structural ritz-based simple-polynomial nonlinear

Structural Ritz-Based Simple-Polynomial Nonlinear

Equivalent Plate Approach - An Assessment

Luciano Demasi∗ and Eli Livne††

University of Washington, Seattle, Washington 98195-2400

The linear equivalent plate approach developed and used for the aeroelastic optimizationof composite wings at the conceptual design stage is a Ritz approach based on simple poly-nomial functions as generalized coordinates. Generalization to the case of geometricallynonlinear structural behavior is presented here in an effort to assess accuracy and perfor-mance of the method in the nonlinear static and dynamic cases. Using the von-Karmanplate formulation for moderately large deformations, 3-dimensional assemblies of thin-platesegments are modeled using a penalty function approach to impose boundary conditionsand compatibility of motion between adjacent segments. Closed form expressions for massand stiffness terms (linear and nonlinear) make numerical integration unnecessary. Nu-merical results obtained by the present method for a variety of plate structures, in bothstatic and dynamic cases, show good correlation with published results and results byother computer codes. The present formulation is also compared with large-deformationUpdated Lagrangian formulation results. Limits of applicability, in terms of the range ofdeformations still captured accurately by the equivalent plate method, are studied in alltest cases.

Nomenclature

t Timex, y, z Coordinate system (local for the wing segments, otherwise global)u, v, w Displacements in x, y and z direction respectivelya Reference length (different for all cases)q0 Pressure amplitudeu, v, w Non-dimensional displacementsP Non-dimensional external pressureh Thickness of the plateρ Material densityE Elastic modulusυ Poisson’s ratioLe External workU Potential energyδ Variational operatorσI , σII Maximum and minimum principal stressesεxx, γxy, εyy In-plane strainsγzx, γzy, εzz Out-of-plane strains

∗Postdoctoral Research Associate, Department of Aeronautics and Astronautics, BOX 352400. Member AIAA.†Professor, Department of Aeronautics and Astronautics, BOX 352400. Associate Fellow AIAA.

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American Institute of Aeronautics and Astronautics

46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference <br>1318 - 21 April 2005, Austin, Texas

AIAA 2005-2093

Copyright © 2005 by L. Demasi and E. Livne. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

DemasiL

Underline

http://attila.sdsu.edu/~demasi/default.htm

DemasiL

Sticky Note

An improved version of this work has been published as "Structural Ritz-Based Simple-Polynomial Nonlinear Equivalent Approach - An Assessment, Journal of Aircraft Vol. 43, No. 6, 2006, pp. 1685-1697"

σxx, τxy, σyy In-plane stressesV VolumeS Surfaceξ, η Natural coordinatesω Angular frequencykx, ky, kz Stiffness of the springs used to impose the boundary conditionsNu, Nv, Nw Number of Ritz functions used in the in-plane expansion of the displacementsJ Jacobian MatrixT Traction force per unit areau, q Physical and generalized acceleration vectorsQ Elastic matrixA, D Plate stiffness matricesσ Stress vectorε Strain vectorεl0 , εl1 Vectors of the linear strainsεnl Vector of the nonlinear strainsF u, F v, F w Vectors containing the Ritz functions used for the in-plane expansion of the displacementsq Generalized displacementsqw Generalized displacements (only the part related to the out-of-plane displacement)Kl0l0 ,Kl1l1 Linear stiffness matricesKl0nl, Knl l0 Nonlinear stiffness matrices with linear dependence on qw

Knl nl Nonlinear stiffness matrix with quadratic dependence on qw

M1, M2 Contributes to the consistent mass matrix

Subscript

0 Referred to the middle plane of the platetop Referred to the top surface of the platebot Referred to the bottom surface of the platemax Referred to the maximum valueu Related to the displacement uv Related to the displacement vw Related to the displacement w,x First derivative with respect to x,y First derivative with respect to y,xx Second derivative with respect to x,yy Second derivative with respect to y,xy Second derivative with respect to x and y

Superscript

T Transposeu Related to the displacement uv Related to the displacement vw Related to the displacement w

Acronym

RVKPF Ritz Von Karman Plate FormulationFSDPT First order Shear Deformation Plate TheoryCPT Classical Plate TheoryFE Finite ElementJW Joined Wing

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I. Introduction

In the case of linear structural / aeroelastic analysis equivalent plate models based on Ritz Series solutionsusing simple polynomials were found to be fast and adequately accurate for design optimization of aircraft

in the early design stages. Among the advantages such models1−6 offer:

a) Ease of model generation, modeling wings of general planforms as assemblages of large p-type trapezoidalstructural elements;

b) Analytical closed-form expressions for all mass and stiffness matrix elements, making numerical integra-tion unnecessary;

c) Continuous, closed form, dependency of deformations (and stresses) over large wing segments, simplifyingthe structural mesh to aerodynamic mesh interpolation problem considerably;

d) Low order system’s equations, leading to low order matrices.

