 1

!"

(1)n+1 cos

(2n 1)ct

2a sin

(2n 1)x

2a(2n 1)2 (18)

0.0 4.0 8.0 12.0 16.0 20.0 ct/a-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2u/Ua

analyticalDS-BEM

Figure 2. One-dimensional bar: potential at boundary node A(a,a/4).

0.0 4.0 8.0 12.0 16.0 20.0 ct/a-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

p/UanalyticalDS-BEM

Figure 3. One-dimensional bar: flux at boundary node B(0,a/4).

Square membrane. In this example a square membrane, defined in the domain 0 x a, 0 y a, is subjected to the initial conditions given by:

u0(x,y) = U x(x a)y(y a) and u.

0(x,y) = 0 (19) The boundary discretization employed 80 elements and the square domain, 800 cells, see Fig. 4.

Figure 4. Square membrane: boundary and domain discretization.

The general analytical solution to this problem, for a rectangular membrane with dimensions a and b, is:

u(x,y,t) = 64Ua2b2

6 m = 1

!"

n = 1

!"

1m3n3 sin

mx

a sin

ny

b cos

#mnct with #mn = m2

a2 + n2

b2 (20)


Results corresponding to the displacement at point A(a/2,a/2) and to the support reaction at boundary node B(a,a/2) are depicted, respectively, in Fig. 5 and Fig. 6. For this example, t = 3/10.

0.0 4.0 8.0 12.0 16.0 20.0 ct/a-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

u/UanalyticalDS-BEM

Figure 5. Square membrane: displacement at point A(a/2,a/2).

0.0 4.0 8.0 12.0 16.0 20.0 ct/a-0.4

-0.2

0.0

0.2

0.4

p/UanalyticalDS-BEM

Figure 6. Square membrane: support reaction at boundary node B(a,a/2).

ConclusionsThe D-BEM formulation presented here, called DS-BEM, confirms the assertion done at the beginning of the work, i.e. the solution of time dependent problems by the BEM offers to researchers a vast range of possibilities. In the proposed approach, the drawback represented by the domain integration is compensated by the easy computational implementation and, mainly, by the accurate results obtained even for large time values, see Figs. 2,3,5,6. Besides, due to the time integration, the initial conditions are imposed directly, with no need of further developments. For this reason, the authors’ conclusion is that the proposed approach looks very promising and, consequently, some research work concerning its development can be done in the near future, which includes a possible extension to elastodynamics.

References[1] J.A.M.Carrer, W.J.Mansur, R.J.Vanzuit. Scalar Wave Equation by the Boundary Element Method: a D-BEM Approach with Non-homogeneous Initial Conditions. Computational Mechanics, 44, 31-44 (2009). [2] G.D.Hatzigeorgiou, D.E.Beskos. Dynamic Elastoplastic Analysis of 3-D Structures by the Domain/Boundary Element Method. Computers & Structures, 80, 339-347 (2002). [3] J.C.Houbolt. A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft. Journal of the Aeronautical Sciences, 17, 540-550 (1950). [4] J.A.M.Carrer and W.J.Mansur. Alternative Time-marching Schemes for Elastodynamic Analysis with the Domain Boundary Element Method Formulation. Computational Mechanics, 34, 387-399 (2004). [5] L.A.Souza, J.A.M.Carrer, C.J.Martins. A Fourth Order Finite Difference Method Applied to Elastodynamics: Finite Element and Boundary Element Formulations. Structural Engineering and Mechanics, 17, 735-749 (2004). [6] C.C.Chien, Y.H.Chen, C.C.Chuang. Dual Reciprocity BEM Analysis of 2D Transient Elastodynamic Problems by Time-Discontinuous Galerkin FEM. Engineering Analysis with Boundary Elements, 27, 611-624 (2003). [7] O.C.Zienkiewicz and K.Morgan. Finite Elements & Approximation. John Wiley & Sons, Inc. (1983). [8] B.A.Finlayson. The Method of Weighted Residuals and Variational Principles. Academic Press (1972). [9] J.A.M.Carrer and W.J.Mansur. Scalar Wave Equation by the Boundary Element Method: a D-BEM Approach with Constant Time-Weighting Functions. International Journal for Numerical Methods in Engineering, 81, 1281-1297 (2010). [10] G.Stephenson. An Introduction to Partial Differential Equations for Science Students. Longman (1970). [11] E.Kreyszig. Advanced Engineering Mathematics. John Wiley & Sons, Inc., 8th edition (1999).


A novel boundary meshless method for radiation and scattering problems Zhuojia Fu1, Wen Chen2

1 Department of Engineering Mechanics, Hohai University, Nanjing 210098, P.R. China, [email protected]

2 Department of Engineering Mechanics, Hohai University, Nanjing 210098, P.R. China, [email protected]

Keywords: Singular boundary method, meshless, singular fundamental solution, unbounded domain, radiation, scattering

Abstract. This paper proposes a novel meshless singular boundary method (SBM) to solve time-harmonic exterior acoustic problems. Compared with the other boundary-type meshless methods, the innovative point of the SBM is to employ a novel inverse interpolation technique to circumvent the singularity of the fundamental solution at origin. The method is mathematically simple, easy-to-program, meshless and integration-free. This study tests the method to three benchmark radiation and scattering problems under unbounded domains. Our numerical experiments reveal that the SBM is a competitive numerical technique to the exterior acoustic problems.

1. Introduction The finite element method (FEM) [3-5] is one of the most popular methods in numerical acoustics but requires the effective treatment of unbounded domains, among which are the local and nonlocal absorbing boundary conditions [6-8], infinite elements [9], and absorbing layers [10,11]. These boundary treatments could be very tricky and arbitrary and are largely based on trial-error experiences. On the other hand, the boundary element method (BEM) [12-17] appears very attractive to handle the unbounded domain problems because its basis function is the fundamental solution which satisfies the governing equation and the Sommerfeld radiation condition at infinity [9]. And no special treatment for unbounded domains is required. However, the treatment of singularity and hyper-singularity [17] is mathematically complex and computationally very expensive. To avoid the singularities of fundamental solutions, the method of fundamental solutions (MFS) [18-20] distributes the boundary knots on a fictitious boundary outside the physical domain, and the location of fictitious boundary is vital for the accuracy and reliability. However, despite great effort of decades, the optimal placement of fictitious boundary is still arbitrary and tricky and is largely based on experiences. Recently, Young et al. [21] proposed an alternative meshless method, called regularized meshless method (RMM) [22], to remedy this drawback. By employing the desingularization of subtracting and adding-back technique, the RMM places the source points on the real physical boundary. In addition, the ill-conditioned interpolation matrix of BEM and MFS is also remedied. However, the original RMM requires the uniform distribution of nodes and severely reduces its applicability to complex-shaped boundary problems. Similar to the RMM, Sarler [23] proposes the modified method of fundamental solution (MMFS) to solve potential flow problems. However, the MMFS demands a complex calculation of the diagonal elements of interpolation matrix. It is worthy of noting that the MFS, RMM and MMFS do not require any mesh and are all truly meshless. This paper proposes a novel numerical method, called singular boundary method (SBM), to calculate the exterior acoustic problems. The SBM is developed to overcome the above-mentioned major shortcomings in the MFS, RMM, and MMFS while retaining their merits. The key point of the SBM is to use a simple numerical approach to calculate diagonal elements when the collocation and source nodes are coincident and are all placed on the physical boundary. This study also examines the efficiency, stability, and accuracy of the proposed technique in testing three benchmark exterior radiation and scattering problems. Based on the results reported here, some remarks will be concluded in section 4.

2. Singular boundary method for exterior Helmholtz problems The problem under consideration is the Helmholtz equation in the domain D exterior to a closed bounded curve


S . To be precise, we consider propagation of time-harmonic acoustic waves in a homogeneous isotropic acoustic medium which is described by the Helmholtz equation

2 2( ) ( ) 0, u x k u x x D , (1) subjected to the boundary conditions:

Du x u x (2a)

( ) N

u xt x t x

n

(2b)

where u is the total acoustic wave ( velocity potential or acoustic pressure), /k c the wave number, the angular frequency, c the wave speed in the exterior acoustic medium D, and n denotes the unit inward normal on physical boundary. ,D N denote the essential boundary (Dirichlet) and the natural boundary (Neumann) conditions, respectively, which construct the whole closed bounded curve S .For the exterior acoustic problems, it requires ensuring the physical requirement that all scattered and radiated waves are outgoing. This is accomplished by imposing an appropriate radiation condition at infinity, which is termed as the Sommerfeld radiation condition [9]:

1 (dim 1)2lim 0

r

ur ikur

, (2c)

where dim is the dimension of the acoustic problems (dim=2 in this study), and 1i .The solution u(x) of the acoustics problem (Eqs. (1) and (2)) can be approximated by a linear combination of the two-dimensional fundamental solution G

1

G , , N

m j m jj

u x x s x D

(3)

where N denotes the number of source points, j is the jth unknown coefficient, and the fundamental solution

(1)0 2

G ,2j jix s H k x s

, (1)nH is the nth order Hankel function of the first kind. We can find that the

fundamental solution G satisfies both the governing equation (1) and the Sommerfeld radiation condition (2c). Thus, the formulation (3) only requires satisfying the boundary conditions (2a) and (2b). If the collocation points xm and source points sj coincide, i.e., xm=sj, we will encounter well-known singularity at

origin, i.e., (1)0G , 0

2m jix s H

. In order to remedy this troublesome problem, the MFS places the source

nodes on an artificial boundary outside the physical domain. However, despite of great effort, the placement of this artificial boundary remains a perplexing issue when dealing with complex-shaped boundary or multiply-connected domain problems. The SBM places all source and boundary collocation nodes on the same physical boundary. Moreover, the source points and the boundary collocation points are the same set of boundary nodes. The SBM formulation is given by

1

G , , ,N

em j m j m m D

ju x x s x x

(4a)

S1,

G , G ( ), N

m j m j m m Dj j m

u x x s m x

(4b)

1

G ,, ,

Nm j e

m j m m Nj

x st x x x

n

(4c)

S

1,

G ,( ) G ( ),

Nm j

m j m m Nj j m

x st x m x

n

(4d)

where GS and SG are defined as the source intensity factors, namely, the diagonal elements of the SBM interpolation matrix. This study employs a simple numerical technique, called the inverse interpolation technique (IIT), to determine the source intensity factors. In the first step, the IIT requires choosing a known sample solution


uS of the Helmholtz acoustic problem and locating some sample points yk inside the physical domain. It is noted that the sample points yk do not coincide with the source points sj, and the sample points number NK should not be fewer than the source node number N on physical boundary. By using the interpolation formula (3), we can then determine the influence coefficients j and j by the following linear equations

,k j j S kG y s u y (5a)

,k j S kj

G y s u yn n

! " ! "

(5b)

Replacing the sample points yk with the boundary collocation points xm, the SBM interpolation matrix of the Helmholtz problem (Eqs. (1) and (2)) can be written as

S 1 2 1

2 1 S 2

1 2 S

G (1) G , G ,G , G (2) G ,

G , G , G ( )

N

Nj S m

N N

x s x sx s x s

u x

x s x s N

# $% &% & % &% &% &' (

(6a)

1 2 1S

2 1 2S

1 2S

G , G ,G (1)

G , G ,G (2)

G , G ,G ( )

N

NS m

j

N N

x s x sn n

x s x s u xn n n

x s x sN

n n

# $% & % & % & % & % & ! "% &

% & % &% & ' (

(6b)

The source intensity factors can be calculated by the following formulations:

1,

G ,( ) ,j m

N

S m j m jj s x

S m j m Dj

u x x sG m x s x

(7a)

1,

G ,

( ) ,j m

Nm jS m

jj s x

S m j m Nj

x su xn n

G m x s x

(7b)

It is stressed that the source intensity factors only depends on the distribution of the source points, the fundamental solution of the governing equation and the boundary conditions. Theoretically speaking, the source intensity factors remain unchanged with different sample solutions in the IIT. Therefore, by employing this novel inverse interpolation technique, we circumvent the singularity of the fundamental solution upon the coincidence of the source and collocation points. It is noted that like the MFS, the SBM does not require considering the Sommerfeld radiation condition (2c) and is a truly meshless numerical technique; unlike the MFS, the SBM avoids the perplexing issue of the fictitious boundary.

3. Numerical results and discussions In this section, the efficiency, accuracy and convergence of the present SBM are tested to the exterior acoustics problems. It is stressed that the boundary conditions are discontinuous in Cases 1 and 2. The present SBM is compared with the exact solution, the RMM and the MFS. Lerr(u) represents the L2 norm error, which are defined as below

2

1

1( ) ,NT

kLerr u u k u k

NT

(8)

where u k and u k are the analytical and numerical solutions at xi, respectively, and NT is the total number


of points in the interest domain which are used to test the solution accuracy, the sample solution (1)2(1)2

( )( , ) cos 2( )s

H kru rH ka

) ) for Dirichlet boundary problem, (1)1(1)1

( )( , )( )

is

H kru r eH ka

)) *

for Neumann boundary

problem. The number of inner sample points is equal to the boundary knots, and the distribution of sample points depends on the shape of the physical domain. In the MFS, according to the boundary shape of the physical domain, we typically place the source points outside physical domain with a parameter d defined as

i i

i

x sdx op

(9)

in which op is the geometric center, namely the origin point in this paper.

3.1 Radiation problems Case 1: Nonuniform radiation problem (Dirichlet boundary condition) for a circular cylinder. We first consider a nonuniform radiation problem (Dirichlet) from a sector of a cylinder as shown in Fig. 1(a). The boundary condition has a constant inhomogeneous value on the arc 2 2 ) + + and vanishing elsewhere. Two discontinuous boundary points can be found on the physical boundary. The analytical solution [8] is

(1) (1)0(1) (1)10

( ) ( )1 sin( , ) cos2 ( ) ( )

n

n n

H kr H krnu r nH ka n H ka

) )

(10)

where (1) ( )nH kr is the first kind Hankel function of the n order. We choose the parameters 532 , ka=1. The

analytical solution is obtained by using 20 terms in the series representations. Fig. 2(a) shows the comparison of the L2 norm errors between the MFS with different fictitious boundary parameters (d=0.01,0.2,0.5) and the present SBM. It can be observed that the arbitrary placing of the off-set boundary points may cause numerical stability. The present SBM avoids such trial-error placement of the fictitious boundary and is more efficient than the MFS with the boundary nodes of the same number.

(a) Dirichlet (Case 1) (b) Neumann (Case 2) Fig. 1 Nonuniform radiation (a) Dirichlet (Case 1) and (b) Neumann (Case 2) problem of a circular cylinder

Case 2: Nonuniform radiation problem (Neumann boundary condition) of a circular cylinder. A nonuniform radiation problem (Neumann) from a sector of a cylinder is considered as shown in Fig. 1(b) [24]. The discontinuous boundary condition is

1, 2 2

,0, otherwise

t a )

) + +

!

(11)

The analytical solution [24] is (1) (1)0(1) (1)10

( ) ( )1 sin( , ) cos2 ( ) ( )

n

nn

H kr H krnu r nk k nH ka H ka

) )

* * (12)

) 1),( )au0),( )au

Drruk )) ),(,0),()( 22

325

) 1),( )au0),( )au

Drruk )) ),(,0),()( 22

325

2 2, ) + +

) 1),( )at0),( )at

Drruk )) ),(,0),()( 22

9

) 1),( )at0),( )at

Drruk )) ),(,0),()( 22

9

2 2, ) + +


Here we choose the parameters 9 , ka=1. The analytical solution is obtained by using 20 terms in the series

representations. Fig. 2(a) shows the convergence curves of the MFS with different fictitious boundary parameters (d=0.01,0.3,0.5) and the present SBM. It can be observed that the fictitious boundary has a big influence on the MFS solution and its optimal placement is problem-dependent. The present SBM can obtain the acceptable results by using only 40 boundary nodes and outperforms the MFS in computational accuracy.

(a) Dirichlet (Case 1) (b) Neumann (Case 2) Fig.2 The accuracy variation of Case 1 and 2 against the number of interpolation knots by the MFS with d=0.01,0.2,0.5 for Case 1 and d=0.01,0.3,0.5 for Case 2 and the present SBM.

3.2 Scattering problems The scattering problem with the incident wave can be divided into two parts, (a) incident wave field and (b) radiation field. And the radiation boundary condition in part (b) can be obtained as the minus value of the incident wave function, i.e. tR= -tI for hard scatter or uR=-uI for soft scatter, where the superscripts R and I denote radiation and incidence, respectively.

Fig. 3 The problem of a plane wave scattered by a rigid infinite circular cylinder (Neumann) in Case 3

Case 3: Scattering problem (Neumann boundary condition) of a rigid infinite circular cylinder We consider a plane wave scattered by a rigid infinite circular cylinder as shown in Fig. 3 [20]. The analytical solution of this scattering field [20] is

(1) (1)00(1) (1)1

0

( ) ( )( , ) ( ) 2 ( ) cos( ) ( )

n nn

nn

J ka J kau r H kr i H kr nH ka H ka

) )

* * * * (13)

The analytical solution in the following figures is calculated by using the first 20 terms in the above series representation (13). Figs. 4(a) and 4(b) plot both the real and imaginary parts of u on r=2a for 4ka by using the SBM and the MFS (d=0.01,0.2) with 100 boundary nodes. It can be found that both the SBM and the MFS


with fictitious boundary parameter d=0.2 agree the analytical solution very well. However, the MFS with d=0.01can not obtain the right result. Thus, the determination of such a parameter d is very tricky and delicate in applications. It is noted that the present SBM avoids the headachy choice of the optimal fictitious boundary and is superior to the MFS. Figs. 5, 6(a) and 6(b) display the contour plot of the real-part potential by using the analytical solution, the present SBM and the MFS with 100 boundary nodes. It can be seen from Fig. 10 that the SBM solution matches the analytical solution very well.

(a) Real part (b) Imaginary part Fig. 4 Plane wave scattered by a rigid infinite circular cylinder (Neumann) in case 3 for 4ka ,r=2a: (a) Real part, (b) Imaginary part

4. Conclusions This study proposes a novel singular boundary method formulation to calculate the exterior radiation and scattering problems. Numerical results demonstrate that the SBM performs more stably than the MFS and more accurate than the RMM, while retaining their merits. The present SBM appears a promising numerical technique to the exterior acoustic problems. In addition, the present SBM is mathematically simple, easy-to-program, accurate, meshless and integration-free and avoids the controversy of the fictitious boundary in the MFS, the uniform boundary node requirement of the RMM, and the expensive calculation of diagonal elements in the MMFS.

Fig. 5 The contour plot of the real-part analytical solution of a plane wave scattered by a rigid infinite circular cylinder in Case 3


(a) SBM solution (b) MFS (d=0.2) solution Fig. 6 The contour plot of the real-part (a) SBM and (b) MFS (d=0.2) solution of a plane wave scattered by a rigid infinite circular cylinder in Case 3

References

[1] P. Morse, K. Ingard, Theoretical acoustics, McGraw-Hill, New York, 1968.

[2] A. Pierce, Acoustics: an introduction to its physical principles and applications. McGraw-Hill series in mechanical engineering, McGraw-Hill, New York, 1981.

[3] F. Ihlenburg, Finite element analysis of acoustic scattering, Applied Mathematical Sciences(132). Springer, New York, 1998.

[4] I. Harari, A survey of finite element methods for time-harmonic acoustics, Comput Methods Appl Mech Eng. 195 (2006) 1594-1607.

[5] L.L. Thompson, A review of finite element methods for time-harmonic acoustics, J Acoust Soc Am. 119 (2006) 1315-1330.

[6] D. Givoli, Recent advances in the DtN finite element method for unbounded domains, Archives of Computational Methods in Engineering 6 (1999) 71-116.

[7] M.J. Grote, C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys. 201 (2004) 630-650.

[8] J.R. Stewart, T.J.R. Hughes, h-adaptive finite element computation of time-harmonic exterior acoustics problems in two dimensions, Comput. Methods Appl. Mech. Eng. 146 (1997) 65-89.

[9] I. Harari, P.E. Barbone, M. Slavutin, R. Shalom, Boundary infinite elements for the Helmholtz equation in exterior domains, Int. J. Numer. Methods Eng. 41 (1998) 1105-1131.

[10] Q. Qi, T.L. Geers, Evaluation of the perfectly matched layer for computational acoustics, J. Comput. Phys. 139 (1998) 166-183.

[11] A. Bermúdez, L. Hervella-Nieto, A. Prieto, R. Rodríguez, An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems, J. Comput. Phys. 223 (2007) 469-488.

[12] C.A. Brebbia, J.C.F. Telles, L.C.L. Wrobel, Boundary element techniques: theory and applications in engineering, Springer, New York, 1984.


[13] J.T. Chen, K.H. Chen, I.L. Chen, L.W. Liu, A new concept of modal participation factor for numerical instability in the dual BEM for exterior acoustics, Mech. Res. Comm. 26 (2003) 161-174.

[14] R.D. Ciskowski, C.A. Brebbia, Boundary element methods in acoustics, Computational mechanics publications, Elsevier Applied Science, 1991.

[15] S. Kirkup, The boundary element method in acoustics, Integrated Sound Software, 1998.

[16] Von Estorff, Boundary elements in acoustics: advances and applications, WIT Press, 2000.

[17] V. Sladek, J. Sladek, M. Tanaka, Optimal transformations of the integration variables in computation of singular integrals in BEM, International Journal for Numerical Methods in Engineering, 47 (2000) 1263-1283.

[18] G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998) 69-95.

[19] G. Fairweather, A. Karageorghis, P.A. Martin, The method of fundamental solutions for scattering and radiation problems, Engineering Analysis with Boundary Elements 27 (2003) 759-769

[20] I.L. Chen, Using the method of fundamental solutions in conjunction with the degenerate kernel in cylindrical acoustic problems, J. Chinese Inst. Engrg. 29 (2006) 445-457.

[21] D.L. Young, K.H. Chen, C.W. Lee, Novel meshless method for solving the potential problems with arbitrary domain, J. Comput. Phys. 209 (2005) 290-321.

[22] D.L. Young, K.H. Chen and C.W. Lee, Singular meshless method using double layer potentials for exterior acoustics, J Acoust Soc Am 119 (2006) 96-107

[23] B. Šarler, Chapter 15: Modified method of fundamental solutions for potential flow problems, in: C.S. Chen, A. Karageorghis, Y.S. Smyrlis (Eds.), The Method of Fundamental Solutions- A Meshless Method, Dynamic Publisher, 2008.

[24] J.T. Chen, C.T. Chen, P.Y. Chen, I.L. Chen, A semi-analytical approach for radiation and scattering problems with circular boundaries, Comput. Meth. Appl. Mech. Engng, 196 (2008) 2751-2764.


Anti-plane shear Green’s function for an isotropic elastic layer on a substrate with a material surface

W. Q. Chen1 and Ch. Zhang2

1 Department of Engineering Mechanics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China, [email protected]

2 Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany,[email protected]

Keywords: Green’s functions, elastic layer, material surface, Gurtin-Murdoch theory.Abstract. The anti-plane shear Green’s function solution for an isotropic elastic layer on a rigid substrate with a material surface subject to a point force is derived using Fourier cosine transform. The elasticity and residual stress of the upper surface of the layer are taken into consideration by adopting the Gurtin-Murdoch theory, while the lower surface of the layer is assumed to be fixed on the rigid substrate. For two limiting cases of a half-plane, analytical expressions are obtained, which coincide with those in literature.

Introduction With the miniaturization of electromechanical devices and systems, the size of structural components becomes smaller and smaller, while the surface-to-volume ratio increases significantly and hence the surface/interface effect becomes more important [1]. Recently, there appear some interesting works in which the significant effect of surface elasticity and residual stress on various mechanical responses of materials and structures has been clarified [2-10]. These works are all based on the continuum theory of a deformable material surface suggested by Gurtin and Murdoch [11] (hereafter, referred as the GM theory) as early in 1975. Gurtin and Murdoch [12] themselves have already shown that the surface stress has an important influence on both the static and dynamical problems of elastic bodies with appropriate surfaces.

Green’s functions play an important role in solving boundary-value problems in elasticity. They are especially very useful in the boundary element method. Based on the GM theory, but with the simplifications that the surface material has the same elastic property as the bulk material and that the bulk material is incompressible, He and Lim [13] obtained the surface Green’s function of a half-space using the double Fourier transforms. Wang and Feng [14] presented analytical solutions of a half-plane with a material boundary subject to uniform as well as concentrated loads acting on the boundary; they used the Fourier transform technique and considered only the effect of residual surface tension. Koguchi [15] derived the surface Green’s functions in complex integral forms for an anisotropic half-space with material surface using Stroh’s formulism. Zhao and Rajapakse [16] investigated the plane problem of a surface-loaded isotropic elastic layer with surface effects by the Fourier transform. Recently, Chen and Zhang [17] derived analytical expressions for the point force solution of an isotropic elastic half-plane with a material surface subject to anti-plane shear deformation. They also showed that the obtained Green’s functions can be utilized to construct appropriate boundary integral equation so that the boundary element method could be used to deal with more complicated problems involving surface effects.

In this paper, we reconsider the layer model in Zhao and Rajapakse [16], but confine ourselves to the case of anti-plane deformation induced by an interior point shear force. The upper boundary of the layer is modeled as a material surface which possesses both residual surface tension and elasticity, for which the GM theory is employed. The lower boundary is fixed on the rigid substrate. Because of the symmetry of the problem, Fourier cosine transform is employed. The Green’s function is obtained and expressed in an integral form. It is found that for two limiting cases, both corresponding to a half-plane model, either classical results for the anti-plane elasticity or that obtained in Ref. [17] are recovered.


Basic formulations Consider the anti-plane shear deformation of an isotropic elastic layer on a rigid substrate as shown in Fig. 1. The layer is subject to a point shear force of magnitude 0p at an interior point, which coincides with the origin of the coordinate system (x, y). The distances between the x-axis and the upper surface and the lower interface are denoted as h1 and h2, respectively, and hence the total thickness of the layer is 1 2H h h . The bulk material is isotropic, obeying the same laws as a conventional elastic material. For the anti-plane deformation, we therefore have the following nonzero strain components

1 1,2 2xz yz

w wx y

(1)

where w is the out-of-plane displacement component which depends on both x and y. The nonzero stress components are calculated from the Hooke’s law as

,xz yzw wx y

(2)

where is the shear modulus. The equilibrium equation is

0yzxz fx y

(3)

where f is the z-component of the body force vector. Substituting Eqs. (1) and (2) into Eq. (3) gives

2 0fw

(4)

where 2 2 2 2 2/ /x y is the two-dimensional Laplacian.

Fig. 1. An elastic layer on a rigid substrate with a material boundary at 1y h .

The upper surface of the layer is assumed to have different material properties from the bulk, and the GM theory [11] is employed. Hence, we have

0szx

yzx

(5)

s szx

wx

, ( )s s s

xzwx

(6)

x

y

h1p01

2

bulk: ,

surface: s,s, s

h2


at 1y h , where the superscript s designates the quantity associated with the surface, s is the residual surface tension, and s is the surface shear modulus. Note that the displacement compatibility between the surface and bulk at 1y h has been implied.

The lower interface is assumed to be fixed, i.e. we have 0w (7)

at 2y h .The configuration in Fig. 1 has been considered by Zhao and Rajapakse [16], who however paid their

attention to the in-plane deformation due to a concentrated force applied on the upper surface only.

Green’s function solution

First, we divide the layer into two regions, region 1 ( 1 0h y ) and region 2 ( 20 y h ), as shown in Fig. 1. The governing equation in each region becomes homogeneous as

2 ( ) 2 ( )

2 2 0w wx y

( 1,2) (8)

where the superscript denotes the respective region. At 0y , we have the following continuity/equilibrium conditions:

(1) (2)w w ,(2) (1)

0 ( )w wp xy y

(9)

The following condition at the material surface ( 1y h ) can be derived from Eqs. (5) and (6): 2 (1) (1)

2 0s w wx y

(10)

On the other hand, the condition at the interface between the layer and substrate is still given by Eq. (7), but with w replaced by (2)w .

We use the following Fourier cosine transform:

0

2( , ) ( , )cos( )dW k y w x y kx x

(11a)

0

2( , ) ( , )cos( )dw x y W k y kx k

(11b)

Now, by applying the Fourier cosine transform defined by Eq. (11a) to Eq. (8) and the conditions (9), (10) and (7), we get

2 ( )2 ( )

2

d 0dW k Wy

( 1,2) (12)

(1)2 (1)d 0

dW r k W

y at 1y h (13)

(1) (2)W W ,(2) (1)

0d 1 dd d2

pW Wy y

at 0y (14)

(2) 0W at 2y h (15) where /sr , the shear modulus ratio between the surface and bulk, has the dimension of length, and can be regarded as an intrinsic length parameter of the problem.

