finite element methods for constrained ...oden/dr._oden_reprints/...international journal for...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING. VOL 18.701-725 (1982)

FINITE ELEMENT METHODS FORCONSTRAINED PROBLEMS IN ELASTICITY

J. T. aDEN,Texas Institute for Computational "'[echanics. Department of Aerospace Engineering and Engineering Mechanics.

The University of Texas at Austin, Austin, Texas, U.S.A.

N. KIKUCHI

DepaT/ment of Mechanical Engineering and Applied Mechanics. The University of Michigan,Alln Arbor, Michigan, U.S.A.

SUMMARY

Several different variational formulations of boundary-value problems with constraints are discussed,with particular reference to constrained problems in elasticity. Special attention is given to exteriorpenalty methods. A discussion of the conditions necessary for penalty methods to provide a basis forstable and convergent finite element methods is given. In particular. the use of reduced integration isdiscussed and criteria on the order of reduced integration rules sufficient to produce stable and convergentschemes are described. Applications of reduced integration-penalty methods to incompressible elasticityproblems and contact problems are described.

INTRODUCfION

In this paper we shall consider several complications that arise in applications of finite clementmethods to equilibrium problems in elasticity in which various constraints arc imposed on themotion. For clarity, special attention will be given to the constraint of incompressibility inlinear elasticity, but some discussion of unilateral constraints for contact problems will alsobe given. Some of the work reported here summarizes recent results of Oden, Kikuchi andSong,14 where a detailed analysis of penalty methods for constrained problems in linearelasticity can be found.

Portions of this paper are expository in naturet and deal with properties of approximationsof somewhat general problems with constraints. We discuss conditions under which suchconstrained problems admit unique solutions and conditions for the existence of multipliersin Lagrange-multiplier formulations. These, in turn, lead to criteria for stability and conver-gence of mixed and penalty methods. We choose equilibrium problems in elasticity as aconvenient area of application of these ideas with constraints.

We give special attention to the Reduced Integration-Penalty methods (RIP) for severalreasons: first, they are very popular and, when they work, can be very effective: second. theyare not well understood; and, third, they constitute the best example of numerical methodsthat cannot be completely evaluated, judged or understood purely on the basis of numericalexperiments-an analysis of their stability and convergence properties is important to theirsuccessful use.

t This paper was presented bv the senior author at the symposium on TIle Unification of Finite Elements, FiniteDifferences and the Calculus of Variations. UnivCfslty of Connecticut. :-"Iay 19!1O.

22b

0029-5981/82/050701-25 S02.50© 1982 by John Wiley & Sons, Ltd.

Receired 7 August 1980Revised 25 June 1981

702 J. T. ODEN AND N. KIKUCHI

(1)

.. - ..

It is not our aim to deal with mathematical issues of a very deep nature here, since we wishto make the ideas and results accessible to a wide audience. Nevertheless, some of themathematical notions we use deserve a brief review and we have, on the advice of colleagueswho read earlier versions of this work, listed some of these in Appendix 1.

CONSTRAINED VARIATIONAL PROBLEMS

We begin by considering a classical minimization problem in the calculus of variations. Given

V = a space of 'admissible functions' (a real Hilbert space),F: V -+ IRa functional defined on V, andK = a nonempty closed subset of V.

find U E K such that, for any v in K, F assumes its minimum value at u:

uEK:F(u)~F(v) 'VvEK

It is well known that there will exist a unique minimizer of F on K whenever the followingfour (minimization) conditions hold:

M.I K is convex; i.e. if u and u belong to K, then Ou + (1- O)v e K, 0 < 8 < 1.M.2 F is strictly convex; Le. for 0<0< I and u ~ v, F(Ou+(I-O)v)<8F(u)+(I-O)F(v).M.3 F is differentiable on K; Le. for each u e K there exists an operator DF(II): V -+ V'

such that

dF(u + 8v)lim -(DF(u), v), 'VUE V

8~O' dO

where V' is the dual space of V, and (. , .) denotes duality pairing on V' x V (i.e.(DF(u), u) is the 'first variation' in F at u in direction v).

M.4 F is coercive; i.e. for v e K,

F(u) -+ +00 as lIullv -+ 00

where 11'lIv is the norm on V.

Virtually all of these conditions can be weakened (with the possible loss of uniqueness);the constraint set K need only be weakly sequentially closed, F need not be convex nordifferentiable. but should be lower semicontinuous in some sense. See Oden and Kikuchi 13

for further details.Now an important aspect of the minimization problem (1) in the case that M.I-M.3 hold,

is that the minimizer u of F can be characterized as the solution of a variational inequality:

UEK: (DF(u), V-II);;:'O 'VuEK (2)

Problems (1) and (2) are equivalent: any solution of (1) satisfies (2) and vice vcrsa (whcnevcrconditions rvU-M.3 hold). In the special case in which K is a linear subspace of V, (2) rcducesto the variational equality

U E K: (DF(u), v)= 0 lit, E K (3)

from which Euler cquations for the variational problem can be derived.In order to simplify the discussion, we shall, from this point onward, consider a restrictcd

class of problems in which the following conventions and assumptions are in force:

CONSTRAINED PROBLEMS IN ELASTICITY

N.! The functional F: V ~ IRis a quadratic functional of the form

F(v)=!a(v,v)-[(v) VEV

703

(4)

(5)

where a (. , .) is a symmetric bilinear form mapping V x V into IRand I is continuouslinear functional on V (i.e. IE V').

N.2 The bilinear form a(· , .) is continuous and V -elliptic; i.e. there exist positive constantsM and m such that

a(u, v) :5;Mllullvllvllv Vu, v E V

a(v,v);;;.mllvllt VVEV

We easily verify that if N.! and N.2 hold, then F is strictly convex and differentiable. Indeed,

(DF(u), v)= a(u, v)-[(v) Vu, v E VMoreover. since

l(v):5; Ilfllv·llvllvwhere 11'llv' is the norm on the dual space V' of V, we must have

(6)

(7)

F(v);;;. ;lllvllt-lI/l1v.lIvllv

for any vE V. Hence, F(v)~+oo as Ilvllv~oo. It follows that whenever conditions N.1 andN.2 hold, conditions M.2-M.4 hold. Thus, conditions N.1 and N.2 are sufficient to guarantccthe existence of a unique solution to the variational incquality (rccall (2) and (6»

ItEK: a(lt, v-It);;;./(v-lt) VVEK (8)

where K is a closed convex subset of V. Again, the solution It of (8) is the unique minimizerof the functional F(v) =!a (v, v) - I(v) in K.

