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FROM THE HU-WASHIZU FORMULATION TO THE AVERAGE NODALSTRAIN FORMULATION

BISHNU P. LAMICHHANE

Abstract. We present a stabilized nite element method for the Hu-Washizu formulation of linearelasticity based on simplicial meshes leading to the stabilized nodal strain formulation or node-based uniformstrain elements. We show that the nite element approximation converges uniformly to the exact solutionfor the nearly incompressible case.

Key words. mixed nite elements, average nodal strain, nearly incompressible elasticity, primal anddual meshes

AMS subject classications. 65N30, 65N15, 74B10

1. Introduction. There are many different mixed formulations for linear elasticity.Among the most popular mixed formulations are HellingerReissner and Hu-Washizu for-

mulations. There are mainly two reasons to use a mixed formulation in elasticity. Onereason is to compute the variables of interest: stress, strain or pressure more accurately andthe other reason is to alleviate the locking effect in the nearly incompressible regime [10, 6].

In this paper, we present a variational approach to obtain the stabilized nodally inte-grated simplicial elements presented in [22]. Recently, the average nodal formulation fordifferent quantities of interest in linear and non-linear elasticity has been of particular inter-est [4, 13, 5, 21, 12]. We have shown that the average nodal pressure formulation originallypresented in [4] can be analyzed within the framework of mixed nite elements [18]. Themixed nite element scheme in [4] is based on displacementpressure formulation, where thedisplacement is discretized by using a standard linear nite element space and the pressureis discretized by using a piecewise constant nite element space on the dual mesh. Usinga similar approach for the Hu-Washizu formulation of linear elasticity, we show that theaverage nodal strain formulation can also be analyzed in the framework of a mixed niteelement method. However, in contrast to the average nodal pressure formulation we have tostabilize the scheme in the average nodal strain formulation to obtain the coercivity. Thisyields a consistent variational framework for the nodally integrated tetrahedral presented in[22].

2. The boundary value problem of linear elasticity. In this section, we introducethe boundary value problem of linear elasticity. Let R d , d {2, 3}, be a bounded domainwith Lipschitz boundary , and let be occupied with a homogeneous isotropic linear elasticmaterial body. For a prescribed body force f L 2 ()d , the governing equilibrium equationis written as

div = f , (2.1)

where is the symmetric Cauchy stress tensor. The stress tensor is a function of thedisplacement u dened as

= C (u ), (2.2)

where (u ) = 12 (u + [u ]t ) is the strain tensor, and C is the fourth-order elasticity tensor,which acts on a tensor d as

= C d := (tr d )1 + 2 d . (2.3) Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National

University, ACT 0200, Canberra, [email protected]

1


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Here 1 is the identity tensor, and and are the Lame parameters, which are positiveand constant in view of the assumption of a homogeneous body. For simplicity, we assumehomogeneous Dirichlet boundary condition on :

u = 0 on . (2.4)We are also interested in the nearly incompressible regime, which corresponds to .

The standard and a stabilized Hu-Washizu formulation.As usual, L 2 () denotes the space of square-integrable functions dened on with the innerproduct and norm being denoted by ( , )0 , and 0 , , respectively. We denote the setof symmetric tensors in by S := {d L 2 ()d d | d is symmetric } having each componentbeing square-integrable. We will use the notation for the dot product between two tensors , d S as : d = di =1

dj =1 ij dij . The space H

10 () consists of functions in H 1 ()

which vanish on the boundary in the sense of traces. To write the weak or variationalformulation of the boundary value problem, we introduce the space V := [H 10 ()]d of dis-placements with inner product ( , )1 , and norm 1 , dened in the standard way, see[11, 9].

We dene the bilinear form A(, ) and the linear functional () by

A : V V R , A(u , v ) := C (u ) : (v ) dx, : V R , (v ) := f v dx.Let V be the space of continuous linear functionals dened on V . Then the standard weakform of linear elasticity problem is as follows: given V , nd u V that satises

A(u , v ) = (v ) , v V . (2.5)

The assumptions on C guarantee that A(, ) is symmetric, continuous, and V -elliptic. Henceby using standard arguments it can be shown that (2.5) has a unique solution u V .Furthermore, we assume that the domain is convex polygonal or polyhedral so that u

[H 2

()]d

V , and there exists a constant C independent of such thatu 2 + div u 1 C f 0 . (2.6)

We refer to [9, 17] for the proof of this regularity estimate. Since we have imposed homoge-neous Dirichlet boundary condition, Cauchy stress satises

: 1 dx = (2 + d )div u dx = (2 + d ) u d = 0 .Thus we restrict the space of stress to S 0 with

S 0 := { S | : 1 dx = 0 }, (2.7)see also [19]. The standard Hu-Washizu formulation [16, 23] is to nd ( u , d , ) V S S 0such that

a ((u , d ), (v , e )) + b((v , e ), ) = (v ), (v , e ) V S ,b((u , d ), ) = 0 , S 0 , (2.8)

where

a ((u , d ), (v , e )) = d : C e dx, and b((u , d ), ) = ( (u ) d ) : dx.