The solutions to Eq. (12) in the two regions are (1) e eky kyW A B , (2) e eky kyW C D (16)


Substituting into Eqs. (13) through (15) yields 1 1 1 12e e ( e e ) 0kh kh kh khAk Bk r k A B

, 0A B C D

0 12

pAk Bk Ck Bk

, 2 2e e 0kh khC D (17)

which in turn gives 122

02

(1 )(e e ) 1[(1 ) (1 )e ] 2 2

khkH

kH

r k pAk r k r k

22

02

(1 )(e 1) 1[(1 ) (1 )e ] 2 2

kh

kH

r k pBk r k r k

120

2

(1 ) (1 )e 1[(1 ) (1 )e ] 2 2

kh

kH

r k r k pCk r k r k

22 2

02

(1 )e (1 )e 1[(1 ) (1 )e ] 2 2

kh kH

kH

r k r k pD

k r k r k

(18)

Substituting into Eq. (16) and in view of Eq. (11b), we get 1

2

22(1) 0

20

[(1 )e e (1 )e ]1 (e 1)cos( )d2 [(1 ) (1 )e ]

kh ky kykh

kH

r k r kpw kx kk r k r k

(19a)

21

22(2) 0

20

1 e e e [(1 ) (1 )e ]cos( )d2 [(1 ) (1 )e ]

khky kykh

kH

pw r k r k kx kk r k r k

(19b)

It seems difficult to obtain the closed-form solution due to the complicated integrands involved in Eqs. (19a) and (19b). However, one may verify that

(1) (2) (2)00

1 1 [e e ]cos( )d2

ky kypw kx k w wk

(20)

Thus, the anti-plane shear displacement in the elastic layer can be written in a unified way as 1

2

220

20

[(1 )e e (1 )e ]1 (e 1)cos( )d2 [(1 ) (1 )e ]

kh ky kykh

kH

r k r kpw kx kk r k r k

(21)

The integration generally can be performed by an appropriate numerical scheme.

Two limiting cases

We now consider two special cases. The first is for 1h , which corresponds to an infinite half-plane with a fixed surface at 2y h subject to a point shear force at 0y . Taking the limit 1h , we obtain from Eq. (21)

2200

2 20 2

2 2

1 e (1 e )cos( )d2

1 ( 2 )ln4

kykhpw kx k

k

p x y hx y

(22)

It is readily seen that when 2y h , 0w , i.e. the boundary condition at the fixed surface is satisfied. As expected, no effect of the material surface is involved in the displacement field given by Eq. (22), from which, the corresponding shear stress components in the half-plane can be calculated according to Eq. (2). These results should be the same as the classical ones in elasticity for anti-plane deformation of an isotropic half-plane with a fixed surface, and hence are not presented in the following.


The second is to let 2h , which corresponds to an infinite half-plane with a material surface at

1y h [17]. In this case, we obtain from Eq. (21) that

120

0

2 2 2 20 01 1

[(1 )e (1 )e e ]1 cos( )d2 (1 )

1 1ln ( )[ ( 2 ) ] ( , 2 ; )4

khky ky

c

r k r kpw kx kk r k

p px y x y h J x y h r

(23)

where a constant, which contributes nothing to the stress/strain field, has been neglected in the above expression, and

10

e( , ; ) cos( )d Re e ( )1

kyz

caJ x y a kx k E z

ak

(24)

where ( i ) /z y x a , and 1( ) (e / )dt

zE z t t

( Arg( )z ) is the exponential integral [18]. The

expression in Eq. (23), which is obtained through the limiting procedure, is identical to that obtained in Chen and Zhang [17].

The first term in Eq. (23) is identical to the classical result of a half-plane with a free surface, while the second is induced by the surface elasticity, which vanishes identically as 0r . In fact, it is the right difference between the classical solution and the one including the surface effect. Thus, the Jc-integral defined in Eq. (24) represents the influence of the surface elasticity. Its characteristic is shown in Fig. 2 for several values of y. As can be seen, the Jc-integral decreases monotonously with x; that is to say, when the field point is away from the load point, the surface effect term in Eq. (23) also diminishes. The reader is referred to Chen and Zhang [17] for a more detailed discussion on the stress field in the half-plane as well as other aspects of the analytical Green’s functions.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5

y=1

y=2

y=4

Fig. 2. Integral ( , ; )cJ x y a for 1a . Since ( , ; ) ( / , / ;1)c cJ x y a J x a y a , the

drawings for 1a can be readily imagined.

x

Jc


SummaryWe present in this paper an analysis of an isotropic elastic layer subject to anti-plane shear deformation under the action of a point shear force. One surface of the layer is assumed to have different material properties from the bulk and the Gurtin-Murdoch theory is therefore employed. The other surface is assumed to be fixed on a rigid substrate. Fourier cosine transform is employed to derive the Green’s function in an integral form. By a limiting analysis, two special cases for a half-plane are discussed, with analytical Green’s functions presented. In particular, in the case of a half-plane with a material surface, the results are found identical to those derived by Chen and Zhang [17].

Acknowledgements The work was sponsored by the National Natural Science Foundation of China (Nos. 10725210 and 10832009). Financial support from the German Research Foundation (DFG, Project-No.: ZH 15/15-1) is also gratefully acknowledged.

References

[1] R.C. Cammarata Progress in Surface Science 46, 1-38 (1994).

[2] R.E. Miller and V.B. Shenoy Nanotechnology 11, 139-147 (2000).

[3] L.H. He, C.W. Lim and B.S. Wu International Journal of Solids and Structures 41, 847-857 (2004).

[4] C.W. Lim and L.H. He International Journal of Mechanical Science 46, 1715-1726 (2004).

[5] C.W. Lim, Z.R. Li and L.H. He International Journal of Solids and Structures 43, 5055-5065 (2006).

[6] L.H. He and Z.R. Li International Journal of Solids and Structures 43, 6208-6219 (2006).

[7] Z.Y. Ou, G.F. Wang and T.J. Wang European Journal of Mechanics, A/Solids 28, 110-120 (2009).

[8] C.F. Lü, C.W. Lim and W.Q. Chen International Journal of Solids and Structures 46, 1176-1185 (2009).

[9] C.F. Lü, W.Q. Chen and C.W. Lim Composites Science and Technology 69, 1124-1130 (2009).

[10] C.I. Kim, P. Schiavone and C.Q. Ru Journal of Applied Mechanics 77, 021011 (2010).

[11] M.E. Gurtin and A.I. Murdoch Archive for Rational Mechanics and Analysis 57, 291-323 (1975).

[12] M.E. Gurtin and A.I. Murdoch International Journal of Solids and Structures 14, 431-440 (1978).

[13] L.H. He and C.W. Lim International Journal of Solids and Structures 43, 132-143 (2006).

[14] G.F. Wang and X.Q. Feng Journal of Applied Physics 101, 013510 (2007).

[15] H. Koguchi Journal of Applied Mechanics 75, 061014 (2008).

[16] X.J. Zhao and R.K.N.D. Rajapakse International Journal of Engineering Science 47, 1433-1444 (2009).

[17] W.Q. Chen and Ch. Zhang International Journal of Solids and Structures 47, DOI: 10.1016/j.ijsolstr.2010.03.007 (2010).

[18] M. Abramovitz and I. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964).


Stress intensity factor formulas for a rectangular interfacial crack in three-dimensional bimaterials

ChunHui Xu1, TaiYan Qin1, Chuanzeng Zhang2, Nao-Aki Noda3

1 College of Science, China Agricultural University, Beijing 100083, PR China,

E-mails: [email protected]; [email protected] 2 Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany,

E-mail: [email protected] 3 Department of Mechanical Engineering, Kyushu Institute of Technology, Kitakyushu 8048550,

Japan, E-mail: [email protected]

Keywords: Boundary integral equations; Fracture mechanics; Stress intensity factors; Interface cracks; Composite materials

Abstract Numerical solutions of hypersingular boundary integral equations (BIEs) are presented for the analysis of a planar rectangular interfacial crack in three-dimensional bimaterials. The problem is formulated as a system of hypersingular BIEs on the basis of the body force method. Based on the numerical results of the BIEs, the stress intensity factor formulas in terms of an area parameter to evaluate rectangular interfacial cracks in three-dimensional bimaterials are considered. Here “area” denotes the projected area of the defects or cracks. In the cases of mode-I and mode-II cracks, the formulas for homogeneous materials are available for interfacial cracks. For the interfacial cracks subjected to tension at the infinity, the stress intensity factors are expressed as a function of the bimaterial constant. For the cracks subjected to shear at the infinity, the stress intensity factors are investigated for varying Poisson's ratio and the aspect ratio of the crack. It is found that the maximum stress intensity factors normalized by the area are always insensitive to the crack aspect ratio. The fitting formulas presented in this paper are useful for engineering applications.

1. Introduction In recent years, composite materials and adhesives or bonded joints are being used in a wide range of engineering sciences. With the rapidly increasing use of composite materials and adhesives, much attention has been paid to the interface because the fracture is usually originated from the interfacial region. Since almost all structural materials contain some types of defects in the form of cracks, cavities, and inclusions, three-dimensional crack solutions may be useful for evaluating the strength of the structures. In the previous studies, stress intensity factor formulas were proposed for evaluating the maximum stress intensity factors for arbitrarily shaped internal cracks subjected to tension z

at infinity for the coordinate system as shown in Fig. 1 [1-6].

For a crack subjected to tension z we have [6, 7]

m 0.50 zI ax SK . (1)

For a crack subjected to shear one has [8-11]

m 0.55 yzII ax SK , (2)

m 0.45 yzIII ax SK , (3) where “ S ” is the projected area of the crack or defect. For example, in Fig. 1(a) area ab , and in Fig. 1(b)

4area ab . However, it should be noted that 220area b when / 5a b , and 220area b when / 0.2a b . To confirm the accuracy of the formulas (1)-(3), the exact maximum stress intensity factors of elliptical [3, 4] and rectangular cracks [5, 6] at A and B subjected to z

and yz

at infinity are shown in Table 1 [5-8].


Evaluation formula Elliptical crack Rectangular crack

*IF 0.5 0.47-0.52 [6] 0.47-0.52 [8] *

IIF 0.55, a/b1 0.46-0.64 [7] 0.47-0.64 [9] *

IIIF 0.45, a/b1 0.32-0.52 [7] 0.39-0.54 [9]

Tab. 1 Maximum stress intensity factors

In Table 1, normalized stress intensity factors are introduced as

* maxII

z

KFS

, * maxIIII

yz

KFS

, * m .III axIII

yz

KF

S

y

xz

2a

2b

B

A

2a

2bA

B

x

y

z

Fig. 1(a) An elliptical crack Fig. 1(b) A rectangular crack

It should be noted that IF

is independent of Poisson’s ratio , but IIF and IIIF

are depending on . Therefore Table 1 shows the range of the maximum stress intensity factors for 0 0.5 . In previous papers [11-13], the authors studied the problem of an interfacial crack under tension or shear loads at the infinity by solving the corresponding hypersigular boundary integral equations for computing the stress intensity factors, obtained fast convergence and high precision of the numerical results. In this paper, we will discuss the applicability of the formulas (1)-(3) for a rectangular interface crack. For this purpose, a boundary element method (BEM) based on hypersingluar BIEs is developed, which are briefly described in the next section.

2. Hypersingular boundary integral equations for a planar interfacial crack

Consider two dissimilar elastic half-spaces bonded together along the x–y plane (see Fig. 2) with a fixed rectangular Cartesian coordinate system ix ( , ,i x y z ). Suppose that the upper half-space is occupied by an elastic medium with constants 1 1( , ) , while the lower half-space by an elastic medium with constants 2 2( , ) . Here, 1 and 2 are shear moduli for space I and space II, and 1 and 2 are Poisson’s ratios for space I and space II. The crack is assumed to be located on the bimaterial interface. Hypersingular intergro-differential equations for three dimensional cracks on a bimateral interface as shown in Fig. 2 were derived by Chen–Noda-Tang [4] and can be expressed as follows

1 21 2 1 1 3

21 2

1 5 5

, 2 1 ( , ) ( , )2

3 ( , ) ( , ) ( , ) ( , ) ( , ),2

zx

S

x y xS S

u x yu dS

x r

x x yu dS u dS p x y

r r

(4a)


1 21 2 1 1 3

21 2

1 5 5

, 2 1 ( , ) ( , )2

3 ( , ) ( , ) ( , ) ( , ) ( , ),2

zy

S

x y yS S

u x yu dS

y r

x y yu dS u dS p x y

r r

(4b)

1 21 1 2 1 3

,, 1 ( , ) ( , ) ,2

yxz z

S

u x yu x yu dS p x y

x y r

!

, (4c)

" #

2 2 21 2

1 2 1 1 2 2 2 12 22

1 1 2 2

, , ,

3 4 , 3 4 , ,

, , ( , ) , ,

r x y

x y S S x y x a y b

$ $

$ $

%

(4d)

2

1

2

1

2

1

1, ( , ,0 ) ( , ,0 ) , ,

1, ( , ,0 ) ( , ,0 ) , ,

1, ( , ,0 ) ( , ,0 ) , .

( 1)

l l

l l

l

l l l

x x x zx

y y y yz

z z z zz

u x y u x y u x y f x y



$

$

&

&

&

(4e)

In Eq. (4), the unknown functions are the crack-opening-displacements, in other words, displacement discontinuities , ,x y zu u u defined in Eq. (4e), which are equivalent to the body force densities

,zxf x y , ,yzf x y , ,zzf x y as given in Eq. (4e). Here, ( , , ) ' is a rectangular coordinate system where the

displacement discontinuities are distributed, and xp , yp , zp are the stresses zx , yz , z at infinity.

Since the integrals have a hypersingularity of the form 3r when x and y , the integrals should be interpreted in a sense of the Hadamard finite-part integrals in the region S. Outside the region of S ,

0, 0, 0x y zu u u , which means that the displacement field is continuous.

Fig. 2 Problem configuration

The hypersingular BIEs (4a)-(4c) have been solved numerically by using the procedure of the BEM. In the numerical solution procedure, it is necessary to express the oscillatory singular stresses, which are present at the crack-tip or on the crack-front of interface cracks [12-14]. Once the crack-opening-displacements have been obtained numerically, the stress intensity factors at a point Q on the crack-front can be computed by using the following relations

Space I

Space II

O

z, (

y, x,

1,1

2,2

(a)

z

z zx

zx

zy

zy Space I

Space II

O

z, (

y, x,

1,1

2,2

(a)

z

z zx

zx

zy

zy


1/ 2 I I

0 0

1/ 2 I

0 0

( ) ( ) ( ) lim 2 ( , ) ( , ) ,

( ) lim 2 ( , ) .

iI II z yzr

III zxr

K Q K Q iK Q r r i r

K Q r r

)

*

*

* *

*

+

+

(5)

3. Maximum stress intensity factors of an interface crack under tension Let us consider a rectangular interface crack under tension at infinity. If a b , the maximum stress intensity factors IK and IIK would appear at points , 0,x y b , . The IIIK values are smaller than the values of

IK and IIK , and the maximum value of IIIK appears at a point which is very close to the corner of the rectangle.

The dimensionless stress intensity factors can be expressed as

max max maxmax max max, , ,I II III

I II IIIz z z

K K KF F F

b b b

max max maxmax max max, , .I II III

I II III

z z z

K K KF F F

S S S

(6)

Suppose that a b , the variations of the stress intensity factors , ,I II IIIF F F and , ,I II IIIF F F are given in Tables 2-4. It is noted that in Tables 2 and 3, the dimensionless stress intensity factors IF and IIF are dependent on the bi-material parameter ) only, but the values of IIIF are determined by Poisson’s ratios also. The numerical errors are given in Table 2; it shows that the maximum error is less than 8%. The numerical error is defined by

m

m

0.5100%.I ax

I ax

FF

-

.

The definition of the bi-material parameter) is

2 1 1

1 2 2

1 ln2

$ ) $

!

, 3 4 ,l lv$ 1,2l . (7)

By applying the least-square method to the numerical results obtained by the BEM, the following approximate or fitting formula can be obtained

2 3m ( ) =0.507+0.030 -2.294 +12.67 I axF ) ) ) ) , 0 0.1) . (8)

The comparing curves of the numerical results and the fitting formula are given in Fig. 3. In particular, when 0) , m 0.507I axF , the value is almost the same as the approximate formula (1), and the following two

approximate formulas can be applied to calculate the values of mII axF

and mIII axF

maxIIF ) / , max max0.5 0.5III IIF F ) . (9)

IF IF *

a/b= 1 a/b = 2 a/b = 4 a/b = 8 a/b = 1 - (%) a/b = 2 - (%) a/b = 4 - (%) a/b = 8 - (%)

) =0[10] 0.753 0.906 0.977 0.995 0.532 5 6.1 0.538 6 7.2 0.488 5 -2.4 0.470 4 -6.3

) =0.02 0.752 8 0.905 2 0.976 0 0.994 7 0.532 3 6.1 0.538 1 7.1 0.488 0 -2.5 0.470 2 -6.3

) =0.04 0.750 9 0.903 8 0.975 0 0.993 8 0.531 0 5.8 0.537 3 6.9 0.487 5 -2.6 0.469 8 -6.4

) =0.06 0.747 8 0.901 3 0.973 0 0.992 0 0.528 8 5.4 0.535 8 6.7 0.486 5 -2.8 0.468 9 -6.6

) =0.08 0.743 3 0.897 5 0.969 9 0.989 1 0.525 7 4.9 0.533 5 6.3 0.484 9 -3.1 0.467 6 -6.9

) =0.10 0.737 3 0.892 1 0.965 4 0.984 8 0.521 4 4.1 0.530 3 5.7 0.482 7 -3.6 0.465 6 -7.4

Tab. 2 Stress intensity factor IF and IF at (0, )b


IIF IIF *

a/b = 1 a/b = 2 a/b = 4 a/b = 8 a/b = 1 a/b = 2 a/b = 4 a/b = 8

) =0 0 0 0 0 0 0 0 0

) =0.02 0.027 4 0.035 2 0.038 8 0.039 7 0.019 0.021 0.019 0.019

) =0.04 0.054 2 0.069 6 0.076 8 0.078 6 0.038 0.041 0.038 0.037

) =0.06 0.079 8 0.102 7 0.113 4 0.116 0 0.056 0.061 0.056 0.055

) =0.08 0.104 0 0.133 8 0.147 9 0.151 5 0.073 0.079 0.074 0.072

) =0.10 0.126 3 0.162 7 0.180 1 0.184 5 0.089 0.096 0.090 0.087

Tab. 3 Stress intensity factor IIF and IIF at (0, )b

IIIF IIIF *

a/b = 1 a/b = 2 a/b = 4 a/b = 8 a/b = 1 a/b = 2 a/b = 4 a/b = 8

) =0 0 0 0 0 0 0 0 0

) =0.02 0.012 0 0.011 9 0.010 1 6.83×10-3 0.008 7.07×10-3 5.05×10-3 3.23×10-3

) =0.04 0.023 8 0.023 5 0.020 0 1.36×10-2 0.017 0.013 9 0.010 6.43×10-3

) =0.06 0.035 1 0.034 5 0.029 5 2.01×10-2 0.025 0.020 5 0.014 8 9.50×10-3

) =0.08 0.045 6 0.044 8 0.038 3 2.63×10-2 0.032 0.026 6 0.019 1 1.24×10-2

) =0.10 0.055 3 0.054 2 0.046 4 3.21×10-2 0.039 0.032 2 0.023 2 1.52×10-2

Tab. 4 Stress intensity factor IIIF and IIIF at / ,0.91*a b b( ) for 1 2 0.3

4. Maximum stress intensity factors of an interface crack under shear The dimensionless stress intensity factors under shear are defined as

max max maxmax max max, , ,I II III

I II IIIyz yz yz

K K KF F F

b b b

max max maxmax max max, , .I II III

I II III

yz yz yz

K K KF F F

S S S

(10)

0.00 0.02 0.04 0.06 0.08 0.100.3

0.4

0.5

0.6

0.7

0.8

F I

0)

(6) Eq.(9)

(5)(1)+(2)+(3)+(4)/4

(4)a/b=8

(3)a/b=4

(2)a/b=2(1)a/b=1

0.00 0.02 0.04 0.06 0.08 0.10

0.500

0.501

0.502

0.503

0.504

0.505

0.506

0.507

0.508

F I

)

(1)+(2)+(3)+(4)/4

FI12314121510)626780)6962:40)5

Fig. 3(a) Stress intensity factors at

)b( 0, for different aspect ratios of the crack subjected to a load z

Fig. 3(b) Comparison of the average values of the stress intensity factors with approximate formula


The stress intensity factors IIK and IIIK are always insensitive to the varying ratio of the shear modulus, and are mainly determined by Poisson's ratio [15]. Table 5 shows the range of the maximum stress intensity factors when 0 0.5 . We found that *F;; varies in the range of 0.47 0.64 0.55 < even when the rectangular shape ratios are changed extremely from / 1a b to /a b + , and the value is almost the same as predicted by the formula (2). However, for IIIF , the range 0.39 0.52 0.45 < is applicable only for

/ 1a b , so we cannot use formula (3) except for / 1a b . When / 1a b = , the derived values of IIIF are less than the case for / 1a b , where formula (3) is not applicable. For the stress intensity factor IF , its values can be approximated by IF ) < . Besides, with the presence of the shear stress, the stress intensity factors are dependent on Poisson’s ratios. When Poisson’s ratios 1v and 1v vary from 0 to 0.5, the errors of *F;; and IIIF are 9.7% and 18% respectively in the case of 1 2 0.3v v . Therefore, we usually assume 1 2 0.3v v in our approximate calculations.

1 2 ) IIF IIIF IF IIF * IIIF * IF *

0 0 0.053 6 0.760 3 0.742 1 0.074 0 0.537 7 0.524 8 0.052 3

0 0.5 0.134 9 0.827 6 0.632 5 0.188 7 0.585 3 0.447 3 0.133 5

0.1 0.1 0.047 5 0.785 8 0.716 0 0.068 2 0.555 7 0.506 4 0.048 2

0.1 0.5 0.115 5 0.840 1 0.626 8 0.167 9 0.594 1 0.443 3 0.118 7

0.2 0.2 0.040 0 0.813 2 0.685 8 0.059 8 0.575 1 0.485 0 0.042 3

0.2 0.5 0.093 5 0.854 0 0.618 5 0.141 6 0.604 0 0.437 4 0.100 1

0.3 0.3 0.030 4 0.842 8 0.650 7 0.047 5 0.596 0 0.460 2 0.033 6

0.3 0.5 0.068 0 0.869 6 0.606 2 0.107 5 0.615 0 0.428 7 0.076 0

0.499 9 0.499 9 0 0.909 8 0.557 0 7 10-6 0.643 4 0.393 9 5 10-6

Tab. 5 Stress intensity factors at )b( 0, for / 1a b

5. Conclusions By using a BEM based on hypersingular boundary integrals equations, the stress intensity factor formulas in terms of an area parameter to evaluate rectangular interfacial cracks in three-dimensional bimaterials are considered in this paper, and two conclusions can be made as follows:

1) When the infinity is subjected to a tensile stress z , the maximum error between the numerical results and formula (1) is less than 8%. Fitting formulas for stress intensity factors with the bimaterial parameter ) as a variable are derived, and they are given by: 2 3

m ( ) =0.507+0.030 -2.294 +12.67 I axF ) ) ) ) ,

maxIIF ) / and max max0.5 0.5III IIF F ) .

2) When the infinity is subjected to a shear stress yz and the rectangular shape ratio varies from / 1a b to

/a b + , the range of IIF is 0.47 0.64 0.55 < , which is close to formula (2). However, for IIIF , the applicable range is only for / 1a b , so we cannot use formula (3) except when / 1a b . When / 1a b = , the derived values of IIIF are less than the case / 1a b , where formula (3) is not applicable. For the stress intensity factor IF , its values can be approximated by IF ) < .

Acknowledgments

The project is supported by the National Natural Science Foundation of China (No. 10872213) and the personnel exchange program of China Scholarship Council (CSC) and German Academic Exchange Service (DAAD).


References[1 ] England F.J., Appl. Mech., 1965, 32(3): 829-836. [2 ] Shibuya T., Koizumi T., Iwamoto T., JSME International Journal: Series A, 1989, 32(4): 485-491. [3 ] Noda N.A., Kagita M., Chen M.C., Int. Solid.& Struct., 2003, 40(24): 6577-6592. [4 ] Chen M.C., Noda N.A., Tang R.J., J. Appl. Mech., 1999, 66(6): 885-890. [5 ] Zhao M.H., Guo C. J., Fang Z.P., J. Mech. Strength, 2002, 24(4): 535-538 (In Chinese). [6 ] Murakami Y.S., Nemat-Nasser S., Engng. Fract. Mech., 1983, 17(3): 193-210. [7 ] Murakami Y., Engng. Frac. Mech., 1985, 22(1): 101-114. [8 ] Irwin G.R., J. Appl. Mech., 1962, 29: 651-654. [9 ] Kassir M.K., Sih G.C., J. Appl. Mech., 1966, 33: 601-611. [10 ] Wang Q., Noda N.A., Honda M.A., Chen M.C., Int. J. Fract., 2001, 108(2): 119-131. [11 ] Noda N.A., Kihara T.A., Archive of Applied Mechanics, 2002, 72(8): 599-614. [12 ] Noda N.A., Xu C.H., Takase Y., JSME,Series A, 2007, 73(4): 379-386 (In Japanese). [13 ] Noda N.A., Xu C,H., Takase Y., JSME,Series A, 2007, 73(4): 468-474 (In Japanese). [14 ] Xu C.H., Noda N.A., Takase Y., JSME,Series A, 2007, 73(7): 768-774 (In Japanese).


Iterative Optimization Methodology for Sound Scattering using the Topological Derivative Approach and the Boundary Element Method

Agustín Sisamon1, Silja C. Beck2, Adrián P. Cisilino3, Sabine Langer4

1 INTEMA, Universidad Nacional de Mar del Plata CONICET. Av. Juan B. Justo 4302, 7600 Mar del Plata, Argentina, asisamon@ fi.mdp.edu.ar, http://www.intema.gov.ar

2 Institut für Angewandte Mechanik, Technische Universität Braunschweig, Spielmannstr. 11, 38106 Braunschweig, Germany, [email protected], http://www.infam.tu-bs.de

3 INTEMA, Universidad Nacional de Mar del Plata CONICET. Av. Juan B. Justo 4302, 7600 Mar del Plata, Argentina, [email protected], http://www.intema.gov.ar

4 Institut für Angewandte Mechanik, Technische Universität Braunschweig, Spielmannstr. 11, 38106 Braunschweig, Germany, [email protected], http://www.infam.tu-bs.de

Keywords: 2D acoustics, topological derivative, iterative optimization process

Abstract. Today, reduction of sound emission plays a vital role while designing objects of any kind.

Desirable aspects might include decreased radiation in certain directions of such an object.

This work shows an approach to iteratively compute the shape of an obstacle which fulfils best to

prescribed design variables using the framework provided by the Topological Derivative and the Boundary

Element Method.

At the beginning of the process an empty design space is defined in which in iterative steps the shape

will be developed. A regular array of points is set over the entire design space. The objective function is

given by a set of prescribed pressure values for the scatter pattern on a circle around this design space.

The object, which acts as a scatterer, is considered acoustically rigid. The shape of the object builds up

cumulatively, adding in each iterative step a rigid inclusion at the position that the Topological Derivative

identifies as the most effective to achieve the prescribed design values. The procedure is repeated until a

given stopping criteria is satisfied.

The proposed method requires the computation of a forward problem and an adjoint problem for each

step. The first is solved using a standard BEM for 2D acoustics, while the latter is solved backwards using

the prescribed pressure values. The insertion of the rigid inclusions in each step is done by removing points

from the design space. The BEM model geometry is updated automatically using a weighted Delaunay

triangularization at those positions where the points have been

eliminated.

The capabilities of the proposed strategy are demonstrated by solving an example.

Introduction

The desire for a quiet environment has lead to including acoustic optimization into the design process of

objects of any kind whether they emit acoustic radiation, transport or receive it. Especially when structures

are to act as noise barriers, a lower sound level in certain areas around these objects is the target.

To determine sound propagation around and along bounded entities the Boundary Element Method

(BEM) is used. It has proven to be effective and has become a well-established technique. Employing the

BEM directly means that the boundary of an object is known in both terms of geometry and boundary

conditions.

