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A Galerkin/least-square finite element formulationfor nearly incompressible elasticity/stokes flow

Kaiming Xia a,*, Haishen Yao b

a Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60607, USAb Department of Mathematics and Computer Science, QCC, The City University of New York, NY 11364, USA

Received 1 May 2005; received in revised form 1 August 2005; accepted 1 November 2005Available online 27 December 2005

Abstract

A Galerkin/least-square finite element formulation (GLS) is used to study mixed displacement-pressure formulation of nearly incompressible elasticity. In order to fully incorporate the effect of the residual-based stabilized term to the weakform, the second derivatives of shape functions were also derived and accounted, which can accurately discretize the resid-ual term and improve the GLS method as well as the Petrov–Galerkin method. The numerical studies show that improvedstabilized method can effectively remove volumetric locking problem for incompressible elasticity and stabilize the pressurefield for stokes flow. When apply GLS to study material nonlinearity, the derivative of tangent modulus at the integrationpoint will be required. Both advantage and disadvantage of using GLS method for nearly incompressible elasticity/stokes

flow were demonstrated.Ó 2005 Elsevier Inc. All rights reserved.

Keywords: Galerkin/least-square; Second derivatives of shape functions; Stabilized term; Incompressible elasticity/stokes flow

1. Introduction

It has been a few decades for scientific researchers to try to develop successful finite element formulationsfor incompressible and nearly incompressible material, which can effectively alleviate or remove the volumetriclocking problem. Mixed displacement-pressure formulations are a suitable alternative because the internal

constraint can be satisfied point-wise. However, not every combination of interpolation functions for pressureand displacements is allowed since they have to satisfy Babuska–Brezzi conditions [1,2] or patch test proposedby Ozienkiewicz and Taylor [3]. While using low-order finite elements, the pressure field might highly oscillateunless the some special stabilization is applied [3], which makes it difficult to use in practical engineering sit-uations. Thus, a lot of efforts have been focused on the development of stabilized finite elements in which the

0307-904X/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2005.11.009

* Corresponding author. Present address: Machine Research, TC-E 852, P.O. Box 1875, Peoria, IL 61656-1875, USA. Tel.: +1 309 5784145; fax: +1 309 578 4277.

E-mail address: [email protected] (K. Xia).

Applied Mathematical Modelling 31 (2007) 513–529

www.elsevier.com/locate/apm

mailto:[email protected]

mailto:[email protected]

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violation of the mixed patch test (or Babuska–Brezzi conditions) can be artificially compensated. Stabilizationmethods have been introduced for fluid mechanics as tools of stabilizing the fluid flow equations [2,4–7], par-ticularly for advective-diffusive model. One of these methods is to introduce nonzero diagonal terms by addinga least-square form to the Galerkin formulation, which was for fluid mechanics [4]. The advantage of stabil-ization methods is to overcome difficulties associated with mixed formulations (such as inappropriate combi-

nation of interpolation fields), which makes it possible to use low-order finite elements.The Galerkin/least-square (GLS) approach presents itself as a modification to constructing a weak form forthe Galerkin form and acted as a means of stabilizing the fluid flow equations. It is mainly applied to incom-pressible Stokes flow problems that coincide with those for incompressible linear elasticity. Additionally, bothGLS and the Petrov–Galerkin method have been used to alleviate volumetric locking problem in solidmechanics [8–10]. For both GLS and Petrov–Galerkin method, the second derivatives of shape functions, withrespect to global Cartesian coordinate system, are required to discretize the residual-based term, which has notbeen used in previous studies. In this paper, the second derivatives of shape functions are presented so that theresidual terms can be accurately accounted.

The remainder of this paper is organized as follows. A Galerkin mixed form of elasticity is presented inSection 2. In Section 3 we present the Galerkin/least-square method for mixed displacement-pressure formu-lation and the second derivatives of shape functions. In Section 4 numerical simulations are given. Concluding

remarks are presented in Section 5.

2. Mixed displacement-pressure formulation

2.1. Strong form

Consider a body X denoted by the open set X & Rndim consisting of material point x 2 Rndim . Its boundary isdenoted by C = oX = oXu [ oXt and B = oXu \ oXt. At any point x, the displacement uðxÞ 2 Rndim and pres-sure p(x) is a scalar.