The introduction of structural nonlinearity makes the generation of structural models and structural analy-sis simulations more complex and computationally intensive.7 It was only natural to hope that extension of

the equivalent plate methodology to the case of nonlinear structural dynamics will yield numerically-efficientadequately-accurate structural simulation capability for conceptual-stage aeroelastic design optimization ofstructurally nonlinear lifting surface configurations.The goal of the proposed paper is to describe the development of such nonlinear equivalent plate capabilityand the assessment of its accuracy and numerical performance.Two variants of the structural plate approach will be described: a MATLAB-based version7 based on theseries summation approach of the classical linear equivalent plate technique, and a FORTRAN-based versionbased on closed-form formulas obtained symbolically.The paper will present the mathematical theory behind both equivalent plate approaches and discuss numer-ical implementation issues. It will then continue to assess accuracy and computational performance of thevarious methods using some simple plate models as well as more challenging plate assemblies and prototypejoined-wing configurations. Both linear and nonlinear results will be presented and compared to linear andfully-nonlinear NASTRAN results to highlight differences between performance of the methods in the linearand nonlinear cases. The paper will conclude with lessons to date regarding the usefulness of equivalentplate wing models in the structurally-nonlinear aeroelasticity case.

II. Description of the Structural Model

The mathematical model is created by assembling nonlinear plate elements (denoted as wing segments).Boundary conditions and compatibility constraints used to connect different segments to one another areimposed by a penalty function method.8 The physical meaning of the weights used for the penalty terms isthat of physical springs. This discretization is shown in Fig. 1. Wing segments are modeled as solid thinplates, with planar cross sections remaining planar and staying perpendicular to the plate’s reference surface.Equivalent plate modeling of real flight vehicle wings must include transverse shear effects.2 However,based on past experience,6 plate models based on Classical Plate Theory (CPT) were used in the presentinvestigation to focus on the comparison between linear and nonlinear structural behavior. They were foundto portray the same behavior in terms of accuracy and convergence with simple-polynomial Ritz functionas plate models based on First order Shear Deformation Plate Theory (FSDPT). Since correlation withFinite Element codes is used to assess the accuracy of the polynomial-based Ritz method, this removesone potential source of difference between different capabilities (that is, the way transverse shear effects

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Figure 1. Joined wing discretization using trapezoidal wing segments.

are modeled). In addition, CPT-based wing and joined-wing modeling is expected to be adequate for themodeling of simple thin-plate aeroelastic wind tunnel models used for experimental studies of fundamentalaeroelastic mechanisms of wings and joined-wings.Strain-displacement assumptions of the von-Karman plate theory for moderately-large displacements areexpressed in:

εxx = u,x + 12w2

,x ,

εyy = v,y + 12w2

,y ,

γxy = 12

(u,y + v,x

)+ 1

2w,xw,y .

(1)

In accordance with Classical Plate Theory, it is also assumed that each layer (lamina) of the plate is in planestrain:

γzx = γzy = εzz = 0. (2)

For the displacement field, CPT (Classical Plate Theory) is based on:

u (x, y, z, t) = u0 (x, y, t)− zw0,x(x, y, t) ,

v (x, y, z, t) = v0 (x, y, t)− zw0,y(x, y, t) ,

w (x, y, z, t) = w0 (x, y, t) .

(3)

Multiplication of the equations of equilibrium by virtual displacements, and integration over the plate’svolume lead to the virtual work principle in the form

∫

V

σijδεij dv +∫

V

ρuiδui dv =∫

S

Tiδui ds, (4)

where Ti is the traction force per unit of area. For the layer-by-layer plane stress case, the last equation canbe written as ∫

V

(σxxδεxx + σyyδεyy + τxyδγxy) dv +∫V

ρ (uδu + vδv + wδw) dv =

=∫S

(Txδu + Tyδv + Tzδw) dS.(5)

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Using a general linear material law for plane-stressa, the previous equation becomes:∫

V

δεT Q εdv +∫

V

ρδuT udv =∫

S

δuT T ds. (6)

The strains can be written as (Eqs. 1 and 3)

ε = εl0 − zεl1 + εnl, (7)

where:

εl0 =

u0,x

v0,y

u0,y + v0,x

, εl1 =

w0,xx

w0,yy

2w0,xy

, εnl =

12w2

0,x

12w2

0,y

w0,xw0,y

. (8)

Carrying out the volume integration, first with respect to z and then with respect to x and y and using thematrices:

A =∫z

Qdz,

D =∫z

z2 Qdz,(9)

one can write, in the case of symmetric layupsb,∫

V

δεT Q εdv =∫

x,y

δεTl0Aεl0dxdy +

∫

x,y

δεTl0Aεnldxdy +

∫

x,y

δεTl1Dεl1dxdy +

∫

x,y

δεTnlAεl0dxdy +

∫

x,y

δεTnlAεnldxdy. (10)

A. Ritz Discretization

Solutions for the displacement field are sought by using a set of Ritz functions as generalized coordinates:

u0 (x, y, t) = Fu1 (x, y) qu1 (t) + Fu

2 (x, y) qu2 (t) + ... + FuNu

(x, y) quNu(t) ,

v0 (x, y, t) = F v1 (x, y) qv1 (t) + F v

2 (x, y) qv2 (t) + ... + F vNv

(x, y) qvNv(t) ,

w0 (x, y, t) = Fw1 (x, y) qw1 (t) + Fw

2 (x, y) qw2 (t) + ... + FwNw

(x, y) qwNw(t) .