For the case of the geometry of an object not being known but at the same time having information about

the field surrounding the object, Feijoó [1] has proposed a method to inversely reconstruct this boundary.

The solution of this inverse scattering problem is based on the topological derivative. An optimization

problem is posed which aims to minimize the difference between the scattering pattern acquired when

placing small scattering objects in the region of interest and the known scattering pattern. The rate of

change of this difference is the topological derivative field. High values in the topological derivate field


indicate positions for placing inclusions to converge to the objective and thus define the sought-after

boundary of the object.

In this work it is intended to combine the BEM and the topological derivative approach to iteratively

identify the optimized shape of an object for a given objective. The framework here is strictly 2D.

In the following sections the topological derivative approach of Feijoó will be outlined, and the iterative

process will be sketched in detail. The example is followed by concluding remarks.

The Topological Derivate Approach

The functional value. The problem outlined in [1] consists of a domain which encloses a scatterer of

unknown shape o (see Fig. 1). On an oblique virtual surface s in a set of values of the wavefield

are known (e. g. from measurements). This set of values corresponds to illuminating the object by an

incident wave from a given angle. The question now is where to put the boundary o of the scattering object

o, so that the results of the wavefield u in the chosen configuration of o fit best to the known wavefield

values um.

Figure 1: The inverse scattering problem.

The solution to this optimization problem is of a least square type:

= arg min (1)

with

= =1

2

2s

(2)

The values of u correspond to the solution of e. g. a plane wave )exp(i)( dxx kuinc(k being the

wavenumber and d the direction of propagation) interacting with the medium and the scattering object. The

solution sinc uuu needs to fulfill 2 + 2 = 0 in 2

o (3)

= 0 on o (4)

lim 1 2 = 0. (5)

Eq. (3) is the Helmholtz equation for a homogeneous medium (without any attenuation: Im(k) = 0), Eq.

(4) is the boundary condition defining the scattering object as being acoustically rigid and Eq. (5) states the

Sommerfeld condition, allowing scattered waves only to travel into infinity.

The topological derivative. The idea is now to place a small circular scattering object of radius into the

domain at point x B (x). This leads to a new domain = \B (x) and a new

functional value .

If extent the new functional value takes a form of

= + DT + o (6)

with

DT = lim 0 as lim 0o

= 0 (7)

s

0

0

n

d

um


The topological derivative DT is an indication for the rate of change in the functional value with

scatterer. When evaluating DT at all points of the domain , a scalar field is

obtained, called the topological derivative. Reconstruction now can be done by placing small scatterers at

points with high values of DT .

The final expression for the topological derivative is of the form (see [1])

DT = Re 2 2 (8)

where u is the solution of the so-called forward problem (see Eq. (3)-(5)), and is the conjugate complex of

the solution of the adjoint problem given by 2 + 2 = 0 in 2

o (9)

= on s (10)

lim 1 2 = 0. (11)

The adjoint problem solves field 0 to be specified on the boundary of the scatterer, o, which results in

the mismatch between the forward problem solution u and the known values um on the virtual surface s.

The Iterative Process

In [1] the topological derivative is computed once to find the most probable shape of the unknown

scatterer. The algorithm proposed in this work allows starting the optimization processes using an initial

geometry for the scatterer. The objective of the optimization is given by prescribed pressure values um on a

virtual surface s that surrounds the design domain (see Fig. 2a).

The iterative algorithm can be summarized as follows:

1) Solve the direct problem (see Eqs. (3) to (5)) for the actual geometry of the scatter using BEM for

the incident planar wave (Fig. 2a)

2) Compute the sound pressure field and its gradient at the internal points and at the points

along the virtual surface s.

3) Compute the mismatch between the forward problem solution u and the known values on the

virtual surface s, .

4) Solve the adjoint problem (see Eqs. (9) to (11)) to find the pressure field on the boundary of the

scatterer.

5) Compute the sound pressure field and its gradient at the internal points.

6) Compute the DT at the internal points using expression (8).

7) Remove the internal points with the maximum values of DT (a few percent of the total number of

points). See Fig. 2b.

8) Remesh the model.

9) Check stopping criterion (typically a limit value for the difference ). If necessary repeat

from step 1)10) At this stage the desired final geometry is obtained.

Figure 2: BEM implementation: (a) Initial BEM model, (b) Elimination of internal points, (c) BEM model

remeshing.

s

um

s

um

s

d

um


Solution of the direct problem (step 1). Since the DT needs the solution of the field and its gradient

, the solution of the direct problem is done using a standard BEM formulation. The geometry of the

scatterer is discretized using two-node linear continuous elements. Sound-hard boundary condition, /

=0, is specified along the complete model boundary. The design space is filled with a regular array of

internal points following the pattern depicted in Fig. 2a.

Solution of the adjoint problem (step 4). The adjoint problem is solved by setting a system of equations

with the nodal values of the pressure field on the boundary of the scatterer as unknowns. To this end, the

points on the virtual surface s are assimilated as internal points, and their values of the mismatch

are expressed in terms of the boundary values of . This results in a system of equations of the form

11 12 1

21 22 2

1 2

1

2 =

1

2 , (12)

where the coefficients contain the integrals of the fundamental solution / on the boundary

elements, are the values of the pressure at the nodes on the boundary and are the values of

the pressure mismatch at the points along the virtual surface s. Following standard procedures (see

reference [2]), the system of equations is set using = 3 , this is, the number of points along the virtual

surface is chosen three times the number of nodes used for the model discretization. The system of Eq. (12)

is solved using a single value decomposition (SVD) algorithm.

Model remeshing (step 8). The removal of internal points is followed by a model remeshing. For this

purpose, the program MeshSuite based on an -shapes algorithm is employed [3]. Upon the input of the

coordinates of the boundary nodes and internal points after each optimization step (see Fig. 2b), MeshSuite

outputs the connectivity of the new model boundary (see Fig. 2c). Thus, those points not used as boundary

nodes are assimilated to internal points in the new discretization for the next iteration. The new boundary

element mesh is checked for multi-connected boundary points and smoothed using a simple relaxation

irregularities on the model boundary as a consequence of the spatial disposition of the internal points.

Further details about the remeshing procedure can be found in reference [4].

Example

The proposed optimization strategy is illustrated by means of an example. We attempt to reconstruct the

shape of a circular scatterer of radius = 2 starting from a square scatterer of side = 2 (see Figure

3a). The radius of the virtual surface s is = 3.5 . The object is illuminated in the direction by a plane

wave with wavenumber = 32 and an amplitude of 1 Pa. The objective values of the wavefield are

specified at = 800 points evenly distributed along s. This large number of points guarantees the

fulfillment of the condition 3 when solving the adjoint problem (see Eq (12)). The distance between

the internal points (which it is also element length of the BEM discretization) is = 0.05 , being

approximately four times smaller than the wave length.

Figures 4a and 4b illustrate contour plots for the pressure solutions and for the direct (Eqs. (3) to (5))

and adjoint problems (Eqs. (9) to (11)) for the initial geometry. The topological derivative result after

computing Eq. (8) is plotted in Figure 4c. It can be seen that maximum values for the DT are at the top and

bottom sides of the object. It is from those zones that the internal points are removed to update the geometry

of the scatterer for the second step. The resulting geometries for the subsequent steps are plotted in Figure

3.

Figure 5 depicts the evolution of the pressure results along the s with the optimization process. In every

case the resulting pressure values are plotted together with the objective values, . It can be seen from

Figs. 3 and 4 that the pressure results converge towards their objective values as the shape of the scatterer

approaches that of a circle.


-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4

(a) (b) (c)

Figure 4: Contour plots for the (a) direct and (b) adjoint pressure fields and (c) the associated topological

derivative result.

Figure 3: Evolution of the shape of the scatterer during the optimization process.

(a) Initial geometry (b) Step #1

(c) Step #6 (d) Step #9

s

Objetive


Conclusions

In this work an iterative optimization strategy based on the topological derivative and BEM has been

presented. The proposed strategy has the ability to compute the shape of an obstacle which fulfils best to a

prescribed sound pressure field specified along a virtual surface which encloses the design space.

The given example shows strategy provides is a promising approach to design the shape of scattering

objects. Its implementation can be easily adapted to optimize the shape of existent objects in order to

decrease its radiation in certain directions.

The discretization and remeshing schemes showed to be flexible and reliable, allowing dealing with

problems of arbitrary shape. However, the effect of the geometric irregularities induced on the model

boundary due to the spatial disposition of the internal points can seriously affect the performance of the

optimization process. Thus, it is important to perform a smoothing of the model boundary in order to

guarantee the performance of the algorithm.

Acknowledgements

This work has been supported by the Project DA0806 sponsored by the MINCYT (Argentina) and the

DAAD (Germany).

References

[1] G. R. Feijoo Inverse

Problems, 20, 1819-1840 (2004).

[2] S. Hampel, S. Langer and A.P. Cisilino Coupling Boundary Elements to a Ray Tracing Procedure

International Journal for Numerical Methods in Engineering. Vol. 73/3, pp. 427-445 (2008).

[3] N. Calvo, S.R. Idelsohn and E. Oñate. The extended Delaunay tessellation Engineering

Computations, 20/5-6 (2003).

[4] L. Carretero Neches and A.P. Cisilino. Topology Optimization of 2D Elastic Structures Using

Boundary Elements. Engineering Analysis with Boundary Elements, Vol. 32, pp. 533-544 (2008).

0 200 400 600 8000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Node

Pre

ssu

re (

Mo

du

le)

0 200 400 600 8000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Node

0 200 400 600 8000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Node

(a) (b) (c)

Figure 5: Evolution of the pressure results: (a) initial geometry, (b) first step (c) ninth step (Objective

function is plotted in black).


A Laplace transform boundary element solution for

the Cahn-Hilliard equation

A. J. Davies and D. Crann

School of Physics, Astronomy and MathematicsUniversity of Hertfordshire, Hatfield, Herts. AL10 9AB, U.K.

Keywords: Laplace transform, boundary elements, Cahn-Hilliard equa-tion, biharmonic diffusion.

Abstract

The Cahn-Hilliard (C-H) equation describes the time development of theconcentration of the components of a fluid during phase separation. Theprocess is described by a non-linear biharmonic diffusion equation. Numer-ical solutions of the C-H equation have used a finite difference approach forthe time variable with finite elements for the space variation.

In previous papers the current authors have shown that the Laplacetransform provides a suitable alternative approach to deal with the timevariation. In particular they have shown that it works very well with theboundary element method for a variety of diffusion problems including non-linear equations and those of biharmonic form. In this paper we shall showhow the introduction of a chemical potential reduces the fourth order par-tial differential equation into a pair of coupled second order equations, one aparabolic equation and the other an elliptic equation. The Laplace transformboundary element method can then be used to solve the coupled system.

Introduction

In a previous paper [1] the authors have considered a Laplace transformboundary element approach to the solution of the biharmonic diffusion equa-tion, ∇4u = 1

α∂u∂t

. The approach involves the introduction of a second de-pendent variable, v, such that ∇2u = v leading to a pair of coupled secondorder partial differential equations.

The C-H equation takes the form

∇4u = −1

α

∂u

∂t+ β∇2φ (1)


whereφ(u) = au + bu2 + cu3 (2)

and a, b and c are constants. Clearly, there are two differences introduced inthe C-H equation: (i) the negative coefficient of ∂u

∂tgives a backward form

of the diffusion equation (ii) the functional form of φ leads to a non-linearequation. In this paper we consider a linearised form in which b = c = 0 toinvestigate the suitability of the Laplace transform approach.

In the modelling of thin oil films [2] the backward biharmonic diffusionequation ∇4u = − 1

α∂u∂t

is developed. In a preliminary study, the currentauthors [3] have shown that, even though the backward Laplacian diffusionproblem ∇2u = − 1

α∂u∂t

is ill-posed [4] e.g. the solution does not exist for mostinitial data and even if a solution exists it is very likely to blow up. Howeverthe equivalent biharmonic problem does not appear to suffer from these dif-ficulties. This leads us to consider a more serious study of the C-H equation.

The Cahn-Hilliard equation

The C-H equation is used to model two-phase fluid flow and was first de-scribed by Cahn and Hilliard [5, 6, 7]. Some mathematical features of theequation have been discussed by, among others, Elliott et al. [8, 9, 10].

We shall consider the linear form of equations (1) and (2) with b = c = 0so we can write equation (1) in the form

∇4u = −1

α

∂u

∂t+ k∇2u (3)

to be solved in some region Ω subject to the boundary conditions on Γ

u = u(s, t) and q ≡∂u

∂n= q(s, t) (4)

and the initial condition

u(x, y, 0) = u0(x, y) (5)

We follow the approach of Toutip et al. [11] and write v = ∇2u so thatequation (3) becomes the pair of coupled equations

∇2u = v (6)

∇2u = −1

α

∂u

∂t+ kv (7)

When applying the boundary conditions we note that we have a pair ofLaplacian operators in equations (6) and (7). To ensure the problem isproperly-posed we must apply either u or q, but not both, at each point ofΓ. We consider Γ comprised of two sections, Γ1 and Γ2 and write

u(s, t) =

u1(s, t) s ǫ Γ1

u2(s, t) s ǫ Γ2


q(s, t) =

q1(s, t) s ǫ Γ1

q2(s, t) s ǫ Γ2

then we chooseu = u1 on Γ1 and q = q2 on Γ2 (8)

p ≡∂v

∂n= p1 ≡ ∇2q1 on Γ1, v = v2 = ∇2u2 on Γ2 (9)

The Laplace transform

We denote by u(x, y;λ) and v(x, y;λ) the Laplace transforms of u(x, y, t)and v(x, y, t) respectively. Then equations (6) and (7) become, in transformspace,

∇2u = v (10)

∇2v = −1

α(λu − u0) + kv (11)

The solutions of equations (10) and (11) are obtained using the followingiterative scheme:

∇2v(n+1) = −1

α(λu(n) − u0) + kv(n) (12)

∇2u(n+1) = v(n+1) (13)

withu(0) = u0 and v(0) = v0 = ∇2u0

We use the dual reciprocity boundary element method to solve the coupledelliptic equations (12) and (13).

Equation (12) is solved subject to the boundary condition

v = v2 on Γ2 and p = p1 on Γ1 (14)

and equation (13) is solved subject to the boundary condition

u = u1 on Γ1 and q = q2 on Γ2 (15)

The dual reciprocity boundary method

We divide Γ into N elements, Γk, and choose L points inside Ω in theusual boundary element manner. Now, we can write each of the equations(12) and (13) in the form

∇2u = b (16)

The integral equation equivalent to equation (16) is

cΓuΓ +

∮Γ

q∗u dΓ −

∮Γ

u∗q dΓ =

∫Ω

bu∗ dΩ (17)


where u∗ = − 12π

ln R is the usual fundamental solution for the Laplacian

operator and q∗ = ∂u∗∂n

.We expand the domain function b as a linear combination of radial basis

functions

b ≈N+L∑j=1

αjfj(R) (18)

where the fj(R) are chosen such that, for some uj , we have [12]

∇2uj = fj(R) (19)

Hence, using equations (18) and (19) together with Green’s theorem,equation (17) becomes

ciui +

N∑k=1

∫Γk

q∗u dΓ −

N∑k=1

∫Γk

u∗q dΓ =

N+L∑j=1

αj

(cj uij +

N∑k=1

∫Γk

q∗uj dΓ −

N∑k=1

∫Γk

u∗qj dΓ

)

for i = 1, . . . ,N . We now write this system of equations in the usual matrixform where we use the subscript B to denote that the corresponding quantityis associated with a boundary node:

HBUB − GBQB =[HBU −GBQ

]α (20)

where the values of the coefficients, αi, are obtained by collocating at theN + L points giving the usual system of equations

b = Fα (21)

and α is given byα = F−1b (22)

Internal values may be obtained in a similar manner [1] and written in theusual matrix form

IUI = GIQB − HIUB +[HIU− GIQ

]α+IUα (23)

Finally, then equations (20) and (23) can be combined in the form

HU − GQ =[HU− GQ

]F−1b (24)

Now we defineS =

[HU− GQ

]F−1 (25)


to obtain the system of equations for the boundary solution UB and QB andthe internal solution UI [

HU− GQ]

= Sb1 (26)

In a similar manner we obtain for v[HV − GP

]= Sb2 (27)

Hence the discrete systems of equations associated with the partial dif-ferential equations (10) and (11) are

HV(n+1)

− GP(n+1)

= b1

(U(n),V(n)

)HU

(n+1)− GQ

(n+1)= b2

(V(n+1)

) (28)

with U(0) = U0 and V(0) = V0

On application of the boundary conditions equation (28) may be writtenin the form

A1x(n+1) = F1

(U

(n), V(n)

)and A2y

(n+1) = F2

(V

(n+1))

where x(n+1) =[V

(n+1)P

(n+1)]T

and y(n+1) =[U

(n+1)Q

(n+1)]T

and the

iteration is terminated using a suitable stopping criterion.Finally then, the approximate transform is inverted to obtain the ap-

proximate solution U.

Numerical inversion of the Laplace transform

We use the Stehfest numerical procedure [13] which has been shown to bewell-suited to the solution of diffusion-type problems [14].

Choose a specific time value, τ , at which we seek a solution and definea set of transform parameters

λj = jln 2

τ: j = 1, 2, . . . ,m; m even

The dual reciprocity boundary element method is used for each λj to obtainsets of approximate boundary and internal values given respectively by

UB, ij; i = 1, . . . ,N ; j = 1, . . . ,m and UI, kj; k = 1, . . . , L; j = 1, . . . ,m

The inverse transforms are then given by

UB, r ≈ln 2

τ

m∑j=1

wjUB,rj and UI, r ≈ln 2

τ

m∑j=1

wjUI,rj


where r = 1, . . . ,N for boundary points and r = 1, . . . , L for internal points.The weights, wj, are given by Stehfest [13] and tabulated in [15].

Results

We illustrate the process with the following example

∇4u = −1

α

∂u

∂t+ k∇2u + h(x, y, t)

in the unit square (x, y); 0 < x < 1, 0 < y < 1 subject to Dirichlet andNeumann boundary conditions appropriate to the exact solution in the caseα = 1

u(x, y, t) = (1 + x4 + y4)e−t

We use 36 linear boundary elements and 9 internal nodes with f(R) = 1+R.In Figure 1 we show the time development at three points and in Figure

2 we show the space variation for three times, in both cases the points arealong the line of symmetry.

In both cases we see that the approximate solutions compare very wellwith the analytic solution. Finally we note here that, as reported in [1], thechoice of Γ1 and Γ2 is somewhat arbitrary and a change in this choice doesnot affect the accuracy of the solutions.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

u(x,y,t)

x=y =0.2

x=y =0.5

x=y =0.8

LT approx.analytic

Figure 1: Time development at three points

Conclusions

The Laplace transform dual reciprocity method has been shown to pro-vide a suitable approach to the solution of a backward biharmonic diffusion


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

l/√

2

u(l, t)

t=0.2t=0.5

t=1.0

t=5.0

LT approx.analytic

Figure 2: Space variation along the line of symmetry, l is the distance fromthe origin

problem, the linearised C-H equation. It is very pleasing to report thateven though it is posed as a backward problem we find no evidence of ill-conditioning. This particular problem is rather stylised since the C-H equa-tion is non-linear. Nevertheless it gives us confidence to move on and tacklethe non-linearity.

References

[1] Davies AJ and Crann D. A Laplace transform solution of the bihar-monic diffusion equation. Boundary Elements XXVIII, 243–252 (2006).

[2] Tanner LH and Berry MV. Dynamics and optics of oil hills andoilscapes. J. Phys. D; Appl. Phys., 18: 1037–1061 (1985).

[3] Crann D and Davies AJ. A Laplace transform boundary element solu-tion for the biharmonic diffusion equation. University of Hertfordshire

Department of Physics, Astronomy and Mathematics Technical Report

97 (2006).

[4] Wilmott P, Howison S and Dewynne J. The mathematics of financial

derivatives. Cambridge University Press (1995).

[5] Cahn JW and Hilliard JE. Free energy of a non-uniform system-I: In-terfacial free energy. J. Chem. Phys. 28: 258–267 (1958).

[6] Cahn JW. Free energy of a non-uniform system-II: thermodynamic ba-sis. J. Chem. Phys. 30: 1121–1124 (1959).


[7] Cahn JW and Hilliard JE. Free energy of a non-uniform system-III:Nucleation in a two-component incompressible fluid. J. Chem. Phys.

31: 688–699 (1959).

[8] Elliott CM and French DA. Numerical studies of the Cahn-Hilliardequation for phase separation. IMA J. App. Math. 38: 97–128 (1987).

[9] Blowey JF and Elliott CM. The Cahn-Hilliard gradient theory for phaseseparation with non-smooth free energy. Part 1: Mathematical Analy-sis. Euro. J. App. Meth. 2: 233–280 (1991).

[10] Blowey JF and Elliott CM. The Cahn-Hilliard gradient theory for phaseseparation with non-smooth free energy. Part 2: Numerical Analysis.Euro. J. App. Meth. 3: 147–179 (1992).

[11] Toutip W, Davies AJ and Kane SJ. The dual reciprocity methodfor solving biharmonic problems. Boundary Elements XXIV, 373-380(2004).

[12] Partridge PW, Brebbia CA and Wrobel LC. The Dual Reciprocity

Boundary Element Method. Computational Mechanics Publications(1992).

[13] Stehfest H. Numerical inversion of Laplace transforms. Comm. ACM.,13: 47–49 and 624 (1970).

[14] Crann D. The Laplace transform boundary element method for

diffusion-type problems. PhD Thesis, University of Hertfordshire (2005).

[15] Davies AJ and Crann D. A handbook of essential mathematical formu-

lae. University of Hertfordshire Press (2008).


Strategy for writing general scalable parallel boundary-element codes

F.C. de Araújo¹, E. F. d'Azevedo²,†, and L.J. Gray²,‡

¹Dept Civil Eng, UFOP, Ouro Preto, MG, Brazil; [email protected] ²CSMD, ORNL, Oak Ridge, P.O. Box 2008, USA; †[email protected]; ‡[email protected]

Keywords: 3D standard BE formulations, parallel processing, CNT-based composites, thin-walled elements,subregion-by-subregion technique.

Abstract. In this work, a strategy based on a generic subregion-by-subregion (SBS) algorithm is employed to develop general scalable BE parallel codes. In this algorithm, the interactions between the subdomains are taken into account only during the solution of the system by a Krylov iterative method. Thereby, significant reduction of memory and CPU-time consumption is achieved as the global system matrix is not explicitly treated. On the other hand, special integration procedures for calculating nearly-strongly-singular integrals make the use of discontinuous boundary elements (in which quasi-singular integrals occur) possible. In fact, discontinuous elements are very useful for establishing BE models for complex heterogeneous domains. A matrix-copy option, useful for modeling systems with repeated parts, as identical fiber reinforcements, is also available. To verify the performance of the code, the 3D microstructural analysis of carbon-nanotube-reinforced composites (CNT composites) is considered. Particularly, mechanical properties of composites are measured. The representative volume elements (RVEs) adopted consist of carbon-nanotubes (shell-like elements) coupled with a polymeric material matrix.

Introduction

The Finite-Element Method (FEM) is still the tool of choice in engineering analysis of structures and solids, however its application to thin-walled solids and composites has been accompanied with a series of issues as element-distortion sensitivity, locking phenomena, and the non-fulfillment of stress continuity between layers or at matrix-fiber interfaces. Especially in the case of composites, the mesh generation itself is a bottleneck in finite-element (FE) analysis. To escape these difficulties, in recent works, the direct application of 3D standard boundary-element formulations has then been considered as an alternative to solve general composites and thin-domain problems [1, 2, 3]. Besides advantages as high accuracy, fulfillment of radiation conditions, and easier mesh generation, the Boundary Element Method (BEM) also presents the following interesting characteristic: it is derived from the exact integral representation of the problem response and does not require any interelement compatibilty (in the FE sense) for assuring solution convergence. This actually allows more flexibility for generating boundary-element models as long as the integrals involved are accurately evaluated. Indeed, this is the basis of discontinuous boundary elements, very useful for the BE subregion-by-subregion (BE-SBS) algorithm [4], considered in this work for the development of the parallel code. In many of the works concerning the development of BE parallel codes [5, 6, 7], either for symmetric multiprocessor (SMP) or massively parallel processor (MPP) architectures, the parallelism has been based on different ways to generate and to scatter the global boundary-element (BE) system of equations onto the available processors, its solution, in most cases, being carried out by applying available high-performance packages as LAPACK or ScaLAPACK. Unlike these works, [8] used a domain decomposition method (DDM) for solving 2D potential problems. They directly scattered the subdomain systems onto the processors and got the solution for the whole problem from the independent solution of each subdomain, wherein iterative schemes were used to introduce the coupling conditions. In [9], a DDM-based strategy is also considered to solve 2D elasticity problems with cracks. This procedure allows the independent assembling of the subdomain matrices, and is based on the condensation of the problem response to the interface tractions. In fact, this strategy may be cumbersome and time-consuming for 3D problems with complex composite morphology. In the present paper, the parallel version of the BE subregion-by-subregion (BE-SBS) algorithm [3, 10], a generic non-overlapping domain decomposition method, is presented. Besides being a fundamental technique in BE formulations, substructuring techniques are a spontaneous way to develop parallel codes, irrespective of the computer architecture. However, differently from displacement-based FE formulations, wherein one should never worry about traction discontinuity at the element corners or edges, BE


formulations are mixed formulations and require simulating traction discontinuity at some corner/edge nodes of the subdomains. Certainly, this is a fact that makes the modeling with subregions, in case of solids with complex internal geometries, a somehow tedious task. This was verified in [11], where continuous boundary elements are used to model 3D frequency-dependent elastodynamic problems. To get rid of this issue, discontinuous boundary elements have been adopted [2, 4, 10]. Doing so, coupling conditions can be directly imposed, and BE-subdomain models are then a lot easier generated. The other issue related to BE-subdomain algorithms (in parallel or serial versions) is how to optimally deal with the highly sparse resulting matrices. Kamiya et al. [8] proposed iterative coupling procedures that perfectly treat the matrix sparsity but are not reliable concerning convergence. In [9], the coupling conditions are directly introduced, but condensing the system unknowns to the interface tractions is awkward, requires additional memory beyond that necessary for allocating the isolated subsystems, and may be time-consuming for complex models. In this work, the BE-SBS algorithm [3, 10] is adopted. Employing some iterative solver, a solution strategy for general coupled problems is derived wherein no explicit global matrix have to be assembled, and only memory space for strictly allocating the subregion subsystems is needed. In this work, a simple diagonal-preconditioned Bi-CG solver is applied. However it is emphasized that the coupling conditions are directly enforced. Structured matrix-vector product (SMVP) and matrix-copy options are also implemented to increase the efficiency the code [4]. 3D simulations of CNT-based composites are carried out to show its performance.

The BE-SBS-based parallel algorithm

The BE parallel code is based on the BE-SBS algorithm detailed in previous papers [3, 10]. This algorithm considers a substructuring technique (non-overlapping domain decomposition method, DDM), and makes use of iterative solvers, similarly as done in element-by-element-based (EBE-based) finite-element formulations, to solve the global BE system of equations without explicitly assembling it. In general, after the boundary conditions have been introduced at each BE subregion separately, a set of sn algebraic systems of equations given by

iii

i

mimimmiim xApGuH

1

1

iii

n

immiimimim yBpGuH

1, sni ,1 , (1)

where sn is the number of subregions, has to be solved by enforcing continuity and equilibrium conditions at the interfaces:

jiij

jiij

pp

uu at ij .

(2)

In Eq. (1), ijH and ijG denote the usual BE matrices obtained for source points pertaining to subregion i

and associated respectively with the boundary vectors iju and ijp at ij . Note that if ji , ij corresponds to the interface between i and j , which denote the i-th and j-th subregion respectively; ii is the outer boundary of i . The global system in Eq. (1) is then conveniently solved by applying an iterative solver. Here particularly, the diagonal-preconditioned biconjugate gradient (J-BiCG) solver. As in the BE SBS algorithm there is no overlapping of coefficients belonging to edges or corners shared by different subregions, as it happens in finite-element models, the data structure in Eq. (1) does not need any further optimization. All zero blocks present in the highly-sparse global system matrix are perfectly excluded. Besides, the following techniques/strategies are especially important for increasing the efficiency of the BE-SBS-based code: discontinuous boundary elements, structured matrix-vector products (SMVP), special integration quadratures, and the matrix-copy option. In the references [2, 3, 4, 10], the BE SBS algorithm is thoroughly described. In Fig. 1, the flowchart of the BE-SBS-based parallel code is presented, wherein it is assumed that kprocesses is considered. In fact, as in the BE SBS algorithm the subdomains are independently treated during the entire analysis, its implementation for running in a parallel-processing platform is immediate. Assembling the algebraic systems needs no information from other processes. Only during its solution, communication between the processes is needed for updating the boundary values in all subregions (Fig. 2).


data input (broadcastfrom ip=0)

data input (broadcastfrom ip=0)

matrix assembly

boundary conditions

coupling search

Krylov solver

consuming parts most-CPU-time-

of the code

parallelprocesses

ip=0 ip=1 ip=k

data input (read)

matrix assembly

matrix assembly

boundary conditions

boundary conditions

coupling search

coupling search

Figure 1. Flowchart of the BE-SBS-based parallel code.

structuredmatrix-vectorproducts (SMVP)

boundary data

SMVP

iter = 1,2,...,until convergence

solution

ip=0 ip=1 ip=k

boundary data

boundary data

SMVP SMVP

solution solution

boundary-data transfer between substructures in different processes

Figure 2. Solution phase (Krylov solver).

