Thompson L - A Stabilized MITC Element for Accurate Wave Response in Reissner-Mindlin Plates

8/12/2019 Thompson L - A Stabilized MITC Element for Accurate Wave Response in Reissner-Mindlin Plates

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A stabilized MITC element for accurate waveresponse in ReissnerMindlin plates q

Lonny L. Thompson * , Sri Ramkumar ThangaveluDepartment of Mechanical Engineering, Clemson University, 102 Fluor Daniel Engineering Innovation Building,

Clemson, SC 29634-0921, USA

Accepted 1 February 2002

Abstract

Residual based nite element methods are developed for accurate time-harmonic wave response of the Reissner Mindlin plate model. The methods are obtained by appending a generalized least-squares term to the mixed variationalform for the nite element approximation. Through judicious selection of the design parameters inherent in the least-squares modication, this formulation provides a consistent and general framework for enhancing the wave accuracy of mixed plate elements. In this paper, the mixed interpolation technique of the well-established MITC4 element is used todevelop a new mixed least-squares (MLS4) four-node quadrilateral plate element with improved wave accuracy.Complex wave number dispersion analysis is used to design optimal mesh parameters, which for a given wave angle,match both propagating and evanescent analytical wave numbers for ReissnerMindlin plates. Numerical resultsdemonstrates the signicantly improved accuracy of the new MLS4 plate element compared to the underlying MITC4element. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Finite element methods; ReissnerMindlin plates; Mixed interpolation; Plate elements; Shell elements

1. Introduction

When modeling the time-harmonic response of elas-tic structures, accurate plate and shell elements areneeded to resolve both propagating and evanescentwaves over a wide range of frequencies and scales. Thepropagating waves are characterized by sinusoidalcomponents with phase speed determined by the mate-rial properties and thickness of the plate, while the ev-anescent waves are characterized by exponential decaywith effects localized near drivers and discontinuities,e.g., near boundary layers. Models based on classicalKirchhoff plate theory agree with the exact theory of

elasticity only in a very limited low range of frequencies;the predicted phase speed at higher frequencies is in-nite, while the exact theory remains bounded [2]. Theinclusion of transverse shear deformation and rotaryinertia effects in the ReissnerMindlin theory accuracypredicts the bounded phase speed of the exact theoryover a large range of frequencies of typical interest [46].The accuracy improvement for intermediate to highfrequencies plays an important role in modeling control structure interactions, dynamic localizations, acousticuidstructure interaction, scattering from inhomoge-neities, and other applications requiring precise model-ing of dynamic characteristics.

The numerical solution of the ReissnerMindlin platemodel for static analysis has been discussed by manyauthors, e.g. [713]. The primary focus has been vari-ous remedies to the well-known shear locking problemfor very thin plates, [14,15]. The locking is most clearlyseen in some low order approximations where an overly

stiff response to bending is exhibited in the solution.

Computers and Structures 80 (2002) 769789www.elsevier.com/locate/compstruc

q Portions of this manuscript originally presented in [1].*

Corresponding author. Tel.: +1-864-656-5631; fax: +1-864-656-4435.

E-mail address: [email protected] (L.L.

Thompson).

0045-7949/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0045-7949(02)00046-9
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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Mathematically, the locking is the result of the lack of stability of the method. Over the last decade, signicantprogress has been made on the mathematical stabilityand error analysis for ReissnerMindlin plate elementsfor static analysis [1628]. Of the low order elements, thepopular bilinear MITC4 element [9] based on mixedinterpolation of shear strains is one of the most attrac-tive. Later, Prathap [10] rederived the same four-nodequadrilateral plate element using the concepts of eld-and edge-consistency. The error analysis [16,17] per-formed on this element showed that it is optimallyconvergent for deections and rotations on regular me-shes. However, for the four-node quadrilateral MITC4element, it is not clear what is the optimal denition of the loading and mass which is consistent with the as-sumed strain eld for dynamic analysis. While elimi-nating shear locking problems for thin plates, what isoften overlooked is the large dispersion error exhibited

in these elements leading to inaccurate resolution of propagating and evanescent wave behavior in dynamicanalysis at intermediate to high frequencies.

To address this problem, a residual-based modica-tion of assumed strain mixed methods for Reissner Mindlin plates is proposed. New plate elements aredeveloped based on a generalized least-squares modi-cation to the total energy for the time-harmonic Reiss-nerMindlin plate model. The least-squares operatorsare proportional to residuals of the governing equationsof motion, and provide a consistent framework for en-hancing the wave accuracy of ReissnerMindlin plateelements for forced vibration and time-harmonic re-sponse. Any of several existing mixed nite element in-terpolation elds which yield plate elements which arefree from shear locking and pass the static patch test maybe used. Here we start from the rm mathematicalfoundation inherent in the shear projection technique of the MITC4 element. A similar generalized least-squaresapproach was used in [29,30,32] to improved the accu-racy of quadrilateral plate elements based on assumedstress elds in a modied HellingerReissner variationalprinciple.