Examples. The important minimization problems of interest hcre are those arising in linearand nonlinear elasticity. Considcr a homogeneous, isotropic linearly elastic body 0 with smoothboundary r subjccted to body forces I; and surface tractions Si on a portion rF and fixed alonga portion rD of r. The strain £i;(V) produced by a displacement field v is. of course, £;;(v) =(avi ax; + a v;!ax; )/2, 1:5; i, j :5;3, and the stress is

<Tij(V)= ( K - 2;)Oi/£kk(V) + 2µ'£jj(v)

= KOaj£kk(V) + 2µ£ ~ (v)

where K is the bulk modulus of the material. µ is the shear modulus, and £ff(v) is the deviatoricstrain,

£p;(V)=£ljh')-~Oji£kk(V) (for !1clR\

In this case, we take for the space of admissible displacements.

V={V=(VI. V2, v))!vjEH1(O), v;=O

a.e. on rD. i= 1,2. 3}

with norm

(9)

(10)


where dx = dX1 dX2 dX3, H1(0) is the standard Sobolev space of functions with square-integrable generalized derivatives, and meas rD >O. The functional F is now the total potentialenergy

F(v)=!a(v,v)-/(v), VE V(11)

a(u, v) = In [2µEP;(U)EP;(V) + KEkk (u)Emm (v)] dx

Here I is the work done by the external forces. Assuming IiE L 2(0), Sj E L 2(rF) (r F being asmooth surface), then

I(v) = J. livi dx + J S;V; dsn rF

As examples of constraints. we mention:

1. Incompressibility. In this case, K is a linear subspace of V,

K =K. ={VE Vldivv=O a.e. in 0}

where div v = Ekk(V) is the divergence of the displacement field v. The problem then is toFind U E K\ such that

a(u,v)=/(v) VveKI

where, whenever u, v E K I,

(12)

(13)

(14)

(15)

~- ...

a(u, v) =t 2µd:(u)£ff(v) dx

2. Unilateral contact. We consider situations in which the body comes in contact with arigid frictionless foundation a (normalized) distance s away from the body in its initialconfiguration. If rc is the contact surface. then

K = K2 ={VE Vlv, n-s ~O a.e. on rd (16)

n being a unit outward normal to rc. In this case, K is a nonempty closed convex subset ofV and v ' n is interpreted in the sense of traces of functions in Von to r.

The statement of the variational boundary value problem for contact of a compressiblcelastic body is

Find u E K2 such that

a(u,v-u)~f(v-u). VVEK\

If the material in the contact problem is also incompressible. we haveFind uE K1 n K2 such that

f. /J /) f \..J 'n2µEi;(u)Ei;(v-u)dx~ (V-U) vVEK,nK2

ALTERNATIVE VARIATIONAL FORMULATIONS

(17)

(IH)

While algorithms can be deviscd to solve the special variational problems (14), (17) and (18)directly, there are two disadvantages in these formulations from a practical point of view: (a)the direct approximation of these variational problems requires that we somehow approximate

CONSTRAINED PROBLEMS I~ ELASTICITY 705

the constraint at sets Kt, K2' or KI nK2 and this is generally quite difficult, and (b) theseparticular formulations do not employ Lagrange multipliers for handling the constraints andin each case the multiplier has a definite physical interpretation. For example, in the case ofthe incompressibility constraint, the Lagrange multiplier is the hydrostatic pressure, and thispressure must be known in order that the stress be determined. In short, in physical problemswith constraints, the multiplicrs associated with thc constraints are often equally as importantas the minimizers of the energy functional themselves. For this reason, it is standard practiceto consider Lagrange multiplier techniques for such problems.

Lagrange multipliers

We consider again the minimization problem (1) with F given by (4), conditions N.l andN.2 in force, and K now defined by

K = {v E VIBv = go} (19)

(22)

where B is the linear continuous constraint operator, B: V ... 0, 0 being a Hilbert space, andgo given data in the range ~(B)c 0 of B. The classical Lagrange multiplier approach to (19)consists of seeking saddle points (u, p) of the Lagrangian

L: V x 0' -.IR; L(v, q) = F(v) - [q, Bv - go] (20)

where 0' is thc dual of 0 and [ . , . ] dcnotcs duality pairing on 0' x O.The saddle point (14, p)will satisfy

L(lt, q) ~ L{It, p) ~ L(v, p) Vq EO', Vv E V (21)

The major issue at this point is: when will there exist n unique saddle point (/I, p) of L suchthat It satisfies (l)? It is not difficult to show (sec. e.g. Ekcland and Tcman") that sufficientconditions for the exislence of a solution to (21) are:

S.l Conditions M.2, M.3, M.4 hold for the functional F for K = V.S.2 There exists a Vo E V such that

L(vo. q) --. -00 as \Iq\lQ' --. 00, Vq EO'Thus, as we noted earlier, whenever N.1 and N.2 hold, condition S.l is satisfied. Moreover,if S.l and S.2 hold, the unique saddle point (u, p) is characterized by the equations

(DF(Il), z;)-[p,Bv]=O VVE V

[q, BIl] = [q, go] Vq E 0'

or, since F is given by (4),

a(u, v)-[p, Bv] = f(v) Vv E V

[q, Bu] = [q, go] Vq EO'

For example, in the case of the incompressibility constraint in linear elasticity, we have

[q,BvJ=foqdiVVdX

(23)


(26)

In this case, V is given by (9), f by (12), Q = e(m = 0'. [. , .]= (. , .) = the e inner product,and B = div: V -+ O(go = 0). If rD = r, conditioJ1S S.l and S.2 do not hold on V x 0; Q mustbe replaced by a certain quotient space.

Perturbed Lagrangian

A major problem with the Lagrange multiplier method is that the coerciveness conditionS.2 may not generally hold. We are then not guaranteed the existence of a unique Lagrangemultiplier p. To overcome this difficulty, one can regularize the problem by introducing aperturbed Lagrangian L: let e be an arbitrary positive number, and define

L,(v,q)=L(v,q)-!ellqll~' (25)

Clearly, S.l and S.2 now hold. Thus for each e >0, there exists a unique saddle point (un p,)of L, characterized by

a(u" V)-[PnBv]=f(v) VVE V

e[q, r1(p,)]+[q, Bu.] = [q, go] Vq E 0'

where r I is the inverse of the Riesz m~p j: 0 -+ 0'.The next issue is to determine if the sequence {(u" Pc)} of solutions to (26) will converge

to a saddle point of L, i.e. to the solution of (23).Since (It .. P.) is a saddle point of L .. we must have

F(It,)-[q, Bu.]-~lIqll~'