Applications and discussions

The performance of the BE SBS-based parallel code detailed above is observed by determining engineering constants for the CNT-based composites shown in Fig. 3, which consider hexagonal fiber-packing patterns for arranging the CNTs inside the matrix material. In all representative volume elements (RVEs), constructed coupling together single unit cells with dimensions nml 101 , and 2l nml 203 (see Fig. 3a), the following phase constants, adopted in [12], are considered:


CNT:2000,1 nm

nNECNT (GPa); 30.0CNT ,

Matrix:2100 nm

nNEm (GPa); 30.0CNT .

12

3

1

2

3

1 2

3

(c)

(a)(b)

(d) (e)

Figure 3. RVEs based on hexagonal-packed CNTs

The long CNT fibers are geometrically defined by cylindrical tubes having outer radius nmr 0.50 , inner radius nmri 6.4 , and length nml f 10 (equal to the RVE thickness). In Table 1, additional data for the models shown in Fig.1 are provided. When needed, discontinuous boundary elements are automatically generated by shifting the nodes interior to the elements a distance of 10.0d (measured in the natural coordinate system). The matrix-copy option is also conveniently considered to replicate physically and geometrically identical subdomains. The boundary element adopted is an 8-node quadrilateral one, and the tolerance for the iterative solver (J-BiCG) is taken as 510 . The analyses were carried out at ORNL Institutional Cluster, consisting of 80 usable nodes, each one having Dual Intel 3.4GHz Xeon EM64T processors, 4GB of memory, and dual Gigabit Ethernet Interconnects. In Table 2, the engineering parameters obtained by employing the present code are confronted with results given in [12], and estimated by the rule of mixture [4, 12, 13]. As seen, values estimated by the rule of mixture and by refined 3D FE models [12] are in very good agreement with the ones calculated with the present method. No significant change in the values is also observed as a function of the number of unit cells per RVE. As sample results for checking the parallel-processing performance, CPU-time and memory-use scalability curves for the 1010 -unit-cell RVE (largest model) under strain state 1 are shown in Fig. 4.


Table 1. Model data for the RVEs

model nsub* nel** nnodes† ndof‡ sparsity (%) 11 6 138 856 2,568 72 22 17 656 3,456 10,368 86 33 34 1,464 7,800 23,400 93 55 86 4,040 21,720 65,160 97 1010 321 16,080 87,040 261,120 99

*n. of subregions; **n. of elements; †n. of functional nodes; ‡n. of degrees of freedom

Table 2. Engineering constants for the hexagonal-packed long-CNT RVEs

model mEE /1 mm EEEE /,/ 32 1312 , 23

11 1.8081 1.0889 0.2943 0.5107 22 1.8074 1.0839 0.2936 0.5107 33 1.8074 1.0916 0.2931 0.5185 55 1.8126 1.0813 0.2927 0.4997 1010 1.8014 1.0805 0.2926 0.5103

rule of mixture† 1.8131 - - - †RVE volume fraction is %9.035fV

20 30 40 50number of processors

0.4

0.45

0.5

0.55

0.6

CP

U ti

me/

nite

r (se

c./it

er.)

measured valueslogarithmic fit

CPU-time scaling (strain state 1)

20 30 40 50number of processors

100

120

140

160

180

200

220

240

used

mem

ory

(real

-val

ued

arra

y, M

byte

s)

measured valueslogarithmic fit

Memory scaling (strain state 1)

(a) (b)Figure 4. Scalability curves for CPU time and storage memory

Concerning memory use, the scalabilty is very good, practically following the logarithmic fit (see Fig. 4b).Concerning the CPU-time, it is observed that the speedup tends to decrease when the number of processors is incremented (e.g. from 30 to 50 processors; see Fig. 4a). An explanation for that is a relative increase on the interprocessor communication compared to the load per processor.

Conclusions

A robust BE-SBS technique is used to derive a general 3D BE parallel code. Particular applications of the code concern the evaluation of effective engineering constants for 3D CNT-reinforced composites. First, it is observed that as a consequence of the special quadratures available in the code, discontinuous and disproportionate boundary elements can be employed. In this way, the modeling of complex coupled solids, as composites, is greatly simplified. In addition, the matrix-copy option, which avoids the repeated mesh generation and calculation of coefficient matrices for identical substructures, considerably facilitates the modeling of very complex periodic composites. In the particular applications shown above, no efficiency


gain has been actually observed during the assembly phase as the corresponding CPU-time measurements were insignificant compared to the solver CPU time (dominant). However, for large identical subregions, this option might increase the computational efficiency. Moreover, for complex composites, boundary-integral-based models are simpler to generate than volume-based ones. Thus, the strategy proposed is believed to be very convenient for analyzing general composites. Among others, a contribution of this study is certainly the proposal of a general strategy for developing parallel-processing BE codes, readily applicable to any BIE-based methods. We notice that the algorithm proposed presents the following interesting general characteristics: (1) the BE models are independently generated, stored, and manipulated (no explicit global matrix assembly takes place), (2) no variable condensing is carried out, avoiding then the calculation of Schur complements, (3) the interface conditions are directly imposed, avoiding then the use of some iterative strategy, (4) discontinuous boundary elements are used to make the generation of coupled models easier, (5) an iterative (Krylov) solver is employed, (6) the high sparsity of the system is perfectly exploited. Obviously, as the models are independently stored, the memory-use scalability of the code is excellent, as we see from the results in the previous section. On the other hand, if the number of processors is incremented, the interprocessor communication will be more intense, decreasing then the processing speedup after a certain critical number of processors. In this work, indeed focused on the BE-SBS-based parallel code, no special attention has been properly paid to the Krylov solver itself. As noted, just a plain diagonal-preconditioned BiCG solver has been employed. In fact, it is known that this particular solver presents irregular convergence behavior, sometimes even not converging, depending on the system-matrix spectrum. Anyway, considering all the development brought about on iterative solvers and preconditioning techniques in the last two decades [14], we do believe that the BE-SBS algorithm is the optimal way to solve complex coupled BE models, and a promising alternative to develop general BE parallel codes, accounting for scalability of memory requirements and processing time.

Acknowledgement

This research was sponsored by the Brazilian Research Council (CNPq), and by the Research Foundation for the State of Minas Gerais (FAPEMIG).

References[1] X.L.Chen and Y.J.Liu Eng. Anal. Boundary Elements, 29, 513-523 (2005).

[2] F.C.Araújo, L.J.Gray Comput. Mechanics, 41, 633-645 (2008).

[3] F.C.Araújo, K.I.Silva, J.C.F.Telles Comm. Num. Methods Eng., 23, 771-785 (2007).

[4] F.C.Araújo, L.J.Gray Comp. Mod. Eng. Sci., 24(2), 103-121 (2008).

[5] R.Natarajan, D.Krishnaswamy Eng. Anal. Boundary Elements, 18, 183-193 (1996).

[6] S.W.Song and R.E.Baddour Eng. Anal. Boundary Elements, 19, 73-84 (1997).

[7] M.T.F.Cunha, J.C.F.Telles, A.L.G.A.Coutinho Adv. Eng. Software, 35, 453–460 (2004).

[8] N.Kamiya, H.lwase, E.Kita, Eng. Anal. Boundary Elements, 18, 209-216 (1996).

[9] X.Lu and W.-L.Wu Eng. Anal. Boundary Elements, 29, 944–952 (2005).

[10] F.C.Araújo, K.I.Silva, J.C.F.Telles Int. J. Numer. Methods Engrg. 68, 448-472 (2006).

[11] F.C.Araújo, C.Dors, C.J. Martins, W.J.Mansur J. Braz. Soc. Mech. Sci. Eng., 26(2), 231-248 (2004).

[12] X.L.Chen and Y.J.Liu Comput. Mat. Sci., 29, 1–11 ( 2004).

[13] M.W.Hyer, Stress Analysis of Fiber-Reinforced Composite Materials, 1st ed., McGraw-Hill (1998).

[14] R.Barett, M.Berry, T. F.Chan, J.Demmel, J.Donato, J.Dongarra, V.Eijkhout, R.Pozo, C.Romine,

H.van der Vorst Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods,

2nd Ed., SIAM (1994).


Incomplete LU preconditioning of BEM systems of equations based upon the generic substructuring algorithm

F.C. de Araújo¹, E. F. d'Azevedo²,†, and L.J. Gray²,‡

¹Dept Civil Eng, UFOP, Ouro Preto, MG, Brazil; [email protected] ²CSMD, ORNL, Oak Ridge, P.O. Box 2008, USA; †[email protected]; ‡[email protected]

keywords: 3D boundary-element models, Krylov solvers, subregion-by-subregion algorithm, incomplete LUpreconditioners.

Abstract. The generic substructuring algorithm, developed in previous works, is directly employed to construct global incomplete LU preconditioners for BEM systems of equations. As the BE matrices for each BE subregion are independently assembled, the corresponding L and U factors are easily calculated, and incomplete-LU-based preconditioners for particular BE models are immediately formed. So as to highlight the efficiency of the preconditioning proposed, the Bi-CG solver, known to present a quite erratic convergence behavior, is considered. Complex 3D representative volume elements (RVEs) of carbon-nanotube (CNT) composites are analyzed to show the performance of the preconditioned iterative solver. The models contain up to several tens of thousands of degrees of freedom. The relevance of the preconditioning technique is also discussed in the context of developing general (parallel) BE codes.

Introduction

The parallelism embedded in fast reliable Krylov solvers combined with the today's parallel computer architectures have definitely contributed for the devising of efficient scalable parallel codes, becoming then in the last decades an appealing alternative for solving large-order engineering problems [1, 2]. In these cases, in general, direct solvers present the following disadvantages: they may be exceedingly CPU time-consuming and memory-consuming, and their parallel implementation is awkward. For general non-symmetric matrices, like BE matrices, the very difficult problem is how to devise reliable iterative solvers. Indeed, reliability regarding convergence in this case (when the matrices are non-symmetric or indefinite), despite the number of outstanding scientific contributions across the last six decades, is a still open question [3]. Knowing that basic iterative solvers, for instance, the Jacobi or Gauss-Seidel methods, are convergent only in very special cases, e.g. when the spectral radius of the corresponding iteration matrix is less than 1, so the possible alternatives for dealing with non-symmetric are then Krylov solvers, which in this case can be subdivided in two broad classes of algorithms: long-recurrence algorithms (GMRES and variants), and short-recurrence ones (Bi-CG and variants). Because of the memory requirements for large problems, and non-rare convergence stagnation in practice, long-term recurrence methods should indeed be avoided, and so our universe of possible efficient iterative solvers for large BE models, at last, reduces to short-recurrence ones such as the Bi-CG. However, the Bi-CG method presents an erratic convergence behavior, and typically fails to find the solution for non-Hermitian systems. Thus, to smooth out possible convergence irregularities, modified hybrid solvers has been to generate by combining the Bi-CG solver with residual-minimization methods, as the GMRES. Following this idea, culminated then in developing solvers such as the transpose-free Bi-CGSTAB(l) [4] and the GPBi-CG (generalized product Bi-CG) [5]. Additionally, preconditioners may be employed to accelerate the iterative process [1]. For BEM solvers, a series of preconditioners have been reported in the technical literature [6-8]. In general, the splitting matrix of basic iterative methods as the Jacobi, Gauss-Seidel or incomplete LU decomposition methods can be used to construct preconditioners. Roughly speaking,


preconditioners are a way to state a relationship between direct and iterative solvers, in the sense that if the preconditioning matrix becomes the system matrix, so the iterative method at hand becomes a direct solver (converging then to the system solution at one single iteration step.) Furthermore, domain decomposition methods (DDM) allied with direct methods may also be employed to construct global preconditioners. This will be very important e.g. to parallelize incomplete LU-based preconditioner, which is among the most efficient ones, but not easily parallelizable. In this paper, the BE substructuring algorithm [9-10] is employed to construct a global incomplete LU preconditioner for the BE model at hand. In other words, a DDM-based technique, applied to decompose a certain problem domain into a generic number of coupled BE models, is considered to define the allowable fill-in positions for the incomplete LU decomposition preconditioner. In the algorithm, the coupling conditions between the subdomains are imposed in a direct (non-iterative) way, and the subsystems are independently assembled, so that the block-diagonal matrices corresponding to each subregion can be easily decomposed in their L and Ufactors. For the applications here, the preconditioning proposed is incorporated into the Bi-CG solver. An important point in this respect is that, as the Bi-CG solver is expected to fail in practice, as commented above, then in the numerical experiments the efficiency of the preconditioner itself will be highlighted. Several models, with thousands of degrees of freedom, employed to simulate complex carbon-nanotube (CNT) composites, are considered to show the performance of the preconditioning. The efficiency and relevance of the preconditioning proposed is also discussed in the context of ideas for developing general scalable BE parallel codes.

2. The BE-SBS algorithm and the associated preconditioner

As well reported in previous works [9, 10], the boundary-element substructuring-by-substructuring (BE-SBS) algorithm is comparable to the element-by-element (EBE) technique, developed to finite-element analysis (FEA) [27] while a subregion or substructure corresponds to a finite element. Thus, if needed, we can have a subregion mesh as fine as a finite-element mesh, and if the BE global system matrix were explicitly assembled, it would be highly sparse as well. It is also noted that the BE-SBS algorithm can be compared to Finite Element Tearing and Interconnecting (FETI) methods [11] as well, where a given problem domain is decomposed (torn) into non-overlapping subdomains, and posteriorly interconnected by imposing the corresponding continuity conditions at the interfaces. The BE-SBS algorithm embeds Krylov iterative solvers, and the global response for a problem is obtained by working exclusively with its local full-populated subsystems of equations. No global explicit system matrix is assembled; no zero blocks are stored or handled. The boundary conditions are introduced during the matrix assembly for each subsystem, and the interface conditions (between the subdomains), given by

jiij

jiij

ppuu

at ij(1)

are directly (not iteratively) imposed in the matrix-vector products during the iterative solution process. For sn subregions, after introducing the boundary conditions, the BE global system of equations is then given by

iii

i

mimimmiim xApGuH

1

1

iii

n

immiimimim

s

yBpGuH 1

, sni ,1 , (2)


where ijH and ijG denote the regular BE matrices obtained for source points pertaining to subregion i and associated respectively with the boundary vectors iju and ijp at ij . Note that if

ji , ij denotes the interface between i and j ; ii is the outer boundary of i . If the system of equations in (2) were, say for 4sn (four subregions), explicitly assembled, it would have the following general aspect:

x 33

x 11

x 22

p31

u12

u13

u24

u14

p21

u23

p32

u34

x 44

p41

p42

p43

G32

A11 H12 H13

A22 H23G21

A33G31

G12 G13

G23H21

H31 H32

H14

H41

H24 G24

H34 G34

G14

H42 H43 A44G41 G42 G43

y1

B 11

B22

B 33

B 44

y2

y3

y4

. (3)

In this system, note that ijH jiH ijG 0G ji if the there is no coupling between i and jsubdomains. However, as commented previously, we do not have any explicit system of equations. Instead, the working subsystems are those ones shown in expression (2). The matrix-vector and transpose-matrix-vector products are then calculated from the separate contributions from each subsystem, while as already commented above, during the solver iterations, the interface conditions are imposed in a direct way. In this study, the allowable fill-in positions for the incomplete LU decomposition are taken as those of the diagonal blocks of the coupled system, i.e., for the particular (explicit) system of equations shown in (3), the subsets of positions highlighted in gray. Inferring from Eq. (3) that, for a generic number of subregions, the diagonal blocks of the coupled system are given by

iniiiiiiii HHAGGQ 1,1,1 , sni ,1 (4)

where the iQ matrices are straightforwardly formed having the subregion matrices of the model at hand, the construction of the global SBS-based ILU preconditioner for the coupled system of equations (3) is then immediate. However, as the subdomain submatrices, this global preconditioner is not explicitly assembled either; it is separately stored per subregion at an additional memory space of the size

)()( ndofnnnondofnnno , where nno is the number of nodes of the model, and ndofn is the number of degrees of freedom per node. In the code, the BE-SBS-based preconditioner is employed to accelerate the Bi-CG iterations.


4. Results and discussions

The performance of the ILU SBS-based preconditioner detailed above is measured by analyzing the complex CNT-based composites shown in Fig. 1, wherein representative volume elements (RVEs) based on 11 , 22 , and 55 unit cells are employed. The long CNT fibers are geometrically defined by cylindrical tubes having outer radius nmr 0.50 and inner radius nmri 6.4 , and length nml f 10 . In general, when needed, discontinuous boundary elements are automatically generated by shifting the nodes interior to the elements a distance of 10.0d (measured in the natural coordinate system). The matrix-copy option is also conveniently considered to replicate physically and geometrically identical subdomains, avoiding then assembling repeatedly their corresponding matrices. The 8-node quadrilateral boundary element is employed, and in all analyses, 88 and 6 integration points are used for evaluating all surface and line integrals involved, respectively, in the special integration quadratures embedded in the code. In all (RVEs), the following pure phase constants are adopted [12]:

CNT:2000,1 nm

nNECNT (GPa); 30.0CNT ,

Matrix:2100 nm

nNEm (GPa); 30.0CNT .

The tolerance for the iterative solver (Bi-CG) is taken as 810 . The diagonal preconditioning (Jacobi) and the preconditioning proposed in this paper (BE-SBS-based ILU decomposition) are then contrasted to show the efficiency brought about by the latter preconditioner. The analyses were carried out at a notebook with dual intel 2.26GHz processor, and 3GB of random access memory. Important model data are provided in Table 1. In Table 2, the engineering parameters extracted from the analysis of all the RVEs shown in Fig. 2 are confronted with results calculated by Liu and Chen [12] via finite-element analysis, and estimated (when possible) by the rules of mixture [12]. As seen, very good agreement between the results is obtained. Furthermore, no significant change in the constant values is also observed as the number of unit cells per RVE increases.

Table 1. Model data for the square-packed long-CNT RVEs model nsub* nel** nnodes† ndof‡ sparsity (%)

11 2 128 608 1,824 29 22 8 512 2,660 7,980 81 55 50 1,344 17,456 52,368 97

*n. of subregions; **n. of elements; †n. of nodes; ‡n. of degrees of freedom

1

(a)

2

3

12

3

(b) 1

2

3

(c)

Figure 1. Square-packed long-CNT-based RVEs.


Table 2. Engineering constants for the square-packed long-CNT RVEs model mEE /1 mm EEEE /,/ 32 1312 , 23

11 1.3227 0.8302 0.2974 0.3595 22 1.3228 0.8319 0.2973 0.3600 55 1.3228 0.8319 0.2972 0.3580

Chen & Liu (3D FE) 1.3255 0.8492 0.3000 0.3799 rule of mixture† 1.3255 - - -

†RVE volume fraction is %617.3fV

Table 3. Performance data for the square-packed long-CNT RVEs; 8100.1tol model system order n. of

iterations (BE SBS-based ILU)

n. of iterations (Jacobi)

CPU time (s)(BE SBS-based ILU)†

CPU time (s) (Jacobi)

1x1 unit cell, strain state 1

1,824 57 561 2 5

1x1 unit cell, strain state 2

1,824 73 621 2 6

2x2 unit cells, strain state 1

7,980 81 2241 11 104


7,980 104 1805 12 84


52,368 116 8920 119 2917


52,368 157 5983 142 2,084

†Including the LU decomposition CPU time

In Table 3, results showing the performance of the preconditioners are presented. Compared to the Jacobi preconditioner, a considerable acceleration of Bi-CG solver is observed when the BE SBS-based ILU one is applied (e.g. the Bi-CG solver becomes about 24 times faster for the 55 -unit-cell RVE under strain state 1). The decaying of the Euclidean residual norm,

2 , as a function of

the iteration order for both preconditioners is also shown in Fig. 2. This graph clearly shows the superiority of the BE SBS-based ILU preconditioning.

0 1000 2000 3000 4000 5000 6000 7000 8000iteration

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

10

100

1000

resi

dual

Euc

lidea

n no

rm

0 1000 2000 3000 4000 5000 6000 7000 8000

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

10

100

1000

PreconditionerJacobiBE SBS-based ILU

(a) Strain state 1

0 1000 2000 3000 4000 5000iteration

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

10

100

1000

resi

dual

Euc

lidea

n no

rm

0 1000 2000 3000 4000 5000

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

10

100

1000

PreconditionerJacobiBE SBS-based ILU

(b) Strain state 2

Figure 2. Residual norm vs. iteration: 55 -unit-cell, square-packed long CNT


5. Conclusions and prospects

The BE SBS technique proposed in previous papers ([9], [10]) is straightforwardly used to construct incomplete-LU-based preconditioners for BE systems of equations. The performance of this preconditioning was verified by analyzing complex composite RVEs. Observing the Table 3, and graphs in Figure 2, we see that the BE-SBS-based ILU preconditioning, compared to the Jacobi (diagonal) one, is considerably more efficient. In fact, the BE-SBS-based ILU preconditioning states a transition (or connection) between direct and iterative solvers, in the sense that the less the number of interfaces, the closer to the global system matrix the preconditioning matrix, Q , is. In addition, knowing that the global coupled system is highly sparse, we can well conclude that the preconditioner proposed will be certainly a good approximation of the global system matrix, which is one of the requirements for finding good preconditioners. Generally speaking, the larger the size of the subsystems, the higher the cost for constructing the preconditioner, however, on the other hand, a better approximation for the global system is achieved, reducing then the number of iterations. Furthermore, being this preconditioner based on the BE-SBS algorithm, its parallelization is immediate. In general, solver-convergence reliability and parallel-processing suitability are attained. Acknowledgements This research was sponsored by the Brazilian Research Council (CNPq), and by the Research Foundation for the State of Minas Gerais (FAPEMIG).

References [1] H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press

(2003).[2] Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied

Mathematics (SIAM), Philadelphia (2003).[3] R. Barrett, M. Berry, J. Dongarra, V. Eijkhout, C. Romine, J. Comp. Appl. Mathematics 74, 91-109

(1996) . [4] G.L.G. Sleijpen, D.R. Fokkema Electronic Trans. Num. Methods Anal., 1, 11-32 (1993). [5] S.-L. Zhang Comp. and Appl. Math. 149, 297–305 (2002). [6] S.A. Vavasis SIAM Journal on Matrix Analysis and Applications 13, 905-925 (1992). [7] K. Davey, S. Bounds Applied Numerical Mathematics 23, 443-456 (1997). [8] M. Merkel, V. Bulgakov, R. Bialecki, G. Kuhn Eng. Anal. Boundary Elements 22, 183-197 (1998). [9] F.C. Araújo, K.I. Silva, J.C.F. Telles Int. J. Numer. Methods Engrg. 68, 448-472 (2006) . [10] F.C. Araújo, L.J. Gray Comp. Mod. Eng. Sci. 24(2), 103-121 (2008). [11] C. Farhat, F.-X. Roux SIAM J. Sci. Statist. Comput. 13, 379–396 (1992). [12] X.L. Chen, Y.J. Liu Comput. Mat. Sci. 29, 1–11 (2004).


Hypersingular BEM analysis of semipermeable cracks inmagnetoelectroelastic solids

R. Rojas-Dıaz1∗, M. Denda2, F. Garcıa-Sanchez3, A. Saez1∗

1 Departamento de Mecanica de los Medios Continuos, Escuela Tecnica Superior de Ingenieros, Caminode los descubrimientos s/n, E41092 Sevilla, SPAIN

[email protected],[email protected] Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98

Brett Rd., Piscataway, NJ 08854-8058, [email protected]

3Departamento de Ingenierıa Civil, de Materiales y Fabricacion, Escuela de Ingenierıas (AmpliacionCampus Teatinos), Universidad de Malaga, C/ Dr. Ortiz Ramos, 29071-Malaga, Spain

[email protected]

Keywords: BEM, magnetoelectroelastic solids, semipermeable cracks, fracture mechanics.

Abstract.In this work, an efficient numerical tool based on the dual formulation of the BEM presented

in [1] is developed for the analysis of different electric/magnetic crack faces boundary conditions inmagnetoelectroelastic solids. A new algorithm for the resolution of multiple semipermeable cracksin magnetoelectroelastic media, based on the one developed in [2] for piezoelectric solids, is designedand implemented. Results for a Griffith crack in a magnetoelectroelastic media will be presented andcompared with the analytical solution, for different mechanical, electric and magnetic loadings.

Introduction

Piezoelectric/piezomagnetic multiphase composites represent a new class of smart materials whichpresent a fully coupling between the mechanical, electric and magnetic fields. In them, a new electro-magnetic coupling which did not appear in each phase is present [3]. Their ability to convert energybetween mechanical, electric and magnetic fields makes them very interesting for their use in smartstructures applications.

The analysis of crack face boundary conditions in magnetoelectroelastic fracture is not a closedtopic. Two ideal conditions (impermeable and permeable) and a more realistic one (semipermeable) areusually considered. Impermeable conditions establishes that the crack is isolated of the electromagneticfields, while permeable condition implies that cracks conduct electric and magnetic fields. However,both assumptions are not completely realistic, and it is possible to say that a consistent crack faceboundary condition will be between them.

In this paper, a new algorithm to solve semipermeable crack problems using the hypersingular BEMformulation and based in the one developed by Denda [2] for piezoelectric materials is presented. Thisformulation will be validated by the comparison with analytical solutions available in literature.

Governing equations for linearl magnetoelectroelasticity

The behavior of a homogeneous and linear magnetoelectroelastic solid under in-plane mechanical,magnetic and electric loading can be described in an elastic-like way by introducing some extendedvariables. In particular, the displacement vector is extended with the electric and magnetic potentials,while the stress tensor is extended with the electric displacements and the magnetic inductions

uI =

⎧⎨⎩

ui, I=1,2φ, I=4ϕ, I=5,

; σiJ =

⎧⎨⎩

σij , J=1,2Di, J=4Bi, J=5,

(1)


where the lowercase subscripts (elastic) vary from 1 to 2, whereas the uppercase ones (extended)take the values 1,2,4,5. An associated generalized traction vector corresponding to a unit normaln = (n1, n2) can also be defined as

pJ = σiJni =

⎧⎨⎩

pj = σijni, J=1,2Dn = Dini, J=4Bn = Bini, J=5,

(2)

And now, the constitutive equations can be expressed as

σiJ = CiJKluK,l (3)

where the material properties have been grouped together into a generalized elasticity tensor definedas

CiJKl =

⎧⎨⎩

cijkl J,K = 1, 2 elij J = 1, 2; K = 4 hlij J = 1, 2; K = 5eikl J = 4; K = 1, 2 −εil J,K = 4 −βil J = 4; K = 5hikl J = 5; K = 1, 2 −βil J = 5; K = 4 −γil J,K = 5

(4)

being cijkl, εil and γil the elastic stiffness, the dielectric permittivities and the magnetic permeabilitiestensors, respectively, whereas elij , hlij and βil are the piezoelectric, piezomagnetic and electromagneticcoupling coefficients, respectively.