(11)

Here1−7 polynomials of the type xsyr as Ritz functions to create complete-polynomials are used.It is well known that simple polynomials lead to ill-conditioning9−11 when used as Ritz functions, as terms

a The Hooke law is:� = Q"

where:� = [σxx σyy τxy]T , " = [εxx εyy γxy ]T ,

Q =

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

.

Notice that Q is layer-dependent.bThe reference plane x, y is the middle plane of the plate.

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with high powers r and s are present alongside low-order terms. However, simple polynomials lead to agreatly simplified problem formulation, and if a configuration is divided into relatively small segments wherelow-order polynomials are used, ill-conditioning is not a problem.Using a more compact notation, the displacement field (Eq. 11) is expressed as:

u0

v0

w0

=

F u T 0v T 0w T

0u T F v T 0w T

0u T 0v T F w T

·

qu

qv

qw

(12)

Substitution into Eqs. 8 and 10 now leads to:∫

V

δεT Q ε dv = δqT Kl0l0q + δqT Kl0nlq + δqT Kl1l1q +

+ δqT Knl l0q + δqT Knl nlq (13)

where Kl0l0 and Kl1l1 are the linear stiffness matrices and Kl0nl, Knl l0 and Knl nl are the nonlinearstiffness matrices. Because of the simple-polynomial nature of the Ritz function used, all volume integralsover trapezoidal plate segments can be carried out analytically without any need for numerical integration.For a symmetric lay-up and uniform density, the inertial terms are:c

∫

V

ρδuT u dv = δqT M1q + δqT M2q. (14)

The virtual work equation (Eqs. 10 and 13) now takes the form:

δqT [Kl0 l0 + Kl0 nl + Kl1 l1 + Knl l0 + Knl nl] q + δqT[M1 + M2

]q = δLe, (15)

where δLe is the external virtual work.

B. Imposition of the Boundary Conditions

Stiffness and mass matrices are calculated for each wing segment separately in local coordinate axes for thesegment. In assembling a configuration from segments, boundary conditions have to be imposed as well ascompatibility conditions between segments.

• Displacements and rotations between two adjacent wing segments must be the samed.c It can be shown that

M1 = ρ h

∫x,y

F uF u T d xd y 0u0v T 0u0w T

0v0u T∫

x,y

F vF v T d xd y 0v0w T

0w0u T 0w0v T∫

x,y

FwFw T d xd y

M2 =ρh3

12

0u0u T 0u0v T 0u0w T

0v0u T 0v0v T 0v0w T

0w0u T 0w0v T∫

x,y

Fw,xF

w T,x d xd y +

∫x,y

Fw,yF

w T,y d xd y

d There is an important exception. In some joined wing study, it could be interesting, for example, to impose only the samedisplacements in the joint and compare the results with a case where the rotations are also imposed.

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• Displacement and/or rotations have to be zero at points or along lines attached to the ground.

One of the essential features of the finite element method is that generalized displacements are actual physicaldisplacements and rotations of element nodes. By adding to the proper locations on the global stiffness andmass matrices stiffness and mass contributions of elements that share the same nodes, compatibility ofmotion is guaranteed at those nodes. But compatibility of motion is not guaranteed along common edges ofelements, and what convergence characteristics ”compatible” or ”incompatible” various plate elements haveis one of the most challenging problems of plate and shell finite elements.When generalized coordinates in the form of global Ritz functions over elements are used, constraint equationsor constraint terms must be used to impose compatibility of displacement and rotation between elements.A penalty function technique8 is used here to impose compatibility along sides of adjacent plate elements.Any order of Ritz functions can be used over a plate segment, while element-to-element attachments areenforced by lines of stiff springs.To do that it is sufficient to place a sufficient number of springs along the edge between two adjacent wingsegments. In order to impose compatibility of the rotations between two wing segments, if only translationalsprings are used, it is sufficient to place translational springs in different positions across the thickness.Consider a simple case where two points, 1A on wing segment A and 1B on wing segment B, have to belinked by springs (Fig. 2). The procedure is based on these steps.

Figure 2. Imposition of boundary conditions by springs.

1. The displacement field at point 1A in the local coordinate system on the wing segment A is considered.

2. The global coordinates of the same point are determined by multiplying by a geometric transformationmatrix from local A axes to global axes.

3. The same procedure is applied to point 1B on the wing segment B.

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4. The potential energy of springs linking the x− y − z motions of points 1A and 1B is written as:

U =12

u1A

v1A

w1A

g

−

u1B

v1B

w1B

g

T

k1x 0 0

0 k1y 0

0 0 k1z

u1A

v1A

w1A

g

−

u1B

v1B

w1B

g

. (16)

The subscripts g in the previous equation indicate that the displacements have to be referred to theglobal coordinate system.