For mixed u À p formulation, the governing equations are given by

r Á r þ b ¼ 0 ðEquilibrium equation in bodyÞ; ð1Þ

r Á u Àp

K ¼ 0 ðVolumetric constitutive equationÞ; ð2Þ

u ¼ g ðDirichlet boundary condition on oXuÞ; ð3Þ

rn ¼ t ðNeumann boundary condition on oXt Þ; ð4Þ

where K is the bulk modulus and defined as K = E /3(1 À 2m).

2.2. Stress decomposition

The stress tensor is also decomposed into two parts: deviatoric stress s and pressure p, and expressed asfollows:

r ¼ s þ p 1 ¼ 2G Idev þ p 1 ¼ Duue þ Dup p ; ð5Þ

where G is shear modulus, 1 is the second-order unit tensor, deviatoric modulus Duu is the deviatoric projec-tion of elastic matrix D and defined as Duu = 2G Idev, Idev is given by Idev ¼ I À 1

31 1, and

Dup ¼ 1 ¼ ½ 1 1 1 0 0 0 T

.

2.3. Weak form

The weak form of the mixed problem can be obtained by using standard weighted residual method. The

corresponding spaces for trial functions and weighting functions are defined as follows:

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S ¼ u j u 2 H 10ðXÞ j u ¼ g; on oXuÈ É

; ð6Þ

p ¼ f p j p 2 L2ðXÞg; ð7Þ

v ¼ fw j w 2 H 1ðXÞ j w ¼ 0; on oXug. ð8Þ

Thus the discretized weighted residual form of the problem can be given as follows:

Z X

rwu : rdX ¼Z X

wu Á b dX þZ C

wu Á t dC. ð9Þ

By substituting Eq. (5), one can further obtainZ X

rwu : ðDuue þ Dup p Þ dX ¼

Z X

wu Á b dX þ

Z C

wu Á t dC. ð10Þ

For volumetric constitutive equation, the weak form is given byZ X

w p r Á u Àp

K

dX ¼ 0; ð11Þ

where wuand w p are weighting functions corresponding to displacement u and pressure p, respectively and canbe chosen as follows:

wu ¼ ðdduÞT

ðNuÞT; ð12Þ

w p ¼ ðdP p ÞT

ðN p ÞT

. ð13Þ

2.4. Matrix form

The spatial approximation of the displacement, pressure and strain are written by

u ¼ Nuu; ð14Þ

p ¼ N p P; ð15Þ

e ¼ symðruÞ ¼ r sNuu ¼ Bu; ð16Þ

where matrix B is given by

B ¼ r sNu ¼ ½ B1 Á Á Á Bnel . ð17Þ

By applying the weighted residual method (Galerkin procedure and substitution of weighting functions Eqs.(12) and (13)) and appropriately approximating the spatial unknowns into Eqs. (10) and (11), the results willbe as follows:

Kuu Kup

KT pu K pp

" #u

P

& '¼

Ru

0

& '; ð18Þ

where

Kuu ¼Z X

BTDuuBu dX; ð19Þ

Kup ¼

Z X

BTDup N p dX; ð20Þ

Ru ¼

Z X

ðNuÞT

Á b dX þ

Z C

ðNuÞT

Á t dC; ð21Þ

K pu ¼

Z X

ðN p ÞTD puB dX; ð22Þ

K pp ¼ À

Z X

ðN p ÞT D pp N p dX; ð23Þ

where D pu ¼ 1 ¼ ½ 1 1 1 0 0 0 T, D pu = 1T = (Dup)T, and D pp = 1/K .

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Eq. (18) is the general mixed displacement-pressure formulation without consideration any stabilizing tech-niques. For low-order elements, the use of low-order interpolation functions for both displacement and pres-sure will give highly oscillated pressure solution. In order to avoid this oscillation, some stabilized techniquesare used here as described below.

3. Galerkin/least-square formulations

3.1. General GLS form

The Galerkin/least-square (GLS) approach presents a modification for constructing a weak form for Galer-kin form, and is as follows [3,4]:Z

X

duTAðuÞ dX þ

Z X

dAðuÞTsAðuÞ dX ¼ 0; ð24Þ

s ¼ Àa0h

2

l

1 0 0

0 1 0

0 0 1

264

375; ð25Þ

where the first term represents the general Galerkin form and the appended term represents the residual-basedterm including an element-dependent stabilized matrix s [4], which is an element-dependent parameter and hasto be selected for good performance. dA(u)T is the corresponding weighting part regarded in GLS method.A(u) represents the residual part of the strong form corresponding to the high-order equation.