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The well-posedness of the saddle point problem (2.8) is analyzed by using the standardsaddle point theory, see [10, 6]. The main difficulty in the discrete setting is to show thatthe bilinear form b(, ) satises a uniform inf-sup condition and the bilinear form a (, ) iscoercive on a suitable kernel space. Using some simple nite element spaces, it is not possibleto satisfy these two conditions simultaneously as the bilinear form a(, ) is not elliptic onthe whole space V S already in the continuous setting. This problem is well-known in thecontext of MindlinReissner plate theory and Darcy equation, see [1, 20, 2].

This gives us a motivation to modify the bilinear form a (, ) consistently by addinga stabilization term so that we obtain the ellipticity on the whole space V S . Themodication of the bilinear form a (, ) is obtained by adding an additional term

( (u ) d ) : ( (v ) e ) dxto the bilinear form a (, ) so that our modied saddle point problem is to nd ( u , d , ) V S S 0 such that

a ((u , d ), (v , e )) + b((v , e ), ) = (v ), (v , e ) V S ,b((u, d ), ) = 0 , S 0 , (2.9)

where

a ((u , d ), (v , e )) = d : C e dx + ( (u ) d ) : ( (v ) e ) dx,b(, ) is dened as above, and > 0 is a parameter.

3. Finite element discretization. We consider a quasi-uniform triangulation T h of the polygonal domain , where T h consists of triangles in d = 2 or tetrahedra in d = 3. Let N h be the set of all vertices of T h , dened as

N h := {i : i is a vertex of element T T h }, and N := # N h .

Each vertex i will also be identied with its co-ordinate vector x i . A dual mesh T h isconstructed based on the primal mesh T h so that the elements of T h are called controlvolumes. The dual mesh for triangular meshes is introduced in the following way. Let x i , x jand xk be three vertices of an element T T h , and x ij , x jk and xki be middle points of the three edges of T . Let cT be the centroid of the triangle T . We connect cT to the threemiddle points of the edges by straight lines to divide the triangle into three quadrilateralsQ i , Q j and Q k , where each quadrilateral Q i shares only one vertex i of the triangle T , andhence corresponds to this vertex of the triangle T . Let T i be the set of triangles having thecommon vertex i . For each vertex i N h , we select a set of quadrilaterals

Q i := {Q : Q corresponds to the vertex i of the triangle T, T T i }.

The control volume V i corresponding to the vertex i is dened as

V i := QQ i Q,

see also [14]. The collection of these control volumes then denes the dual mesh, see Figure3.1. We can follow a similar construction for a tetrahedron T selecting centroids of fourfaces, middle points of four edges and a centroid cT of T .


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000111

00110 00 01 11 1

000111

000111000111

x i

x j

V i

V j

cT

T

x i

x j

xk

Q iQ j

Q k

x ij

x jk

xki

Fig. 3.1 . Primal and dual meshes with vertices x i and x j and the triangle T divided into three quadrilaterals in the two-dimensional case

We call the control volume mesh T h regular or quasi-uniform if there exists a positiveconstant C > 0 such that

Ch d | V i | hd , V i T h ,

where h is the maximum diameter of all elements T T h .It can be shown that, if T h is locally regular, i.e., there is a constant C such that

Ch dT | T | h dT , T T h

with diam( T ) = h T for all elements T T h , then this dual mesh T h is also locally regular.However, the analysis can be extended to other dual meshes. Let S h be the standard linearnite element space dened on the triangulation T h ,

S h := {v H 1 ()| v| T P 1 (T ), T T h },

and its dual volume element space M h ,

M h := { pL 2 ()| p| V P 0 (V ), V T h }.

Let ph M h and uh S h such that u h =N i =1 u i i and ph =

N i =1 pi i , where i is

the standard nodal basis function associated with the node i , and i is the characteristicfunction of the volume set V i dened by

i (x) =1 if x V i ,0 else.