Dual BEM formulation and implementation

The dual BEM formulation for bidimensional problems in MEE solids was presented in [1]. If cracksare self-equilibrated and the extended displacements and tractions in the crack are expressed as

∆uJ = u+J − u−

J ; ∆pJ = p+J + p−J (5)

then, the hypersingular formulation of the BEM consists in the application of the extended displace-ment boundary equation (EDBIE), equation (6), for collocation points which does not belong to thecrack, and the application of the extended traction boundary equation (ETBIE), equation (7), whichcan be obtained by derivation of the EDBIE, when the collocation point belongs to the crack. Bothboundary integral equations can be then expressed as

cIJuJ +∫

ΓB

p∗IJuJdΓ +∫

Γ+

p∗IJ∆uJdΓ =∫

ΓB

u∗IJpJdΓ (6)

pJ + Nr

∫ΓB

s∗rIJuJdΓ + Nr

∫Γ+

s∗rIJ∆uJdΓ = Nr

∫ΓB

d∗rIJpJdΓ (7)

where ΓB is any external boundary and Γ+ is one of the crack surfaces. cIJ are the so-called free termsand u∗

IJ and p∗IJ are the Green’s functions corresponding to the response of an infinite homogeneous2-D linear magnetoelectroelastic solid due to the application of a static generalized point force, whichis available in the literature [4] and is expressed in terms of the extended Stroh’s formalism. In ETBIE,N denotes the outward unit normal to the boundary at the collocation point and the kernels s∗rIJ andd∗rIJ are obtained by differentiation of p∗IJ and u∗

IJ , respectively, with the following expressions

d∗rIJ = CrIMl u∗MJ,l ; s∗rIJ = CrIMl p∗MJ,l (8)

Meshing strategy and integration of the hypesingular kernels in equations (6-7) follow the approachdeveloped by [1]. In particular, discontinuous quadratic elements are used to mesh the cracks in orderto fulfill the C1 continuity of the displacements required to compute the ETBIE, and quarter pointdiscontinuous elements are used to ensure a proper representation of the square-root behavior ofthe displacements around the crack tip. And, as done in [1], extended field intensity factors (stressintensity factors -SIF-, electric displacement intensity factor -EDIF- and magnetic induction intensity


factor -MIIF) can be directly obtained from the nodal values of the crack opening displacements andthe electric and magnetic potential jumps across the crack from⎛

⎜⎜⎝KII

KI

KIV

KV

⎞⎟⎟⎠ =

√π

8rY−1

⎛⎜⎜⎝

∆u1

∆u2

∆φ∆ϕ

⎞⎟⎟⎠ , (9)

where r is the distance between the crack tip and the point where extended crack opening displace-ments are evaluated and Y is the Irwin Matrix, which depends on the material constants [1].

Crack faces boundary conditions

In fracture mechanics analysis of magnetoelectroelastic solids, three different boundary conditionson open crack surfaces can be considered. While the mechanical boundary conditions on the cracksurface is always traction free, the electric and magnetic ones comes in different degrees of shieldingthe electric displacement and magnetic induction defined, respectively, by the electric permittivityand by the magnetic permeability. Thus, a crack along the x1−axis can be considered as

(i) Fully impermeable crack, if the normal electric displacement and magnetic induction on the cracksurfaces are zero, so

D+2 = D−

2 = 0B+

2 = B−2 = 0

(10)

which means that the crack is extended tractions free on its surface.

(ii) Fully permeable crack if the crack does not obstruct any electric or magnetic field. This conditioncan be expressed as

D+2 = D−

2 ; φ+ − φ− = 0B+

2 = B−2 ; ϕ+ − ϕ− = 0

(11)

(iii) Semipermeable crack. This condition, which gives a more realistic boundary condition for openedcracks, was proposed in [5] as an extension of the one previously proposed [6] for piezoelectricsolids.

D+2 = D−

2 ; D+2 (u+

2 − u−2 ) = −εc(φ+ − φ−)

B+2 = B−

2 ; B+2 (u+

2 − u−2 ) = −γc(ϕ+ − ϕ−)

(12)

where εc is the permittivity of the medium between the crack faces and γc, the permeability.

The solution of the problem in which either impermeable or permeable (ideal) crack faces boundaryconditions are considered, is carried out by a direct evaluation of both boundary integral equations (6-7), but imposing different boundary conditions in the crack. Thus, if a permeable crack is assumed,∆u4 = 0 and ∆u5 = 0 must be imposed and, in the other hand, if impermeable crack faces boundarycondition is considered, the conditions p4 = 0 and p5 = 0 must be applied to both boundary inte-gral equations. In the next section, it will be introduced a procedure to analyze cracks under thesemipermeable crack face boundary condition assumption.

Numerical solution algorithm for semipermeable cracks

The more realistic semipermeable condition is given by a non-linear equation so, for solving thatproblem, an iterative algorithm will be proposed and implemented. This algorithm is a generalizationof the one proposed by [2] for piezoelectric cracked solids.

Let us call the jumps of the electric and magnetic potentials in the crack as ∆u4 = (φ+ − φ−)and ∆u5 = (ϕ+ − ϕ−). The semipermeable solution implies that the electromagnetic potentials, theelectric displacement and the magnetic induction on the crack faces are generally different to zero.Since the semipermeable crack solution is somewhere in between the two ideal crack surface boundarycondition, then the following iteration procedure for multiple cracks problem is proposed.


1. Get the impermeable solution ∆u[0]4 , ∆u

[0]5 , which will be used as the starting point of the

iteration procedure. The number between brackets denotes number of iteration step.

2. Define, for each crack k, two pairs of proportionality parameters hkie and hki

m (i=1,2), which varyin the interval (0,1)

3. (a) Take, for each crack, hk1e and hk1

m slightly bigger than zero (what would correspond to thequasi-permeable solution and, thus, values for εc and γc tend to infinity) and hk2

e and hk2m

slightly lower than one (what would correspond to the quasi-impermeable solution). Then,set ∆u

k[1]4 = hk1

e ∆u[0]4 , ∆u

k[2]4 = hk2

e ∆u[0]4 , ∆u

k[1]5 = hk1

m ∆u[0]5 and ∆u

k[2]5 = hk2

m ∆u[0]5 .

(b) Calculate the mechanical crack opening displacement, the electric displacement and themagnetic induction based on the set values of the previous item.

(c) Calculate for each crack at M sample collocation points ξj

εkij = Dki

n (ξj)∆uki

2 (ξj)∆uki

4 (ξj); γki

j = Bkin (ξj)

∆uki2 (ξj)

∆uki5 (ξj)

(13)

which are obtained by the substitution in equation (12) of the corresponding ECOD andthe electric (Dn) and magnetic (Bn) tractions previously obtained in step (3b).

(d) Calculate the averages for each crack and each pair of parameters hkie and hki

m of the param-eteres defined in section (3c).

εki =

∑Mj=1 εki

j

M; γki =

∑Mj=1 γki

j

M(14)

This parameter are the so-called electric permittivity in the crack and magnetic perme-ability in the crack, respectively.

(e) While the electric permittivity and magnetic permeability of any crack is not equal to thevalues for the medium between the crack surfaces, ε0 and γ0 iterate using a procedure tosolve non-linear equations, until a pair of values h

k[n]e and h

k[n]m for each crack is obtained.

Let us remark that all those values may be different.

4. After setting ∆uk[n]4 = h

ki[n]e ∆u

[0]4 and ∆u

k[n]5 = h

ki[n]m ∆u

[0]5 , solve the problem required to get

the semipermeable solution searched.

Validation of the algorithm

In this section, a single horizontal crack of length 2a in an infinite magnetoelectroelastic domain willbe analyzed for different crack faces boundary conditions and loading combinations. A BaTiO3 −CoFe2O4 magnetoelectroelastic solid with a Vf = 0.5, which properties can be found, e.g., in [1],will be considered. In all cases, the medium between both crack faces is air, what implies thatthe electric permittivity and magnetic permeability are, respectively, ε0 = 8.8542 · 10−12N/V 2 andγ0 = 4π · 10−7N/A2.

The analytical solution for this problem, was first obtained by Wang and Mai [5], and will be nowbriefly presented.

The extended crack opening displacements ∆uI , I = 1, ..., 5; are given by

∆uI = u+I − u−

I = 2YIJ(σ∞J2 − σc

J2)√

a2 − x21 (15)

where Y is the compliance (Irwin) matrix, σ∞J2 are the components of the extended stress tensor

applied at infinity, σcJ2 are the components of the extended stress tensor on the crack surfaces, and the

summation rule over repeated indices is applied. The different crack face boundary conditions thatmay be considered for a crack along the x1-axis.


(i) Fully impermeable crack. In this case, the crack is extended traction free, what implies that

Dc2 = 0 ; Bc

2 = 0 (16)

where, since D+2 = D−

2 and B+2 = B−

2 , the upperindex c has been used to denote either of thecrack surfaces.

(ii) Fully permeable crack. For fully permeable cracks no jump in the electromagnetic potentialappear. This condition can be expressed as

∆u4 = 0 ; ∆u5 = 0 (17)

The substitution of that condition in (15) will lead to a system of equation whose solutionprovides the analytical expressions of the extended tractions on the crack faces

Dc2 =

Y4JY55 − Y5JY45

Y44Y55 − Y54Y45σ∞

J2 ; Bc2 =

Y5JY44 − Y4JY54

Y44Y55 − Y54Y45σ∞

J2 (18)

(iii) Semipermeable crack. The semipermeable crack conditions are

Dc2∆u2 = −εc∆u4 ; Bc

2∆u2 = −γc∆u5 (19)

Substituting now (19) in (15) and operating a non-linear system of equations which defines theextended tractions in a semipermeable crack, it will be obtained.

Dc2 = −εc

Y4Jσ∞J2 − Y44D

c2 − Y45B

c2

Y2Jσ∞J2 − Y24Dc

2 − Y25Bc2

; Bc2 = −γc

Y5Jσ∞J2 − Y54D

c2 − Y55B

c2

Y2Jσ∞J2 − Y24Dc

2 − Y25Bc2

(20)

where the summation rule over repeated is applied.

The analytical solution previously deduced will be compared with the results obtained with theproposed formulation. In figure 1 the mechanical opening displacement are shown for the case inwhich only a mechanical loading is applied and in the case in which a combination of loads defined byσ∞

22 = 1N/m2, D∞2 = 10−9C/N and B∞

2 = 10−8A−1 · m is applied. The analytical solution is plottedin lines, comparing them with the results obtained numerically (points), and those magnitudes arenormalized with their respective value under permeable conditions in x1 = 0.

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1/a

∆ u 2/∆

u2pe

rm| x 1=0

ImpermeableSemipermeablePermeable

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x1/a

∆ u 2/∆

u2pe

rm| x 1=0


Figure 1: Crack opening displacement when only a mechanical loading is applied (left) and a fullcombination of electromagnetomechanic loading is applied (right).


In figure 2 the analytically obtained jumps in the electric and magnetic potentials are comparedwith the results obtained numerically (points). Those magnitudes are normalized with their respectivevalues under impermeable conditions in the center of the crack (x1 = 0). In all cases, an excellentagreement between analytical and numerical results is observed

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

x1/a

∆ u 4/|∆

u4im

p | x 1=0


−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

x1/a

∆ u 5/|∆

u5im

p | x 1=0


Figure 2: Jumps in the electric (left) and magnetic (right) potentials on the crack.


A new algorithm for the study of the different crack faces boundary conditions in magnetoelectroe-lastic solids has been designed and implemented in a hypersingular boundary elements code. Theaccuracy of this numerical tool has been proved by comparing with analytical results available inliterature.

References

[1] F. Garcıa-Sanchez, R. Rojas-Dıaz, A. Saez, and Ch. Zhang. Fracture of magnetoelectroelasticcomposite materials using boundary element method (BEM). Theoretical and Applied FractureMechanics, 47(3):192–204, 2007.

[2] M. Denda. BEM analysis of semipermeable piezoelectric cracks. Key Engineering Materials,383:67–84, 2008.

[3] Y. Beneviste. Magnetoelectric effect in fibrous composites with piezoelectric and piezomagneticphases. Physical Review, B 51:16424–16427, 1995.

[4] J.X. Liu, X. Liu, and Y. Zhao. Green’s functions for anisotropic magnetoelectroelastic solids withan elliptical cavity or a crack. International Journal of Engineering Science, 39:1405–1418, 2001.

[5] B. Wang and Y.W. Mai. Applicability of the crack-face boundary conditions for fracture of mag-netoelectroelastic materials. International Journal of Solids and Structures, 44:387–398, 2006.

[6] V.Z. Parton and B.A Kudryatsev. Electromagnetoelasticity. Gordon and Breach science publisher,New York, 1988.


Boundary Element Analysis of Cracked Transversely Isotropic and

Inhomogeneous Materials

C.Y. Dong1, X Yang1 and E Pan2

1Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, China

2Department of Civil Engineering, University of Akron, Akron, Ohio, USA

Keywords boundary element method, transversely isotropic materials, inclusion, crack, stress intensity factors

Abstract. In this paper, the boundary element method (BEM) is utilized to study the effect of the transversely isotropic inclusion on the mixed-mode stress intensity factors (SIFs) of a rectangular crack. Both conventional and three special nine-node quadrilateral elements are used to discretize the inclusion-matrix interface and the square crack surface. Displacement and traction integral equations are applied to the inclusion-matrix interface and the square crack surface. Once the displacements and tractions over the interface and the crack opening displacements over the crack are obtained, the mixed-mode SIFs are calculated using a well-known formulation. In the numerical calculation, the distance between the inclusion and the crack as well as the orientation of the isotropic plane of transversely isotropic medium are varied to show their influences on the mixed-mode SIFs along the crack fronts.

Introduction The mechanical behavior of heterogeneous materials of isotropy such as composites, rock structures, porous and cracked medium has been widely investigated using various methods, e.g. finite element method [1] and integral equation method [2]. However, only a few studies exist when the inhomogeneous material is of anisotropy, e.g., transverse isotropy. Huang and Liu [3] used the eigenstrain method to obtain the elastic fields around the inclusion and further studied the interactive energy between the inclusion and the applied loads. Pan and Young [4] investigated the fracture mechanics problems in three-dimensional (3D) anisotropic solids using the combined displacement and traction integral representations. Ariza and Dominguez [5] obtained the boundary traction integral equation for cracked 3D transversely isotropic bodies in which explicit expressions for the fundamental solution traction derivatives are presented. Yue et al. [6] calculated the 3D stress intensity factors (SIFs) of an inclined square crack within a bi-material cuboid using the dual BEM. Chen et al. [7] studied the fracture behavior of a cracked transversely isotropic cuboid using 3D BEM. Benedetti et al. [8] presented a fast dual BEM for cracked 3D problems.

In existing literature, the interaction between the inclusions and cracks embedded in a transversely isotropic medium has not been researched yet. Therefore, in this paper, the effect of a spherical inclusion on the SIFs of a square-shaped crack, both being embedded in a transversely isotropic matrix, is studied using the single-domain BEM. The influence of the distance between the inclusion and the square-shaped crack and the material orientation on the SIFs of the crack fronts


is discussed.

Boundary integral equations We consider a transversely isotropic inclusion embedded in a cracked infinite matrix of transverse isotropy. In order to study the effect of the inclusion on the SIFs of the rectangular crack, the single-domain dual BEM is used. In other words, the displacement and traction boundary integral equations for a cracked medium of transverse isotropy are as follows [4]

0

( ) ( , ) ( ) ( ) ( , ) ( ) ( )

( , )[ ( ) ( )] ( ) ( )

ij j S ij S S j S S ij S S j S SS S

ij S j j i S

b u y U y x t x dS x T y x u x dS x

T y x u x u x d x u y

(1)

,

,

* 0 0,

[ ( ) ( )] / 2 ( ) ( , ) ( ) ( )

( ) ( , )[ ( ) ( )] ( )

( ) ( , ) ( ) ( ) [ ( ) ( )] / 2

l l m lmik ij k S j S SS

m lmik ij k j j

m lmik ij k S j S S l lS

t y t y n y c T y x u x dS x

n y c T y x u x u x d x

n y c U y x t x dS x t y t y

(2)

where ijb are coefficients that depend only on the local geometry of the inclusion–matrix interface

S at Sy . A point on the positive (or negative) side of the crack is denoted by x (or x

), and on

the inclusion–matrix interface S by both Sx and Sy ; mn is the unit outward normal of the positive

side of the crack surface at y ; lmikc is the fourth-order stiffness tensor of the material transverse

isotropy; 0i su y is the displacement component along the i-direction at point Sy caused by a

given remote uniform loading, and 0lt y

and 0lt y

are the corresponding traction

components along l-direction at the points y and y

; iu and it are the displacements and

tractions on the inclusion–matrix interface S (or the crack surface ); ijU and ijT are the Green’s

functions of displacements and tractions; ,ij kU and ,ij kT are, respectively, the derivatives of the

Green’s displacements and tractions respect to the source point.

The displacement integral equation for a transversely isotropic inclusion is as follows


( ) ( , ) ( ) ( ) ( , ) ( ) ( )ij j S ij S S j S S ij S S j S SS S

b u y U y x t x dS x T y x u x dS x (3)

Equations (1), (2) and (3) can be used to investigate the effect of the inclusion on the SIFs of the crack embedded in a transversely isotropic medium. In discretization of these equations, we apply nine-node quadrilateral curved elements as shown in Fig.1 to the inclusion-matrix interface and the crack surface in which the crack front surface is meshed into special elements. Taking each node in turn as the collocation point, and performing various kinds of integrals, we finally obtain the compact forms of the discretized equations from Eqs. (1), (2) and (3) as

1 2 3

654

987

1 2 3

654

987

2/3

crack front

1 2 3

654

987

2/3

crack front

2/3

Type I Type II

Type III

1 2 3

654

987

2/3crack front

2/3

Type IV

Fig. 1: Four element types for the crack surface.

11 12 1 11 12

21 22 2 21 22

m m

c c

U TH H B G GU TH H B G G

(4)

and

i i i iH U G T (5)

where the subscripts i and m represent the inclusion and the matrix, respectively; H and G are


respectively the influence coefficient matrices containing integrals of the fundamental Green’s solutions; B1 and B2 are the displacement and traction vectors, respectively, induced by the remote

loading; mU ( iU ) and mT ( iT ) are the node displacement and traction vectors, respectively, over

the matrix side (inclusion side) of the inclusion-matrix interface; cU and cT are, respectively, the

discontinuous displacement and traction vectors over the crack surface. In this paper, we assume

that the tractions on both sides of the crack are equal and opposite. Therefore cT is equal to zero.

Using the continuity condition of the displacement and traction vectors along the interface between the inclusion and matrix, we can combine Eqs. (4) and (5) into

1111 11 12

1221 21 22

mi i

ci i

U BH G G H HU BH G G H H

(6)

Therefore, once the unknowns mU and cU are solved, the SIFs along the crack front can be

evaluated using the following equation [4, 9]

11

2

3

22II

I

III

u Kru K

u K

L (7)

where r is the distance behind the crack front; L is the Barnett-Lothe tensor [9] which depends

only on the anisotropic properties of the solid in the local crack-front coordinates; and 1u ,

2u and 3u are the relative crack opening displacements in the local crack-front coordinates.

Numerical examples An embedded spherical inclusion and a square-shaped crack in an infinite matrix are shown in Fig. 2. The radius of the sphere is R=1.0m and it is made of transversely isotropic marble with the

following elastic properties: 90XE GPa , 55ZE GPa , 0.3XY YZ , 21YZG GPa . The side

length of the square is 2a (=2.0m). The matrix material properties

are 12XE GPa , 4ZE GPa , 0.3XY YZ , 1.6YZG GPa . Here, we should note that all these

material properties are with respect to the local coordinates and that the local X, Y and Z representthe longitudinal, transverse and normal directions, respectively. The plane X-Y is the plane of isotropy. On the other hand, x, y and z refer to the space-fixed global coordinates, and the relation

between these two coordinates (X,Y,Z) and (x,y,z) is shown in Fig. 3 with and being the


orientation and inclined angles. The crack center and the inclusion center are located on the x-axis,

separated by a distance d. A far-field stress 1.0GPa is applied in the z-direction.

R=1m

d

L=2m

zz=1GPa

x

z

Fig.2 (a): A spherical inclusion and a square shaped crack in an infinite domain

x

y

A B

CD

d

Fig.2 (b): Planeform in x-y plane.

x

y

z

Y

X

Z

Fig. 3: The relationship between the principal material coordinates (X,Y,Z) and the space-fixed global coordinates (x,y,z) [4]

Twenty-four nine-node quadrilateral elements with 98 nodes (Fig. 4(a)) and one hundred nine-node quadrilateral elements with 441 nodes (Fig. 4(b)) are used to discretize the inclusion-matrix

interface and the crack surface, respectively. For various values of d and fixed 00

and 00 for both the inclusion and matrix, the SIF /IKI K a along the crack fronts

AB, BC, CD and DA of the square is shown in Fig. 5. It is obvious that as d decreases, the SIF of the crack front DA close to the inclusion is significantly decreased, while the SIFs of the other


crack fronts AB, BC and CD are nearly insensitive to d. For fixed d/R=0.5 and fixed 00

and 00 of the inclusion but different angles and of the matrix, the SIF

/IKI K a along the crack fronts AB, BC, CD and DA of the square is shown in Fig. 6.

We can observe that with increasing angle , the SIF KI of the crack fronts AB and CD decreases, while the SIF KI along the crack fronts BC and DA increases. The maximum value of KI appears at the middle point of crack fronts BC and is approximately equal to 0.9, while minimum value of KI appears at the middle point of the crack fronts AB and CD, approximately equal to 0.6. For

fixed d/R=0.5, fixed 00 and 00 of the inclusion and fixed 00 and 045 of the

matrix, the SIFs /IIKII K a and /IIIKIII K a along the crack fronts AB,

BC, CD and DA of the square are shown in Fig. 7. It is observed that the variation of the SIF KIIand KIII along the crack front is much more complicated than the SIF KI.

Fig.4(a): Discretization of the interface of the spherical inclusion and its matrix

Fig. 4(b): Discretization of the square shaped surface


Fig.5: Normalized mode I SIF along the square shaped crack fronts AB, BC, CD and DA for

different values of d with fixed 00 and 00 for both the inclusion and matrix.

Fig.6: Normalized mode I SIF along the square shaped crack fronts AB, BC, CD and DA for

different values of and of the matrix with fixed d=0.5m, and fixed 00 and 00 for the

inclusion.


Fig.7: Normalized mode II and III SIFs along the square shaped crack fronts AB, BC, CD and DA

for fixed d=0.5m, 00 and 00 of the inclusion, and fixed 00 and 045 of the

matrix.

ConclusionsA BEM formulation is developed to study the fracture problem in a transversely isotropic and heterogeneous medium. In the numerical analysis, three kinds of nine-node quadrilateral elements are used to discretize the inclusion-matrix interface and the square crack surface. The mixed-mode stress intensity factors were calculated from the solved discontinuous displacements over the crack surface. The effect of the distance between the inclusion and crack as well as the material anisotropy on the SIFs of crack fronts is investigated.

Acknowledgements

The support of the National Natural Science Foundation of China under Grant no. 10772030 is gratefully acknowledged.

References [1] J. Zhang and N. Katsube, Finite Elements in Analysis and Design, 19, 45-55, (1995). [2] C.Y. Dong, S.H. Lo and Y.K. Cheung, Computational Mechanics, 30, 119-130, (2003). [3] J.H. Huang and H.K. Liu, Int J Eng Sci, 36, 143-155, 1998. [4] E. Pan and F.G. Yuan, Int J Num Meth Eng, 48, 211-237, (2000). [5] M.P. Ariza and J. Dominguez, Int J Num Meth Eng, 60, 719-753, (2004). [6] Z.Q. Yue, H.T. and E. Pan, Eng Anal Bound Elem, 31, 50-60, (2007). [7] C.S. Chen, C.H. Chen and E. Pan, Eng Anal Bound Elem, 33, 128-136, (2009). [8] I. Benedetti, A. Milazzo and M.H. Aliabadi, Int J Num Meth Eng, 80, 1356-1378, (2009). [9] T.C.T. Ting, Anisotropic Elasticity: theory and Applications, Oxford University Press, New York, (1996).


A Family of 2D and 3D Hybrid Finite Elements for Strain Gradient Elasticity

Ney Augusto Dumont and Daniel Huamán Mosqueira

Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, 22453-900, Brazil.e-mail: [email protected]

Keywords: Gradient elasticity, variational methods, hybrid finite element, Hellinger-Reissner potential.Abstract. The present paper starts from Mindlin’s theory of the strain gradient theory, based on three additionalconstants for homogeneous materials (besides the Lamé’s constants), to arrive at a proposition made by Aifantiswith just one additional parameter. It is shown that the hybrid finite element formulation – as proposed byPian and generalized by Dumont for finite and boundary elements – provides a natural conceptual frameworkto properly deal with the interelement compatibility of the normal displacement gradients, in which “cornernodes” are not an issue. Nonsingular fundamental solutions – domain interpolation functions – are presentedfor 2D and 3D problems, with the generation of families of finite elements that may be implemented in astraightforward way. Since the experimental data available in the technical literature are still scarce and thenumerical results are mostly questionable, consistency is assessed by means of patch tests and by investigatingthe spectral properties of the matrices derived for some 2D plane strain elements, as well as for truss and beamelements.

Introduction

The mathematical modeling of microdevices, in which structure and microstructure have approximately thesame scale of magnitude, as well as of macrostructures of markedly granular or crystal nature (microcom-posites), demands a nonlocal approach for strains and stresses. In no detriment to developments due to otherresearchers [4, 28, 13], Mindlin’s works in the 1960s [18, 19] may be accounted the basis of the strain gradi-ent theory. It has recently become the subject of a large number of analytical and experimental investigationsmotivated by the development of new structural materials together with the increasing use of micromechanicaldevices in the industry. Starting in the 1990s, Aifantis and coworkers [1] managed to develop a simplifiedstrain gradient theory based on only one additional elasticity constant, which opened up a series of interestingpractical applications [24, 17, 20]. On the other hand, developments that take into account nonlocal residu-als by means of an integral operator were proposed in the 1970s [14] and have been ever since the subject ofinvestigation [27].Some recent works done by Beskos and collaborators have largely extended the field of applications of Aifantis’propositions [29, 25, 23, 21]. Since Toupin and Mindlin’s time, investigations have been under developmentto establish the variational basis of the theory and to formulate equilibrium and kinematic boundary conditionsconsistently [2, 15]. The non-singular formulation for 2D and 3D elasticity problems [2] has also been de-veloped, which enables the construction of hybrid finite and boundary element families of general shape andnumber of degrees of freedom, as already done in the classical elasticity [6, 12, 7, 8]. All these formulationsare the subject of the present investigations, including conceptual studies of the simplest conceivable rod, beamand 2D finite elements implementations, as a sequel of References [10, 11]. Although all developments havealready been extended to the frequency-domain analysis of time-dependent problems [16], one restricts thepresent outline to the static analysis, which actually involves the most relevant concepts.

Problem Formulation

Throughout this paper, repeated indices stand for summation and ( ),i denotes a derivative with respect to thei-th direction. Following equations are given by Aifantis as a development of Mindlin’s work:

εi j = εi j + cε∇2εi j; σi j = σi j + cσ∇2σi j; σi j = λεkkδi j + 2µεi j (1)

where, quoting Aifantis [1], “(σi j, εi j) denote the stress and strain tensors for elastic deformation. The quantities(λ, µ) are the usual Lamé constants. The gradient coefficients c’s are new phenomenological coefficients. . . . (Infact, the simplest form of gradient elasticity theory corresponds to the case cσ = 0).”


Independently from Mindlin’s works, a nonlocal elasticity theory [14] has also been developed, in which thestress field (in equilibrium with applied forces in the domain Ω) is expressed in terms of a nonlocal attenuationfunction α (|x − x′|, τ), which incorporates into the constitutive equations the nonlocal effects at the referencepoint x produced by local strain at the source x′, as

σi j(x) =∫Ω

α(|x − x′|, τ)Ci jklεkl(x′)dΩ (2)

where Ci jkl is the elastic module tensor of the classical isotropic elasticity, |x − x′| is the Euclidean distance,and τ is a material property that depends on external and internal characteristic lengths. According to [3], eq(1) represents “gradient elastic models (the stress is defined explicitly from the local strain and its derivatives”,whereas eq (2) represents “integral elastic models (the stress is obtained implicitly from an integral operator ofthe local strain)”. Moreover, “(g)radient models can be considered as a ‘weakly’nonlocal model, whereas (the)integral elastic model can be classified as (a) ‘strongly’nonlocal model”.In either gradient or nonlocal formulation, the crucial issue is the determination of the material characteristiclength(s). When consistently applied, both formulations should lead to similar results. In fact, there may beno strict differentiation between the formulations when it comes to a computational implementation [26]. It isworth pointing out two basic requirements that a gradient or nonlocal formulation should fulfill for consistency:

1. As proposed by Mindlin for a general three-dimensional problem, the normal component n jui, j of thedisplacement gradient is independent of ui on the boundary, with double tractions Ti j performing (virtual)work along Γ on the normal gradient

n jnlui,l ≡ ui, j − (δ jl − n jnl)ui,l (3)

In this equation, ni are the Cartesian projections of the outward unit normal to Γ and δ jl is the Kroneckerdelta. A mechanical interpretation of the non-symmetric tensor Ti j, including its correlation with theCosserat couple-stress vector, is given by Mindlin [18].