3.2. Decomposition of least-square part

As required for our current mixed displacement-pressure formulation, the corresponding least-square partcan be modified as follows:Z

X

dAðuÞTsAðuÞ dX ¼

Z X

ðr Á rðwu;w p ÞÞTsðr Á rðu; pÞ þ bÞ dX. ð26Þ

From Eqs. (5), (12) and (13), we can obtain that weighting term

ðr Á rðwu;w p ÞÞT

¼ ðr Á ðDuu~e þ 1~ p ÞÞT; ð27Þ

where

~e ¼ ðdduÞTðr sNuÞT

; ð28Þ

~ p ¼ ðdP p ÞTðN p ÞT. ð29Þ

Thus, Eq. (27) can be rewritten as follows:

ðr Á rðwu;w p ÞÞT ¼ ðdduÞTð r Á ðDuur sNuÞÞT þ ðdP p ÞTð r Á ð1N p ÞÞT ¼ ðdduÞTLT

uu þ ðdP p ÞTLTup . ð30Þ

Also, the residual of the strong form can be rewritten as

r Á rðu; p Þ þ b ¼ r Á ðDuur sNuuÞ þ r Á ð1N p PÞ þ b ¼ Luuu þ Lup P þ b; ð31Þ

where

Luu ¼ LðDuur sNuÞ; ð32Þ

Lup ¼ LðDup N p Þ. ð33Þ

When using the GLS scheme, some difficult points should be mentioned. The derivation of matrix Luu and Lup

seems very complicated. Detailed derivations are provided below.

For isotropic elasticity, deviatoric modulus is a symmetric matrix and can be given as follows:


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Duu ¼ 2lIdev ¼2l

3

2 À1 À1 0 0 0

À1 2 À1 0 0 0

À1 À1 2 0 0 0

0 0 0 3 0 0

0 0 0 0 3 00 0 0 0 0 3

26666666664

37777777775

¼

c11 c12 c13 0 0 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 00 0 0 0 0 c66

26666666664

37777777775

. ð34Þ

Then we can obtain matrix Luu as follows:

Luu ¼ LðDuur sNuÞ ¼

Á Á Á l1;3iþ1 l1;3iþ2 l1;3iþ3 Á Á Á



264

375

3Â3nel

; ð35Þ

where

l1;3iþ1 ¼ c11

o2 N i

o x

2þ c44

o2 N i

o y

2þ c66

o2 N i

o z

2; ð36Þ

l1;3iþ2 ¼ l23iþ1 ¼ ðc12 þ c44Þo

2 N i

o xo y ; ð37Þ

l1;3iþ3 ¼ l3;3iþ1 ¼ ðc13 þ c66Þo

2 N i

o xo z ; ð38Þ

l2;3iþ2 ¼ c44

o2 N i

o x2þ c22

o2 N i

o y 2þ c55

o2 N i

o z 2; ð39Þ

l2;3iþ3 ¼ l3;3iþ2 ¼ ðc23 þ c55Þo

2 N i

o y o z ; ð40Þ

l3;

3iþ3 ¼c

66

o2 N i

o x2 þc

55

o2 N i

o y 2 þc

33

o2 N i

o z 2.

ð41

Þ

For elastoplasticity, consistent tangent modulus will vary from point to point and be updated for each incre-mental step. The gradients of components of the consistent tangent modulus will be required to accuratelycompute Lup. This will bring additional difficulty to the nonlinear computational solid mechanics. This mightbe the intrinsic difficulty of GLS method.

Lup ¼ L Dup N p ð Þ ¼

o N 1

o xÁ Á Á

o N nel

o x;

o N 1

o y Á Á Á

o N nel

o y ;

o N 1

o z Á Á Á

o N nel

o z

26666664

37777775

3Ânel

. ð42Þ

3.3. Mixed displacement-pressure formulations for GLS

Then Eq. (26) can be further decomposed into two parts as follows:Z X

d AðuÞTsAðuÞ dX ¼

Z X

ðdduÞT

LTuu þ ðdP p Þ

TLTup

sðLuuu þ Lup P þ bÞ dX

¼

Z X

ðdduÞT

LTuusðLuuu þ Lup P þ bÞ dX þ

Z X

ðdP p ÞT

LTup sðLuuu þ Lup P þ bÞ dX. ð43Þ

By substituting Eq. (12) into Eq. (10), we can obtain the general weak form (Galerkin form) for the equilib-

rium equation as follows:


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Z X

ðdduÞT

rT s Nu: ðDuur sNu þ Dup N p PÞ dX ¼

Z X

ðdduÞT

ðNuÞT

Á b dX þ

Z C

ðdduÞT

ðNuÞT

Á t dC. ð44Þ

Then, by appending the first part of Eq. (43) to Eq. (44), we can obtain the Galerkin/least-square form for ourmixed displacement-pressure formulation

Z X

BTDuuB dXu þZ X

BTDup N p dXP þZ X

LTuusLuu dXu þ

Z X

LTuus Lup dXP

¼

Z X

ðNuÞT

Á b dX þ

Z C

ðNuÞT

Á t dC À

Z X

LTuusb dX. ð45Þ

Eq. (45) can be rewritten as follows:

Kuu þ KG uuÀ Á

u þ Kup þ KG up

P ¼ Ru þ RG ; ð46Þ

where

K

G

uu ¼Z X L

T

uus

Luu dX;

ð47Þ

KG up ¼

Z X

LTuusLup dX; ð48Þ

RG u ¼ À

Z X

LTuusb dX. ð49Þ

By taking the same procedure and substitute Eq. (13) into Eq. (11), one will getZ X

ðdP p ÞTðN p ÞT r Á u Àp

K

dX ¼ 0. ð50Þ

Next, append the second part of Eq. (43) to Eq. (50), and we can obtain the weak form for the volumetric

governing equation corresponding to Galerkin/least-square approachZ X

ðdP p ÞTðN p ÞT r Á u Àp

K

dX þ

Z X

ðdP p ÞTLTup sðLuuu þ Lup P þ bÞ dX ¼ 0. ð51Þ

Removing dP p on both sides of Eq. (51), and insert Eq. (15), thus we obtain

Kup þ KG up

u þ K pp þ KG pp

P ¼ RG p ; ð52Þ

KG pu ¼

Z X

LTup sLuu dX; ð53Þ

K

G

up ¼Z X L

T

up s

Lup dX;

ð54Þ

RG p ¼ À

Z X

LTup sb dX. ð55Þ

Combining Eq. (46) and Eq. (52) together, we rewrite the mixed displacement-pressure formulation in matrixform as follows:

Kuu þ KG uu Kup þ KG up

Kup þ KG up

T

K pp þ KG pp

24

35 u

P

& '¼

Ru þ RG u

RG p

( ). ð56Þ

The above Eq. (56) is the Galerkin/least-square formulation for the mixed form of nearly incompressible elas-

ticity/stokes flow.


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3.4. Second derivatives of shape functions

As we described earlier, the important part to completely implementing the Galerkin/least-square approachis to use the second derivatives of shape functions. The key formulations for calculation of second derivativesof shape functions are given as follows:

o2 N I

o x2

o2 N I

o y 2

o2 N I

o z 2

o2 N I

o xo y

o

2

N I o y o z

o2 N I

o z o x

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;

¼ À½ M 2½ M 1 J À1

o N I

on

o N I

og

o N I

of

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

þ ½ M 2

o2 N I

on2

o2 N I

og2

o2 N I

of2

o2 N I

onog

o2

N I

ogof

o2 N I

ofon

8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>;

; ð57Þ

where matrix [M 2] and [M 1] are given by

½ M 2 ¼

j211 j212 j213 2 j11 j12 2 j12 j13 2 j13 j11;

j221 j222 j223 2 j21 j22 2 j22 j23 2 j23 j21;

j231 j232 j233 2 j31 j32 2 j32 j33 2 j33 j31;

j11 j21 j12 j22 j13 j23 j11 j22 þ j13 j22 j12 j23 þ j13 j22 j11 j23 þ j13 j21;

j21 j31 j22 j32 j23 j33 j21 j32 þ j22 j31 j22 j33 þ j23 j32 j21 j33 þ j23 j31;