Weighted residuals of the governing EulerLagrangeequations in least-squares form were rst used to stabi-

lize the pathologies exhibited by the classical Galerkinmethod for the numerical solution of advectiondiffu-sion problems [33]. These so-called stabilized methodshave been successfully employed in a wide variety of applications where enhanced stability and accuracyproperties are needed, including problems governed byNavierStokes and the compressible Euler equations of uid mechanics, [34]. Generalized methods based on thegradient of the residuals in least-squares form were rstused by Franca and do Carmo [35] for the advection diffusion equation. In [36,37], Hughes et al. established arelationship between various stabilized methods and

variational multiscale methods. Residual-based methods

have since been extended to the scalar Helmholtzequation governing steady-state vibration and time-harmonic wave propagation, (e.g. acoustics), by Harariand Hughes, [38,39], and Thompson and Pinsky [40]. In[40,41], nite element dispersion analysis was used toselect mesh parameters in the least-squares modicationto the Galerkin method, resulting in improved phaseaccuracy for both two- and three-dimensional problems.In Oberai and Pinsky [42], variable mesh parameters andresiduals on inter-element boundaries are included toreduce the directional dependence of dispersion error.Other numerical methods designed to improve the ac-curacy of the scalar Helmholtz equation can be found ine.g. [43,44].

The rst use of residual based methods for staticanalysis of plate structures was the stabilized mixedformulations by Hughes and Franca [20] where sym-metric forms of the equilibrium equations were appended

to the standard Galerkin equations to improve transverseshear accuracy. In [27], the stabilized formulations of [20]are combined with the shear interpolation of the MITCplate bending element for static analysis. Grosh andPinsky applied a generalization of the Galerkin gradientleast squares (GGLS) method of Franca and do Carmo[35] to improve the accuracy of displacement basedTimoshenko beam elements for steady-state vibration[45]. An important feature of this GGLS element, is thatin the zero frequency limit, the mesh parameters modifythe shear strain approximation in the stiffness matrix,reverting to selective-reduced-integration (SRI) in thestatic case. As mentioned in [45], the extension of thisGGLS formulation for 1-D Timoshenko beams to 2-DReissnerMindlin plate elements based on bilinear dis-placement interpolation failed to produce a quadrilateralelement which is free from shear locking.

In this work, we combine the mixed interpolation of the MITC4 plate element with residual-based methodsto develop a mixed least squares (MLS) quadrilateralelement for accurate time-harmonic wave response of the ReissnerMindlin plate model. A key feature of ourmethod is that we require the mesh parameters to vanishin the static limit of zero frequency, thus retaining thelocking-free behavior of the underlying MITC quadri-

lateral element. Using complex wave number dispersionanalysis [46], we design optimal mesh parameters, which,for a given wave angle relative to a uniform nite ele-ment mesh, match both propagating and evanescentanalytic wave numbers for ReissnerMindlin plates.This strategy for designing mesh parameters is similar tothat used in the displacement based GGLS Timoshenkobeam element proposed in [45], here extended to arbi-trary quadrilateral plates. In general, the direction of wave propagation is not known a priori. However,similar to [40], we can select a wave angle in the deni-tions for the mesh parameters to minimize dispersion

error over the entire range of possible angles.

770 L.L. Thompson, S.R. Thangavelu / Computers and Structures 80 (2002) 769789


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2. ReissnerMindlin plate equations

We consider the ReissnerMindlin plate bendingmodel [46] with thickness t, two-dimensional midsur-face A R 2, boundary o A, and transverse coordinate z.The distributed load q

x; y

is restricted to the direction

normal to the midsurface dened by the unit vector e z .Without loss of generality, we assume that the plate isclamped along its boundary. The deformation at anypoint is given by the three-dimensional displacementvector dened by

u z h x; y w x; y e z ; 1where h h x; h y

T

2 H 10 A2 denotes the two-dimen-

sional vector of section rotations, such that h?e z , andw 2 H 10 A is the vertical deection of the midsurface.The components h x and h y are the section rotationsabout the y and x axes respectively. As a consequence of the kinematic assumptions, the in-plane bending straintensor e z j , is linearly related to the tensor of cur-vatures j , through the symmetric part of the rotationgradient,

j h : 12rh rhT

: 2Using rst-order shear deformation theory, the trans-verse shear strains are dened by the angle between theslope of the midsurface after deformation and the sec-tion angle, c rw h.For a homogeneous, isotropic plate with linear elasticmaterial properties, the constitutive relations for thebending moment and shear resultants are