~F(v)-[p" Bv]-~I\p,llt,

for any q E 0' and v E V. Without loss of generality, we assume throughout this study thatker B ,c {O}. Then it is possible to choose va,c 0 such that Bvo = O. Choosing also q = 0, we have

(27)

But since F is coercive (recall M.4), this bound implies that a constant C, independent of e,exist such that

l\u,l\v"'C Ve>O

However (see item 2 in Appendix I), this guarantees the existence of a subsequence {II,.} ofsolutions 3nd an element U E V such that li,' converges weakly to tI as E' tends to zero:

11,'- II

Observe that by (26h,

[p" Btl, - go] = -E[p" rIc p,)] = -Fllp,1I6.o; 0

Hence, since L, (II" p,)'" L, (v. p,) for arbitrary t·, we have

F(II,)'" F( t·) +[p" BII, - KnJ ~ F( t·) V L' E K

We easilY show that the weak limit II of the subsequcnce {li,·} E K. Hence It is prccisely theminimizer of the functional F (since F(u) "'liminf ,'_II F( 11,').0; Fl t·) V l' E K).

CONSTRAINED PROBLEMS IN ELASTICITY 707

Troubles with the multipliers-the Babuska-Brezzi condition

To obtain a multiplier p as the weak limit of the subsequence {Pe.}, we must also show thatIIp.,llo', is uniformly bounded in e'. Unfortunately, this is generally impossible, for reasons weshall now explain, and it is necessary to add another condition to our theory to overcomesuch difficulties. We continue to choose go = O.

Let B*: Q' -+ V' denote the transpose of the constraint operator B:

and set

(B*q,v)=[B,Bv] VveV. VqeQ'

ker B* = {q e Q'I(B*q, v) = 0 Vve V}

(28)

(29)

If ker B* ~ 0, we can never expect the solution p to (23) to be unique, since the additionto p of any element in ker B* would also satisfy these equations. To overcome this difficulty,we introduce the quotient space Q'/ker B*, the elements of which are equivalence classes(cosets) defined by

p = {q e Q'I p - q e ker B *}

The space Q'/ker B* is a Banach space with norm

IIpllO'/kcr n- = inf lip + qllo'q"ker UO

(30)

(31)

Since [p, Bv] = [p +q, Bv] Vq E ker B" I we will use the notation [p, Bv] for representing [p +q, Bv] for evcry q e ker B*. That is, the tnUIliplier pin (23) can only be determined to withinan element in ker B*. HoweL'er, tire equivalence class p ma)' be unique in Q' I ker B*. Thus wcwrite

- .. -.

a(u,v)-[p,Bv]=f(v). VveV

[q,Bu]=O, VqeQ'

a(ur, v)-[P.,Bv]=f(v), VVE V

[q,er1(p,)+Bu.]=0, VqeQ'

(32)

in place of (23) and (26).We still must show that the sequences {P.} are uniformly bounded in E in some sense. For

this purpose, we shall introduce the so-called Babuska-Brezzi condition, which plays a funda-mental role in the theory oLelliptic equations: t

There exists a constant Q' > 0 such that

[p,IJv] ,.s;sup ~II 'tip E Q

liL V(33)

t The importance of conditions of this type in the theory of elliptic equations 1--1 ;Ind thcir approximation was 11rstdemonstrated by Babuska. who used this circle of ideas III many diverse applications. Further work on this areawas done in a well known and important paper bl{ llrczzi.5 A similar condition for the special case of Stokes'problem wllh the incompressibility condition di~ \. = 0 was studied somewhat carlier hy Ladvszhenskaya. and thecondition in connection with incompressible tlows has been referred to as the I.BB ILadyszhenskaya-Babuska-Brezzi)condition in that context.


Clearly, when (32) and (33) hold,

II' II [p" Bv] a(u" v)-f(v)a P. O'/ker B':SO; sup II II - sup II II""v v v ""v V v

:so;M(lIu,llv +1If1lv')

:so;C = constant (34)

Thus, there exists a subsequence of functions p" which converge weakly in Q'/ker B* toa unique element p. One can easily show the weak limit (u, p) of (u,', Pc') is the unique solutionof problem (32)1'

Strong convergence. When conditions N.!, N.2 and (33) hold, the situation is actually muchbetter than that implied above. Indeed, subtracting (32h from (32h. gives

a(u-u .. v)=[p-p .. Bv] VVE VThus, from (33),

Likewise, from (5h and (35),

mllu -utll~:so; a(u - u" u - u,) = [p -p" Bu - Bu,]

= e[p - Pro r I(p,)]

On the other hand, inequalities in (27) imply

IIPrll~':so;~(F(vn) - F(u,»e

(35)

(36)

(37)

for Vn E V such that BVn = O. Since u, is uniformly bounded in £ in V, it follows from theinequality that

for a proper positive number C >O. Thus we have

mllu - u,II~:so;ellp ~ P,1I0'/ker B,llp.llo'

:so;CJ(e )llp":' p,lIo'/ker n',

since r1 is an isometry. Combining (36) and (39), we have

lIu -1I,llv :so;C1 J£ and lip":' p,llo'/ker B' :so;c2Je

(38)

(39)

(40)

where C, and C: are positive numbers independent of e, Thus, II, --+ II in V and p, --+ Ii inQ'/ker B* strongly as e --+ O.

If ker B* = {O}, it is possible to obtain e-convergence for (40) instead of J e-convergence.Indeed, (36) and (37) are changed by

and

1711111 -1I,lIt:so; dp - p" r'(P.)]

:so;£[p-p"rl(p)]

(36a)

(37a)

Then we have

instead of (40).


(40a)

Penalty methods

In view of (32}4, the perturbed multiplier can be eliminated from (32h to give the variationalproblem

a(u, l')+ E-1[j(Bu. - go}, Bl'] = f(l') 'til' e V

This is precisely the characterization of the minimizer of the penalty functional F.: V -. IR

(41)

(42)

Thus, the perturbed Lagrange method is completely cquivalent to the following (extcrior)penalty method:

(i) For each E >0, find a minimizer of F, over all of V

(43)

(ii) Under conditions N.! and N.2, we will have a uniquc solution II. of (43) for cvery E >0and. morcover, II. -. II as E -.0, whcre II is thc unique solution to the minimization problem (1).(iii) To obtain an approximation of the Lagrangc multiplicr p, take

1 . .p, = - - J(BII, - go).

E

(iv) If the Babuska-Brezzi condition (33) holds, thcn we arc assured that subsequences P. -. Pin 0' Iker B* as E -. O.

