a stabilised petrov-galerkin formulation for linear tetrahedral elements in compressible, nearly...

Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions

A Stabilised Petrov-Galerkin Formulation For LinearTetrahedral Elements In Compressible, Nearly Incompressible

and Truly Incompressible Fast Dynamics

Chun Hean Lee1, Antonio J. Gil2, Javier Bonet3, Miquel Aguirre4

Zienkiewicz Centre for Computational Engineering (ZC2E)College of Engineering, Swansea University, UK

Advances in Finite Element Methods for Tetrahedral Mesh Computations I (MS209A)

11th World Congress on Computational Mechanics (WCCM XI)

1 https://www.researchgate.net/profile/Chun_Hean_Lee2/2 http://www.swansea.ac.uk/staff/academic/engineering/gilantonio/

3 http://www.swansea.ac.uk/staff/academic/engineering/bonetjavier/4 https://www.researchgate.net/profile/Miquel_Aguirre

CHL-AJG-JB-MA (MS209A: Advances in Finite Element Methods for Tetrahedral Mesh Computations I) 20th - 25th July 2014

https://www.researchgate.net/profile/Chun_Hean_Lee2/

http://www.swansea.ac.uk/staff/academic/engineering/gilantonio/

http://www.swansea.ac.uk/staff/academic/engineering/bonetjavier/

https://www.researchgate.net/profile/Miquel_Aguirre


Outline

1 Motivation

2 Reversible elastodynamicsBalance principlesConservation laws

3 Petrov-Galerkin formulationPetrov-Galerkin spatial discretisationPerturbed test function spaceTemporal discretisationIncompressible and nearly incompressible formulationFractional-step formulation

4 Numerical results

5 Conclusions and further research



Motivation

• Standard solid dynamic formulations:

× Linear tetrahedral elements behave poorly in nearly incompressibleand bending dominated scenarios

× Uniform and selective reduced integrated linear hexahedral elementssuffer from respected hourglassing and pressure instabilities

× Convergence of stresses and strains is only first order

× Shock capturing technologies are poorly developed

X Time integrators are robust and preserve angular momentum

X Extensive availability of commercial packages (ANSYS, AltairHyperWorks, LS-DYNA, ABAQUS, . . .)

• Mixed conservation law formulation:

X Express as first order conservation laws enabling the use ofstandard CFD discretisation process

X Permits the use of linear tetrahedra, as well as enhanced linearhexahedra, for solid dynamics without locking difficulties

X Achieves optimal convergence with equal orders in velocities andstresses

X Take advantage of the conservative formulation to introducestate-of-the-art discontinuity-capturing operator

× Enhance existing time integrators to preserve angular momentum



Outline

1 Motivation



4 Numerical results




Balance principles

First order conservation formulation

• Consider the standard dynamic equilibrium equation:

∂ρ0v∂t− DIVP(F , J) = ρ0b

• To alleviate bending difficulty, the conservation law for the deformationgradient can be incorporated:

∂F∂t− DIV (v ⊗ I) = 0

• To avoid volumetric locking, the conservation law for the Jacobian can beadded:

∂J∂t− DIV

(HT

F v)

= 0; HF = (detF )F−T

Constitutive model is needed to complete the coupled system



Governing equation

Conservation laws

• The mixed equations can be written as a system of first order conservation laws:

∂

∂t

ρ0vFJ

+ DIV

−P(F , J)−v ⊗ I−HT

F v

=

ρ0b00

• More generally, if the energy equation is added:

∂

∂t

ρ0vFJ

ET

+ DIV

−P(F , J)−v ⊗ I−HT

F vQ − PT v

=

ρ0b

00s

• Or in standard form:

∂U∂t

+DIVF(U) = S; U =

ρ0vFJ

ET

; F =

−P(F , J)−v ⊗ I−HT

F vQ − PT v

; S =

ρ0b

00s

Our aim is to develop a library of second order numerical schemes for a mixedconservation law formulation of fast solid dynamics using existing CFD technologies



Governing equation

CFD formulations for fast solid dynamics

• Following are the stabilised numerical methodologies recently developed for fastsolid dynamics using mixed formulation:

Swansea University Research Group (Led by Prof. Javier Bonet and Dr. Antonio J.Gil)

· Two-Step Taylor-Galerkin (2TG) Formulation [Karim, Lee, Gil and Bonet, 2011]

· Total Variation Diminishing (TVD) Upwind Cell Centred Finite Volume Method(CCFVM) [Lee, Gil and Bonet, 2012]