5. Substitution of the Ritz function series approximations for the displacements of 1A and 1B in axes Aand B, respectively (Eqs. 11, 12), and using the subscripts A and B for the generalized coordinateson the wing segments A and B, the potential energy can be rewritten as:

U =12

[qT

A

(Ks1

AA

)qA + qT

A

(Ks1

AB

)qB + qT

B

(Ks1

BA

)qA + qT

B

(Ks1

BB

)qB

]. (17)

The quantities Ks1AA, Ks1

AB , Ks1BA, Ks1

BB are the stiffness matrices that have to be added to the totalstiffness matrix in locations corresponding to generalized coordinates qA and qB (Fig. 3).

Figure 3. Example of imposition of boundary conditions by springs.

C. Analytical Integration

An advantage of the usage of the Ritz functions in the form xsyr is that it is possible to use analyticalintegration to calculate the integrals over a trapezoidal domain (the middle surface of the wing segment).Consider a wing segment in which two edges are parallel to the local x axise, as shown in Fig. 4. Thefront and rear (aft) lines depend on the coordinates of the vertices of the trapezoid and are defined by theequations:

xF (y) = F1y + F2; xA (y) = A1y + A2 (18)

where:F1 = xF R−xF L

yR−yL; F2 = xF LyR−xF RyL

yR−yL;

A1 = xAR−xAL

yR−yL; A2 = xALyR−xARyL

yR−yL.

(19)

eIt is always possible to divide the wing into segments with this characteristic without losing generality.

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Figure 4. Analytical integration.

The integral of any simple-polynomial term with powers r and s over the wing is of the type:

ITR (r, s) =y=yR∫y=yL

yrx=xA(y)∫x=xF (y)

xsdxd y = 1s+1

y=yR∫y=yL

yr[xA (y)s+1 − xF (y)s+1

]dy =

= 1s+1

y=yR∫y=yL

yr (A1y + A2)s+1 dy − 1

s+1

y=yR∫y=yL

yr (F1y + F2)s+1 dy =

= 1S

y=yR∫y=yL

yR (A1y + A2)S dy − 1

S

y=yR∫y=yL

yR (F1y + F2)S dy.

(20)

The last two integrals can be calculated by using recursive formulas:

∫yR (A1y + A2)

S dy =yR+1 (A1y + A2)

S

R + S + 1+

SA2

R + S + 1

∫yR (A1y + A2)

S−1 dy, (21)

∫yR (F1y + F2)

S dy =yR+1 (F1y + F2)

S

R + S + 1+

SF2

R + S + 1

∫yR (F1y + F2)

S−1 dy. (22)

A table of integrals of simple-polynomial functions of increasing order is prepared once for each trapezoidalsegment at the start of a simulation run up to the highest order needed. Integrals of xs yr over the trapezoidare then taken from such tables in the stiffness and mass matrix assembly process. Different runs usingdifferent order polynomials for the displacement fields can then be carried out using the same table ofintegrals prepared initially. This approach has been used in the MATLAB-based version of the presentstructural plate model. However, it is possible to fix the Ritz functions that are used and, therefore, allintegrals can be calculated once and a priori using a symbolic commercial code such as MATHEMATICA.Using this procedure, all mass, linear and nonlinear stiffness matrices can be calculated in closed-formreducing the number of operations that have to be performed at each iteration in the nonlinear solutionprocess. MATHEMATICA-derived closed-form matrices have been used in the FORTRAN-based version of

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the code. Note that while terms of the linear stiffness matrix are independent of generalized displacements,q, the nonlinear parts of the stiffness matrix do depend on qf, and closed-form expressions for these termscontain dependency on the generalized displacements. These generalized displacements vary under load.

D. Solution of the Nonlinear system

Several methods are presented in literature for the integration and solution of fully nonlinear structuralanalysis problems.12−19 In the formulation used here for the moderately-small deformations of plates, theNewton Raphson method and the direct method have been adopted. In the Newton Raphson method, onlythe tangent matrix has been used, while in the direct method only the secant matrix20 has been used. Whilethe linear part of the stiffness matrix is calculated once, the nonlinear parts, which depend on the motionq, have to be repetitively calculated during simulation (in the FORTRAN-based version of the code thematrices are obtained in closed-form using MATHEMATICA).

III. The Selected Test Problems

A number of test cases were selected for evaluation of the nonlinear structural plate analysis describedhere. Results of the present capability were compared with reported results from the literature and withMSC-NASTRAN solutions. In all MSC-NASTRAN cases CQUAD4 plate elements were used. The followingcases were analyzed (see Figs. 5, 6 and 7):

Case 1: Square Plate Simply-Supported along its Perimeter

Case 2: Square Cantilevered Wing

Case 3: Swept Cantilevered Wing

Case 4: Joined Wing (First Model)

Case 5: Joined Wing (Second Model)

Case 6: Joined Wing (Third Model)

In all cases, the load was represented by an uniform pressure q0 with direction +z.

Figure 5. Case 1,2 and 3: geometry and notations.

fMore precisely, the nonlinear matrices depend on qw only. If the tangent matrix is calculated, it depends also on the othercomponents of the vector q

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IV. Results

Results of displacement predictions, natural frequencies (for the linear case), and stresses obtained bythe Ritz-based plate approach (indicated with RVKPF) by finite elements (NASTRAN) are discussed in thefollowing paragraphs.