Similar formulations to (41) can be obtaincd for problcms involving constraints representedby inequalities. Let B be a linear continuous operator from V into 0, and let go be a givendata in O. Using the partial ordering rclation .~' defined on the space 0, the constrained setK is supposed to be given by

K={VE VIBv-go~O}

Then we have

(Lagrangian Multiplier Method)(u,p)e VxM:

(DF(Il), v)-[p, Bv]=O, Vve V

[q-p, Bu-go]~O, 'tIqelv!

(Perturbed Lagrange Multiplier Method)(u"p,)e VXA-f:

(DF(II,), v)-[p" Bu]=O, VI: E V

[q-p"Er'(PF)+BIIF-go]~O, VqeAf(Penalty Method)

UF e V:

(DF(IIF), v)+.!.[j(BII-go)', Bt']=O, VVE VE

(44)

(45)

(46)

(47)

710 J. T. aDEN AND N. KIKUCHI

Here (DF(u,), v) = a (u" v) - f(v) and '+' is the generalization in the function space Q of theoperation defined on IR:4> + = (4) + 14>1)/2 for 4> E IR,and the set M is given by

M ={q E Q'\q ~O}

We note that the penalty functional Ft: V ... IRis now defined by

F.(v) = F(v)+ 21

eII(Bv - gotll~

PENALTY METHODS AND REDUCED INTEGRATION

(48)

(49)

(50)",Joo. _ ••

Of all the variational methods discussed in the previous section, the penalty method is, perhaps,the most attractive as a basis for computational methods for the following reasons:

1. It involves minimization of a functional F, on the entire linear space V rather than on theconstraint set K. Thus, it is not necessary to construct special approximations of the constraints.2. For each e >0, only u. is unknown; approximations P. of Lagrange multipliers can beobtained by an independent and direct calculation, P. = -e -lj(Bl/, - go). Thus, the number ofunknowns in a discretization of the penalty formulation is substantially less than a saddle-pointformulation based on Lagrange multipliers.3. The solutions (II" P.) to thc penalty problem will convcrge to a solution (l/, p) of thesaddle-point problem as e'" 0 under the conditions listed in the previous section. However,the penalty method will always yield a unique solution p, for the approximate multiplierswhile the direct saddlc-point approximation of p may not be unique.

There are, howevcr, some major ditlic\lltics in applying the penalty method due to therequirement (33). To demonstrate the key idcas, let us again consider the problem of incom-pressible elasticity. In this case, the penalized functional is

F,(v) = !a(v, v) - f(v) +LIidiv vll~

where a ( . , .) is defined by (24) and 11·110 is the L2 -norm (IIvll~ = Sn v 2 dx). The associatedvariational boundary-value problem is then to find II, E V such that

a (u" v) + e -I(div u" div v) = f(v) Vv E V (51)

For the problcm of incompressible elasticity (or Stokes' flow problem), the Babuska-Brezzicondition (33), which is the key assumption for showing the convergence of the penalty method,has been verified by Babuska and Aziz (Reference 4, Lemma 5.4.2, pp. 172-174) for a smoothdomain nc IR~,and by Temam (Reference 16. Lemma 1.2.4, p. 32) for the Lipschitz open-bounded domain n in RH

• Thus the convergence results shown in the previous section can beapplied to the penalty method (50) or equivalently (51).

To construct a finite clcment approximation of (51), we ctevclop. in the usual way, a family{VI,} of finitc-dimensional subspaces of \' using ell-piecewise polynomial basis functions ona scquence of meshes. The mesh size II identifies a space V" c V obtained through regular.uniform refinements of the mesh. We then approximate (51) on Vh with one provision: Inanticipation of some ditliculties to be described below. we shall evaluate the penalty term(2e )-lildiv "II~using numerical quadrature. In particular, let 1(') denote the quadrature rule

CONSTRAINED PROBLnlS IN ELASTICITY

(for a continuous function f)

711

EI(f) = L I,(f);

,=1

G

I,(f) = L W;f(~nj ... I

(52)

where E is the number of finite elements, Wj the quadrature weights, and ~j the integrationpoints in element n, c n. Then approximation of the penalty method (51) consists of seekingu~e Vh such that

a(u;,vh)+E-II(divu~divvh)=f(v") Vvhe V" (53)

Of course, if the order G of the integration rule (52) is sufficiently large, we can expect I(.)to yield exact integration: l(fg) = (f, g) = in fg d.\'.

It is a remarkable fact that (53), in general, provides physically unacceptable numericalresults under certain boundary conditions if exact integration is used. This has led manyinvestigators to use 'reduced integration' for finite element methods based on penalty formula-tions-by which is meant the use of an integration rule 1(') of order G lower than that whichis necessary to integrate the penalty terms exactly. Such devices have been advocated byZienkiewicz, Taylor and TOO,17 Hughes,S Malkus and Hughes,12 and others: for a completeanalysis of such methods and additional references, see Oden, Kikuchi and Song.I.US

A key to understanding penalty methods and reduccd integration is thc rcalization that,for fixed mesh size h, the solution to the discrete penalty problem should converge, in somesense, to a mixed finite elemcnt appr'oximation of (23). Thus, thcre must be inherent in anydiscrete penalty method a finite-dimensional spacc OJ. approximating the space 0' of Lagrangemultipliers. With this in mind, we choose the space OJ, to satisfy the following thrcc conditionsfor reduced-intcgration/penalty (RIP) methods:

R.1 Oh is such that Vqh e O~ and Vv" e Vt ..

(q", div Vi,) = l(qh div v") (54)

R.2 Let Bh: V" ~ 0", B~: OJ. ~ V;' be defined by

(55)

(i.e. Bil is the discrete approximation of the operator div defined by our choice of integrationrule l). Then OJ. must have the property that a constant ai, > 0 exists such that the followingdiscrete Babuska-Brezzi conditions hold:

I( J. d' I,)"j q IV" ",a"lIq 'llo/kefll~<$; sup 11,1'11 Vq E OJ. (56)

,hEVh \ v

R.3 There exists a unique p~ e 0;' such that

p~(W=-E-l divll~(~j) (57)

for 1 ~ e <$;E, 1.;:;; j <$;G (effectively, the nodal points of elements defining 01. are the integrationpoints given in the definition of 1(' ), and the notation

1I{(I'lIu/k"f /IT. = ".inf lilli' +(ql')*lIu (58)1.<, I .- kef U~

is applied wherc ker n~ is thc kerncl of thc map Bt. If kcr B,~c: kcr B*, we have

Iltt'lIn/kef IIi, = 1I1;1'1I"!k<'fW (59)