Dening the space of bubble functions

B h := {bT H 1 (T ) : bT | T = 0 , and T bT dx > 0, T T h },we introduce our nite element space for the displacement as V h = [S h B h ]d . Denotingthe d + 1 barycentric co-ordinates of an element T T h by i , 1 i d + 1, the cubicbubble function bT associated with the element T can be written as bT (x) = cbd +1i =1 T i (x),where the constant cb is determined by using the fact that bT evaluated at the centroid of the element T is one.

Then the nite element approximation of (2.8) is dened as a solution to the followingproblem: nd ( u h , d h , h ) V h S h M h such that

a ((u h , d h ), (v h , e h )) + b((v h , e h ), h ) = (v h ), (v h , e h ) V h S h ,b((u h , d h ), h ) = 0 , h M h ,

(3.1)


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where

M h := { h [M h ]d d | h is symmetric, and h : 1 dx = 0 }.4. An a priori error estimate. In order to show that the nite element approxima-

tion converges uniformly to the continuous solution, we introduce an orthogonal projectionoperator: h : L 2 () M h . Using the denition of h , we write the strain as

d h = h ( (u h )) ,

where operator h is applied to tensor (u h ) componentwise. The denition of dual meshesallows us to write the action of operator h on a function v L 2 () as

h v =n

i =1

ci i , (4.1)

whereci =

1|V i | V i v dx.

It is easy to see that operator h is local. Using this expression of d h , we can now eliminatethe stress h and the strain d h from our discrete problem to obtain the reduced problem of nding u h V h such that

Ah (u h , v h ) = (v h ), v h V h ,

where

Ah (u h , v h ) =

h (u h ) : C h (v h ) dx +

( (u h ) h (u h )) : ( (v h ) h (v h )) dx.

(4.2)In the following, we will use a generic constant C , which will take different values at differentplaces but will be always independent of the mesh-size h and Lame parameter . Since his stable in the L 2 -norm, we have

h v 0 , C v 0 , , v L 2 (). (4.3)

Furthermore, the following approximation property follows by using the fact that M h con-tains piecewise constant functions with respect to the dual mesh T h :

v h v 0 , Ch v 1 , , v H 1 (). (4.4)

Using the fact that h is a projection operator, and h C ( (u h )) = C h ( (u h )), we cansimplify the expression for Ah (, ):

Ah (u h , v h ) = h (u h ) : C (v h ) dx + ( (u h ) h (u h )) : (v h ) dx. (4.5)Remark 4.1. When = 0 , we recover the node-based uniform strain elements for

simplices introduced in [13]. The nodally integrated tetrahedral element analyzed in [22] is obtained by replacing by some constant fourth-order tensor. However, the consistent derivation shows that it is not necessary to use a fourth-order tensor.


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Lemma 4.2. The bilinear from Ah (, ) is coercive on V h V h uniformly with respect to , i.e., there exists a constant C independent of such that

Ah (u h , u h ) C u h 21 , . (4.6)

Proof . In the rst step, we apply the Korns inequality to obtain

u h21 , C K (u h )

20 , .

An application of the triangle inequality gives

u h21 , C K (u h ) h (u h )

20 , + h (u h )

20 ,

maxC K

,C K 2

(u h ) h (u h ) 20 , + 2 h (u h )20 , + tr h (u h )

20 ,

= maxC K

,C K 2

Ah (u h , u h ).

Hence

Ah (u h , u h ) C u h 21 ,

with

C =1

max C K ,C K2

.

Thus the coercivity constant C depends on and but does not depend on .Remark 4.3. Here the parameter > 0 can be arbitrary as we have a consistent sta-

bilization. However, the smaller value of decreases the coercivity constant in the previous lemma, and the larger value of can pollute the approximation. The choice of the parameter > 0 can be utilized for accelerating the solver as in an augmented Lagrangian formulation [3]. As suggested by the previous lemma, one natural choice of the parameter is 2 [22].Since we do not focus on this aspect of the problem, we simply put = 1 in the rest of the paper.

The projection property of h also allows us to write Ah (u h , v h ) as

Ah (u h , v h ) = (h C (u h ) + (u h ) h (u h )) : (v h ) dx orAh (u h , v h ) = (h C (v h ) + (v h ) h (v h )) : (u h ) dx. (4.7)

Thus we pose the positive denite formulation: nd u h V h such that

Ah (u h , v h ) = (v h ), v h V h . (4.8)

Remark 4.4. As h is not a uniformly coercive operator with respect to the L 2 -norm even on the space of piecewise constant functions, we cannot show the coercivity of Ah (, )without adding the stabilization term. This is closely related to the instability of P 1 P 1discretization of Stokes problem, see [15, 18].