· Jameson-Schmidt-Turkel (JST) Vertex Centred Finite Volume Method (VCFVM)[Aguirre, Gil, Bonet and Carreño, 2013]

· Stabilised Petrov-Galerkin (PG) Finite Element Method [Lee, Gil and Bonet, 2013]

· Fractional-Step Petrov-Galerkin (PG) Framework [Gil, Lee, Bonet and Aguirre, 2014]

M.I.T Research Group (Led by Prof. Jaime Peraire)

· Hybridizable Discontinuous Galerkin (HDG) Method [Nguyen and Peraire, 2012]



Outline

1 Motivation



4 Numerical results




Petrov-Galerkin spatial discretisation

Stabilised Petrov-Galerkin formulation

• Variational statement of Bubnov-Galerkin formulation (unstable):

∫V0

δV ·R dV = 0; R =∂U∂t

+ DIVF − S; δV =

δvδPδq

• Integration by parts gives:

∫V0

δV ·∂U∂t

dV =

∫V0

F : ∇0δV dV −∫∂V0

δV ·FN dA +

∫V0

δV · S dV

• Define stabilised Petrov-Galerkin (PG) formulation satisfying Second Law ofThermodynamics:

∫V0

δVst ·R dV = 0; δVst =

δvst

δPst

δqst



Perturbed test function space

Perturbation

• Stabilised test function space is generally defined by

δVst = δV + τT(∂F I

∂U

)T ∂δV∂XI︸︷︷︸

Perturbation

• Define flux Jacobian matrix:

∂F I

∂U=

03×3 −CI −κ [HF ]I− 1ρ0

I I 09×9 09×1

− 1ρ0

H I − ∂(v·[HF ]I)∂F 0

• Assuming τ (intrinsic time scale) a diagonal matrix for simplicity:

δVst :=

δvst

δPst

δqst

=

δv − τpFρ0

DIVδP − τpJρ0

HF∇0δq

δP − τFpC : ∇0δv − τFJ (v ⊗∇0δq) : ∂HF∂F

δq − τJpκHF : ∇0δv

; δP = C : δF

• Bubnov-Galerkin is recovered by setting stabilisation τ = 0




Petrov-Galerkin stabilisation

• Weak statement of stabilised Petrov-Galerkin (PG) formulation:

0 =

∫V0

[δV +

(∂F∂U

τ

)T∇0δV

]·R dV

=

∫V0

δV ·R dV︸︷︷︸Bubnov-Galerkin

+

∫V0

[∂F∂U

τR]

: ∇0δV dV︸︷︷︸Petrov-Galerkin stabilisation

• Integration by parts gives:

∫V0

δV·∂U∂t

dV =

∫V0

[F −

∂F∂U

τR]

︸︷︷︸Fst

: ∇0δV dV−∫∂V0

δV·FN dA+

∫V0

δV·S dV

• The stabilised flux Fst can be more generally defined as (equivalent toVariational Multi-Scale (VMS) stabilisation):

Fst = F(Ust ); Ust = U + U ′; U ′ = −τR




Finite element discretisation

• Using standard linear finite element interpolation for velocity, deformation gradientand Jacobian renders:

v =∑

avaNa; F =

∑a

F aNa; J =∑

aJaNa

∑b

Mabpb =

∫∂V0

NatB dA +

∫V0

Naρ0b dV −∫

V0

P(F st , Jst )∇0Na dV

∑b

MabFb

=

∫∂V0

Na(vB ⊗ N) dA−∫

V0

vstF ⊗∇0Na dV

∑b

Mab Jb =

∫∂V0

Na(vB · HF N) dA−∫

V0

vstJ · HF∇0Na dV

• By construction the stabilised deformation gradient, Jacobian and velocities are:

F st = F + τFp

(∇0v − F

); Jst = J + τJp

[DIV

(HT

F v)− J]

vstF = v +

τpF

ρ0(DIVP + ρ0b − p) ; vst

J = v +τpJ

ρ0(DIVP + ρ0b − p)

• To reduce implicitness of the resulting formulation additional time-integratedresidual-based artificial diffusions can be added




Finite element discretisation

• Using standard linear finite element interpolation for velocity, deformation gradientand Jacobian renders:

v =∑

avaNa; F =

∑a

F aNa; J =∑

aJaNa

∑b

Mabpb =

∫∂V0

NatB dA +

∫V0

Naρ0b dV −∫

V0

P(F st , Jst )∇0Na dV

• By construction the stabilised deformation gradient, Jacobian and velocities are:

F st = F + τFp

(∇0v − F

)+ ξF (∇0x − F )

Jst = J + τJp

[DIV

(HT

F v)− J]