j31 j31 j32 j32 j33 j33 j11 j32 þ j12 j31 j12 j33 þ j13 j32 j11 j33 þ j13 j31

26666666666664

37777777777775

ð58Þ

and

½ M 1 ¼

P o2 N I

on2x I

P o2 N I

on2y I

P o2 N I

on2z I

P o2 N I

og2 x I P o

2 N I

og2 y I P o

2 N I

og2 z I

P o2 N I

of2x I

P o2 N I

of2y I

P o2 N I

of2z I

P o2 N I

onog x I

P o2 N I

onog y I

P o2 N I

onog z I

P o2 N I

ogof x I

P o2 N I

ogof y I

P o2 N I

ogof z I

P o2 N I

ofon

x I P o2 N I

ofon

y I P o2 N I

ofon

z I

2666666666666666666666666664

3777777777777777777777777775

; ð59Þ


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J ¼

o x

on

o y

on

o z

on

o x

og

o y

og

o z

og

o x

of

o y

of

o z

of

266666664

377777775

¼

P o N I

onx I

P o N i

ony I

P o N i

onz I

P o N i

ogx I

P o N i

ogy I

P o N i

ogz I

P o N i

ofx I

P o N i

ofy I

P o N i

ofz I

266666664

377777775

¼

J 11 J 12 J 13

J 21 J 22 J 23

J 31 J 32 J 33

264

375

; ð60Þ

J À1 ¼1

det½J

J 22 J 33 À J 32 J 23 J 13 J 32 À J 12 J 33 J 12 J 23 À J 13 J 22



264

375

¼

j11 j12 j13

j21 j22 j23

j31 j32 j33

264

375. ð61Þ

4. Numerical examples

4.1. Convergence rate study

For convergence study, the developed method is applied to study a widely used cantilever beam with ana-lytical solutions [11]. The beam is of length 10 m, height 2m, and thickness 1 m and subjected to a parabolicshear traction at the free end as shown in Fig. 1. The material properties for Fig. 1 are Young’s modulusE = 7.5E+07 N/m2 and Poisson’s ratio m = 0.4999. The load is P = 2560 N.

Figs. 2–4 shows the results of the convergence rate study for the 3 node triangle and the 4 node quadrilat-eral, which was performed using uniform meshes of 10 · 2, 20 · 4, 40 · 8, and 80 · 8. Totally four different

Fig. 1. Diagram of parabolic shear-loaded beam.

-6

-5

-4

-3

-2

-1

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

Log (1/h)

L o g ( e r r o r n o r m o

f d i s p l a c

e m e n t )

GLS, k=1.78 for T3

GLS, k=1.80 for Q4

Fig. 2. Convergence rate for the L2 norm of the displacement.


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0.5

0.6

0.7

0.8

0.9

1.0

1.1

10 100 1000 10000

Number of nodes

N o r m a l i z e d d i s p l a c e m e n t

GLS for T3

GLS for Q4

Fig. 5. Tip deflection convergence for plane stress.

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

Log (1/h)

0

1

2

3

4

5

L o g ( e r r o r n o r m

o f p r e s s u r e )

GLS, k=1.81 for T3

GLS, k=1.44 for Q4

Fig. 4. Convergence rate for the L2 norm of the pressure field.

-1.0

-0.5

0.0

0.5

1.0

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

Log (1/h)

L o g ( e r r o r n o r m o

f e n e r g y )

GLS, k=0.98 for T3

GLS, k=1.0 for Q4

Fig. 3. Convergence rate for the energy norm.


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10 100 1000 10000

Number of nodes

0.5

0.6

0.7

0.8

0.9

1.0

1.1

N o r m a l i z e d s t r e s s

GLS for T3

GLS for Q4

Fig. 6. Stress convergence for plane stress.

0.6

0.7

0.8

0.9

1.0

1.1

10 100 1000 10000

Number of nodes

N o r m a l i z e d d i s p l a c e m e n t

GLS for T3

GLS for Q4

Fig. 7. Tip deflection convergence for plane strain.

10 100 1000 10000

Number of nodes

0.5

0.6

0.7

0.8

0.9

1.0

1.1

N o r m a l i z e d s t r e s s

GLS for T3

GLS for Q4

Fig. 8. Stress convergence for plane strain.


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mesh sizes h were used here. This paper presents the mathematical rate of convergence study for displacementsand pressure in the L2(X) norm and the energy norm. As described above, GLS method requires numeric teststo obtain the optimal stabilized parameter. In this paper, we provide all the results corresponding to theparameter a0 in Eq. (25) a0 = 0.05 for 4-node quadrilateral element. For the linear elements of the 3-node tri-angle and 4-node quadrilateral, the theoretical rate of convergence for the displacement and pressure in theL2(X) norm is 2 and the energy norm is 1. The results of the convergence rate study were carried out forthe plane stress problem and shown in Figs. 2–4.