A. Case 1: Square Plate Simply-Supported along its Perimeter

The following NASTRAN meshes have been used:

MESH1 CQUAD4 4× 4;

MESH2 CQUAD4 10× 10;

MESH3 CQUAD4 20× 20;

MESH4 CTRIA3 4× 4;

MESH5 CTRIA3 10× 10;

MESH6 CTRIA3 20× 20.

Frequency parameter ω1

√a4ρEh2 ω2

√a4ρEh2 ω3

√a4ρEh2 ω4

√a4ρEh2

NASTRAN (MESH 1) 5.610 13.733 13.733 19.924NASTRAN (MESH 2) 5.887 14.645 14.645 22.841NASTRAN (MESH 3) 5.937 14.824 14.824 23.508NASTRAN (MESH 4) 5.910 14.554 14.554 20.778NASTRAN (MESH 5) 5.938 14.902 14.902 23.779NASTRAN (MESH 6) 5.951 14.900 14.900 23.800RVKPF (Order 3) 4061.570 4078.181 4078.181 5716.770RVKPF (Order 4) 6.347 461.542 461.542 5572.076RVKPF (Order 5) 6.347 17.849 17.849 461.542RVKPF (Order 6) 5.975 17.849 17.849 28.004RVKPF (Order 7) 5.975 14.980 14.982 28.005RVKPF (Order 8) 5.973 14.981 14.981 24.026RVKPF (Order 9) 5.973 14.927 14.931 24.014

Table 1. Case 1. Frequency parameter. Comparison RVKPF vs NASTRAN.

Natural frequencies for the simply-supported square plate are listed in Table 1, and show less than 1% errorin the predicted 4th frequency with 8th order Ritz polynomials compared to the converged NASTRAN result.Normalized vertical displacement at the center of the plate is shown versus the normalized load in Fig. 8, andthe correlation between the RVKPF approach and the Finite Element (FE) approach is excellent throughoutthe range of loading considered. Comparison with other results available in the literature21−23 is also made.

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Figure 8. Case 1. Non-dimensional maximum deflection.

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B. Case 2: Square Cantilevered Wing

The same NASTRAN meshes as in the case 1 have been used. Normalized natural frequencies of thecantilevered square plate are shown in Table 2. With 7th-order Ritz polynomials the RVKPF approachleads to frequency values for the 4th natural frequency that are about 1.5% different than correspondingFE frequencies. In-plane (Fig. 9) and out-of-plane (Fig. 10) displacements and principal stresses (Figs. 11and 12) at selected points on the plate show good correlation between Ritz and FE predictions. Note thesmaller loading range covered in this case compared to the simply-supported case, the larger out-of-planedisplacement (in terms of plate thickness h), and the non-linear behavior of the in-plane displacement whichstarts at relatively low loading values.


√a4ρEh2 ω2

√a4ρEh2 ω3

√a4ρEh2 ω4

√a4ρEh2



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Figure 9. Case 2. Non-dimensional displacement u.

Figure 10. Case 2. Non-dimensional displacement w.

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Figure 11. Case 2. Non-dimensional maximum principal stress.

Figure 12. Case 2. Non-dimensional minimum principal stress.

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C. Case 3: Swept Cantilevered Wing

The meshes are the same as the mesh used in cases 1 and 2. Normalized natural frequencies for theswept cantilevered plate are listed in Table 3, with about 2% difference between 7th order Ritz-plate andFE prediction for the 4th natural frequency. Excellent correlation between RVKPF-based on 7th orderpolynomials and the FE result is evident in Fig. 13 for the out-of-plane displacement. In-plane displacementsobtained by the Ritz-based plate method are less accurate compared to the FE results, but they capture thenonlinear behavior very well (Figs. 14 and 15). As to principal stresses on the top surface of the plate (Figs.16 and 17) the smaller principal stress is captured well. In the case of the larger principal stress (Fig. 16),large errors begin to accumulate even at relatively low loading and reach about 20% at the highest loadingtested. Note that stresses in Figs. 16 and 17 are calculated at a point not too far away from the rear pointon the root of the wing - a region known for its steep stress gradients in the case of swept-back wings. Inthis area FE stress results become quite sensitive to the FE mesh used.


√a4ρEh2 ω2

√a4ρEh2 ω3

√a4ρEh2 ω4

√a4ρEh2



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Figure 15. Case 3. Non-dimensional displacement v.


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D. Case 4: Joined Wing (First Model)

The following meshes have been used:

MESH 7 CQUAD4 10× 10 for each wing;

MESH 8 CQUAD4 20× 20 for each wing.

Table 4 shows normalized natural frequencies for the simple Joined-Wing (JW) model of test case 4. Adifference of about 5% is obtained between the Ritz-based plate approach and the NASTRAN FE resultwith just two large plate segments (an upper one and a lower one) with 8th order polynomial displacementson each. The Joined-Wing case is challenging because of its 3D geometry, and the fact that under upwardspressure loads the upper wing is under compression. Indeed, it has been found that attention had to bepaid to the FE mesh in this case to obtain converged results (Figs. 18-21). Displacement (Figs. 22-25)and stress accuracy (Figs. 26-29) of the nonlinear Ritz-based plate method was excellent up to normalizedloads of about 30, and was quite good up to the full loads tested. The Ritz plate results in Figs. 26-29 wereobtained using 8 large plate segments (4 on the top and 4 on the bottom plates) with 8th order polynomialRitz functions on each segment.