712 J. T. aDEN AND N. KIKUCHI

(60)

and qO" may be identified with the coset q" with respect to the kernel of B*. Oden. Kikuchiand Songl4 have shown that if conditions R.I. R2 and R.3 hold, then the following can beconcluded:

(i) For every E > 0 and h > 0, there exists a unique solution u= to (53).(ii) The sequence {(u~, p=)}, where p~ is given by (58), converges to (u", pO") as E -.0 for fixedh, where (1/", pO") is a solution of the mixed finite element approximation:

(" ") 0"''') ") "au, v - I(p diV V = t(v Vv E V"

l(q/ div u") = 0 Vq" E O~

(iii) For fixed E >0 and, Jz >0, the following error estimate holds

Ilu-u/llv ~ C1(1 +J...){' Jnf Ilu-v"lIva" v eVh

~CJ( 1+ :J"u-u~llv,where Cit C2, CJ are constants independent of E and It, and u = 1 if ker Bt = {Ol, u = 112 ifker Bt ~ {O},(iv) According to (60), every RIP method is related to a mixed method. However. thcre isone major difference: the direct approximation (60) may not have a unique solution! On theother hand, by construction, a unique (I/~. fJ~) solution to (60) is obtained for cach E > O. Thepoint here is that, in general, kcr Bt ~ kcr B* and ker Bt ~ {O}, particularly for rectangularelements. Thus, the conventional mixed finite element method may produce approximatemultipliers p" with components in kcr Bt. Then the system (60) is singular! However, anyRIP method satisfying R1-R3 will have

p;(~j)=-E-ldivu~(~j) (62)' ... -.-

Thus, let q~ E ker Bt ; i.e.

l( I·d· ") (B*" h) 0 'I.J I, v:q 0 IV V = q 0, v = v v E " (63)

Then, from

l( p~. q:;) = 0 Vq~ E ker Bt (64)

In other words, p~ is orthogonal to kcr Bt with respect to the discrete inner product lUg).Indeed, if (54) holds, this orthogonality is with rcspect the L2(0) inner product.

A PATCH TEST FOR RIP METHODS

dim Oh =111,dim V,. = II,

The numerical stability of the RIP mcthod is governed by the discretc Babuska-Brezzi condition(56). If it is not satisfied. there may occur oscillations in the approximatc pressure p~ as h -+ 0which quickly become unbounded. It is obvious that a necessary condition for (56) to hold isthat the matrix B approximating the constraint of full rank. i.e. if

II,HI

(lid' I,) \' "q. IV V = ~ B q,/:,'._\~l


where q" v, are the degrees-of-freedom of the approximate pressures and displacements,respectively. Then we must have

rank (Rrs) = m (65)

if (56) is to hold.However, (65) is not enough! For an unconditionally stable method, we must generally have

a" in (56) independent of h.We shall now describe a method for testing the LEE condition for specific choices of

elements and integration rules that is based on some techniques of GirauIt and Raviart 7 formixed methods. The idea is to find a function V" in the finite element space V" such that, forany ve V,

(66)

Indiv(VI'-v)q"dx=O Vq"eOi.

IIV"llv ~ Cllvllv

Then it can be shown (cf. Oden, Kikuchi and Songl4) that whenever (66) holds, the discrete

LEE condition (56) is satisfied with a constant a" independent of It.From these facts, we can construct a 'patch test' for verifying the stability of RIP methods:

1. Pick an arbitrary w" e V".2. Construct the patch integral

Ip = J div (V" _W")qh dx = - J Vq"· (Vh -w") dx +L f n· (Vh -w")q" ds~ ~ .~

for a patch nh of elements with intcrelement boundaries an. (n a unit outward normal to an.)with q" It ker Rt.3. Determine, by direct calculation, if there exists a Vi, such that Ip = 0 and IIV/llv ~ Cllwl'llv.4. For arbitrary ve V, set w" = Ph'i, where P" is the V -orthogonal projection of V on to Vh.Determine C, independent of It, such that Ip = 0 and IIV"llv ~ Cllvllv.

If these four steps prove to be possible for a given choice of Vh and 1('), then (56) holds foran a" independent of It.

There are actually only a limited number of elements which have been shown to satisfysuch patch tests. For two-dimensional problems with rectangular domains and Dirichletboundary conditions, the following choices satisfy the discrete LEE condition:

Piccewise constants/integration

Piecewisc linear

Piecewise constants/integration

OhPiecewise constants

'3-point' integration

'I-point' integration

II-point Gaussianquadrature'I-point' integration

Vh

1. 6-node. quadratic triangles

2. 9-node, biquadraticrectangle

3, 8-node, seredipidityelement

4. Composite of four equal4-node bilinear elements

5. 9-node, biquadratic '3-point' integration Piecewise linear

In these examples, ker B~ c ker B* and the analysis is based on a uniform mesh on arectangular domain. By 'I-point' and '3-point' integration in the examples. we mean that the


(67)

method is equivalent to some I-point and 3-point rule, but that the actual calculations aredone using a perturbed-Lagrangian formulation; e.g. for a typical element we have

Ku. - Up, = f+O'

BTU, + el\lpc = 0

where K and f are the element stiffness matrix and load vector 0' the 'connecting vector' offorces (which vanish upon assembling the elements) and n is the 'constraint matrix'. In (67),M is the mass matrix corresponding to the local pressure approximations assumed here to bediscontinuous across interelement boundaries. If we use piecewise constant or linear approxi-mations for Pro we then calculate

Pc = -e -IM-1STU. (68)

Then (67h results in a penalty method

Kuc+e-1SM-1BTu, =f+O'

even though no integration scheme has actually been identified. It is such schemes that areunderstood to be used in the examples above.

Numerical results indicate that the Babuska-Brezzi constant ah for schemes 1-5 is indepen-dent of h:

a/. = ao = constant> 0

Then, these schemes are numerically stable. However, therc is a price to be paid for schemes1-3: the low inregralioll rule leads to a loss of One complere order in tlte asympTOric rales ofCOli vergence of these merhods. Thus, whilc stable, they converge slowly. Schemes 4 and 5appear to be much better and are theoretically sound, but their analyses remains incomplete.

We mention two other examplcs which arc popular in thc literaturc:

In both of these cases (see Reference 15), we obtain for mixed boundary conditions

ah = aoh = O(h )

....... - ..

Vh

6. 4-node bilinear7. 9-node bilinear

II-point Gaussian2-point Gaussian quadrature

OhPiecewise constantPiecewise bilinear

and for Dirichlet boundary conditions

ali = aoh 2 = 0(112)

Thus, these methods are unstable. The convergence of such methods for rectangularuniform meshes has been explained in analyses by Johnson and Pitkaranta Q and in anindependent study by the authors 15 using a different approach. Nevertheless, the pressuremay diverge in L 2(0) (due largely to the presence of spurious components in ker B;~). Likewise.method 7 may have a suboptimal rare of convergence for displacemenrs in V and (cad ro pressureapproximations which diverge ill L 2(0).