+ ξJ (det∇0x − J)

• To reduce implicitness of the resulting formulation additional time-integratedresidual-based artificial diffusions can be added



Explicit time marching scheme

Time Integration

• Integration in time is achieved by means of an explicit two-stage Total VariationDiminishing Runge-Kutta (TVD-RK) time integrator:

U (1)n+1 = Un + ∆tUn

U (2)n+2 = U (1)

n+1 + ∆tU (1)n+1

Un+1 =12

(Un + U (2)

n+2

)

together with a stability constraint

∆t = αCFLhmin

Unmax

; Unmax = max

a

(Un

p,a)

• Introduce Lagrange multiplier correction to preserve the angular momentum [Lee,Gil, Bonet, 2013]



Fractional-step formulation

Incompressible and nearly incompressible formulation

• Time steps are very small given the presence of a very large value of Poisson’sratio in near incompressible solids

• Fully incompressible limit cannot be modelled

• Using standard fractional-step formulation renders:

ρ0(v int − vn)

∆t− DIVPn

dev − DIV(pnHF n

)− ρ0bn = 0

F n+1 − F n

∆t−∇0vn = 0

ρ0(vn+1 − v int)

∆t− DIV

[(pn+1 − pn

)HF n

]= 0

• Incompressiblity constraint gives

pn+1 − pn

κ∆t− HF n : ∇0v int −

∆tρ0

HF n : ∇0

[DIV

(pn+1 − pn

)HF n

]= 0



Fractional-step formulation

Finite element discretisation: fractional-step formulation

• Using Petrov Galerkin stabilisation the predictor step becomes:

∑b

Mab

(F n+1

b − F nb

)∆t

=

∫∂V0

Na(vB ⊗ N) dA−∫

V0

vn ⊗∇0Na dV

∑b

Mabρ0(v int

b − vnb

)∆t

=

∫∂V0

NatBdA +

∫V0

Naρ0bndV −∫

V0

Pn(

F st , pst)∇0NadV

• Project the velocity onto a space of divergence-free:

∑b

[1κMab +

∆t2

ρ0Kab

](pn+1

b − pnb

∆t

)dV =

∫∂V0

(HT

F n vB

)·NNa dV−

∫V0

(HT

F n vst)·∇0NadV

• Update velocity:∑b

Mab

ρ0

(vn+1

b − v intb

)∆t

=

∫V0

Na

(DIV

[(pn+1 − pn

)HT

F n

])dV



Outline

1 Motivation



4 Numerical results




1D Cable

1D mesh convergence

· Problem description: L = 10m, ρ0 = 1Kg/m3, E = 1Pa, ν = 0, αCFL = 0.5, P = 1 × 10−3EXP(−0.1(t − 13)2)N,τFp = 0.5∆t , τpF = ξF = 0

2

1

2

1

1D convergence analysis by means of the L2 norm has been carried out at t = 40s

Demonstrates the expected accuracy of the available schemes for all variables

The use of both slope limiter and lumped mass matrix maintains the expected order ofconvergence



3D L-shaped block

Angular momentum preserving example

· Problem description: L-shaped block, ρ0 = 1000Kg/m3, E = 5.005 × 104Pa, ν = 0.3, αCFL = 0.3, τFp = 0.5∆t ,

τpJ = 0.2∆t , ξJ = 0.5µκ

, τpF = τJp = τFJ = ξF = 0, lumped mass contribution

1X

2X

3X

T(3,3,3)

T(0,10,3)

T(6,0,0)

)t(1F

)t(2F

J. C. Simo, N. Tarnow, K. K. Wong. Exact energy-momentumconserving algorithms and symplectic schemes for nonlinear

dynamics, CMAME 100, 63-116 (1992)

• Imposed external forces at faces X1 = 6,X2 = 10 described as

· F1(t) = − F2(t) = η(t) (150, 300, 450)T

η(t) =

t, 0 ≤ t < 2.55 − t, 2.5 ≤ t < 50, t ≥ 5

• Free BC at all sides

• Suitable for long term dynamic response

· Angular Momentum· Total energy (summation of kinetic and

potential energies)

[MOVIE]

Study the conservation properties of the proposed formulation



3D Slender beam

Detrimental locking effects

· Problem description: Column 1 × 1 × 20, ρ0 = 1.1Mg/m3, E = 0.017GPa, ν = 0.49999, αCFL = 0.3, linear variation of

velocity field V0 = 2m/s, τFp = 0.5∆t , τpJ = 0.2∆t , ξJ = 0.5µκ

, τpF = τJp = τFJ = ξF = 0, lumped masscontribution

J. Bonet, H. Marriott, O. Hassan. An averaged nodal deformationgradient linear tetrahedral for large strain explicit dynamic

applications, COMMUN NUMER METH EN 17, 551-561 (2001)