2. A constant strain state must yield the same results of the classical elasticity [27]. This requirement hasnot been observed by some researchers, with inconsistent results.

Moreover, consistency of the formulation for applied rigid body displacements as well as for simple displace-ment fields (patch tests) should be checked, as done in the present paper. The implementation of the formulationin terms of finite elements is also not straightforward, as unique relations between local and global non-classicalquantities must be ensured. These issues are dealt with later on in the paper.

The Hellinger-Reissner potential applied to gradient elasticity

The Hellinger-Reissner potential [5] is applied in the next Sections, as split up into two virtual work principlesin order to clarify the conceptual aspects affected by the gradient elasticity. Owing to its conceptual relevancy,one basically repeats in this Section the initial outline of Reference [11]. One starts by stating the two-fieldassumptions necessary to develop all matrix equations in terms of n∗ and nd stress and displacement parameters.

Stress assumption. The stresses are approximated in the domain Ω of a generic three-dimensional elasticbody by the Cauchy stress τs

i j and the double stress µski j, where the superscript ( )s stands for stress assumption.

For the sake of checking equilibrium, τsi j and µs

ki j are split into two parts, τsi j = τ

∗i j + τ

pi j and µs

ki j = µ∗ki j + µ

pki j,

where the superscript ( )∗ denotes the homogeneous solution of the differential equilibrium equation and thesuperscript ( )p stands for some arbitrary, particular solution for the applied body force fi defined per unitvolume in Ω (no “body double force” [2] is prescribed). All quantities are supposed to vary in Ω as functionsof the coordinate x, y, z. Forces are prescribed on part Γσ of the boundary: “Classical” traction forces Pi, whichmay perform virtual work on displacements δui, as well as “double” traction forces Ri = T jin j, which mayperform virtual work on normal displacement gradients δqi ≡ nlδui,l. Except for analyticity, no concern is madeat present about equilibrium on Γσ. According to the technical literature on the gradient elasticity, one definesthe tensor of total stresses σs

i jσs

i j = τsi j − µs

ki j,k (4)


where τsi j is the Cauchy stress tensor and µs

ki j,k is the “couple” stress tensor. This definition will be arrived atnaturally in the frame of the virtual work statement below, as well as the equilibrium statement:

σsji, j + fi = 0 in Ω (5)

σsi j, τ

si j and µs

ki j are symmetric with respect to the directions i and j. According to Aifantis’ proposition, thecouple stress tensor is related to the Cauchy stress tensor by means of the “material characteristic length” g2:

µski j = g2τs

i j,k ⇒ σsi j = τ

si j − g2τs

i j,kk (6)

This stress field description for the gradient elasticity corresponds to a “Type II” description of three equivalentforms of the strain energy density [19, 2]. The present notation is a simplified version of the notation proposedby Mindlin and Eshel, with the equivalences τs

i j ≡ σsi j and µs

ki j ≡ µki j. The strain field corresponding to τsji is

ε si j =

12 (us

i, j + usj,i) ≡ εi j, and the strain gradient corresponding to µs

k ji is 12 (us

i, jk + usj,ik) = κki j.

The homogeneous part τ∗i j of τsi j is interpolated in Ω, in the frame of the present variational formulation, by a

sufficiently large number of “fundamental solutions” τ∗i jm:

τ∗i j ≡ τ∗i j(x, y, z) = τ∗i jm(x, y, z)p∗m ≡ τ∗i jm p∗m ⇒ µ∗ki j ≡ µ∗ki j(x, y, z) = µ∗ki jm(x, y, z)p∗m ≡ µ∗ki jm p∗m in Ω (7)

The set of n∗ parameters p∗m (whose physical meaning is context-dependent) are, together with the nodal dis-placement parameters of the next Paragraph, the primary unknowns of the problem one is about to formulate.Non-singular fundamental solutions τ∗i jm are known in the literature for truss, beam and boundary elements, ingeneral, and for some specific finite elements [2]. General non-singular solutions for 2D and 3D finite elementshave already been developed by the authors [10, 16], as presented below for elastostatics problems.Displacement solutions us

i are obtained from τsi j, µ

ski j in terms of the fundamental solutions of eq (7) and of the

particular solution τpi j, µ

pki j, which include arbitrary rigid body displacements:

usi = u∗im p∗m + up

i in Ω (8)

Displacement assumption. The displacement is approximated in Ω by an independent field udi , where the

superscript ( )d stands for displacement assumption. udi satisfies the required boundary continuity conditions,

that is, udi = ui on part Γu of the boundary with prescribed displacement ui. This is the only requirement on

udi that is made in the present frame of the Hellinger-Reissner potential. Except for analyticity, no assumption

is explicitly made for the displacement udi in Ω whether or not concerning gradient elasticity. According to

the requirement number (1) in the Problem Formulation, a normal gradient field qdi must be postulated on Γ

independently from udi, jn j (which is actually not defined) and such that qd

i = qi on part Γq of the boundary withprescribed normal gradients qi. Then, for Γ described in terms of the parametric variables (ξ, η),

udi ≡ ud

i (ξ, η) = uin(ξ, η)dn ≡ uindnqd

i ≡ qdi (ξ, η) = uin(ξ, η)qn ≡ uinqn

on Γ (9)

where uin are interpolation functions of a total of nd parameters dn and qn (not to be confounded with qi) thatare nodal displacement and displacement gradient values defined – together with p∗m of eq (7) – as primaryunknowns of the variational problem. The same sets of functions uin are used to interpolate both displacements(through the components ud

x, udy and ud

z ) and displacement gradients (through the components qdx, qd

y and qdz )

along each subregion of Γ (a “boundary element”) in the present variational context – consistently with theisoparametric description of the body’s geometry. In the case of non-smooth surfaces, the nodal parameters qnmust be independently prescribed for each normal direction of the surfaces adjacent to a node.

Displacement virtual work for equilibrium checking

The following statement is found in similar form in several developments on gradient elasticity: δW int = δWext,as the authors of References [19, 2] imply the equivalence of the variation of internal and external works,in terms of virtual displacements that should be referred to herein as δus

i . However, in the present context,δus

i δudi (except for trivial cases as for truss and beam elements [10]).


Equilibrium of the stress field is weakly enforced by means of the displacement virtual work statement∫Ω

(τs

jiδudi, j + µ

sk jiδu

di, jk

)dΩ =

∫Ω

fiδudi dΩ +

∫Γ

Piδudi dΓ +

∫Γ

Riδqdi dΓ (10)

The virtual displacement field is assumed as simply as possible, with the virtual strain gradient, on which thedouble stress µs

k ji = µski j performs work, given as the double derivative δud

i, jk regardless of material properties.Integrating by parts the terms at the left-hand side of eq (10) and applying the divergence theorem, one obtains

−∫Ω

(σs

ji, j + fi)δud

i dΩ +∫Γ

(σs

jin j − Pi)δud

i dΓ +∫Γ

µsk jinkδud

i, jdΓ −∫Γ

Riδqdi dΓ = 0 (11)

The first two integral terms on the left are already expressed by means of the total stress σsji, defined in eq (4) –

similarly to the classical elasticity theory. The domain integral is void, according to eq (5). Modifying the thirdintegral term, according to the identity of eq (3), one obtains from eq (11)∫

Γ

(σs

jin j − Pi)δud

i dΓ +∫Γ

(µs

k jinkn j − Ri)δqd

i dΓ +∫Γ

µsk jink

(δ jl − n jnl

)δud

i,ldΓ = 0 (12)

In this equation, δqdi substitutes for nlδud

i,l, according to Mindlin’s proposition that the normal gradient onΓ should be independent from the displacement derivatives. This is a natural assumption in the frame of ahybrid variational formulation, and consistently with the boundary-only integral statement of eq (12). In fact,ui is defined only on Γ, according to eq (9), in terms of boundary parametric variables, with uin ≡ uin(ξ) for2D problems or uin ≡ uin(ξ, η) for 3D problems. There is no possibility of defining how ui varies across theboundary, except by explicitly introducing an independent field qd

i , as done in eq (9).Mindlin [18], followed by other researchers, proposed a manipulation of the latter integral term in eq (12) in or-der to enable the implementation of a numerical model, which ends up requiring the computation of “jumping”terms, for a non-smooth boundary. Remarkably, several authors show numerical implementations that havebecome feasible only by artificially smoothing the boundary around corner points. 3D implementations seemto be extremely complicated in such a framework [2, 25]. However, Mindlin’s proposition is naturally circum-vented in the frame of the Hellinger-Reissner potential, as outlined in the following. Firstly, one recognizes thatthe term δ jl − n jnl in eq (12) is orthogonally projecting δud

i,l onto the tangent plane to Γ at a given point ξ, for2D problems, or (ξ, η), for 3D problems. As a result, only the projection δqd

i = nlδudi,l of δud

i,l onto the normalto Γ requires an independent discretization, which is done according to eq (9).In fact, 2D numerical models may be formulated from eq (12) directly as∫

Γ

(σs

jin j − Pi)δud

i dΓ +∫Γ

(µs


i dΓ +∫Γ

µsk jink|J|−2t jδud

i,ξdΓ = 0 (13)

where δudi,ξ ≡ δdud

i /dξ = u′inδdn on Γ comes from tlδudi,l ≡ δdud

i /dξ and δ jl − n jnl = |J|−2t jtl, in which onedefines the tangent vector t = [dx/dξ dy/dξ]T and the Jacobian |J| ≡ |t|, always making use of eq (9).3D numerical models are also formulated directly from eq (12) as∫

Γ

(σs

jin j − Pi)δud

i dΓ +∫Γ

(µs


i dΓ +∫Γ

µsk jink|J|−2 t jsδud

i,sdΓ = 0 (14)

To arrive at this equation, one firstly defines the tangent vectors in terms of uin ≡ uin(ξ, η) in eq (9):

u = [∂x/∂ξ ∂y/∂ξ ∂z/∂ξ]T (15)v = [∂x/∂η ∂y/∂η ∂z/∂η]T (16)

For |J|2 = |u × v| and defining the matrices t = [u v] and t⊥ = [v − u], one obtains that

δ jl − n jnl = |J|−2t jrt⊥mrt⊥mstls (17)

The indices r and s vary from 1 to 2, as they refer to ξ and η. One identifies in eq (14) that δudi,s stands for

tlsδudi,s = tlsuin,sδdn, where ( ),s denotes derivatives with respect to ξ and η. According to eq (17), t js = t jrt⊥mrt⊥ms.


Taking eq (13), for 2D problems, as just a particular case of eq (14), in terms of notation, one writes thenumerical expression of the virtual work principle of eq (10), for stresses and displacements approximatedaccording to eqs (7, 8, 9), as:

〈δdn δqn〉⎛⎜⎜⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎢⎢⎣

∫Γ

(σ∗jimn juin + µ

∗k jimnk|J|−2 t jsuin,s

)dΓ∫

Γµ∗k jimnkn juindΓ

⎤⎥⎥⎥⎥⎥⎥⎥⎦ p∗m(18)

+

⎧⎪⎪⎪⎨⎪⎪⎪⎩∫Γ

(σ

pjin juin + µ

pk jink|J|−2 t jsuin,s

)dΓ∫

Γµ

pk jinkn juindΓ

⎫⎪⎪⎪⎬⎪⎪⎪⎭ −⎧⎪⎪⎨⎪⎪⎩

∫Γ

PiuindΓ∫Γ

RiuindΓ

⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎟⎟⎟⎟⎟⎠ = 0

Since this equation holds for arbitrary δdn and δqn, one obtains the matrix equilibrium system

HTp∗ = p − pp (19)

in which the equilibrium matrix HT as well as the vectors of nodal forces p and pp that are equivalent to theboundary and domain traction forces, respectively, are inferred from eq (18).

Stress virtual work for displacement compatibility checking

The second statement that comes from the Hellinger-Reissner potential may be obtained by weakly enforcingcompatibility of the displacements us

i and udi in terms of the stress virtual work equation

∫Ω

(us

i, j − udi, j

)δτ∗jidΩ +

∫Ω

(us

i, jk − udi, jk

)δµ∗k jidΩ = 0 (20)

Integrating by parts the terms of this equation, applying the divergence theorem and using eq (4), one arrives at

−∫Ω

δσ∗ji, j(us

i − udi

)dΩ +

∫Γ

δσ∗jin j(us

i − udi

)dΓ +

∫Γ

δµ∗k jink(us

i, j − udi, j

)dΓ = 0 (21)

The domain integral of this equation is void, according to eq (5). As done in the preceding Section, one splits thedisplacement gradient into normal and tangential contributions, thus obtaining, in the notation for 3D problems,

∫Γ

δσ∗jin j(us

i − udi

)dΓ +

∫Γ

δµ∗k jinkusi, jdΓ −

∫Γ

δµ∗k jinkn jqdi dΓ −

∫Γ

δµ∗k jink|J|−2 t jsudi,sdΓ = 0 (22)

Introducing the approximations for stresses and displacements given in eqs (7, 8, 9), one writes the numericalexpression of the virtual work principle of eq (20) as

⟨δp∗m

⟩ ([∫Γ

(σ∗jimn ju∗in + µ

∗k jimnku∗in, j

)dΓ

] p∗m

(23)

−[∫Γ

(σ∗jimn juin + µ

∗k jimnk|J|−2 t jsuin,s

)dΓ

∫Γ

µ∗k jimnkn juindΓ]

dnqn

+

∫Γ

(σ∗jimn jup

i + µ∗k jimnkup

i, j

)dΓ

)= 0

The latter integral is void for upi corresponding to rigid body displacements (translations and rotations), as an

equilibrium requirement for the fundamental solution of eq (7). Since δp∗m is arbitrary, one obtains the matrixcompatibility system

F∗p∗ = Hd − b (24)

where the flexibility matrix F∗, the displacement transformation matrix H as well as the vectors of nodal dis-placements d and equivalent nodal displacements b are inferred from eq (23).


Stiffness matrix of the finite/boundary element formulation

Solving for p∗ in the latter equation and substituting into eq (19), one obtains the nodal equilibrium equation

HTF∗(−1)Hd = p − pp +HTF∗(−1)b (25)

In this equation, one recognizes that a stiffness matrix K = HTF∗(−1)H has been naturally arrived at. The termson the right account for the action of boundary and domain forces, according to the assumptions previouslymade. The vector d comprises both nodal displacements dn and nodal normal gradients qn. The flexibilitymatrix F∗ is demonstrably symmetric [16]. For elastostatics problems, F∗ is singular whether coming froma formulation with singular (boundary elements) or non-singular (finite elements) fundamental solutions. Infact, F∗ and H feature spectral properties of the classical elasticity formulation [6, 10]. In the finite elementimplementation of eq (25), one constructs the kinematic matrix H of eqs (18, 19) with n∗ rows and nd columnsand rank equal to nd − nrig, where nrig is the number of columns of W = N(HTH) that span the space ofrigid body displacements. One also constructs the flexibility matrix F∗ of eqs (23, 24) of order n∗ and rankn∗−nrig, where V = N(F∗) is a matrix with nrig columns. For homogeneous materials, the generalized boundarydisplacements ud

i and qdi of eq (9) are linear combinations of u∗im in eq (8), and V = N(F∗) ⇒ HTV = 0.

As already developed in the frame of the hybrid finite/boundary element method [6, 12, 9], the generalizedinversion of F∗ in eq (25) is carried out by simply replacing F∗(−1) with

(F∗ + VVT

)−1.

Application to truss and beam elements

Truss element. The homogeneous differential equation of a truss element of length L, constant cross sectionA and elasticity modulus E is

d2u∗

dx2 − g2 d4u∗

dx4 = 0, with solutions u∗ =⟨ex/g e−x/g x 1

⟩(26)

As outlined in Reference [10], the corresponding stiffness matrix, for the degrees of freedom given in the firstscheme of Figure 1 and substituting g, C and S for g/L, cosh(L/g) and sinh(L/g), is

K =EA/L

S + 2g − 2Cg

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S g(C − 1) −S g(C − 1)

g(C − 1) gL(C − S g) g(1 −C) gL(S g − 1)

−S g(1 −C) S g(1 −C)

g(C − 1) gL(S g − 1) g(1 −C) gL(C − S g)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(27)

The null space of this stiffness matrix is the vector of rigid body displacements W = 〈1 0 1 0〉T. One checksthat a constant strain state given by the vector of displacements 〈−1/2 1/L 1/2 1/L〉 yields only classical forcesEAL 〈−1 0 1 0〉 as a response (which also clarifies the meaning of the non-classical displacement degrees of

freedom in terms of the global coordinates schematically represented in Figure 1). Moreover, if one assemblesthe stiffness matrices of two elements of lengths αL and (1 − α)L, for any 0 ≤ α ≤ 1, and performs staticcondensation of the internal degrees of freedom, the above stiffness matrix K is retrieved.

Slender beam element. The application of the Euler-Bernoulli hypothesis to thin beams, plates and shells inthe frame of the strain gradient elasticity is an oxymoron, to say the least. An attempt to consistently assessgeneral bending problems in the present framework is in progress [16]. Nevertheless, it may be academi-cally worth applying the present developments to a slender beam, which has already been the subject of manyresearch works [3, 22, 21, 23, 26]. The homogeneous differential equation is

d4u∗

dx4 − g2 d6u∗

dx6 = 0, with solutions u∗ =⟨g4ex/g g4e−x/g x3 x2 x 1

⟩(28)


For a constant moment of inertia I, the corresponding stiffness matrix is [16]

K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

K11 K11L/2 K13 −K11 K11L/2 −K13

sym K22 K23 −K11L/2 K11L2/2 − K22 K23 − K13L

sym sym K33 −K13 K13L − K23 −K36

sym sym sym K11 −K11L/2 K13

sym sym sym sym K22 −K23

sym sym sym sym sym K33

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

K11 = 12(2C − 2 − S/g)∆, K22 = 4L2(3C − 3S g − S/g)∆K33 = L4

(4S g −C − 24(S −Cg + g)g3 + 12g2

)∆, K13 = 6L2

(4(1 −C)g2 + 4S g − 1 −C

)∆,

K23 = 2L3(12(1 −C)g2 − 1 + 9S g − 2C

)∆, K36 = L4

(2S g + 1 + 24(S −Cg + g)g3 − 12Cg2

)∆

(30)

with ∆ = (8C + 24g(Cg − S − g) − S/g + 4) EI/L3, for g, C and S as before. The null space of this matrix isspanned by the vectors of rigid body displacements 〈1 0 0 1 0 0〉 and 〈0 1 0 L 1 0〉. Resorting to the secondscheme of Figure 1 for the global orientation of the degrees of freedom, one checks that classical pure bendingEIL 〈0 − 1 0 0 1 0〉 is obtained from the vector of constant curvature 〈0 − 1/2 1/L 0 1/2 1/L〉. Moreover,

if one assembles the stiffness matrices of two elements of lengths αL and (1 − α)L, for any 0 ≤ α ≤ 1, andperforms static condensation of the internal degrees of freedom, the above stiffness matrix K is also retrieved.

Figure 1: Schematic representation of the generalized,globally oriented, displacement and force degrees offreedom of a truss and a beam elements.

Figure 2: Assemblage of three quadrilateral elements with linearboundary interpolation functions.

2D and 3D finite elements

Basic formulation. The homogeneous differential equation to be solved is

(1 − g2∇

) (u∗im,kk +

11 − 2ν

u∗km,ki

)= 0 (31)

Expressing u∗im in terms of a potential function, u∗im = δimΦ0,kk−Φ0,im/(2−2ν), one arrives at the basic differentialequation (1 − g2∇2)(∇4Φ0) = 0, which must provide n∗ solutions. The subscript of Φ0 is justified because a


general solution Φ for time-dependent problems in the frequency domain has also been obtained [16]. Thesolution of Φ0 for 2D problems is expressed in polar coordinates (r, θ) as

Φ0 = rn [C1n cos(nθ) +C2n sin(nθ)] + In(r/g) [C3n cos(nθ) +C4n sin(nθ)] (32)

where In(r/g) is the modified Bessel function of the first kind and order n, with argument (r/g). The trigono-metric terms in θ correspond to n∗ = 4(2n + 1) complete polynomials or order n, rigid body displacementsincluded. For 3D problems, the solution is expressed in spherical coordinates (r, θ, φ) as

Φ0 = Pn(cos θ)

rn

[(r2 + 4n + 6)g2 + 4n

])[C1 cos(φ) +C2 sin(φ)

](33)

+I 12+n(r/g)

[C3 cos(φ) +C4 sin(φ)

]/√

r

where the Bessel function I 12+n(r/g) of fractional order is actually a polynomial and Pn

(cos θ) is the associated

Legendre function of first kind, degree n and order with argument (cos θ), also a polynomial in Cartesiancoordinates. Pn

(cosθ) exists only for ≤ n. There are 4n + 2 solutions in eq (33), with a total of n∗ = 6(n + 1)2

solutions u∗im comprised by a complete polynomial of degree n, rigid body displacements included.

Finite element implementation and results. To make sure that the formulation is always well posed andthe final expression of the stiffness matrix K in eq (25) is independent from the location of the finite elementin the Cartesian coordinate frame, the number n∗ of internal parameters must correspond to a complete poly-nomial, whose order n is chosen in such a way that n∗ ≥ nd. Table 1 illustrates the results for some commonfinite element patterns (one obtains n∗ = nd only for CST and Te4 elements in classical elasticity [12], butthis also happens for the H8 element shown in the Table). Since the evaluation of H and F∗ is carried outonly in terms of boundary integrals, according to eqs (18) and (23), one may construct finite elements of anyshape, provided that n∗ is not too large, as ill-conditioning will certainly arise (differently from which occursin a hybrid boundary element formulation that is based on singular fundamental solutions). Figure 2 illustrates

2D problems 3D problemsPolynomial order n 2 3 4 5 2 3 4 5

n∗ internal 4(2n + 1) 6(n + 1)2

degrees of freedom 20 28 36 42 54 96 150 216Element type CST Q4 T6 T8 Te4 H8 Te10 H20

nd external d.o.f. 18 24 30 40 48 96 102 204

Table 1: Illustration of the number n∗ of solutions necessary for the implementation of some 2D and 3D finite elements (CST, T6 = 3,6 node triangles; Q4, Q8 = 4, 8 node quadrilaterals; Te4, Te10 = 4, 10 node tetrahedrons; H8, H20 = 8, 20 node hexahedrons).

the assemblage of three quadrilateral elements with linear boundary interpolation functions for udi and qd

i (Q8).Two classical degrees of freedom are represented at each node (numbered from 1 to 14). Pairs of non-classical,globally oriented, degrees of freedom are also schematically represented at each edge extremity (in the draw-ing, they are shown at a small distance from the extremities only to characterize to which edge they actuallycorrespond). While the classical degrees of freedom are nodal attributes, the non-classical ones are either edgeor face attributes, for 2D or 3D elements. This accounts for the huge number of extra degrees of freedomthat must be taken into account in a finite element implementation, as illustrated in Table 1 and in the Figure.As already mentioned for the truss and beam elements, the non-classical degrees of freedom, for the stiffnessmatrix obtained as in eq (25), are negatively oriented at negative faces or edges. Since one can tell that the leftedges of the truss and beam elements are the negative ones, for the schemes of Figure 1, the expressions of Kin eqs (27) and (29) already take this fact into account, so that elements can be assembled in a straightforwardway. For general 2D and 3D finite elements, however, which can be of any shape and orientation, one mustalways decide which edges or faces are negative (in terms of their outward normals) and multiply the corre-sponding rows and columns of the non-classical degrees of freedom by (−1) before assemblage. Such a needof reversing orientation was already mentioned by Mindlin [18], although not in the context of a finite elementimplementation.


The matrix W of rigid body displacements, as introduced in the paragraph after eq (25), depends only on thestructure’s geometry. The vector of rigid body rotations include coefficients of the normal gradient displace-ments qn that render only tangential components, which means that their normal projections, as defined in eq(3), are void in such a case. This subtle spectral property of the matrix H was checked in all finite element im-plementations. The matrix V, required in the inversion of F∗, can be constructed directly from the fundamentalsolutions derived from the potential functions of eqs (32) and (33), independently from geometry [16].The numerical code presently implemented applies to 2D finite elements of any shape and any number ofstraight or curved edges with linear, quadratic or cubic boundary interpolation functions. However, owing tothe conceptual difficulty of interpreting results, only the elements with linear interpolation functions have beentested. A rectangular element was tested with the length increasingly larger than the hight. The coefficients ofthe stiffness matrix corresponding to longitudinal degrees of freedom tended to the coefficients of the stiffnessmatrix of a truss element of corresponding mechanical properties, eq (27), drawing the authors’ attention tothe fact that negative faces require reverse orientation of the non-classical degrees of freedom. A second runconsisted in evaluating the stiffness matrices of several irregular quadrilaterals, as shown in Figure 2, for a seriesof patch tests. Equivalent nodal forces p were evaluated according to eqs (18) and (19) for linear, quadratic andcubic fields comprised by the fundamental solutions u∗im. Equation (24) was also checked for the correspondingnodal vectors d that comprise both displacements and normal gradients. For the linear field, complete agreementwas achieved, also leading to only classical forces, as required. The quadratic and cubic fields are not exactlyrepresented by the Q8 elements. However, the assemblage of the elements led to zero forces for classical andnon-classical resultants along the edges between two elements, and particularly at the node that joins all threeelements in Figure 2, which is a strict consistency check of the whole formulation.

Conclusions

This paper presents a concise hybrid finite/boundary element formulation of gradient elasticity problems basedon two virtual work principles that stem from the Hellinger-Reissner potential. General non-singular funda-mental solutions – the homogeneous solutions of eq (5) – are derived in a comprehensive framework [16].The most important contribution is the evidence that the proposed hybrid formulation naturally approximatesnormal displacement gradients along Γ independently from displacements, which is a step forward from theproposition made by Mindlin [19]. The finite element implementation of gradient elasticity requires a hugenumber of degrees of freedom, which still lack a complete mechanical interpretation. One shows that a nu-merical implementation can be undertaken that systematically leads to simple patch tests, consistency checksand the correlation between local and global physical quantities, so that meaningful non-classical boundaryconditions can be established.

Acknowledgments

This project was supported by the Brazilian agencies CAPES, CNPq and FAPERJ.

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Transient thermoelastic crack analysis in functionally graded materials by a BDEM

A. Ekhlakov, O. Khay and Ch. Zhang Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany,

[email protected], [email protected], [email protected]

Keywords: Functionally graded materials, boundary-domain element method, dynamic stress intensity factors, thermal shock, transient thermoelastic analysis, Laplace-transform

Abstract. A transient thermoelastic crack analysis in two-dimensional, isotropic, non-homogeneous and linear elastic functionally graded materials subjected to a thermal shock is presented. For this purporse, a boundary-domain element method based on a boundary-domain integral equation formulation is developed. Fundamental solutions of linear coupled thermoelasticity for the corresponding homogeneous, isotropic and linear elastic materials are employed to derive boundary-domain integral equations in the Laplace-transformed domain. The numerical implementation is performed by using a collocation method for the spatial discretization. The final time-dependent solutions are subsequently obtained by the Stehfest’s algorithm. Numerical results for an exponential gradation of the material parameters are presented to verify the accuracy and efficiency of the boundary-domain element method. The influences of the material gradation on the thermal stress intensity factors are investigated.