M EI

1 m j hn m1 m div hI o; 3

Q G st rw h: 4Here, I t 3=12, with Youngs modulus E , Poissonsratio m, shear modulus G , and j is a shear correctionfactor, G s j G . In the above, div stands for diver-gence, i.e., div h h x; x h y ; y , and I is the unit tensor.We assume time-harmonic motion with time-depen-dence e ix t ; x is the circular frequency measured inrad/s. The variational problem is to minimize the totalenergy functional with respect to the generalized dis-placements v w; h. For the ReissnerMindlin model,the total energy may be expressed as F M v P M v x 2

12 Z Aq t w2 q I h2 d A Z A wq d A:

5In the above, qt is the mass density per unit area, q I is

the rotary inertia, P M is the internal strain energy split

into bending and shear parts

P Mv 12

Bh; hG st 2 Z Arw h2 d A; 6

Bh; h : EI

1 mZ A j h : j hh m1 m div h2id A:7The symmetric tensor inner product is dened by, j :

j j 2 x j 2 y 2j 2 xy .The dynamic EulerLagrange equations corre-sponding to this variational problem are:

R1 : div Q q t x 2w q 0; 8R 2 : div M Q q I x 2h 0; 9In the above, R1 is a scalar residual associated with shearequilibrium, and R 2 R2 x; R2 y

T is a vector residual as-sociated with moment equilibrium. Applying the diver-gence operator to the vector Eq. (9), i.e. div R 2, andwriting the bending and shear resultants in terms of displacements M M h and Q Q w; h, via (3) and(4), the residuals can be restated in terms of the twoscalar equations,

R1v : D sdiv c q t x 2w q 0 10 R2v : div R 2 Dbr 2 q I x 2 div h D sdiv c 0

11where Db EI =1 m2, D s G st , r 2 div r , andc r

w h.

2.1. Wave numberfrequency dispersion relation

The homogeneous plate equations of motion admitsolutions of the form

w w0eik m x; h h0m eik m x; div h ik h0eik m x 12In the above, i ffiffiffiffiffiffiffi1p , k is the wave number, m cos u ; sin u denes a unit vector in the direction of wavepropagation, with wave vector k k m k cos u ; sin u .Conditions for the allowed waves are obtained by sub-stituting the assumed exponentials (12) into the homo-

geneous equations of motion (10) and (11) with q 0.The result is the dispersion equation relating frequencyx to wave number k :

D k ; x : k 4 k 2 s k 2 p k 2 k 2 p k 2 s k 4b 0; 13k p x =c p ; k s x =c s; k b q t x 2= Db

1=4;

c p E q1 m2

1=2

; c s G s

q 1=2:Wave number solutions occur in pairs: k 1 and k 2.

The character of these solutions are well known [2,3]. At

L.L. Thompson, S.R. Thangavelu / Computers and Structures 80 (2002) 769789 771


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frequencies below a cut-off frequency, the wave numberpair k 1 occurs as purely real, while the pair k 2 ispurely imaginary. The real wave number pair corre-sponds to propagating waves while the imaginary paircorresponds to evanescent waves characterized by ex-ponential decay.

3. Generalized nite element formulation

Consider a nite element mesh obtained by parti-tioning A into convex quadrilateral elements. Let Aedenote the area for a typical element number e. We thendene M h [e Ae as the union of element interiors. Thediscrete total energy for the plate equations of motionwith assumed strain ch and trial displacements vh wh; h

h

may be stated as

F M vh

: P M vh

x2 1

2 Z A qt wh

2

h q I hh

2

id AZ A

whq d A; 14

P M vh 12

Bhh; hh

G st 2 Z Ach2 d A: 15

Remark. The variational equations associated with theabove functional with the kind of interpolations de-scribed in the following section, should also include theshear force resultants Q h as additional dependent vari-ables [19]. However, imposing an orthogonality condi-

tion [19], the shear force is eliminated from the nalform.

To develop a residual-based formulation with en-hanced wave number accuracy, we start with the totalenergy functional and then add weighted differentialleast-squares operators proportional to the governingdynamic equations of motion. Our modied functionalcan be written as

F MLS vh F M vh F LS vh 16with generalized least-squares term,

F LS vh

12 X Ae2M h Z Ae s

1 r Rh1

2

n s2 R

h2

2

od A: 17In the above,

Rh1 D sdiv ch q t x 2wh q; 18 Rh2 div R h2 Dbr 2 q I x 2 div h

h

D sdiv ch 19are discrete residual functions for the dynamic plateequations. The functions s1x and s 2x are frequencydependent local mesh parameters determined from dis-persion analysis and designed to match the analytical

wave numberfrequency relation for Mindlin plates.