• Imposed linear variation in velocity fielddescribed as

· v(X ) = (V0X3/L, 0, 0)T ; V0 = 2m/s

• Thin structures in bending-dominatedscenario

• Nearly incompressible material behaviourwith Poisson ratio ν = 0.49999

• Eliminate shear and volumetric lockingeffects and the appearance of pressureinstabilities

[MOVIE]

Assess the performance of the PG formulation in the case of near incompressibility



3D Twisting column

Fractional step formulation· Problem description: Column 1 × 1 × 6, ρ0 = 1.1Mg/m3, E = 0.017GPa, ν = 0.499, αCFL = 0.3, sinusoidal rotational

velocity field Ω = 100m/s, lumped mass contribution

p-F PG p-F -J PG Fractional step

Assess the performance of the fractional step method in the case of near incompressibility



3D Taylor impact bar

Classical benchmark impact problem

· Problem description: Copper bar, L0 = 3.24 cm, r0 = 0.32 cm, v0 = (0, 0,−227) m/s. von Mises hyperelastic-plastic

material with ρ0 = 8930Kg/m3, E = 117GPa, ν = 0.35, τ0y = 0.4GPa, H = 0.1GPa, αCFL = 0.3, 1361 nodes, lumped

mass contribution

0V

= 03X

0L

0r

Radius and length (in cm) at t = 80µs

Methods Radius LengthStandard 4-Node Tet. 0.555 -8-Node Hex. (P1/P0) 0.695 2.1484-Node ANP Tet. (P1/P1-projection) 0.699 -4-Node Mixed Tet. (P1/P1-stabilised) 0.700 2.156

J. Bonet, A. Burton. A simple average nodal pressure tetrahedral element forincompressible and nearly incompressible dynamic explicit applications, COMMUN

NUMER METH EN 14, 437-449 (1998)

O. C. Zienkiewicz, J. Rojek, R. L. Taylor, M Pastor. Triangles and tetrahedra in explicitdynamic codes for solids, INT J NUMER METH ENG 43, 565-583 (1998)

[MOVIE]

Assess the performance within the context of contact/impact mechanics



Outline

1 Motivation



4 Numerical results




Conclusions and further research

Conclusions• A stabilised Petrov-Galerkin formulation is presented for the numerical simulations

of fast dynamics in large deformations

• Linear tetrahedral elements can be used without usual volumetric and bendingdifficulties

• Velocities (or displacements) and stresses display the same rate of convergence

On-going works• Standard CFD techniques for discontinuity capturing operator can be

incorporated [Scovazzi et al., 2007]

• Sophisticated constitutive models (Mie-Gruneisen) can be employed [Aguirre et al.,Under review]

• Industrial applications including crash, impact analysis and explosion modelling



Publications

Journal publications

· C. H. Lee, A. J. Gil and J. Bonet. Development of a cell centred upwind finite volume algorithmfor a new conservation law formulation in structural dynamics, Computers and Structures 118(2013) 13-38.

· I. A. Karim, C. H. Lee, A. J. Gil and J. Bonet. A Two-Step Taylor Galerkin formulation for fastdynamics, Engineering Computations 31 (2014) 366-387.

· C. H. Lee, A. J. Gil and J. Bonet. Development of a stabilised Petrov-Galerkin formulation for amixed conservation law formulation in fast solid dynamics, CMAME 268 (2013) 40-64.

· M. Aguirre, A. J. Gil, J. Bonet and A. Arranz Carreño. A vertex centred Finite VolumeJameson-Schmidt-Turkel (JST) algorithm for a mixed conservation formulation in soliddynamics, JCP 259 (2014) 672-699.

· A. J. Gil, C. H. Lee, J. Bonet and M. Aguirre. A stabilised Petrov-Galerkin formulation for lineartetrahedral elements in compressible, nearly incompressible and truly incompressible fastdynamics, CMAME 276 (2014) 659-690.

Under review

· M. Aguirre, A. J. Gil, J. Bonet and C. H. Lee. An edge based vertex centred upwind finitevolume method for Lagrangian solid dynamics. JCP. Under review.

· J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. A first order hyperbolic frameworkfor large strain computational solid dynamics: Part 1 Total Lagrangian Isothermal Elasticity.CMAME. Under review.


a stabilised petrov-galerkin formulation for linear tetrahedral elements in compressible, nearly...

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