Introduction

Functionally graded materials (FGMs) receive increasingly growing research interest in material and engineering sciences due to their improved mechanical and thermal properties [1]. The absence of any interfaces is a main advantage of FGMs over conventional composites, which results in a continuous transition in physical, mechanical and thermal properties. The material parameters depend on spatial position in FGMs. A representative example of FGMs is the ceramic/metal FGMs, which are compositionally graded from a ceramic phase to a metal phase. Ceramic/metal FGMs possess the desirable properties of metals such as high toughness, high mechanical strength and bonding capability as well as high heat, wear and corrosion resistances of ceramics. They have a great potential in thermal, structural and biomedical applications. Because of the inherent brittle nature of ceramics cracks may develop in the manufacturing phase or during their services. Therefore, the fracture and damage analyses of FGMs under extreme thermal and mechanical loadings are important to their thermal and mechanical integrity, functionality, reliability and durability in engineering applications. Mathematically speaking the initial-boundary value problems of transient thermoelastic crack analysis in FGMs are described by a system of coupled partial differential equations with variable coefficients supplemented with prescribed boundary and initial conditions. Due to the high mathematical complexity of these problems, analytical methods can be employed only for very special simple geometry and loading conditions. Therefore, in general case numerical methods have to be used. Boundary element method or boundary integral equation method has been successfully applied to solving crack problems in homogeneous and linear elastic materials. Since the required fundamental solutions for FGMs are either not available or mathematically extremely complicated, their extensions and applications to continuously non-homogeneous FGMs are very restricted. For this reason, it is essential to develop efficient numerical methods for fracture analysis in FGMs. In this paper, a boundary-domain element method (BDEM) for a thermoelastic crack analysis in isotropic, continuously non-homogeneous and linear elastic FGMs is developed. An edge crack in a two-dimensional (2-D) finite domain subjected to a thermal shock is presented. The equations of motion and the thermal balance equation constitute the governing equations of the transient coupled linear thermoelasticity. To eliminate the time-dependence in these equations the Laplace-transform technique is applied. Boundary-domain integral representations are derived from the reciprocity theorem in FGMs in conjunction with the fundamental solutions for homogeneous, isotropic and linear thermoelastic solids [2], from which boundary-domain integral equations (BDIEs) are obtained for the unknown fields. They include additional terms represented by domain integrals, which appear due to the material’s non-homogeneity. A collocation method is implemented for the spatial discretization by using continuous quadratic boundary and quadrilateral domain elements. The arising boundary


and domain integrals are computed after a special regularization procedure by the standard Gaussian quadrature. To obtain time-dependent solutions an inverse Laplace-transform is performed by using the Stehfest’s inversion method [3]. Numerical results are given to show the effects of the material gradation on the dynamic stress intensity factors (SIFs).

Boundary-domain integral equations

Consider continuously non-homogeneous, isotropic and linear elastic FGMs in a 2-D domain Ω . In the absence of body forces and heat sources, the equations of motion and the generalized heat-conduction equation in transient coupled thermoelasticity [2,4] can be written as

, ( , ) ( ) ( , ) ,0ij j it u t =σ −x x x (1)

, ,,( ) ( , ) ( ) ( ) ( , ) ( ) ( , ) 0,i k ki

k t c t u t θ − θ − η = x x x x x x x (2)

where ijσ , iu and θ are the stresses, displacements and the temperature difference. Hereafter, the conventional

summation rule over double indices is implied, a comma after a quantity indicates spatial derivatives, a dot over a quantity denotes time derivative, and Latin indices vary from 1 to 2 unless otherwise stated. The material parameters (mass density ( )x , Young’s modulus ( )E x , thermal conductivity ( )k x , specific heat

( )c x , etc.) are assumed to depend on the Cartesian coordinates and Poisson’s ratio ν is constant. In this case,

the elasticity tensor ( )ijklc x can be expressed in terms of the elasticity tensor 0ijklc of the homogeneous materials

as

0( ) ( ),ijklijklc c= µx x (3)

with

0

1 2 ij kl ki lj kij j liklc δ δ δ δ + δν

+− ν

δ2

= and ( )( )

( ),

2 1

Eµ =

+ ν

xx

where ( )µ x is the shear modulus, ijδ denotes the Kronecker symbol. The Duhamel-Neumann constitutive

relations for the stress components ( , )ij tσ x can be written in the form

( ) ( ) ( ) ( ) ( )0, , ,, kij ijkl l ijc tt u tσ = − δ θµ γx x xx x (4)

with

( )( )

( )2 1

,1 2 t

+ νγ = α µ

− νx x

where ( )γ x is the stress-temperature modulus and tα is the linear thermal expansion coefficient.

The following essential and natural boundary conditions for the mechanical and thermal quantities are prescribed as

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

, , on , , , on

, , o

,

,n , , on ,

i i u i i t

q

u t u t t t t t

t t q t q tθ

= Γ = Γ

θ = θ Γ = Γ

x x x x

x x x x

(5)

where ( , ) ( , ) ( )i ij jt t t n= σx x x are the components of the traction vector, ,( )( , ) ( , )i ikt t nq = θx xx is the heat flux

and in denotes the components of the outward unit normal vector. Here, uΓ and tΓ are the parts of the external

boundary exΓ ( ex u tΓ = Γ ∪Γ ) with prescribed displacements and tractions, respectively, while on θΓ and qΓ

( ex qθΓ = Γ ∪Γ ) the temperature and the heat flux are given. The crack-faces C±Γ are assumed to be free of

mechanical and thermal loadings


( ) ( ), 0, , 0 on .Cit t q t ±= = Γx x (6)

Initial conditions at 0t = are assumed for the mechanical and the thermal quantities as

( ) ( ) ( ) ( )0 0 00,, , , 0, , 0.i k kit t tt

u t u t u t t= = === = = θ =x x x x (7)

Applying the Laplace-transform to the initial-boundary value problem (1), (2) and (5)-(7) yields

( ) ( )0 2, , , , , 0,ijkl k lj j k l i i ic u u p uµ + µ − γ θ + γ θ − = (8)

2

,, , , 0,iii i k k

k cp pu

k k

θ + θ − θ − η =

(9)

with the boundary conditions on the external boundary exΓ and the crack-faces C±Γ ( exC±Γ = Γ ∪Γ )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

, , on , , , on

, ,

,

on , , , on

, 0, , n ,

,

0 o

i i u i i t

q

Ci

u p u p t p t p

p p q p q p

t p q p

θ

±

= Γ = Γ

θ = θ Γ = Γ

= = Γ

x x x x

x x x x

x x

(10)

where all quantities in the Laplace-transformed domain are denoted by an over-bar, p is the complex Laplace-transform parameter. Integral representations of the displacements ju and the temperature θ at an arbitrary point ∈Ωx are derived

from the generalized Somigliana identity for FGMs by using the fundamental solutions of the transient linear coupled thermoelasticity for homogeneous materials [2]. After moving the observation point to the boundary ∈Γx the following system of BDIEs for the mechanical and the thermal fields at the boundary and internal

points is obtained as

( ) ( ) ( )( )

( ) ( )

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

( ) ( )

( ) ( )( )

( ) ( )

0

0 0

0

0

1, , , , , ,

1, , , , , , 0,

, , , , , ,

1, ,

1

, , | |,

i ij i ij

uj

i i

j

j

i

j

i

u p u p T p t p U p d

p Z p q p U p d F p

pp u p T p t p U p d

p F p q p T p d

Γ

Γ

Γ

+ − − Γ −

µ

−κ θ − − Γ − =

η

κ ηθ − − − Γ +

γ µ

+κ θ − − Γ

η

x x x

x x x

x x x

x x

( ) ( ), 0,F pθ

Γ

− = x

(11)

where

( ) ( )( )( )

( ) ( )

( )( )

( ) ( )( ) ( )

( ) ( )

( )( ) ( )

( )( )

( )( )

( ) ( )

20

, 20 02

, ,, , ,2

, 1 , ,

1, , 1 , ,

11 , , , ,

ui ij

kik ij

i ii j i i

j

j

j

F p p u p U p d

p U p d p U p d

kp U p d p U p d

Ω

Ω Ω

Ω Ω

= − − − Ω +

µ

+ σ − Ω + κ β − θ − Ω +

η κ

η+ − θ − Ω − + θ − Ω

η η µ

µ

µ

y

y

y

y y

yx y y x

y

yy y x y y x

y y y

y yy y x y y x

y y y

, !

(12)


( ) ( )( )( )

( ) ( )

( )( )

( ) ( )( ) ( )

( ) ( )

( )( ) ( )

( )( )

( )( )

20 00

0

, 20 02

, ,, , ,2

, 1 , ,

1, , 1 , | |,

11 , | |,

i i

kik i

i ii i

pF p p u p U p d

p U p d p T p d

kp T p d

θ

Ω

Ω Ω

Ω

κ η = − − Ω −

γ µ

µ − σ − Ω − κ β − θ − Ω +

µ η κ !

η+ − θ − Ω − + θ

η η µ

y

y y

y

yx y y x

y

yy y x y y x

y y y

y yy y x

y y y

( ) ( ), | |, .i p T p dΩ

− Ω

! yy y x

(13)

Here, a tilde over a quantity denotes the ratio of a non-homogeneous to a homogeneous quantity that is designated by a zero subscript. The BDIEs (11) contain boundary and domain integrals with strongly and weakly singular kernels. The strongly singular integrals can be understood in the sense of Cauchy principal values. The regularization of the strong singularities to weak singularities is realized by using the singularity subtraction technique [5].

Numerical solution procedure

An efficient BDEM is developed to solve the BDIEs (11). A collocation method is employed for the spatial discretization of the BDIEs in the Laplace-transformed domain. The boundary Γ and the domain Ω are divided into bN quadratic boundary and dN quadrilateral domain elements, respectively, and the total number of nodes is b dN N N= + . A system of linear algebraic equations follows from the collocation of the BDIEs (11) by using quadratic approximations of the unknown primary ( iu and θ ) and secondary ( it and q ) field quantities. After numerical integrations, applying the prescribed boundary conditions (10) and a rearrangement of the equations, we obtain a system of 3N linear algebraic equations that can be written in matrix form as

b b b b

i i i i

− =

A 0 D x y

A I D u y. (14)

Here, bx is the 3 bN vector of the unknown values of the displacements iu , the tractions it , the temperature θ

and the heat flux q at the boundary collocation points, iu is the 3 dN vector of the unknown displacements iu

and temperature θ at the internal collocation points, by and iy denote the 3 bN and 3 dN vectors composed of

the prescribed boundary conditions. The sizes of the matrices bA , iA , bD and iD are 3 3b bN N× , 3 3d bN N× , 3 3bN N× and 3 3dN N× , respectively, and I is the identity matrix. The system of linear algebraic equations (14) is solved numerically for discrete values of the Laplace-transform parameter p to obtain the boundary

unknowns bx and the interior primary field quantities iu . The time-dependent solutions can be calculated by an inverse transform that is an ill-posed problem, because small truncation errors may be accumulated in the inversion process. There are many numerical inversion methods and their comparative analysis can be found in [6]. In the present paper, the sophisticated Stehfest’s algorithm is employed [3], from which an approximated value af of a function f for a specific time t can be computed as

1

ln 2 ln 2( )

sN

a ki

f t v f it t=

=

" (15)

where

min( , /2) /2

/( 2)

[( 1)/2]

(2 )!( 1)

( / 2 )! ! ( 1)!( )! (2.

)!

s s

s

i N NN i

ik i

k kv

N k k k i k k i+

= +

= −− − − −

"


For each time instant we solve sN boundary value problems in the Laplace-transformed domain for the

corresponding transform parameter ln 2 /p i t= . Stehfest has suggested to use 10sN = for single and 18sN =

for double precision arithmetic [3]. In the present analysis 18sN = was adopted and we obtained numerical results comparable with those computed by Durbin’s method [7] that requires a complex arithmetic and more computing time to achieve the convergence.

Computation of SIFs

Dynamic SIFs can be conveniently computed by the extrapolation technique following directly from the asymptotic expansions of the displacements in the vicinity of the crack-tip [8]. The asymptotic crack-tip fields in continuously non-homogeneous and linear elastic FGM is the same as in homogeneous and linear elastic materials [9, 10]. For a crack located on the 1x − axis, the dynamic mode-I and mode-II SIFs are related to the

crack-opening-displacements ( , )iu t∆ x by

( )( )

( )( )

I 2tip

0II 1

,2 1lim ,

,1

K t u t

K t u taε→

∆ επ = µ

∆ εκ + − ε ! ! (16)

with

3 4 , for plane strain,

3, for plane stress,

1

− ν

κ = − ν + ν

where tipµ is the shear modulus at the crack-tip, ε is a small distance from the crack-tip to the considered node

on the crack-faces and the crack-opening displacements ( ),iu t∆ x is defined by

( ) ( ) ( ), , , .C Ci i iu t u t u t+ −∈Γ ∈Γ

∆ = −x x

x x x (17)

Numerical results

An edge crack in a rectangular, isotropic, continuously non-homogeneous and linear elastic FGM plate is considered. The plate geometry is described by the width 1w = , length 2 3l w= and the crack length 0.4a w=

(Fig. 1). The cracked plate is subjected to a thermal shock 2

00( , ) H( )

xt t

=θ = −θx at the lateral side, where 0θ is

the constant amplitude and H( )t is the Heaviside step function.

The material gradation in the ix − direction is given by an exponential law

( ) ( ) ( ) ( ) ( ) ( )0 0 0

0 0 0

exp | | , exp | | , exp | | ,

1 1 1ln , ln , ln ,

i i i i i i

b b b

b b b

E x E x k x k x c x c x

E k c

E k c

= α = β = γ

α = β = γ =

(18)

where 0 (0)E E= , 0 (0)k k= , 0 (0)c c= , a quantity with the subscript b denotes the material parameter on the

opposite external boundary of the plate, and b is equal w for the 1x − and l for the 2x − direction. The mass

density, the Poisson’s ratio and the linear thermal expansion coefficient are taken as constant ( ) 1=x , 0.25ν =

and 0.02tα = , respectively. Plane strain condition is assumed in the numerical analysis. Due to the selected material gradations, the symmetry of the loading conditions and the plate geometry only one half of the plate is considered. In this case, the mode-I dynamic SIF occurs whereas the mode-II SIF is


identically zero. In the plate discretization, 80 boundary and 341 interior nodes are used. The displacement and the temperature fields are modelled by 40 three-node quadratic boundary and 100 eight-node quadrilateral domain elements.

Fig. 1: An edge crack in a rectangular FGM plate subjected to a sudden cooling

The temporal variations of the normalized mode-I SIF I I 0/ tK K a= α θ π for the homogeneous plate, for the

graded material properties parallel to the crack-line ( 1x − direction) and perpendicular to the crack-line

( 2x − direction) are presented in Figs. 2-4, respectively, where 2 /Lt t c c k= is the dimensionless time and Lc is

the velocity of the longitudinal elastic wave. Numerical results in Figs. 3a and 4a are calculated for the material gradient parameters 0.7α = β = γ = − , while the curves in Figs. 3b and 4b correspond to the values

0.7α = β = γ = . Mode-I SIFs IK provided by the BDEM (solid lines) are compared with those obtained by the FEM (dashed lines). They show a quite good agreement of the present numerical results with the FEM results.

Fig. 2: Normalized dynamic SIF for a homogeneous plate

-0.4

-0.2

0.0

0.2

0.4

0.0 1.2 2.4 3.6 4.8

IK

homogeneous

: FEM

: BDEM

t


Fig. 3: Normalized dynamic SIF for a material gradation in the 1x − direction

Fig. 4: Normalized dynamic SIF for a material gradation in the 2x − direction

It can be seen that the first peak of the mode-I SIF increases with increasing value of the gradient parameters regardless of the directions of the material gradation (Figs. 3 and 4). The peak values of the normalized mode-I dynamic SIF and the time instants, at which they occur, depend on the material gradient parameters. The maximum values of the mode-I SIF IK for the negative gradient parameters α , β and γ in the case of a material gradation parallel to the crack-line (Fig. 3a) arrive at smaller time instants and are larger than those for the material gradation perpendicular to the crack-line (Fig. 4a). In addition, the negative gradient parameters stipulate a reduction of the mode-I SIF IK in comparison with the homogeneous material case. In contrast, the

first peaks of the mode-I SIF IK for the positive gradient parameters α , β and γ in the case of the

-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 1.2 2.4 3.6 4.8-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 1.2 2.4 3.6 4.8

0.7α = β = γ = − 0.7α = β = γ =

a) b): FEM: homogeneous

: BDEM: FEM: homogeneous

: BDEMIKIK

t t

-0.4

-0.2

0.0

0.2

0.4

0.0 1.2 2.4 3.6 4.8-0.4

-0.2

0.0

0.2

0.4

0.0 1.2 2.4 3.6 4.8

0.7α = β = γ = − 0.7α = β = γ =

a) b): FEM: homogeneous

: BDEM: FEM: homogeneous

: BDEMIKIK

t t


1x − gradation (Fig. 3b) reach at larger time instants and are smaller than those for the case if the 2x − gradation

(Fig. 4b). Positive gradient parameters cause an increase in the mode-I SIF IK compared to the corresponding homogeneous material case. This implies that the direction of the material gradation has significant influences on the peak values of the SIFs.

Summary

A 2-D transient thermoelastic crack analysis in isotropic, non-homogeneous and linear elastic FGMs subjected to a thermal shock is presented in this paper. Fundamental solutions of linear coupled thermoelasticity for the corresponding homogeneous, isotropic and linear elastic materials are implemented in the developed BDEM. The material non-homogeneity is described by additional domain integrals, which require a special regularization and domain discretization. A collocation-based BDEM is developed in the Laplace-transformed domain. The Stehfest’s algorithm is applied to obtain the final time-dependent numerical solutions. Dynamic SIFs are computed by using a displacement extrapolation technique and compared with those obtained by FEM. Numerical results show that the material gradation (i.e., magnitude of the gradient parameters and directions) has significant influences on the dynamic SIFs.

Acknowledgement

This work is supported by the German Research Foundation (DFG, project no.: ZH 15/10-1 and ZH 15/10-2), which is gratefully acknowledged.

References

[1] S. Suresh and A. Mortensen Fundamentals of Functionally Gradient Materials, London: The Institute of Materials (1998).

[2] J. Balas, J. Sladek and V. Sladek Stress Analysis by Boundary Element Methods, Elsevier (1989). [3] H. Stehfest Algorithm 368: Numerical inversion of Laplace transform, Communications of the ACM 13,

47-49 (1970). [4] W. Nowacki Thermoelasticity, Pergamon Press (1986). [5] L. Gaul, M. Kögl and M. Wagner Boundary Element Methods for Engineers and Scientists, Springer

(2003). [6] B. Davies and B. Martin Numerical inversion of the Laplace transform: A survey and comparison of

methods, Journal of Computational Physics 33, 1-32 (1979). [7] F. Durbin Numerical inversion of Laplace transforms: An efficient improvement to Dubner and Abate's

Method, The Computer Journal 17, 371-376 (1974). [8] J. Sladek, V. Sladek and Ch. Zhang An advanced numerical method for computing elastodynamic fracture

parameters in functionally graded materials, Computational Materials Science 32, 532-543 (2005). [9] J.W. Eischen Fracture of nonhomogeneous materials, International Journal of Fracture 34, 3-22 (1987). [10] F. Erdogan Fracture mechanics of functionally graded materials, Composites Engineering 5, 753-770

(1995).


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HEDD-FS Method for Numerical Analysis of Cracks in 2D Finite Smart Materials

CuiYing Fan1, GuangTao Xu2 and MingHao Zhao3

1,2,3The School of Mechanical Engineering, Zhengzhou University, No. 100 Science Road, Zhengzhou, 450001, China

E-mail: [email protected], [email protected], [email protected]

Abstract In this paper, the Hybrid Extended Displacement Discontinuity-Fundamental Solution Method (HEDD-FSM) is presented for numerical analysis of cracks in two-dimensional (2D) finite smart media by combing the extended displacement discontinuity method (EDDM) and the fundamental solution method (FSM). In the HEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with the extended displacement discontinuities placed on the crack. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the prescribed conditions on the boundary of the domain and on the crack face. The HEDD-FSM and the calculation of the extended intensity factors at crack tips are revisited for elastic, piezoelectric and magnetoelectroelastic problems.

Keywords: HEDD-FSM, smart material, crack, extended displacement discontinuity method, fundamental solution method, stress intensity factor

1. Introduction

Piezoelectric and magnetoelectroelastic materials, two kinds of smart materials, are finding more and more applications in smart structures and systems due to the mechanical-electrical-magnetic coupling effect. Defects, such as cracks, inclusions and voids in the materials greatly influence the performance of the structures and systems. Many efforts have been made to study the fracture problems of smart material [1-8].

Analytical solutions are usually difficult to obtain especially for cracks in finite domains. It is necessary to resort to numerical methods. As is well known, the Extended Displacement Discontinuity Method (EDDM) [9] is the preferable technique in dealing with problems involving stress singularities and has proved to be one of the most powerful methods in fracture mechanics of purely elastic media, piezoelectric media, as well as magnetoelectroelastic media[10-15]. On the other hand, the Fundamental Solution Method (FSM) is similar to the Charge Simulation Method (CSM). They share all the advantages of the BEM over domain discretization methods and have been used to solve various problems [16-21].

Combing the EDDM and the FSM, the Hybrid Extended Displacement Discontinuity-Fundamental Solution Method (HEDD-FSM) is proposed to solve the fracture problems of finite piezoelectric media [19,20] and magnetoelectroelastic media [21]. In this paper, we summarize briefly the HEDD-FSM for fracture analysis of two-dimensional (2D) problems of finite elastic, piezoelectric and magnetoelectroelastic media.

2. Basic Equations

In the absence of body force, electric charge and electric current, the governing equations for a three-dimensional magnetoelectroelastic medium are given by

,0,0,0 ,,, iiiijij BD ),,(3,2,1, zyxji , (1b) where iij D, and denote the stress, electric displacement and magnetic induction components, respectively, and a subscript comma denotes the partial differentiation with respect to the coordinate.

iB

The constitutive equations can be expressed in terms of the displacement components (iu yx uuuu 21 , and ), the electric potential zuu 3 and the magnetic potential

,2/)( ,,,, kkijkkijkllkijklij feuuc (2a)


,2/)( ,,,, kikkikkllkikli guueD (2b) ,2/)( ,,,, kikkikkllkikli guufB (2c)

where ijijijijij gfec ,,,, and ij are the elastic constants, piezoelectric constants, piezomagnetic constants, dielectric permittivity, electromagnetic constants and magnetic permeability, respectively.

On letting and0,0 ijij gf 0ij , the first two equations in eq (1) and eq (2) are reduced to the governing equations for a piezoelectric medium.

3. Boundary Conditions

S

1 2

3

4

56

7

8

N1

z

xo

-a a

N2

Fig.1 Source and collocation points for a finite domain of smart plate

There is an arbitrarily shaped plate occupying finite planar domain V bounded by , as show in Fig. 1. The Cartesian coordinate system oxz is set up, such that the polarization direction of the smart material is along the z-axis. There is a line crack S on the x-axis. Because electric field and magnetic field can exist everywhere, the electric and magnetic boundary conditions on crack faces are more complex compared with those in purely elastic problems. 3.1 Boundary conditions for piezoelectric material

For piezoelectric material, there are two kinds of boundary conditions on boundary and on crack faces S. One is the mechanical condition and the other the electric condition. The mechanical boundary conditions on boundary and on crack face S are given by

,, zzxx uuuu or ,, zzzzxxzzxzxzxxxx tnnttnnt (3) where the over bar “-” denotes the prescribed values on the boundary, is the traction on the boundary, and

is the unit outward normal vector on boundary and on crack face S. it

inThe electric boundary conditions on boundary is given by

, or zzxx nDnD , (4) where is the electric displacement boundary value. And the electric boundary condition on crack faces S takes one of the following electric boundary conditions [3], i.e.,

, zz DD (5a)

for electrically impermeable condition, where superscripts “+” and “-” denote the quantities on the upper and lower crack faces, respectively, or

,,cc zzzz DDDD (5b) for electrically permeable condition, where denotes the electric displacement in the crack cavity in the z-axis direction, or

czD

,cc zzzz DDDD ],/[][cc zzz uuD (5c)

for crack opening model of piezoelectric media, where is the dielectric constant of the material in the crack cavity.

c

3.2 Boundary conditions for magnetoelectroelastic material There are three categories of boundary condition on boundary and on crack faces S, namely the

mechanical condition, the electric condition and the magnetic condition. The mechanical boundary conditions have the same form as that for piezoelectric material given in eq (3).

The electric and magnetic boundary conditions on boundary are given by zzxx nDnDor, and ,or, zzxx nBnB (6)

where is the magnetic induction boundary value. However, there are five kinds of electric and magnetic boundary condition on the crack faces S as

described below [14], ,

zz DD and , zz BB (7a)

for electrically and magnetically impermeable condition, or ,,and,, cccc zzzzzzzz BBBBDDDD (7b)


for electrically and magnetically permeable condition, or ,,and, cc zzzzzz BBBBDD (7c)

for electrically impermeable and magnetically permeable condition, or ,and,,cc

zzzzzz BBDDDD (7d) for electrically permeable and magnetically impermeable condition, or

],/[][,

],/[][,cccc

cccc

zzzzzzz

zzzzzzz

uuBBBBB

uuDDDDD

(7e)

for crack opening model of magnetoelectroelastic media, where is the magnetic induction in the z-direction in the crack cavity and is the magnetic permittivity of the material in the crack cavity.

czB

c

4. HEDD-FSM for Analysis of Cracks in Finite Magnetoelectroelastic Media

The HEDD-FSM for smart media [19-21] is used to analyze the crack in a finite magnetoelectroelstic medium, as shown in Fig 1. First, N1 collocation points are selected on the boundary, and an equal number of source points N1 are taken accordingly outside the domain V. An unknown extended concentrated load

is applied at source point k , where and are the mechanical loads along the x- and z-axis, respectively, and and denote the point electric charge and electric current, respectively. Secondly, the crack S is divided into N

)41;2,1( 1 jNkPkj 1kP 2kP

3kP 4kP2 elements represented by the middle point. Constant

elements are used and unknown extended displacement discontinuities )...,,2,1( 2Nkuuu kjkjkj are

uniformly distributed on each element. Using the extended point force fundamental solutions given in [8], the extended Crouch fundamental

solutions given in [15], and the superposition principle, the extended displacement and the extended stress at any field point X can be expressed by [19-21]

),4,3,2,1(,),(),()(21

1

4

1

c

1

4

1

*

iuXXuPXXuXu kjS

N

k jkjkjP

N

k jiji

(8a)

),7,2,1(,),(),()(21

1

4

1

c

1

4

1

*

iuXXPXXX kjS

N

k jijkjP

N

k jiji (8b)

where denote the extended displacements iu ,, zx uu and , respectively; i denote the extended stresses

xzxxzxx BDD ,,,, zz , and , respectively; and are the fundamental solutions corresponding to extended point forces given in [8], and and are the extended Crouch fundamental solutions given in [15]. X

zB *iju

cij

ij*

ciju

P and XS respectively denote the source point outside the domain and the source point on crack. On letting eq (8) satisfy the given boundary conditions at the collocation points on boundary and

every element on the crack, one obtains 2(N1+N2) linear algebraic equations for purely elastic media, 3(N1+N2) linear algebraic equations for piezoelectric media and 4(N1+N2) linear algebraic equations for magnetoelectroelastic media to determine the unknown extended loads and the unknown extended discontinuity displacements

kiP

kju .

Solving the equations gives the unknown quantities. Fitting the calculated extended displacement discontinuity using the corresponding values of the Nc- and (Nc+1)-th elements from the crack tip, the asymptotic behavior of the extended displacement discontinuity near the crack tip can be expressed as

2/32

2/11 rkrku iii , (9)

where r is the distance of the point on crack faces from the crack tip, and are the fitting coefficients. Furthermore, one has

1ik 2ik

10/lim iir

kru

. (10)

Finally, the extended intensity factor can be calculated by using the extended displacement discontinuity ,][lim2,lim2,][lim2 42410D110II32310I rLuLKruLKrLuLK zrxrzr

(11a)

for cracks in piezoelectric media [3], and


,][lim2,][lim2

,lim2,][lim2

5352510B4342410D

110II3332310I

rLLuLKrLLuLK

ruLKrLLuLK

zrzr

xrzr

(11b)

for cracks in magnetoelectroelastic media [14], where are material constants given in [3,14]. ijL

5. HEDD-FSM for the Crack Opening Model in Finite Magnetoelectroelastic Media

Based on eq (8) and the boundary condition in eq (7e), 4(N1+N2) algebraic equations can be obtained to determine the unknown extended loads acting at the source points and the unknown extended discontinuity displacements on a crack face in a magnetoelectroelastic media, while the electric and magnetic boundary conditions on crack faces are expressed as

.),(),(

,),(),(

c

1

4

17

1

4

1

*7

c

1

4

15

1

4

1

*5

21

21

zkjS

N

k j

cjkjP

N

k jj

zkjS

N

k j

cjkjP

N

k jj

BuXXPXX

DuXXPXX (12)

Eq (5c) and eq (7e) are nonlinear algebraic equations, and the solution is difficult to obtain. The iterative method [14] is adopted here to solve the nonlinear problem. At first, the crack is treated as an impermeable one with the boundary condition given in eq (5a) and eq (7a):

,0),(,0),( )0(c)0(c zxBzxD zz (13) where the number “l” in the parenthesis in superscript “(l)” denotes the l-th iteration as described below. Solving the 4(N1+N2) nonlinear algebraic equations, we obtain the solution of the extended loads acting at source points and the extended discontinuity displacement on crack face denoted by )1(

iu . Then substituting the value of the obtained extended discontinuity displacement into eq (7e), new values of electric displacement and magnetic induction in the crack cavity can be calculated:

./,/ )1()1(c)1(c)1()1(c)1(czzzz uBuD (14)

The first round iteration has been completed. The iteration continues until the final solution is obtained when the preset accuracy )2,1( ii is satisfied

./,/ 2)(c)1(c)(c

1)(c)1(c)(c l

zl

zl

zl

zl

zl

z BBBDDD (15)

Substituting the extended discontinuity displacements of the final iteration results into eq (11), one can obtain the extended intensity factors, and the electric displacement and the magnetic induction fields in the crack cavity in the crack opening model.