Setting s1 s2 0, reverts to the underlying assumedstrain formulation. The residual-based least-squaresterms are constructed to maintain symmetry of the un-derlying energy functional for isotropic materials. Theuse of derivatives on the residuals is necessary to sim-plify the formulation for elements with low-order ap-proximations.

A slightly simplied form results if we neglect a cross-coupling term R2 x; x R2 y ; y , resulting in the alternativeform, F LS vh 12 X Ae2M h Z Ae s1 r Rh1

2

s2 Rh2 x; x 2 Rh2 y ; y 2 d A: 20Both forms (17) and (20) may be recast in a more gen-eral expression for the least-squares operator, see [31].

Any of several existing mixed nite element approxi-mation elds which give rise to spaces which avoid shearlocking and pass the static patch test may be used witheither least-squares functional (17) or (20). In this paper,we use the eld- and edge-consistent interpolations of the popular MITC4 plate bending element originallyproposed by Bathe and Dvorkin [9]. In [30], least-squarestabilizing operators similar to (20), but with residualsdened by independent moment and shear resultants,were used to modify the discrete HellingerReissnerfunctional in an assumed stress hybrid element formu-lation. The difference here is that the stress resultants arewritten as dependent functions of generalized displace-ments and assumed shear strains.

In the following, we denote four-node quadrilateralelements based on the functional forms (17) and (20) asMLS4-1 and MLS4-2, respectively.

3.1. Finite element interpolations

We dene the nite element subspaces for the ap-proximation of the deection wh and rotation vector hh

as

W h wh 2 H 10 A; whj Ae 2 Q1 Ae ;8 Ae 2M h

; 21

V h fhh 2 H 10 A; hhj Ae 2 Q1 Ae2;8 Ae 2M hg; 22

where Q1 Ae is the set of low-order polynomials of degree 6 1 in each variable dened on Ae , and Ae is thecurrent element in the discretization. This space of polynomials provides for equal order basis functions forthe deection and both components of the rotation. Thenite element interpolation of the element domain Ae ,together with the displacement eld wh , and hh , followsthe standard isoparametric procedure [14]. We denen n; g to be natural coordinates on the referencebiunit square

^

A A dened by the interval 1; 12

. The



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reference domain is then mapped onto the physical ele-ment domain Ae with cartesian coordinates x x; y parameterized by,xn; g X

4

i1 N in; gx i; 23

where x i xi; y i 2R 2 are nodal coordinates, and N i arebilinear shape functions, N in; g 1 nin1 gig=4; i 1; . . . ;4 24with nodal coordinates ni; gi 2 f1; 1; 1; 1;1; 1; 1; 1g. The displacements are constructed usingthe same bilinear functions:whn; g X

4

i1 N in; gwi; hhn; g X

4

i1 N in; ghi;

25where wi are nodal deections and hi h

i x; h

i y , are nodal

rotations. We let [ J ] be the Jacobian transformationmatrix of the mapping x : ^ A A ! Ae , i.e. ^rr J

T

r , where

J : x ;n x;n x;g y ;n y ;g : 26

Here ^rr stands for the gradient operator with respect tothe n and g variables.3.2. Assumed shear strain eld

To eliminate locking, the shear energy is dened

in terms of the assumed covariant transverse shearstrain eld of the MITC4 mixed interpolation [9]. Theassumed strain ch is dened by a reduction operatorR h : H 1 Ae

2

! Ch Ae, which maps the shear straininterpolants evaluated from the spaces W h and V h to theassumed strain space Ch , [1618], i.e.,

ch R hrwh hh

rwh R hhh

rwh J

T R ^ A AJ T hh:

The assumed strain space may be dened as [18]:

Ch Ae cj Ae 2 S h

Ae; sc s it 0;on E i; i 1; 2; 3; 4 27

with continuous tangential shear strains across elementedges. Here E i are the edges of the quadrilateral element Ae , s i are tangent vectors to the edge E i , and s t denotesthe jump in a quantity across an element interface. S h isthe rectangular rotated RaviartThomas space [48],

S h Ae fc J T cc; cc 2 S

h

^ A Ag;S h^ A A fcc cn; cg; jcn a1 a2g; cg b1 b2ng:For completeness, we review the MITC4 strain inter-

polation.