√a4ρEh2 ω2

√a4ρEh2 ω3

√a4ρEh2 ω4

√a4ρEh2

NASTRAN (MESH 7) 4.466 5.947 6.584 7.727NASTRAN (MESH 8) 4.463 6.011 6.579 7.814RVKPF (Order 8) 4.481 6.088 6.846 8.228RVKPF (Order 9) 4.479 6.085 6.854 8.175


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Figure 18. Case 4. Non-dimensional displacement w. Convergence test (NASTRAN)

Figure 19. Case 4. Non-dimensional displacement u. Convergence test (NASTRAN)

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Figure 20. Case 4. Non-dimensional displacement w. Convergence test (NASTRAN)

Figure 21. Case 4. Non-dimensional displacement u. Convergence test (NASTRAN)

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Figure 22. Case 4. Non-dimensional displacement w. RVKPF vs NASTRAN.

Figure 23. Case 4. Non-dimensional displacement u. RVKPF vs NASTRAN.

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Figure 24. Case 4. Non-dimensional displacement w. RVKPF vs NASTRAN.

Figure 25. Case 4. Non-dimensional displacement u. RVKPF vs NASTRAN.

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E. Case 5: Joined Wing (Second Model)

The following mesh has been used in the NASTRAN computations:


When the JW configuration is swept so that the upper wing is swept forward and the lower wing is sweptback the case becomes even more challenging. Now steep stress gradients appear near the rear corner rootpoint of the lower wing and the front corner root point of the upper wing. The simple polynomial Ritzfunctions have to capture displacement behavior over distorted wing segments in space. As Figs. 30-35show, displacement are captured well by the Ritz-based plate method up to a normalized load of about25. Out-of-plane displacements are predicted well beyond that load, but in-plane displacements become lessaccurate (Fig. 34) as the load increases (though their trends are captured well). Wing twist results areshown in Figs. 36-38, while principal stresses are compared in Figs. 39-40. Note the excellent correlation ofprincipal stresses along leading edge of upper wing. As the root is approached, the well known stress rise isevident.


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Figure 36. Case 5. Wing twist.

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F. Case 6: Joined Wing (Third Model)

The following mesh has been used in the NASTRAN model:


Finally, the 3rd Joined-Wing model, made of two higher aspect ration segments compared to the JW of theprevious case, is used to assess the accuracy of the Ritz-based plate method. Results are shown in Figs. 41-48. The RVKPF approach captures displacements, angle of attack (twist), and stresses well. Deteriorationof accuracy of the Ritz-plate stress prediction is observed in Figs. 47-48 above a normalized load of 1.In all nonlinear cases studied computational efficiency of the NASTRAN FE solution was at least an orderof magnitude higher than that of any of the Ritz-plate approaches testedg. While the Ritz-based plateapproach leads to a smaller number of degrees of freedom in the resulting model, the sparseness of theFE matrices allows solutions of the matrix equations to be fast and efficient. This becomes even morepronounced in the case of nonlinear structural analysis, where nonlinear stiffness equations have to be solvedrepetitively. The calculation of nonlinear stiffness terms with the Ritz-plate approach is done in closed form,but it involves time consuming high order do-loops, and that, when high-order Ritz polynomials are used,contributes significantly to the increase in computing time, compared to the FE method, which utilizes lowerorder polynomials.

Figure 41. Case 6. Non-dimensional tip displacement u/h.

gThe MATLAB-based version of the code was compiled using the internal MATLAB C/C++ compiler MCC.

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Figure 42. Case 6. Non-dimensional tip displacement v/h.

Figure 43. Case 6. Non-dimensional displacement w/h.

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V. Conclusions

Considering the results obtained here, the following conclusions can be made.

• The Von Karman Ritz Plate Formulation is very effective when the geometry of the structure isrelatively simple (square, rectangular, swept but not by too much) and the resulting motion is not toolarge relative to the thickness of the plates. In such conditions, a single wing segment can describe wellthe displacement field and stresses at all points of the plate. Linear, nonlinear, static and dynamiccases can be effectively studied.

• When the geometry/load configuration is complex (3D, swept and distorted wing segments, in-planecompression or tension in segments), a single wing segment is not enough in order to obtain a goodaccuracy. Expecting displacement gradients in areas of abrupt geometric changes, or segment-to-segment joints more wing segments have to be used and the computation can be considerably slowcompared with NASTRAN-FE capability. Because of this slow computation and the large number ofplates needed, the FE approach is recommended for the nonlinear structural and aeroelastic studiesof Joined-Wing configurations. The authors studied joined wing24 using an Updated Lagrangian For-mulation25−26) and they found a better correlation with NASTRAN and much better computationalspeed compared to their Ritz-plate capabilities described here.

• One of the questions to be addressed when extending the Ritz-based plate approach to the nonlinearcase was sensitivity of boundary conditions constraints to the spring values used.7−8 It has beendemonstrated7 that the convergence is excellent when the spring stiffness is of the order of the YoungModulus E (for isotropic materials, but similar conclusions can be found for an orthotropic material).Moreover, if the spring stiffness is changed, no significant change in the results has been found.