It is true. however, that 'filtering' schemes can be devised for these elements which product.:pressurc approximations which are stable and converge in L~(O). According to the results in14. scheme 4 might be interpreted as the filtering strategy for scheme 6. However, methods6 and 7 sti!l appear to be delicate: they are sensitive to singularities and distortions in themesh geometry. Despite their popularity, these observations indicate that they should be usedwith great care.


CHECKERBOARDS AND NUMBERS

Several authors (e.g. Lee et al.11) have observed that in mixed finite elements approximations

of certain constrained problems, particularly Stokes problems or incompressible elasticityproblems with constraints such as div u = 0, spurious checkerboard patterns for the pressuresare present which appear to be superimposed over solutions which are physically reasonable.These spurious patterns are explained without difficulty in the theory outlined previously forthe case of square uniform meshes.

Observe that for RIP methods, the approximation p~ of the hydrostatic pressure may includecomponents in ker B~, even though it is clear that Bt p~' ¥ O.

Let us attempt to characterize ker B~ for a representative finite element for which spuriousnodes have been detected. In particular, in the case of a square domain on which Dirichletboundary conditions have been imposed. we consider a uniform mesh of 4-node bilinearelements for approximating the displacement Vi, and piecewise constants for the pressures q".This corresponds to I-point Gauss integration for the rule l(.) in the RIP method discussedearlier in the fourth section of the paper. For a mesh of N elements of width It, it can beshown 15 that

N(q", div v") = 211 L [Vkl(qk +q~ -q~ -qtc) + Vu(qk -q~ -q~ +qtc)) (69)

k-1

where Vkl is the ;th component of v" at node k (i = 1,2), and q'k is the value of the (constant)pressure q" in cell 111 surrounding node k, 111 = 1,2,3,4, with 111 = I for the lower left quadrantand the rest numbered in counterclockwise sequence around node k. Let us make theidentification, 0" = 0;,. If q"e ker Bt, then q" is in the range of B" and one can find a Vi, E V"such that B" v" = q". Hence, if q" E ker Bt, we must have

123" 123"qk+qk-qk-qk=O, qk-qk-qk+qk=O

This is precisely the checkerboard pattern indicated in Figure 1. Thus, the spurious modesobserved in mixed finite element calculations are merely the components of the approximatepressure in ker Br These spurious components should not appear in any RIP method forwhich conditions R.l, R.2, and R.3 hold, provided there are no singularities in the solution.However, as will be shown in examples, spurious components are sometimes observed in thepressure p~ locally, e.g. in certain subsels of the mesh. This generally occurs in methods forwhich a" = 0(1t"'), u > 0, in which case the method may be unstable in L 2(0).

Spurious components in p~ by, for example, scheme 6 (4-node bilinear and I-point integra-tion) can be eliminated by taking the filtering operation used in convergence analysis and also


(73)

(74)

(75)

introduced in the work. \I Indeed, the component of 'local' ker Bt is defined in a compositeof four equal 4-node bilinear elements as

c=l(p;-p;+p~-p~), (70)

where p~= p~In:, 1 ~ i ~ 4, and n~is the ith 4-node element within'the e th composite element.Then the pressure p~ is filtered by p~ defined by

-1 I C -2 2 C ·3 3 C -4 4 C (71)P.=p.-, P.=P.+, Pt=Pt-, Pt=Pt+

within each composite element. The pressure p~ is then with linear function which correspondsto the case of scheme 4. As shown in the following example, p~ does lIot include any spuriousmode. .

It is further noted that

lUi" div Vi,) = 0, Vv" if and only if q" = constant (72)

if q" is obtained from qh by using the filtering scheme (71). That is, the kernel of Bt in thespace 01. modified from 01. becomes same to ker B*. For the uniform mesh defined above,the following identity

J(qh div vI.) = J(qh div vI.)

also holds. Thus, the mixed formulation corresponding to (53) can be written as

a (u~, vI.) - /(p~ div vI.) = f(vh), Vv" E VI.

e/(q"p:) + l(qh div u;) = 0, Vqh E 0;'

where p~ is filtered by (71). The limit of (74) as e -+0 is then

a(u". Vh)_J(ph div vI.) = f(vh), VVE VI.

l(qh div u") = 0, Vq" E 0;'

by applying the notations under the heading 'Troubles with the multipliers-the Babuska-Brezzi conditions' in the third section of this paper.

Example. One example involves a Dirichlet problem of plane strain of a square slab ofincompressible linearly elastic material subjected to a constant body force F = (800, 800)applied over a square domain no. Let Young's modulus be E = 103

. We use the rather coarsemesh of 16 elements shown in Figure 2 and employ

02 elements (9-node biquadratics) for V"1(' ) - 2 x 2 Gaussian quadrature

Figure 3 shows computed hydrostatic pressure along section A-A.For other choice of data, we observe the checkerboard modes in hydrostatic pressure.

Indeed, if a point load f = 200(8(x - i, Y - f). 8(x - i, y - y)) is applied at point Ii,y) En,then the checkerboard modes in ker Bt appear to be activated as numerical results in Figure4 show. It is noted that data consisting of a point force inside the domain is not compatiblewith the variational formulation discussed so far, since it does not bclong to the dual spaceof V c 111(0).

The second example is a mixed boundary-value problem of plane strain of a rectangularslab of incompressihle linearly elastic material indented by a flat rigid frictionless punch, asshown in Figure 5. We checked convergence of the penalty method for 01 elements with the


I-

2 A

1

2

Figure 2. Finite element model of a Dirichlet problem

PressureDistribution

..........

A

-300 t-200

.100 1

100

200

A

Figure 3. Pressure distribution of the cross-section A-A by 2 x 2 Gaussian rule for a smooth applied body force

PressureDistribution

-400j-300

i-2001

(-l . _Ioo-~I (-)~_ (-) 1-

(~~--- --(~

lOOt

I

Figure 4, Pressure distribution of the cross-section A-A by2 x 2 Gaussian rule for a singular applied body force

\I

718

1

y

I

J, T. ODEN AND N. KIKUCHI

Depth of Indentation (d' 0,1)

i Rigid Flat Punch, , ,

I

f! 4-node ElementsI 1- point Integration: ~

~I # ~

E' 1000

I ,

Figure S. Finite element model of a mixed problem

,0.25

.... " -...