For the two-dimensional plate element with bilinearmapping (23), covariant basis vectors are dened interms of the in-plane tangent vectors:

t n : x ;n x;n; y ;nT ; t g : x ;g x;g ; y ;g

T : 28The complimentary contravariant basis vectors,

g n 1 J y ;g ; x;g

T ; g g 1 J y ;n ; x;n

T

29satisfy the orthogonality conditions, t n g n 1, t g g n 0, and t g g g 1, t n g g 0, (see [15]). In the above, J det J x;n y ;g x;g y ;n , is the element Jacobian.Using this basis, the covariant shear strain tensor com-ponents may be written in vector form as [47]:

cc ^rr w J T h J

T

rw h; 30where

cc cn; cg T ; ^rr w w;n; w;g T ; h h x; h y T : 31The covariant strains are transformed to cartesian co-ordinates, using the rotation matrix, c J

T cc, i.e.,

c xz c yz J T cncg ; 32

where J T is the inverse of J T :

J T

g n e x g g e x g n e y g g e y

1 J

y ;g y ;n x;g x;n : 33

Following Bathe and Dvorkin [9], the assumed covari-ant transverse shear strain eld is dened by the linearinterpolation between mid-points of the element edges.The essential assumption is to assume the transversalshear interpolation in local convective co-ordinates to belinear in g direction for cn , and linear in n direction forcg ,

chng 12 1 gc

Bn

12 1 gc

Dn ; 34

chgn 12 1 nc

Ag

12 1 nc

C g : 35

Evaluating the covariant transverse shear strains collo-cated at the midpoints of the element boundaries, resultsin the assumed strain eld, cch 2 S

h

^ A A,chng

14 1 gw2 w1 x

B;n h2 h1

14 1 gw3 w4 x

D;n h3 h4; 36

chgn 14 1 nw4 w1 x

A;g h4 h1

1

4 1

n

w3

w2

x C ;g

h3

h2

;

37



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where

x A;g x 1;g x 4;g x 4 x 1=2;xC ;g x 2;g x 3;g x 3 x 2=2;x B;n x 1;n x 2;n x 2 x 1=2;x

D;n x

3;n x

4;n x 3 x 4=2:

Making use of N i;n ni1 gig=4, N i;g gi1 nin=4, it follows that the assumed covariant straincomponents may also be expressed as

chng X4

i1 N i;nwi X

4

i1 N

inx i;n hi; 38

chgn X4

i1 N i;gwi X

4

i1 N

igx i;g hi; 39

where t ni x i;n and t gi x i;g are the covariant basisevaluated at the node points, and N

1g N

4g 1 n=4; N

2g N

3g 1 n=4; 40

N 1n N

2n 1 g=4; N

3n N

4n 1 g=4: 41

In this form, it is clear that the the assumed covariantstrain in natural coordinates may be interpreted as areduction operation R ^ A A: H 1^ A A

2

! S h

^ A A, cch ^rr whR ^ A AJ

T hh , which interpolates piecewise smooth functionsinto the space of linear functions, cch 2 S h^ A A.

It is this linear strain property that will be used tosimplify the residuals Rh1 and Rh2 appearing in the F LS

functionals given in (17) and (20). In particular, thefollowing important properties of the rotated Raviart Thomas space for the assumed shear strain interpolationelds are used,

ddn

chng 0; ddg chgnh i 0: 42Remarks

(i) The form of the MITC4 interpolation for shearstrains given here avoids computation of the square-

roots appearing in kx ;nk x2;n y

2;n

1=2

, kx ;gk x2;g y 2;g

1=2 , and used in the original implementation given in[9].

(ii) Following the approach given in Prathap [10],the construction of the nite element space S h for theassumed strains expressed in (38) and (39), can also beinterpreted as a eld consistent interpolation between^rr wh and assumed covariant section rotations h

hn , h

hg ,

interpolated with the smoothing functions given in (40)and (41), i.e.,

hhn

X4

i

1

N inhni; h

hg

X4

i

1

N ighgi: 43

To maintain edge consistency (continuous tangentialshear strains), the nodal values hni and hgi aretransformed to Cartesian coordinate denitions of therotations h xi and h y i using the Jacobian transforma-tion given in (26) evaluated at the nodes,

hnihgi J i h xih y i : 44

Dening the assumed covariant strains by

chng wh;n hhn; c

hgn wh;g h

hg 45

and using (43) and (44), leads to the expressions given in(38) and (39), or equivalently (36) and (38). In [10], thesmoothing functions N

iare derived using a least-squares

t of the covariant transverse shear strains within anelement. The equivalence between the eld and edgeconsistent development of [10] and the original MITCdevelopment of [9] does not seem to be recognized in theopen literature, see e.g. [49,50]. The equivalence betweendifferent cures for shear locking phenomena is discussedin [25].