• The compatibility of the rotations (between wing segments or the boundary) can be imposed usingtranslational springs disposed across the thickness. No significant difference has been found by changingthe distance between the springs.

• Once the convergence has been achieved, the displacements and their spatial derivatives are wellapproximated. Therefore, the stresses can be calculated without problems even where the springsare located.

• If the aspect ratio of the wing segments is quite small (for example, if the ratio is 0.2), an ill conditioningappears at low order of the polynomials used in the Ritz functions. This problem can partially beavoided by using more wing segments. However, the best approach would be the usage of orthogonalpolynomials9−11 (e.g., Legendre polynomials), but this last method has not been used here.

• It can be concluded that despite the attractiveness of the simple-polynomial Ritz approach and itssimple (theoretically if only a few large wing segments are used) structural-aerodynamic mesh inter-action, the Ritz-based plate approach is not recommended for the study of non-planar wings (such asJoined Wings). Recent results27 for the case of nonlinear aeroelasticity of a delta wing show bettercorrelation with experimental results when general nonlinear finite elements were used compared to aplate approach based on von-Karman’s assumptions and theory. When nonlinear effects become im-portant on cantilevered and 3D wing assemblies, out-of-plane displacements can be larger than a fewtimes the thickness of the plates. It is the panel flutter case, where panel boundaries are constrainedand out-of-plane displacements are small, that nonlinearity begins at relatively low loads and can bewell captured by the Ritz-based plate theory.28−29

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Appendix A - Details About the Nonlinear Matrices

In this appendix one term of the Knl nl matrix is expressly shown. That matrix can be written as

Knl nl = K1nl nl + K2

nl nl + K3nl nl + K4



nl nl + K9nl nl. (23)

The term K9nl nl is:

K9nl nl =

A66

2

0u0u T 0u0v T 0u0w T


0w0u T 0w0v T∫

x,y

[F w

,yF w T,x + F w

,xF w T,y

]qwqT

w

[F w

,xF w T,y + F w

,yF w T,x

]dxd y

. (24)

Appendix B - Transformation of the Integrals to Performtheir Analytical Computation

The integrals that compare in the nonlinear stiffness matrices have to be transformed in order to performthe analytical calculation. This operation is required if the table of integrals calculated once is used. Toillustrate the method, consider one of the integrals in the first term K1

nl nl:∫

x,y

F w,xF w T

,x qwqTwF w

,xF w T,x dxdy. (25)

It is easy to see that the termsF w T

,x qw,

qTwF w

,x ,(26)

are scalar quantities. For this reason, they can be written as:

F w T,x qw = qT

wF w,x ,

qTwF w,x = F w T

,x qw.(27)

By using these expressions, the integrals become:

∫x,y

F w,xF w T

,x qwqTwF w

,xF w T,x dx dy =

∫x,y

F w,x

[F w T

,x qw

] [qT

wF w,x

]F w T

,x dxdy =∫

x,y

[F w T

,x qw

]F w

,xF w T,x

[qT

wF w,x

]dxdy =

=∫

x,y

[qT

wF w,x

]F w

,xF w T,x

[F w T

,x qw

]dxdy =

∫x,y

[qw kwFw

,x kw

]Fw

,x iwFw

,x jw

[Fw

,x lwqw lw

]dxdy =

=∫

x,y

qw kwFw

,x kwFw

,x iwFw

,x jwFw

,x lwqw lwdx dy = qw kw

qw lw

∫x,y

Fw,x kw

Fw,x iw

Fw,x jw

Fw,x lw

dxdy kw, lw, iw, jw = 1, ..., Nw.

(28)This expression is useful because in the integrals only Ritz functions of the type xsyr appear and it is possibleto use the analytical integration as seen above.

Appendix C - Numerical Integration

Even though area integrals in the Ritz-based simple-polynomial method can be obtained analytically, itwas found that numerical integration led to much shorter computing times. The Gauss rule to integrate the

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Figure 49. Numerical integration: transformation of coordinate system

functions over the wing area can be used. Therefore, a transformation (between physical coordinates andnatural coordinates) is required. The equations of lines 1 and 2 (see figure 49) are:

y−yL

yR−yL= x−xF L

xF R−xF L

y−yL

yR−yL= x−xAL

xAR−xAL

(29)

It can be observed that the lines 3 and 4 are parallel to the x axis, so for this particular case, y is a functiononly of η:

η + 1y − yL

=2

yR − yL⇒ y =

η + 12

(yR − yL) + yL = y(η) (30)

The lines 1 and 2 are not parallel to the y axis, therefore, a nonlinear relation between x and ξ, η has to befound. Using Figure 49:

ξ + 1xP − xP ′

=2

xP ′′ − xP ′(31)

And from equation 29:xP ′ = y−yL

yR−yL(xFR − xFL) + xFL

xP ′′ = y−yL

yR−yL(xAR − xAL) + xAL

(32)

substituting these last relations into equation 31 and using equation 30:

x = 14

[ξ [(xAL − xFL) + (xAR − xFR)] + η [(xAR − xFL)− (xAL − xFR)]+

+ ξη [(xAR − xFR)− (xAL − xFL)] + xFL + xFR + xAR + xAL

]= x (ξ, η)