I-point Gaussian integration rule (scheme 6 in the previous section). The relation betweenthe square root of the mesh size and the inverse of the L2 norm of pressure is plotted inFigure 6. The plot suggcsts that thc L2 norm of pressurc converges with the optimal convergentrate O(h 1/2), since the solution u to the present problem is expected to be in H3/2

-6(0), where

8 is an arbitrary positivc number. It is nOled, however. that spurious modes can be observedin the pressure distribution, even though their size of oscillation decays as h -+ 0, as shown inFigure 7. If the filtering scheme (71) is applied, a smooth distribution of the pressure p~ canbe obtained for any choice of h > O.

-3~IO

A:&~01{)E -'ozv,-'

N...J

.. ~=-

o

L--1.....--.l---J-. __l -L

0125Mesh Size (h)

Figurc 6. Convcrltcnce of the L' norm of lhe pressure tield

9

B

7

6

5

4

3

2

CONSTRAINED PROBLBIS IN ELASTICITY

Y' -I

Figure 7. Pressute distributions on the line x = O'25

y.Q

719

CONT ACf PROBLEMS IN ELASTICITY

Results similar to thosc described above can bc obtained for problems with inequality con-straints such as that encountered in contact problcms in linear elasticity. Let us return toproblem (14) with the constraint sct K being given by the set K2 of (12):

K2={VE vlv· n-s~O a.e. on fc} (76)

~- .. Then the contact problem for compressible materials can be expressed as the variationalinequality

whereas its corresponding penalty formulation is

1{ . • )u, E V: a(u., v)+- (/(U,)" -5) ,v,,)={(\'e

'r/VE V

(77)

(78)

whcre (, , . ) denotes duality pairing on lV' x W, where W is the Sobolev space

lin == II . /I (/I' being a unit outward normal to rc), and (rb)- is the positive part of the functionrb (e.g. cb .. = max \0, cb) relative to an orderillg defined by rb ... 0 => cb (x) ... 0 a.e. on r c).

The corresponding Babuska-Brczzi condition (33) in the present case is given by

. (7'. L',,){3>O: {3lHrf/2.r •.oo;;sl~P-11 'II

l'~ ~ \ I

720 J, T. ODEN AND N, KIKVCHI

where 11·IIT/2.fc is the norm on the dual space (HI/2(r c)y of H1/2(r C). The existence of such

a constant may be shown by solving the problem

c/J E H1(D.): In {Vc/J . Vcb +c/J . cb} dx = (T, J) 'TIJ E HI(f1)

Then, by putting cb == c/J in the above equation we have

(T, c/J) = 1lc/J1l~for the solution c/J. Further we .have

Thus

I/Tllfl2.rc.s;; sup (T, c/J)¢eH'(Ol Ilc/JIII

To approximate (78), we use the RIP method:

\I" E V,,: a(u", v"} +.!.J[«U~)n - s)"' u~J = f(v") 'TIv" E Vit, E

(79)

where V" is, as before, a finite-dimensional subspace of V and J is a quadrature rule forintegration rc; e.g.

E' 0'

J(f) = ~ ~ Oif(TlI).-If-I

(80)

where Oi are quadrature weights and TIl quadrature points on rc.The method is stable and convergent whenever J defines a finite-dimensional space WIt of

approximate contact pressure u~ such that

R.l 'TIT",.;.I'E WIt,

J( ItAI,) f. II AI, dTT = TT Src

R.2 There exists a unique u~E Wit such that

(81)

(82)

R.3 There is a constant {3o> 0, independent of h (as h -+ 0) such that 13" = 130h 1/2 and

{3 II 1.'1 IJ(T\'~:)I I, (83)"T O.f"c~ sup II hll 'TIT E Wh

v·"V. U I

Again, the discrete Babuska-Brczzi condition (83) is the key to numerical stability of theapproximations of the contact pressures.

Example. As an example. we consider the plane strain problem of indentation of a rigidcylindrical punch into a rectangular block of incompressible linearly elastic foundation whichhas a modulus of elasticity of E = 10J. We model half the domain with 4 x 8 in. mesh ofrectangular elements, as shown in Figure 8.

'.

CONSTRAINED PROBLEMS IN ELASTICITY

R=8

T•4

E = 1000

111= .4999

',<

I8 "\~

Figure 8. Finite element model of a rigid punch problem

ContactPressure

Goot500

721

400

300

200

100

I"

Unit of Lenqth

Indentation Depthd" 0,8"

o

o

o

oo

a

o

o

Figure 9. Numerical results by Simpson's rule

__ un_ Simpson's Rule

- 2-Point Gaussian Rule

J. T. ODEN AND N. KIKUCHI

," " ,,,,

\\\\\\\\\

722

ContactPressure

1\ 600

500

F~,----400

300

200

100

Figure 10. Pressure distribution by 2-point Gaussian rule

... _~ - ..

Figure 9 shows results obtained using

Q2 elements (9-node biqu,adratic) for V"

1(' ) - 2 x 2 Gaussian quadrature

J( . ) - Simpson '5 rule

Computed results are quite satisfactory. However, if we use

J(.) - 2-point Gaussian rule

the distribution of contact pressure is no longer smooth, as shown in Figure 10.These unstable pressure distributions can again be explained by the LBB condition (83).

Indeed, for Simpson's rule, it is shown in Reference 15 that

J 1/2 0{3" = {3o I >However, for 2-point Gaussian rule, we have

a {3 I 1/2 J 1/2/-f" = 0 I - {31 I

Thus, this is not suitable for the Babusak-Brezzi condition (83).Therefore, for certain models. {3,. may be zero, in which case the discrete problem IS

ill-posed.

ACKNOWLEDGEMENTS

The support of this work by the U.S. Air Force Office of Scientific Research under ContractF-49620-78-C-0083 and the National Science Foundation under Grant NSF-ENG 75-07846is gratefully acknowledged. The authors also thank Dr. Y. J. Song for his help with numericalcomputations.

APPENDIX I: SOME MATHEMATICAL PRELIMINARIES

This appendix is provided for a brief revicw of several of the mathematical ideas usedthroughout this paper. For additional details, see standard texts on functional analyscs, e.g.

18Taylor and Lay.