Remark 4.5. We have to use V h = [S h B h ]d for the nite element space for the displacement eld to obtain the uniform convergence of the nite element approximation in


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the incompressible limit. However, we obtain the coercivity of Ah (, ) also on the space [S h ]das can be seen from Lemma 4.2. Hence the formulation is stable in the compressible regime even if we use [S h ]d to discretize displacement in the Hu-Washizu formulation.

Now we focus to show the uniform convergence of the nite element solutions of thestatically condensed formulation (4.8) in the incompressible limit. The main ingredients of the proof are Fortin interpolation operator and Strangs rst lemma. A similar idea hasbeen used in [19] to prove the uniform approximation of the Hu-Washizu formulation in thenearly incompressible regime using a class of quadrilateral meshes.

We know that the pair of spaces ( V h , M h ) satises the uniform inf-sup condition [18].There exists a constant > 0 independent of the mesh-size such that

inf qh M h

supv h V h div v h q hv h 1 , q h 0 , .

Therefore, the Fortins interpolation operator [10, Section II] or [7, 4.8 & 4.9] I F h : [H 2 ()

H 10 ()]d V h can be dened for the problem of nding ( I F h w , ph ) V h M h such that

I F h w : z h dx + div z h ph dx = w : z h dx, z h V h , div I

F h w q h dx = div w q h dx, q h M h .

(4.9)

The approximation property of the Fortins interpolation operator is shown in the followinglemma. For proofs, see [8, 19].

Lemma 4.6. Under the regularity assumption (2.6), the operator I F h satises the ap-proximation property

u I F h u 1 + div u h (div I F h u ) 0 Ch f 0 .

Lemma 4.7. Assume that u is the solution of (2.5) , and regularity estimate (2.6) holds.Then there exists a v h V h so that

|A(w h , u ) Ah (w h , v h )| C |w h |1 , h f 0 , , w h V h .

Proof . Using the simplied expression of the bilinear form Ah (, ) from (4.7), a combi-nation of CauchySchwartz and triangle inequality yields

|A(w h , u ) Ah (w h , v h )|

= (w h ) : (C (u ) C h (v h ) (v h ) + h (v h )) dx C |w h |1 , ( C (u ) h C (v h ) 0 , + (v h ) h (v h ) 0 , ) (4.10) C |w h |1 , G (u , v h ), (4.11)

where

G (u , v h ) = C (u ) h C (v h ) 0 , + (v h ) h (v h ) 0 , . (4.12)

We use the expression (2.3) of the action of C on a tensor and the fact that h C (v h ) =C h (v h ) as well as a triangle inequality to write

G (u , v h ) 2 (u ) h (v h ) 0 , + div u h div v h 0 ,+ (v h ) (u ) 0 , + (u ) h (v h ) 0 , .


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Using again a triangle inequality, the approximation property of h given by (4.4) andstability of h in the L 2 -norm from (4.3), we obtain

G (u , v h ) C (h u 2 , + u v h 1 , + div u h div v h 0 , ) . (4.13)

The result now follows by choosing v h = I F h u and using Lemma 4.6.Now we formulate the main result of this paper.Theorem 4.8. Assume that u and u h be the solutions of (2.5) and (4.8) , respectively,

and regularity estimate (2.6) holds. Then, we obtain an optimal a priori estimate for the discretization error in the displacement

u u h 1 , Ch f 0 , . (4.14)

where C < is independent of and h .Proof . We use the rst lemma of Strang, see [6], to prove this theorem. Using the

coercivity of Ah (, ), Lemma 4.7 and (2.5), we nd that for v h := I F h u

u h v h21 , C |Ah (u h v h , u h v h )| = C |Ah (u h v h , u h ) Ah (u h v h , v h )|

= C |A(u h v h , u ) Ah (u h v h , v h )| C u h v h 1 , h f 0 , .

In terms of the triangle inequality, we obtain

u u h 1 , C ( u v h 1 , + v h u h 1 , ) Ch f 0 , .

5. Conclusion. We have presented a nite element method for a stabilized Hu-Washizuformulation using primal and dual meshes. The approach yields a consistent variationalframework for recently introduced average nodal strain formulation [13, 5, 21, 12, 22] ex-plaining the necessity of stabilization. However, we show that it is necessary to enrich thespace of displacement with the bubble functions in order to achieve the uniform convergenceof the nite element approximation in the incompressible regime.