6. Numerical Examples

Consider a center crack of length 2a in a smart plate of 2h length and 2b width, as shown in Fig.2. The poling direction is along the z-axis.

b

a

h

h

A A1

B B1

z

S

O x

a

,,zp

b

Fig.2 A center crack in a smart plate

Based on the HEDD-FSM [19-21], N1=200 collocation points are on the boundary of domain V, and an equal number of N1=200 source points are taken outside the domain V. And N2=100 elements are on the line crack. The ratio of AA1/AB is 2.3, where AA1 is the distance between collocation point A on boundary and the corresponding source point A1 outside the domain, while AB is the distance between collocation point A and its adjacent collocation point B.

By using the HEDD-FS method, the intensity factor of a center crack in an elastic plate is calculated under MPa100zp and plotted versus crack length a/b in Fig. 3, where the normalized stress intensity factors Fe and FI respectively based on the analytical solution [22] and the HEDD-FSM are given by

)2/(sece baF , )./(I apKF z (16)


It can be seen that the HEDD-FSM is effective to study the fracture problems with finite domain.

a/b0.0 0.2 0.4 0.6 0.8 1.0

Nor

mal

ized

inte

nsity

fact

ors

F I, F e

0.8

1.0

1.2

1.4

1.6

1.8

2.0

HEDD-FSM FI

Analytical solution Fe

Fig. 3 Intensity factor comparison in elastic plate

between HEDD-FSM and analytical solution

Fig.4a and Fig.4b show the dimensionless intensity factors by the impermeable crack model and crack opening model for center cracks in piezoelectric medium PZT-6B under the applied load MPa100zp and

, where the normalized electric displacement intensity factor F

2C/m10D is given by

)./(DD aKF (17) The results demonstrate FI and FD increase with increasing a/b, but the effect of the geometric size on FI and FD is different under different electric boundary condition.

Under the combing mechanical-electric-magnetic loading C/m0.1,MPa100 zp and 10N/Am ,

the dimensionless intensity factors of a center crack in the magnetoelectroelastic medium are displayed in Fig.5a and Fig.5b by using respectively the impermeable crack model and the crack opening model, where the normalized magnetic induction intensity factor FB is given by

a/b0.0 0.2 0.4 0.6 0.8

Nor

mal

ized

inte

nsity

fact

or F

I,FD

0.8

1.0

1.2

1.4

1.6

1.8

FI impermeable crack modelFD impermeable crack model

2C/m1.0 MPa,100 zp

Fig.4a Intensity factors versus crack length for

imba /

permeable crack in a piezoelectric plate

a/b0.0 0.2 0.4 0.6 0.8

Nor

mal

ized

inte

nsity

fact

or F

I,FD

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

FI crack opening modelFD crack opening model

2C/m1.0 MPa,100 zp

Fig.4b Intensity factors versus crack length a for

crack opening model in a piezoelectric plate b/

)./(BB aKF (18) The magnetoelectroelastic medium is a composite material made of BaTiO3 and CoFe2O3 with the volume fraction of the piezoelectric inclusion . It can be seen that the three normalized intensity factors increase with crack length. For impermeable boundary condition, the three normalized intensity factors are equal for a small crack. However, the three normalized intensity factors are unequal considering the crack opening and the electric and magnetic fields in the crack cavity.

5.0i V

a/b0.0 0.2 0.4 0.6 0.8N

orm

aliz

ed in

tens

ity fa

ctor

s F I,F

D,F

B

0.6

0.8

1.0

1.2

1.4

1.6

1.8

FI crack opening modelFD crack opening modelFB crack opening model

10N/AmC/m,0.1,MPa100 zp

Fig.5b Intensity factors versus crack length for crack opening model in a magnetoelectroelastic plate

ba /a/b

0.0 0.2 0.4 0.6 0.8Nor

mal

ized

inte

nsity

fact

ors

F I,FD,F

B

0.6

0.8

1.0

1.2

1.4

1.6

1.8

FI impermeable crack modelFD impermeable crack modelFB impermeable crack model

10N/AmC/m,0.1,MPa100 zp

Fig.5a Intensity factors versus crack length for impermeable crack in a magnetoelectroelastic plate

ba /


Similarly, the intensity factors for other electric and magnetic boundary conditions can be obtained by using the HEDD-FSM. The results demonstrate that the electric and magnetic boundary conditions, as well as the geometry of the medium, greatly influence the solutions.

7. Conclusion

The HEDD-FSM composes the merits of both the extended displacement discontinuity method and the charge simulation method in analyzing cracks in 2D finite piezoelectric and magnetoelectroelastic media. Different electric and magnetic boundary conditions on crack face, such as the impermeable, permeable and semi-permeable conditions, can be easily incorporated in this method.

This method is of higher computing speed and more efficiency for analysis of cracks in two-dimensional smart media. Numerical examples show that the freedom used in this method is two orders less than that used in the finite element method for analyzing cracks in smart plates. This method can be used to solve problems including multi-cracks, mixed-mode cracks, and can be further extended to solve crack problems of 3D finite smart media.

Acknowledgement

The work was supported by the National Natural Science Foundation of China (10872184), the Innovation Scientists Technicians Troop Construction Projects of Henan Province (084200510004) and the Program for Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN013).

References

[1] Z. Suo, C.M. Kuo, D.M. Barnett and J.R.Willis Journal of the Mechanics and Physics of Solids, 40,739-765 (1992).

[2] Q.H. Qin Southampyon: WIT Press (2001). [3] T.Y. Zhang, M.H. Zhao and P.Tong Advances in Applied Mechanics, 38, 147-289 (2002). [4] K. Meinhard Engineering Fracture Mechanics, 77, 309-326 (2010). [5] G.C. Sih, R. Jones and Z.F. Song Theoretical and Applied Fracture Mechanics, 40, 161-186 (2003). [6] B.L. Wang and Y.W. Mai European Journal of Mechanics, A/Solids, 22, 591-602 (2003). [7] X. Wang and Y.P. Shen International Journal of Engineering Science, 40, 1069-1080 (2002). [8] H.J. Ding, A.M. Jiang, P.F. Hou and W.Q. Chen Engineering Analysis with Boundary Elements, 29,

551-561 (2005). [9] S.L. Crouch International Journal for Numerical Methods in Engineering, 10, 301-343 (1976). [10] E. Pan Engineering Analysis with Boundary Elements, 23, 67-76 (1999). [11] C.Y. Dong, S.H. Lo and H. Antes Computational Mechanics, 41, 207-217 (2008). [12] B.J. Zhu and T.Y. Qin Theoretical and Applied Fracture Mechanics, 47, 219-232 (2007). [13] J.A. Sanz, M.P. Ariza and J. Dominguez Engineering Analysis with Boundary Elements, 2, 586-596

(2005). [14] M.H. Zhao, C.Y. Fan, F. Yang and T. Liu International Journal of Solids and Structures, 44, 4505-

4523 (2007). [15] M.H. Zhao, C.Y. Fan, T. Liu and F. Yang Engineering Analysis with Boundary Elements, 31, 547-558

(2007). [16] J.R. Berger and A. Karageorghis Engineering Analysis with Boundary Elements, 25, 877-886 (2001). [17] S. Chantasiriwan, B.T. Johansson and D. Lesnic Engineering Analysis with Boundary Elements, 33,

529-538 (2009). [18] X.C. Li and W.A. Yao Engineering Analysis with Boundary Elements, 30, 709-717 (2006). [19] C.Y. Fan, M.H. Zhao and Y.H. Zhou Journal of the Mechanics and Physics of Solids, 57, 1527-1544

(2009). [20] M.H. Zhao, G.T. Xu and C.Y. Fan Engineering Analysis with Boundary Elements, 33, 592–600 (2009). [21] M.H. Zhao, G.T. Xu and C.Y. Fan Computational Mechanics, 45, 401-413 (2010). [22] H. Tada, P.C. Paris and G.R. Irwin Del Research Corp, Hellertown, PA, 1973.


Recent developments of radial integration boundary element method in solving nonlinear and nonhomogeneous multi-size problems

X. W. Gao 1, M. Cui 1 and Ch. Zhang 2

1 School of Aeronautics and Astronautics, State Key Laboratory of Structural Analysis for Industrial

Equipment, Dalian University of Technology, PR China,

E-mails: [email protected]; [email protected] 2 Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany,

E-mail: [email protected]

Keywords: Boundary element method; Radial integration method; Nearly singular integrals, Thin structures

Abstract This paper presents some recent developments of the meshless boundary element method based on radial integration method (RIM) for solving 2D and 3D heat transfer and thermoelasticity problems. Special attention is paid to the consideration of the effects such as structural multi-sizes, nonlinear, non-homogeneous, and anisotropic problems. Firstly, general boundary-domain integral equations for heat conduction and stress analysis are derived using the weighted residual method based on the source point isolation technique, in which fundamental solutions for the corresponding linear homogeneous problems are adopted. The use of linear isotropic fundamental solutions for anisotropic, nonlinear and nonhomogeneous problems results in domain integrals appearing in the basic integral equations. The domain integrals appearing in the integral equations, then, are transformed into equivalent boundary integrals using RIM, resulting in a pure boundary element analysis algorithm without the need of internal cells. The thermal and mechanical material properties can be the functions of both temperature (resulting in nonlinear heat transfer) and spatial coordinates (for non-homogeneous materials). The Newton-Raphson iteration scheme is applied to solve the resulting nonlinear equation set. The nearly singular boundary integrals stemming from treating thin-structures using BEM are evaluated using the non-equally spaced element sub-division technique. The three-step solver of multi-domain BEM is employed to solve composite structural problems consisting of different materials. Finally, numerical examples are given to demonstrate the accuracy and efficiency of the presented method.

1. Introduction Thin structures are frequently used in aerospace engineering [1], such as multi-layered coatings, laminated structures, honeycomb structures etc. The investigation shows that the thermal stresses induced in laminated structures are the main cause of structural failure [2]. Therefore, the thermal stress analysis of composite structures is significantly important in aerospace engineering. The boundary element method (BEM) has distinctive advantages in solving problems of fracture mechanics [3] and thin structural problems [4], since it only needs to discretize the boundary of the problem into elements. However, the conventional BEM is not so attractive in solving nonhomogeneous, nonlinear and thermoelasticity problems, since domain integrals are inevitably introduced in the resulting integral equations [5]. A direct evaluation of domain integrals requires the discretization of the domain into internal cells. This severely eliminates the advantage of BEM. To overcome this disadvantage, Nardini and Brebbia [6] developed the dual reciprocity method (DRM) to transform the domain integrals into equivalent boundary integrals. To avoid using particular solutions required in the DRM, Gao proposed the radial integration method (RIM) [7] which can transform any domain integrals to the boundary based on a pure mathematical manipulation. RIM has been successfully applied to solve thermoelasticity [8], elastic inclusion [9], and creep damage mechanics problems [10]. In view of the robustness and simplicity of RIM in evaluating domain integrals without using internal cells, Hematiyan [11] gave a very good assessment to RIM, and Albuquerque et al. [12] compared RIM to DRM numerically through applications to dynamic problems with a more positive conclusion. Although thermoelasticity problems with constant material properties have been solved using the boundary-only element method based on RIM [8], this methodology has yet not been applied to solve heat conduction


and thermoelastic thin structure problems with varying material properties. This paper is an attempt for this purpose. First, boundary-domain integral equations for temperature and displacements are derived from the weighted residual forms of governing equations. Then, the domain integrals arising in the integral equations are transformed into equivalent boundary integrals using RIM, resulting in a pure boundary element solution algorithm. Material properties are allowed to be any type of functions of spatial coordinates. The treatment of nearly singular integrals is a challenge issue in solving thin structural problems using BEM [13]. A non-equally spaced element sub-division technique is presented for evaluating the nearly singular integrals. Numerical examples are given to demonstrate the correctness and efficiency of the presented method.

2. Boundary-domain integral equations for general nonlinear and nonhomogenous heat conduction problems

2.1. Formulations for general heat conduction problems The governing equation for general heat conduction problems can be expressed as

0ij

Q

xTk

x ji

, (1)

where ijk and Q are the thermal property tensor and the source term, respectively, and T denotes the temperature. ijk and T both may be functions of spatial coordinates for non-homogeneous problems or functions of the temperature for non-linear problems. It is noted that the thermal property tensor is symmetric, i.e., jiij kk . The repeated subscripts in eq (1) represent summation over the ranges of their values. Using a weight function G to multiply both sides of eq (1) and integrating over the entire domain , the following weak-form can be written.

0ij

QdGdxTk

xG

ji

. (2)

The first integral can be manipulated as follows

ij ij i ij

ij

,

i j j i j

j i

T T G TG k d Gk n d k dx x x x x

GGqd q Td k Tdx x

Gqd q Td I

(3)

where

jijiij , nkxGqn

xTkq

ij

, (4)

TdxGk

xI

ijij , (5)

in which denotes the boundary of the domain , in is the i-th component of the outward normal vector to , and q is the heat flux. It is noted that the domain integral in eq (5) may be strongly singular (depending on the choice of G) and, therefore, a different integral symbol is used to denote this. We assume that the weight function G is a fundamental solution of either isotropic or anisotropic problems. Usually, it is a function of the distance r between the source point p and the field point q [14]. When 0r , G may be singular and, therefore, an infinitesimal circular domain centered at the source point p with radius can be isolated from (Fig. 1).


p

Fig. 1 An infinitesimal domain isolated from

The last term in eq (3) now can be written as

ij ij ij0 0

ij ij0

lim lim

( ) lim ( ) ( ) ,

j i j i j i

j i j i

G G GI k Td k Td k Tdx x x x x x

G GT p k d k Td k p T p VTdx x x x

(6)

where

ij ij ij0 0 0( ) lim lim ( ) lim ,i j

i j j i

G G Gk p k d k n d k p n dx x x x

, (7)

jijiji xxGk

xG

xk

xGk

xV

2

ijij

ij . (8)

It is noted that the last domain integral in eq (6) is interpreted in the Cauchy principal value sense. Substituting eq (6) into eq (3) and the result into eq (2) yields

dQGTdVdqGTdqkT . (9)

Equation (9) is a boundary-domain integral equation valid for isotropic, anisotropic, linear and nonlinear heat conduction problems. The weight function G can be any type of functions. If G is a regular function, the coefficient k as determined by eq (7) will be zero since the radius 0 ; if G is chosen as the Green’s function [14] which is weakly singular when the source point p approaches the field point q under integration, k has a finite value; and if G is chosen as a higher singular function than the Green’s function, k is infinite and, therefore, this type of G doesn’t make sense. Once the weight function G and thermal property tensor ijk are given, all coefficients and kernel functions in eq (9) are known, and the unknown quantities be computed using the standard BEM discretization procedure [5]. It is noted that although eq (9) is derived for an internal source point p, it can also be used for boundary nodes since it is actually not necessary to compute the coefficient k directly using eq (7). This is based on the fact that the contribution of k to the final system of equations is in the diagonal term, which can be determined using a more efficient way, i.e., the “rigid body motion condition” [5].

2.2 Using isotropic fundamental solutions for general anisotropic heat conduction problemsIn principle, the weight function G can be any function. However, the simplest way is to choose G as the Green’s function for isotropic heat conduction problems, i.e.,

1 1ln( ), 2 ,21 , 3 ,

4

Dr

GD

r

(10)


where r is the distance between the source point p and field point q. The derivatives of G can be expressed as

ii

rrx

G,2

1

, (11)

)(2

1,,

2

jiijji

rrrxx

G

, (12)

where ii xrr /, , =2 for 2D and =3 for 3D problems, and =-1. Since is a circle (2D) or a sphere

(3D), we have jj rn ,, . Thus, from eq (7) it follows that

drrrk

drnrk

k jiijjiij

,,,

22. (13)

For an internal point, using the following relationship [5]

, ,

, 2 ,

4 , 3 ,3

iji j

ij

Dr r

dDr

(14)

equation (13) can be integrated as /iikk . (15)

It can be seen that k is the average value of the diagonal term of ijk . This is helpful for understanding the coefficient k in eq (9). It is also pointed out that if the problem is isotropic, the last term in eq (8) is zero and eq (9) is reduced to the result in [14].

3. Boundary-domain integral equations for thermoelasticity with variable coefficients The governing equations of the thermoelasticity problems can be expressed as [8]

TuC ijlkijklij ~,

0 , (16) where

21)1(2~

, (17)

jkiljlikklijijklC

212

0 , (18)

in which represents the shear modulus, the Poisson’s ratio, the thermal expansion coefficient, and

ku the displacement components. Both and are functions of temperature and spatial coordinates. Through applying the weighted residual method [8,15] to eq (16), the following boundary-domain integral equations for the displacements and the stresses can be obtained

,

( ) ( , ) ( ) ( ) ( , ) ( ) ( )

ˆ ( , ) ( ) ( ) ( , ) ( ) ( ),

i ij j ij j

ij j ij j

u U t d T u d

V u d U T d

y x y x x x y x x

x y x x x y x x (19)

( ) ( , ) ( ) ( ) ( , ) ( ) ( )

ˆ ˆ( , ) ( ) ( ) ( , )[ ( ) ( )] ( )

ˆ ˆ ( ) ln ( , ) ( ) ( ) ( ) ( ),

ij ijk k ijk k

ijk k ij

ij ij ijk k

U t d T u d

V u d T T d

rT r r d hT F un

y x y x x x y x x

x y x x x y x y x

y x y x y y y

(20)


where ijU , ijT , ijV , ijkU , ijkT , ijkV and ijkF can be found in [15], and other quantities are given by

jj uu ~ , (21)

TT

21)1(2ˆ

, (22)

)()1(2)21(

,, jiijij rrr

, (23)

)1(6)21)(1(

h . (24)

Equations (19) and (20) are only suitable for internal points. For boundary nodes, a limiting process is needed to establish the boundary integral equations from eq (19), and the “Traction-Recovery Method” [5] is adopted to compute the boundary stresses.

4. Evaluation of nearly singular integrals using a non-equally spaced element sub-division technique

When solving thin-structure problems using BEM, the treatment of nearly singular integrals is a challenge issue [4-6,13]. The element sub-division technique is a simple and robust technique in handling such problems [16] with the advantage of treating various orders of singularities using a unified way. Gao and Davies [5] proposed an equally-spaced element sub-division technique for evaluating the nearly singular integrals. The technique is simple, however, when the source point is very close to the element under integration, the number of sub-elements is huge and the computational time is intolerable. In this study, a non-equally spaced element sub-division technique is presented, which is able to reduce the computational time by several magnitude of orders for the same accuracy. The technique is based on a relationship proposed by Gao and Davies [16] for Gaussian quadrature regarding the number of Gaussian points, the minimum distance to element and the element size. The Gaussian quadrature formula for surface integrals can be expressed as [17]:

21211 1

21 ),(1 2

EEfwwI jim

i

m

jji

, (25)

where ),( 21ji are the Gaussian point coordinates, 1

iw and 2jw are the weighting factors, 1m and 2m are

the numbers of Gaussian points, and 1E and 2E are the integration errors, i.e.,

!i

im

ii

im

ii f

mmmLE 2

3

412

)!2()12()!(

, (26)

in which imf 2 denotes the im2 -th derivative of the function f, iL is the length of the element in the i-th direction. From eq (25) it can be seen that the integration error relies on the number of Gaussian points and the element length. Therefore, to ensure a desired accuracy, a big element must be divided into small sub-elements. Based on the analysis of an upper bound of the relative error, Mustoe [17] presented the following approximate formula for the specified accuracy tolerance e

epmpm

RL

i

im

ii

"

)!1()!2()!12(

42

2

, (27)

where p is the singularity order of the integral kernel characterized by pr , R is the minimum distance from the source point to the element. Equation (27) shows that, to retain the specified accuracy e and the number of Gaussian points cannot exceed a given number, the value of RLi / needs to be reduced by dividing the big element into a number of sub-elements. Based on the numerical investigation, Gao and Davies [16, 5] proposed the following formula for determining the number of Gaussian points.


!)4/(log2)2/(log'

RLepm

ie

ei , (28)

where

52

32' pp . (29)

After a rearrangement eq (28) yields

imp

ieRL

2'

24

. (30)

Equation (28) can be used to determine the minimum number of Gaussian points for a given accuracy tolerance e, while eq (30) can be used to determine the length of each sub-element for the specified values of allowed maximum number of Gaussian points, singularity order and the minimum distance. The non-equally spaced element sub-division technique can be summarized as follows: 1) Compute length iL of the boundary element under integration and the minimum distance R from the

source point y to the element. Detailed Fortran subroutines for determining iL and R can be found in [5]. 2) Calculate the required number of Gaussian points im using eq (28) in terms of the values of iL and R. 3) If maxmmi " ( maxm being the specified maximum number of Gaussian points), evaluate integrals using

Gaussian quadrature formulas.

4) If maxmmi # , let maxmmi and compute the length niL of each sub-element along the integration

direction i using the value of R and eq (30). 5) As shown in Fig. 2, mark the graduations in two integration directions in terms of nL1 and nL2 and form

all sub-elements (enclosed by dashed lines) from these graduations. 6) Evaluate integrals over each sub-element using Gaussian quadrature formulas. For understanding this process easily, Fig. 2 gives the schematic show of a big boundary element divided into 12 sub-elements by partitioning the line along 1 direction into 4 segments and the line along 2 direction into 3 segments.

R

11L 2

1L31L

41L

12L

22L

32L

y

Fig. 2 Schematic show of the element sub-division

5. Numerical examples Based on the method described in the paper, a computer code named BERIM (Boundary Element analysis based on Radial Integration Method) has been developed. In the code, all domain integrals appearing in eqs (9), (19) and (20) have been transformed into boundary integrals using RIM [7], resulting in a pure boundary element analysis algorithm without need of internal cells. When the material properties are functions of the temperature, the Newton-Raphson iteration scheme is applied to solve the nonlinear heat conduction equations. To solve composite structure problems, the three-step multi-domain BEM (MDBEM) solver proposed in [18] is adopted for both heat conduction and thermoelasticity problems. Corner and edge points are treated using the discontinuous elements to model the discontinuity of the heat flux and the traction. Two


numerical examples are presented in the following.

5.1 Thermal stress analysis over a honeycomb structure The first example is a honeycomb structure which is commonly used in thermal protection system (TPS) [1]. The structure consists of upper and lower cover plates with a thickness of 0.125mm. The honeycomb core has a wall thickness of 0.035mm, a wall height of 7.11mm, and a width of 4.76mm. The structure has a total of 100 honeycombs with the global dimension of 49.98mm$ 42.56mm$ 7.36mm. Figure 3 shows the BEM model consisting of 8946 four-noded boundary and interface elements with 8484 nodes. In the BEM model, the upper and lower plates, the honeycomb wall, and the hollow volume filling with air are treated as different sub-domains. The heat conductivities are 7.8$ 510 W/(mm%K) for the upper and lower plates, 1.7$ 410 W/(mm%K) for the honeycomb wall, and 2.3$ 510 W/(mm%K) for the filled air. The thermal boundary conditions are given as follows: the top and bottom surfaces are specified with the temperature distribution of

20036.7

60098.49

7536.798.49

225),,( $

zxxzzyxT (K) and the side surfaces are adiabatic.

Firstly, the heat conduction computation is performed to obtain the temperature distribution in the structure, and then the thermoelasticity computation is carried out using the obtained temperature. In the thermoelasticity computation, the material parameters are taken as =280GPa, =0.25, and the thermal expansion coefficient is mm/K102.47 -6$ . The top surface is uniformly imposed by traction conditions of x& =0.05MPa and z& =-0.5MPa, the bottom surface is fixed, and the side surface is traction-free. Figure 5 shows the computed temperatures at points shown in Fig. 4, which are located below the inner surface of the upper plate with a distance of 0.01mm to the inner surface. The computed heat flux zq , displacement xu and stress xx at these selected points are presented in Figs. 6, 7 and 8, respectively.

Mpax 05.0&Mpaz 5.0&

T=800K T=500K

T=200K

T=125K

q=0x

z

Fig. 3 BEM model and boundary conditions Fig. 4 Points for results plotting

x-coordinate (mm)

Tem

pera

ture

(K)

x-coordinate (mm)

Hea

t flu

x q z

(W/m

m2 )

Fig. 5 Distribution of temperature Fig. 6 Distribution of heat flux zq


x-coordinate (mm)

Dis

plac

emen

t u x (

mm

)

x-coordinate (mm)

xx

(a)

Fig. 7 Displacement xu Fig. 8 Stress xx

From Fig. 6, we can see that the heat flux zq has a much larger value on the honeycomb side wall than in the hollow air. This important phenomenon captured in the example is attributed to the discretization of the two surfaces of the honeycomb wall into boundary elements. Figure 8 shows that the computed stress xx is much larger than the imposed tractions. This indicates that the thermal stress is an important factor in TPS failure analysis.

5.2 Rectangular plate with a crack under tensile loading The second example analyzed is a rectangular plate with an edge crack, which is subjected to a uniform tensile loading as depicted in Fig. 9. The geometry of the cracked plate is described by: plate width b=10, plate length 2h=30 and crack-length a=0.4b. To demonstrate the capability of the presented method to treat the crack problem, a single computational domain is used in our computation. The upper and lower surfaces of the crack is very close, but not completely in contact, measured with the width ratio of the opening distance to the crack-length a. The boundary of the plate including the crack surfaces is discretized into 115 quadratic boundary elements with 254 boundary nodes. Two nodes are defined at the crack-tip for utilizing the discontinuous element [18] to model the discontinuity of the traction across the tip. Plain strain condition is assumed in our computation.

400025.0

Fig. 9 A plate with an edge crack Fig. 10 Deformed plate

Figure 10 shows the deformed plate plotted using the computed displacements multiplied by a factor of 200 for the case of the width ratio being 0.1%. To verify the correctness, this problem is also computed using the multi-domain BEM code [18], in which lower and upper parts of the plate are treated as two sub-domains along the crack. The displacements in y-direction computed using the present single domain method denoted by “1-Domain” and using the multi-domain BEM denoted by “2-Domain” are listed in Table 1 for three corner nodes as shown in Fig. 10. Comparison of the results from the two methods shows that the relative errors are rather small. To investigate the convergence of the computational results to the width of the crack, computations are carried out using different ratios of the crack-width to the crack-length. The computed results for the three selected corner nodes (see Fig. 10) are shown in Fig. 11. From Fig. 11, it can be seen


that the convergence is achieved when the width ratio is larger than 0.02% which is small enough to model a real crack-width. Figure 11 also shows that the usual single domain BEM combined with the non-equally spaced element sub-division technique described in this paper can solve the crack problems efficiently without the use of other complicated methods [3, 18].

Table 1 Computed displacement yu at three selected nodes Lower Upper Top

1-Domain 8.66874E-4 5.7889E-3 6.6526E-3 2-Domain 8.75949E-4 5.7325E-3 6.6045E-3 Error (%) -1.04 0.984 0.728

Width ratio of crack (%)

Disp

lace

men

t u y

(X10

-2)

LowerUpperTop

Fig. 11 Displacement computed using different values of the crack-width

6. ConclusionsA boundary element technique is presented for solving 2D and 3D nonlinear and non-homogeneous heat transfer and thermoelasticity problems. A non-equally spaced element sub-division technique is proposed for evaluating nearly singular boundary integrals in the analysis of thin structures using BEM. Numerical results show that the presented method can not only effectively solve the thin-wall structure problems, but can also solve crack problems in the usual way. This is very convenient to solve complicated engineering problems.

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[10] Gun H., 3D boundary element analysis of creep continuum damage mechanics problems. EngineeringAnalysis with Boundary Elements, 2005, 29: 749-755.

[11] Hematiyan M.R., A general method for evaluation of 2D and 3D domain integrals without domain discretization and its application in BEM. Computational Mechanics, 2007, 39: 509-520.

[12] Albuquerque E.L., Sollero P., Paiva W.P., The radial integration method applied to dynamic problems of anisotropic plates. Commun. Numer. Meth. Eng., 2007, 23: 805-818.

[13] Niu Z., Cheng C., Zhou H., Hu Z., Analytic formulations for calculating nearly singular integrals in two-dimensional BEM. Engineering Analysis with Boundary Elements, 2007, 31: 949-964.

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