The engineering convergence study was presented for both plane stress and plane strain problems, the nor-malized tip deflection convergence study is given at point (10,0) while normalized stress convergence is selectedat point (0, À10). The engineering convergence studies can be seen in Figs. 5–8, which show the numericalsolutions will converge to exact solutions with refined meshes and convergent elements were obtained.

4.2. Plane strain Cook’s membrane problem

The Cook’s membrane beam problem has been widely used as a benchmarking test to check the performanceof developed finite element formulations. Here it was used to demonstrate the performance of the stabilized

Fig. 9. Plane strain Cook’s membrane problem.

0

4

8

12

16

20

0 4 8 12 16 20

Elements/side

T o p c o r n e r d i s p l a c e m e n t

GLS method

Displacement-based

Fig. 10. Plane strain Cook’s membrane problem: convergence for incompressible elasticity.


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finite element formulation on the alleviation of volumetric locking issue. We consider a tapered panel, clampedon one end and subjected to a shearing load at the free end. The geometry of plane strain Cook’s membranebeam problem is shown in Fig. 9. The plane strain problem can be considered as a three-dimensional problemwith the fixed displacement boundary conditions on the front and back surfaces, which provides the mosthighly constrained problem and has volumetric locking issue in solid mechanics. In order to test the conver-

gence behavior of the GLS formulation, the problem has been discretized into 2·

2, 4·

4, 8·

8, 16·

16 finiteelement meshes. Fig. 10 shows that the displacement will converge quickly to the exact solution for nearlyincompressible elasticity (Young modulus E = 250, Poisson ratio m = 0.4999) while element size decreases.Fig. 11 shows that the GLS formulations can effectively remove locking phenomena while the standard dis-placement-based formulation will exhibit locking effect. Fig. 12 shows that spurious unstablized pressure fieldwill be obtained if Galerkin method was used. Fig. 13 shows the pressure field can be stabilized if Galerkin/least-square formulation was used with distorted mesh. Also, Fig. 14 shows the pressure field can be stabilized

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5 0.6

Poisson ratio

T o p c o r n e r v e r t i c a l d i s p l a c e m e

n t

Displacement-based method

GLS method

Fig. 11. Plane strain Cook’s membrane problem: displacement versus Poisson ratio.

Fig. 12. Unstabilized pressure field with 4-node Q4 for Galerkin method.


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with distorted composite mesh (3-node triangle and 4-node quadrilateral). Therefore this example demonstratesthe stabilization method is very effective in suppressing the oscillation of pressure field.

4.3. Stokes flow analogy

Since the equations of Stokes flow are similar to the equations of isotropic nearly incompressible elasticity.The only difference is in the interpretation of the variables. For Stokes flow, u will be regarded as the velocityof the fluid. Stokes flow governs highly viscous phenomena. Simulations of the incompressible Stokes flow

with the classical Galerkin method may suffer from spurious oscillations arising from the source, which has

Fig. 13. Stabilized pressure feld with 4-node Q4 for GLS method.

Fig. 14. Stabilized pressure field with composite mesh for GLS method.


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to do with the mixed formulation character of the equations and is limited by the choice of equal linear order

finite element interpolations used to approximate the velocity and pressure fields. A two-dimensional case is

Fig. 15. Lid-driven cavity flow analogy: geometry and boundary conditions.

Fig. 16. Oscillated pressure field by Galerkin method.

Fig. 17. Oscillated pressure field by GLS (a = 0.01).


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Fig. 18. Stabilized pressure field by GLS (a = 0.1).

Fig. 19. Stabilized pressure field by GLS (a = 0.5).

-15

-10

-5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0 1.2

X-coordinate

P r e s s u r e

α=0.01

α=0.1

α=0.5

Fig. 20. Driven cavity flow problem: pressure distribution at y = 0.35, 20 · 20 mesh.


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considered for a square domain with unit side lengths. The boundary conditions are as specified in Fig. 15. Thematerial properties are the same as those of incompressible elasticity used for the convergence study.

Fig. 16 shows that the pressure field by the Galerkin method will be highly oscillated. Figs. 17–19 shows thepressure field can be gradually stabilized by using an appropriate stabilization parameter a, which is includedin the stabilization matrix s . If a is too small, oscillations remain in the pressure field. In the other extreme, if a

is too large, the stabilization will be too strong and the pressure field will turn out to be too smooth and mightfail to capture the correct solution in the corners. Therefore an optimum stabilization parameter has to beobtained when using Galerkin/least-square method as well as Petrov–Galerkin method [8,9]. Fig. 20 showsthe pressure distribution along the horizontal line ( y = 0.25).