(33)

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This expression and equation 30 can be simplified by introducing the quantities:

X1 = 14 [(xAL − xFL) + (xAR − xFR)]

X2 = 14 [(xAR − xFL)− (xAL − xFR)]

X3 = 14 [(xAR − xFR)− (xAL − xFL)]

X4 = 14 [xFL + xFR + xAR + xAL]

Y1 = yR−yL

2

Y2 = yR+yL

2

(34)

Finally, the relations are:y = Y1η + Y2

x = X1ξ + X2η + X3ξη + X4

(35)

The inverse of the previous expression is:

ξ =x−X2

(y−Y2

Y1

)−X4

X1+X3

(y−Y2

Y1

)

η = y−Y2Y1

(36)

To use the Gauss rule, the Jacobian matrix of the geometrical transformation shown in figure 49 is needed:

J =

∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

(37)

And using equation 35, the Jacobian matrix becomes:

J =

X1 + X3η 0

X2 + X3ξ Y1

(38)

The inverse of Jacobian matrix can easily be written as:

J−1 =

1X1+X3η 0

− X2+X3ξ(X1+X3η)Y1

1Y1

(39)

Recalling the definition of Jacobian, the relation with the derivative operators is:

∂∂ξ

∂∂η

=

∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

·

∂∂x

∂∂y

= J

∂∂x

∂∂y

(40)

In the considered case, the transformation is always possible. Because of that, the inverse of the previousrelation can be written as:

∂∂x

∂∂y

=

∂ξ∂x

∂η∂x

∂ξ∂y

∂η∂y

·

∂∂ξ

∂∂η

= J−1

∂∂ξ

∂∂η

(41)

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The determinant of the Jacobian matrix is:

detJ = J11J22 − J12J21 =∂x

∂ξ

∂y

∂η− ∂y

∂ξ

∂x

∂η(42)

And using relation 38:detJ = (X1 + X3η)Y1 (43)

Next focus is on the application of the numerical integration. For example, consider the matrix

K2l0l0 = A12

0u0u T∫

x,y

F u,xF v T

,y d xd y 0u0w T


0w0u T 0w0v T 0w0w T

(44)

The nonzero term is:

∫

x,y

F u,xF v T

,y dxd y =∫

x,y

Fui,x

F vj,x

d xd y =∫

x,y

∂Fui

∂x

∂F vj

∂ydxd y = Iij

i = 1, ..., Nu

j = 1, ..., Nv

(45)

By using equation 41, the partial derivatives can be written as

∂F ui (x,y)∂x = J−1

11∂F u

i (ξ,η)∂ξ + J−1

12∂F u

i (ξ,η)∂η = 1

X1+X3η∂F u

i (ξ,η)∂ξ

∂F vj (x,y)

∂y = J−121

∂F uj (ξ,η)

∂ξ + J−122

∂F uj (ξ,η)

∂η = − X2+X3ξ(X1+X3η)Y1

∂F uj (ξ,η)

∂ξ + 1Y1

∂F uj (ξ,η)

∂η

(46)

With this relation, and recalling the geometrical transformation, the integral becomes:

Iij =∫

ξ,η

1X1 + X3η

∂Fui (ξ, η)∂ξ

(− X2 + X3ξ

(X1 + X3η) Y1

∂Fuj (ξ, η)∂ξ

+1Y1

∂Fuj (ξ, η)∂η

)det J (47)

Appendix D - Transformation of the Integrals to Performtheir Numerical Computation

The integrals that compare in the nonlinear stiffness matrices have to be transformed in order to performthe numerical calculation and improve the speed of the code. It is possible to avoid the four loops in thenonlinear matrices. To illustrate the method, consider one of the integrals in the first term K1

nl nl. Inappendix B it has been proven that

∫x,y

F w,xF w T

,x qwqTwF w

,xF w T,x dx dy = qw kwqw lw

∫x,y

Fw,x kw

Fw,x iw

Fw,x jw

Fw,x lw

dx dy kw, lw, iw, jw = 1, ..., Nw.

(48)which lets one to easily write the derivatives with respect to the generalized coordinates and, therefore,to write the tangent matrix. This approach has been followed by the authors, but it is a computationallyexpensive method. The reason is that a cycle with four loops has to be performed (see the integral in which4 indexes appear). It is possible to avoid this problem by using only the secant matrix20 in the nonlineariterative solution. The convergence ratio is well known to be worse than the Newton Raphson approach, butthe secant matrix can be modified in order to have a cycle with only two loops (and, thus, much more fast).This operation is performed here by using the relation:

w0,x = qTwF w

,x = F w T,x qw (49)

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Substituting inside the equation representing the integral (equation 48):∫

x,y

F w,xF w T

,x qwqTwF w

,xF w T,x dxdy = w2

0,x

∫

x,y

F w,xF w T

,x dx dy (50)

The other integrals can be treated similarly and they can be numerically calculated as explained above inAppendix C.

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structural ritz-based simple-polynomial nonlinear

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