4. Evaluating element parameters

In this section, we determine optimal parameterss1 and s2 appearing in the MLS terms (17) and (20),for the four-node quadrilateral element with assumedtransverse shear strain given by the MITC4 mixed in-terpolation. The dispersion analysis follows the sameprocedures used in [30] to determine optimal mesh pa-rameters for an assumed stress hybrid least-squares plateelement. We begin by simplifying the residuals appear-ing in the generalized least-squares functionals F MLS andevaluate the repetitive difference stencil associated with auniform nite element mesh. Using this stencil we obtainthe numerical dispersion equation relating wave num-bers to real frequency input. Solving this equation givestwo root pairs similar to that found for the analyticaldispersion relation one real propagating wave number

k 1, and one purely imaginary wave number

k 2 . We

evaluate the parameters s1 and s2 by matching the niteelement wave number pairs to the analytical wavenumber pairs for a given free wave angle u . A similardesign criterion for matching the single real valued wavenumber for the scalar Helmholtz equation in two di-mensions was used in [40].

To determine the stencil, we consider a uniform meshM h , of four-node quadrilateral elements with elementside lengths h jD xj jD y j. In this mesh we label eachnode m; n, where m and n are integers. Thus the co-ordinates of each node m; n are given by xm; y n mh; nh. The nodal degrees-of-freedom at these nodesare denoted w

hm;n w

h

xm; y n, and hhm;n h

h

xm; y n.



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For a general four-node quadrilateral element char-acterized by the parametric mapping, r J

T ^rr , theLaplacian and divergence operators in natural coordi-nates may be expressed as

r2

1

J 2 a2

o 2

o n2 b2

o 2

o g2

2c2

J 2o 2

o no g x ;ng J T ^rr ; 46

div c 1 J 2

a2cn;n b2cg;g c2

J 2 cn;g cg;n 2x ;ng J

T cc ; 47

div h 1 J

a h;n b h;g ; 48where x ;ng constant,a dcurlcurl y y ;g; y ;n

T ; b dcurlcurl x x;g ; x;nT

49and a ak k, b bk k, c2 x;n y ;n x;g y ;g .For square element geometries, x;n y ;g h=2, and x;g y ;n 0, so that a2 b2 h=2, c2 0, and J h2=4, simplifying the above expressions. The Laplacianreduces to

r2

1

J

^

rr2

1

J

o 2

o n2

o 2

og

2 :

50

From the eld-consistent property given in (42) for therotated RaviartThomas space cch 2 S

h

^ A A, the diver-gence of the MITC4 interpolated shear strains vanisheswithin the element,

div ch 1 J

^rr cch 1 J

cn;n cg;g 0; 8 ch 2 Ch: 51Furthermore, the divergence of the section rotations

simplies to

div hh

1

ffiffiffi J p ^

rrhh

1

ffiffiffi J p hh

x;n hh

y ;g:

52

Since hh 2 Q1^ A A, then div hh 2 P 1^ A A / j/ c1 f

c2n c3gg, and therefore,

r 2div hh

1 J

^rr 2 1

ffiffiffi J p ^rr h

h 0; 8hh 2 V h 53Using (51) and (53), the residuals in the generalizedleast-squares functional reduce to,

r Rh1 q t x 2rwh f ; Rh2 q I x 2r hh

54and (17) becomes,

F LS 12 X Ae2M h Z Aefr 1rwh f rwh f

r 2r hh

2

gd A; 55where

r 1 s1 qt x 2 2; r 2 s2 q I x 2

2; f q= qt x 2 :Similarly, the simplied form (20) reduces for square

elements with MITC4 interpolation to,

F LS 12 X Ae2M h Z Ae r 1rwhn f rwh f

r 2 hh x; x

2h hh y ; y 2iod A: 56A similar least-squares stabilizing operator was obtainedby the assumed stress hybrid formulation given in [30].In that case, the simplication arises from the require-ment that the assumed stress-eld satises static equi-librium within Ae .

Substituting the bilinear interpolations for wh andhh , together with the assumed strain ch dened by theMITC4 interpolation, into the reduced MLS functionalsand imposing stationary conditions with respect to wh

and hh , results in the following system of linear algebraicequations for each element,

K e x 2M e r 1x M M e1 r 2x M M e2 d e

f e

57Here, d e is the 12 1 vector of element nodal displace-

ments derived from vh

x i wi; hi; i 1; . . .

;4f g, andK e and M e are the element stiffness and mass matrices,respectively. The frequency independent stabilizationmatrices resulting from (55) are dened by

M M e1 Z Ae N Tw; xN w; xn N Tw; y N w; y od A; 58M M e2 Z Ae N h x; x N h y ; y T N h x; x N h y ; y d A; 59where N w, N h x and N h y are row vectors of bilinear basisfunctions dened by the interpolations (25) written in

vector form,

wh N wd e; hh x N h xd

e ; hh y N h y d e : 60

Here, we have assumed r i , i 1; 2 are constant within Ae , although variable r i are possible. Alternatively, thesimplied form resulting from (56) may be used, with themodied matrix,

fM M e2 Z Ae N Th x; xN h x; xn N Th y ; y N h y ; y od A: 61For square elements, the stabilization matrix for the

simplied form,



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fM M ex r 1x M M e1 r 2x fM M e2 62can be written in nodal block form as

fM M e

1

6

A B C DB A D C

C D A B D C B A

26664

37775

63

with diagonal nodal blocks,

A diag 4 r 1; 2r 2; 2r 2 ; C diag 2r 1; r 2; r 2;B diag r 1; 2r 2; r 2; D diag r 1; r 2; 2r 2:

A similar closed form expression can be obtainedusing (59).