5. Concluding remarks

In this paper, we have derived the second derivatives of shape functions for the stabilized formulation,which is important for the stabilized Galerkin/least-square method and Petrov–Galerkin method. The secondderivatives of shape functions were used to revisit the GLS method. The numerical results confirm that theGalerkin/least-square method can effectively stabilize the pressure field and the volumetric locking can auto-matically removed. The convergence studies show that GLS method promises convergent elements and allows

the use of equal low-order interpolations for both displacement and pressure fields. However, GLS methodstill has an intrinsic difficulty to study material nonlinearity, which has not been verified before. In the past,the residual-based terms involved with second derivatives were always neglected without using the secondderivatives of shape functions. For the inelastic problem, accurately accounting for the residual-based termwill require the calculations of the derivatives of the tangent modulus at each integral point, which might posea difficulty even though GLS and Petrov–Galerkin method have been used in solid mechanics. Hopefully thispaper with the second derivatives of shape functions can provide some further insight for the applications of GLS and Petrov–Galerkin method in solid mechanics.

Appendix A

For a three-dimensional problem, one can directly calculate the four modulus Duu

, Dup

, D pu

and D pp

basedon the formulations shown above. For plane stress and plane strain problems, the calculation should be mod-ified based on the assumptions of plane stress and plane strain. The corresponding tangent modulus arederived and provided below.

A.1. Plane stress

For plane stress problems, based on the assumption that the stresses in third direction are zero and uniformstrain along the thickness, the tangent modulus is as follows:

Duu ¼ E 1 À m2

2 À m

3

2 À m

30

2 À m3

2 À m3

0

0 01 À m

2

266666664

377777775

; ð62Þ

D pp ¼1

K

1 À m

1 À 2m. ð63Þ

A.2. Plane strain

For plane strain problems, the strains in the third direction are zero, thus we have the corresponding tan-

gent modulus as follows:


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Duu ¼E

1 þ mð Þ 1 À 2mð Þ

2 À 4m

3

2m À 1

30

2m À 1

3

2 À 4m

30

0 01 À 2m

2

26666664

37777775

; ð64Þ

D pp ¼1

K . ð65Þ

References

[1] I. Babuska, The finite element method with lagrange multipliers, Numer. Math. 20 (1973) 179–192.[2] T.J.R. Hughes, L.P. Franca, M.A. Balestra, A new finite element formulation for computational fluid dynamics: V. Circumventing

the Babuska–Brezzi condition: a stable Petrov–Galerkin formulation of the stokes problem accommodating equal-orderinterpolations, Comput. Meth. Appl. Mech. Eng. 59 (1986) 85–99.

[3] O.C. Ozienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1, fourth ed., McGraw Hill, New York, 1994.[4] T.J.R. Hughes, L.P. Franca, M.A. Balestra, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/

least square methods for advective-diffusive equations, Comput. Meth. Appl. Mech. Eng. 73 (1989) 173–189.[5] F. Brezzi, M.O. Bristeau, L. Franca, M. Mallet, G. Roge, A relationship between stabilized finite element methods and the Galerkin

method with bubble functions, Comput. Meth. Appl. Mech. Eng. 96 (1992) 117–129.[6] C. Baiocchi, F. Brezzi, L. Franca, Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S), Comput. Meth. Appl. Mech.

Eng. 105 (1993) 125–141.[7] E. Onate, Derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems, Comput.

Meth. Appl. Mech. Eng. 151 (1998) 233–265.[8] O. Klaas, A. Maniatty, M. Shephard, A stabilized mixed finite element method for finite elasticity. Formulation for linear

displacement and pressure interpolation, Comput. Meth. Appl. Mech. Eng. 180 (1999) 65–79.[9] A. Maniatty, Y. Liu, O. Klass, M.S. Shephard, Higher order stabilized finite element method for hyperelastic finite deformation,

Comput. Meth. Appl. Mech. Eng. 191 (2002) 1491–1503.[10] S. Commend, A. Truty, T. Zimmermann, Stabilized finite elements applied to elastoplasticity: I. Mixed displacement-pressure

formulation, Comput. Meth. Appl. Mech. Eng. 193 (2003) 3559–3586.[11] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewoods Cliffs,

NJ, 1987.


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