REFERENCES

[1] D.N. Arnold and F. Brezzi. Some new elements for the ReissnerMindlin plate model. In Boundary Value Problems for Partial Differerntial Equations and Applications , pages 287292. Masson,Paris, 1993.

[2] P.B. Bochev and C.R. Dohrmann. A computational study of stabilized, low order C 0 nite elementapproximations of darcy equations. Computational Mechanics , 38:323333, 2006.

[3] D. Boffi and C. Lovadina. Analysis of new augmented lagrangian formulations for mixed nite elementschemes. Numerische Mathematik , 75:405419, 1997.

[4] J. Bonet and A. J. Burton. A simple average nodal pressure tetrahedral element for incompressibleand nearly incompressible dynamic explicit applications. Communications in Numerical Methodsin Engineering , 14:437449, 1998.

[5] J. Bonet, H. Marriott, and O. Hassan. Stability and comparison of different linear tetrahedral formula-tions for nearly incompressible explicit dynamic applications. International Journal for Numerical Methods in Engineering , 50:119133, 2001.

[6] D. Braess. Finite Elements. Theory, fast solver, and applications in solid mechanics . CambridgeUniversity Press, Second Edition, 2001.

[7] D. Braess. Finite Elements. Theory, fast solver, and applications in solid mechanics . CambridgeUniversity Press, Second Edition, 2007.

[8] D. Braess, C. Carstensen, and B.D. Reddy. Uniform convergence and a posteriori error estimators forthe enhanced strain nite element method. Numerische Mathematik , 96:461479, 2004.

[9] S.C. Brenner and L. Sung. Linear nite element methods for planar linear elasticity. Mathematics of Computation , 59:321338, 1992.


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[10] F. Brezzi and M. Fortin. Mixed and hybrid nite element methods . SpringerVerlag, New York, 1991.[11] P.G Ciarlet. The nite element method for elliptic problems . North Holland, Amsterdam, 1978.[12] E. A. de Souza Neto, F. M. Andrade Pires, and D. R. J. Owen. F-bar-based linear triangles and tetrahe-

dra for nite strain analysis of nearly incompressible solids. part I: formulation and benchmarking.International Journal for Numerical Methods in Engineering , 62:353383, 2005.

[13] C.R. Dohrmann, M.W. Heinstein, J. Jung, S.W. Key, and W.R. Witkowski. Node-based uniform strainelements for three-node triangular and four-node tetrahedral meshes. International Journal for Numerical Methods in Engineering , 47:15491568, 2000.

[14] R. E. Ewing, T. Lin, and Y. Lin. On the accuracy of the nite volume element method based onpiecewise linear polynomials. SIAM Journal on Numerical Analysis , 39:18651888, 2002.

[15] V. Girault and P.-A. Raviart. Finite Element Methods for Navier-Stokes Equations . Springer-Verlag,Berlin, 1986.

[16] H. Hu. On some variational principles in the theory of elasticity and the theory of plasticity. Scientia Sinica , 4:3354, 1955.

[17] V.A. Kozlov, V.G. Mazya, and J. Rossmann. Spectral Problems Associated with Corner Singular-ities of Solutions to Elliptic Equations . Mathematical Surveys and Monographs 85. AmericanMathematical Society, Providence, RI, 2001.

[18] B.P. Lamichhane. Inf-sup stable nite element pairs based on dual meshes and bases for nearly incom-pressible elasticity. IMA Journal of Numerical Analysis , 29:404420, 2009.

[19] B.P. Lamichhane, B.D. Reddy, and B.I. Wohlmuth. Convergence in the incompressible limit of nite

element approximations based on the Hu-Washizu formulation. Numerische Mathematik , 104:151175, 2006.[20] A. Masud and T.J.R. Hughes. A stabilized mixed nite element method for darcy ow. Computer

Methods in Applied Mechanics and Engineering , 191:43414370, 2002.[21] F. M. Andrade Pires, E. A. de Souza Neto, and J. L. de la Cuesta Padilla. An assessment of the

average nodal volume formulation for the analysis of nearly incompressible solids under nitestrains. Communications in Numerical Methods in Engineering , 20:569583, 2004.

[22] M. A. Puso and J. Solberg. A stabilized nodally integrated tetrahedral. International Journal for Numerical Methods in Engineering , 67:841867, 2006.

[23] K. Washizu. Variational methods in elasticity and plasticity . Pergamon Press, 3rd edition, 1982.

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