Finite element difference relations associated with atypical node location ( xm; y n ) are obtained by assemblinga patch of four elements. The result is a coupled system

of three, 27-term difference stencils associated with thethree nodal degrees-of-freedom and nine connectednodes centered at node m; n. Let E p x and E q y be denedby the directional shift operations: E p xd m;n : d m p ;n; E q y d m;n : d m;nq: 64

Then the stencil associated with the solution d m;n vh xm; y n, may be expressed in the form,

X1

p 1 X1

q 1D pq E p x E q y d m;n 0f g; 65

where

D pq are 3 3 nodal partitions dened by the

nine-point block difference star associated with the nineconnected nodes.

4.1. Finite element dispersion relation

To obtain the nite element dispersion relation as-sociated with this stencil, a plane wave solution is as-sumed for the nodal displacements, similar in form tothe analytical solution to the homogeneous problem:

d m;n w0

h0m eik h x hmeik h y hn; 66where k h x k h cos u , k h y k h sin u are components of thewave vector kh k hm k hcos u ; sin u . Substitution of (66) into the stencil equations (65), leads to the niteelement dispersion relation for the plate expressed as

D k h; x : G G 11 G G 22 G G 212 0: 67For the MLS4 element, the frequency dependent coef-cients take the form,

G G 11 G 11 r 1 H 11 ; G G 22 G 22 r 2 ^ H H 22; G G 12 G 12 :

68

The functions G 11 , G 22 and G 12 depend on the stiffnessand mass matrix coefficients K eij and M

eij , the frequency

x , wave number k h , and wave angle u . The form of thesefunctions are dened in [30], with the stiffness and massmatrices replaced with those arising from the MITC4interpolation.The functions resulting from the least-squares stabilization matrices (58) and (59) are denedby,

H 11 4 c x c y 2c xc y =2;^ H H 22 a1 a2=2 3a3=4:

a1 1 c x cos2 u c y sin2 u ;

a2 c y cos 2 u c x sin2 u c xc y ;

a3 s x s y sin 2u :

c x

cos k hh cos u

; c

y cos k hh sin u

;

s x sin k h h cos u ; s y sin k hh sin u : 69For the simplied least-squares stabilization matrix de-ned in (62), ^ H H 22 , reduces to ^ H H 22 a 1 a2=2. This sim-plied coefficient is identical to that found in the hybridleast-squares (HLS4) element in [30]. This result followsfrom the fact that both MLS4 and HLS4 use bilinearinterpolation of section rotations hh with simpliedmatrix in the form (62). Additional functions related tothe mesh parameter r 1 are present in the HLS4 elementdue to the cross-coupling of the nodal deections andsection rotations in the vertical displacement approxi-mation.

The nite element dispersion equation D x ; k hh;u ; K ij ; M ij ; r 1; r 2 dened in (67) relates frequency x , tothe numerical wave number k hh and u , and dependson the stiffness and mass coefficients K ij K e ij , and M ij M e ij , and mesh parameters r 1, r 2. Similar to theanalytic dispersion relation, there are two pairs of nu-meric wave numbers k h1 and k

h2 that satisfy (67) which

correspond to propagating and evanescent waves, re-spectively. For waves directed along mesh lines corre-sponding to u 0, then, a3 0, so that ^ H H 22 1 c x c y c xc y =2, and as expected the dispersion relation forboth MLS4-1 and MLS4-2 are the same.

4.2. Selection of optimal design parameters

Following the procedures employed in [30], meshparameters r 1 and r 2 are determined such that the niteelement wave number pairs match the analytical wavenumber pairs k 1 and k 2 for a given orientationu u 0. In particular, we set k h k 1x and k h k 2x in the nite element dispersion relation (67). In partic-ular, we replace k hh cos u ; k hh sin u , in (69) withk 1h cos u ; k 1h sin u , and k 2h cos u ; k 2h sin u , respec-tively. This results in two equations for r 1 and r 2:



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c11 c12 r 1 c13 r 2 c14 r 1r 2 0; 70c21 c22 r 1 c23 r 2 c24 r 1r 2 0; 71with coefficients c1i cik 1; u , and c2i c ik 2; u , i 1,2, 3, 4, dened by substituting k 1 and k 2 into the func-tions,c1 G 11 G 22 G 12G 12;c2 G 22 H 11 ;c3 G 11 ^ H H 22;c4 H 11 ^ H H 22:

Eliminating r 2 from (70) and (71), allows the designparameter r 1 to be obtained in closed-form by solvingthe quadratic equation

e2r 21 e1r 1 e0 0; 72where e l elc ij, l 0, 1, 2, are dened by,e2 c24c12 c14c22 ;e1 c23c12 c13c22 c11 c24 c21c14;e0 c11 c23 c21c13 :For the MLS4 element, solution of the quadratic equa-tion (72) results in two real negative roots. We select thelargest root to determine r 1, as this root matches theanalytical dispersion relations. The other design pa-rameter can then be written in terms of the rst,

r 2

c21r 1 c11c31 c41r 1

:

73

Remark. For the HLS4 element derived in [30], a cubicequation in r 1 results from cross-coupling of sectionrotations in the deection approximation, thus requiringmore work to compute roots.

The design parameters r l r l K ij ; M ij ; x ; h; u ,l 1; 2 are obtained in terms of the stiffness and masscoefficients in the underlying MITC4 element, the fre-quency dependent wave numbers satisfying the analyti-cal dispersion relation, and u . In general, the directionof wave propagation u is not known a priori. However,similar to [40,30], we can select an angle in the deni-

tions for r 1 and r 2 which minimizes dispersion error overthe periodic interval 0 6 u 6 p=4. With the choice u 0in the denitions for r 1 and r 2, then, as expected, thedispersion relations for our MLS4 plate element spe-cialize to the relations for the GGLS 1-D Timo-shenko beam element described in [45], with EI Et 3=121 m2.4.3. Distorted elements

For distorted quadrilateral nite element geometries,the simplications indicated in (54) are no longer strictly

valid. For the MITC4 interpolations on distorted bilin-

ear quadrilateral elements, the divergence of the as-sumed strain ch 2 Ch Ae and Laplacian operator actingon the divergence of the section rotations hh , are notnecessarily zero. In this case, and with the MITC4 in-terpolation cch 2 S h^ A A, the gradient of the shear residualwithin an element takes the form,

r Rh1 J T ^rr qt x 2 wh f

c2

J 2 chn;g chg;n

2x ;ng J T cch ; 74

where chn;g constant, chg;n constant. However, in im-plementing our MLS method on nonuniform meshes, weneglect the effect of the relatively small mixed derivativesand Laplacian on the residuals, and revert to (54). Thusfor distorted elements, we retain the form of the stabi-lization matrices (58) and (59), with constant element

jacobian J h2eI , consistent with the mesh parameterdenitions for r 1 and r 2 . We dene the element length he

by either a local size determined by the square root of the element area, he ffiffiffiffiffi Aep , or by an average elementlength have computed over a patch of similarly sized el-ements. While our denition for the mesh parameters r 1and r 2 were derived from a dispersion relation on auniform mesh, with constant element length he , accuratesolutions on nonuniform meshes are shown to be rela-tively insensitive to the precise denitions used.

5. Dispersion accuracy

For a range of frequencies x , and wave angles u ,relative to uniform mesh lines, the wave number accu-racy for our residual-based MLS4 four-node element iscompared with the underlying MITC4 element [9], andthe SRI4 element [7]. Results are presented for a steelplate with properties: E 210 1010 dynes/cm 2, m 0:29, q 7:8 g/cm 2, plate thickness t 0:15 cm, andshear correction factor j 5=6. The node spacing is h D x D y 1:0 cm, resulting in a ratio of plate thicknessto element length of t =h 3=20. Both dispersion anal-ysis and numerical examples show similar solutionsusing the original divergence form F LS dened in (55)and the simplied form dened in (56). For this reason,results for our four-node quadrilateral element MLS4are reported for the simplied residual-based form (56).

We begin with a dispersion analysis of the underlyingMITC4 element. The resulting numerical wave numbersk h1 and k

h2 for the MITC4 element are compared to the

analytical wave numberfrequency relation in Fig. 1. Wenote that the dispersion curves for the MITC4 elementfor uniform meshes are nearly identical to the results forthe SRI4 element, see [29,30]. In the frequency rangeplotted, both the analytical and numerical wave number

exhibit one real wave number k 1, corresponding to a



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propagating wave, and one purely imaginary wavenumber k 2, corresponding to an evanescent decayingwave. Results are plotted for equally spaced angles u 0 , 15 , 30 , 45 . Due to symmetry, results are boundedby the extreme angles of 0 and 45 , corresponding towaves directed along mesh lines and mesh diagonals,respectively. The values are plotted over the range 0

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Thompson L - A Stabilized MITC Element for Accurate Wave Response in Reissner-Mindlin Plates