1. Strong and weak convergence

The spaces V of admissible functions described earlier, we recall, are assumed to be realHilbert spaces equipped with inner products (. , ')v and norm 11'llv (e.g. when V is given by(4), (0, v)v = (0, vh = In Ui./Vi.; dx, lIuliv = lIull:= (0, D)~/2). When we say that a sequence Ii", E Vconverges to an elemcnt U E V, we mean that

lim Ilu", - ullv = 0",_00

This type of convergence is called strong convergence; to be specific, we may say that Ii",

converges strongly to U and we write

u", -. u (as m -. 00)

For each space Vane has its (topological) dual space V' which consists of all continuouslinear functionals defined on V. Thus, V' consists of linear functions I which map V con-tinuously into real numbers. As noted earlier, we use the notation

"1(1i)=(f,u) IE V', UE V

to denote the value of I at Ii.

Whenever a, sequence Um E V has the property that

lim 1(lim)=/(Ii) VIE V'",_00

we say that lim converges weakly to Ii and we write

11m - II (as In -. 00)

Every strongly convergent sequence converges weakly, but the converse is not true. Forexample, consider the Fourier series representation of functions gin L2(0, 1):

00

g(x)= L gm1'",(x);1'm(x)=J2sinm7Tx;O<x<1m=\

1

g", = (g, 1'",) = fa g1'm dx

The sequence gm of real numbers converges to zero as m tends to infinity. Hence, the sequenceof functions 1'm converges weakly to zero:

lim (g, 1'",) = lim gm = 0 Vg E L2(0,1)m ....OO m_w

However, 1I1'IIL'IO,n= 1 so that the 'Ym does not converge strongly to zero.

2. Weak compactness

In Hilbert spaces it can be shown that every scquence bounded in norm has a weaklyconvergent subsequence. In other words, if ll", is a sequence in V with the property that

IIltmllv"; C = constant

then we are guaranteed the existence of a subsequence lim' and an element Ii E V such thatlim' .... U. This property also prevails in rellexive Banach spaces.


3. Riesz map

The Riesz representation theorem establishes that for every linear functional! in the dualV' of a Hilbert space V, there exists a unique element Uf such that

<!,V}=(Uf'V}v 'v'VEV

In other words, the action of any linear functional! on V can be reproduced by an mnerproduct of the elements of V by an element uf uniquely determined by every f.

The correspondence between! and Uf can be characterized by a mapping iv: V ... V' knownas the Riesz map for V. The operator iv is linear, continuous, invertible, and is an isometryfrom Von to V' (i.e·llullv = lIivullv.).

4. Ouotient space

Let M and N denote linear subspaces of a linear vector space V such that M ('\ N = to}and every v E V is of the form v = m + II, with mE M and n EN. We then say that V is thedirect sum of M and N, written V = M + N, and that M and N are complementary subspacesof V; in particular, N is a complement of M and vice versa.

We introduce an equivalence relation R on V according to which vectors VI and V2 aresaid to be equivalent modulo M if VI - V2 E M. An equivalence class corresponding to R isthen the set

V· = {u E Vlv - u EM}

The set of all such equivalence classes is denoted VIM. When endowed with the operationsU' + V· = (It + v)· and au' = (au)', VIM is a linear vector space, called the quotient space ofV modulo M. If V is a Banach space with norm 1I'llv then VIM is a Banach space whenprovided the norm

Ilv'lIvlM = inf IIv - ullvuE~1

If N is any complement of M, then VIM is isomorphic to N (indeed, the map y: V ... VIM,y(v) = V', is an isomorphism). If A1 is a closed linear subspace of a Banach space V, then thedual space (VIM)' is isometrically isomorphic to MJ., where Ml. = {v* E v'lv*(v) = 0 if v EM}.Thus, (VIM)' and Ml. are both algebraically and metrically (and, therefore, topologically)equivalent, and we write (VIM)' =:,; Ml..

REFERENCES

I. J. Babuska. 'The finite element method with Lagrange multipliers', NlIm. ,\,fa,. 20 11973).2, J. Babuska, 'Error bounds lor fini1e elemcnt method', NWlIer. Alatl1, 16, 322-333 (1971),3. J. Babuska, 'The finite element method for elliptic equations with discontinuous coefficients', Compurillg, 5,

207-213 ([ 970).4. I. Babuska and A. K. Aziz. 'Survey lectures on the mathematical foundations of the finite element me1hod. , :,

in .\falltemarical FOllndations of lite Fini'e EI.·mrnt Alelhod wilh Applications to Parrial Differelttial Eqllll/ians(Ed. A, K, Azizl Academic Press. N,y', 1972, pp. 1-354.

5. F. 13rezzi, 'On the existence. uniqueness and approximation of saddle-point problems arising from Lagrangianmultipliers', R.A.I.R,O.1l ()9741.

6. I. Ekeland and R. Temam. Com'ex AI/all'sis ami Variational Problems. North-Holland. Amsterdam. 197o,7. V. Girault and P. A. Raviart, Fini'e Eleme,,' Ml',hoci~ lor rite Nuvier Srokes EqlJations, Lecture ;o.;otes in

Mathematics, vol. 749. Springcr· Verlag. Bc.rlin. 1980.8. T, J. R. Hughes. 'Equivalence oi finite elements for nearly incompressible elasticity',], Appl. Mech. (March 1977).

CONSTRAISED PROBLE~1S IN ELASTICITY 725

.. ~- ...

9. C. Johnson and J. Pitkaranta, 'Analysis of somc mixed finite elemcnt mcthods related to reduced integration',Research Report, Dept. of Computer Science, Chalmers University of Technology and the University of Goteborg,Fall,1980.

10. O. A. Ladyszhenskaya, The MalJremalical17rtory of Viscolls Incompressible Flows, Gordon & Breach, New York.1969 .

11. R. L. Lee, P. M. Gresho and R. L. Sani, 'Smoolhing techniques for ccrtain primitive variables solutions of theNavier-Stokes equations', Inl. J. nllm. Melh. Engng, 14. 1785-1804 (1979),

12. D. S. Malkus and T. J. R. Hughes, 'Mixed finite element me1hods-reduced and selective in1egration techniques-aunification of concepts', Complller Melh. Appl. Mech. En gr. 15 (1978).

13. J, T, Oden and N, Kikuchi, 'Existence theory for a class of problems in nonlinear elasticity', TICOM Reporl78-3. The University of Texas at Austin. Auslin. Texas, 1978.

14. J. T. Oden, N. Kikuchi and Y. J. Song, 'Reduced integration and exterior penalty methods for finite elementapproximations of contact problems in incompressible elasticity', TICOM Report 80-2, The University of Texasat Austin, Austin, Texas, 1980.

15. y, J. Song, J. T. Odcn and N. Kikuchi. 'RIP-methods for contact problems in incompressible elasticity', TICOA,[Report 80-7, The University of Texas at Austin, Austin. Texas, 1980.

16. R, Temam, Navier-Stokes Eqllalions, NOflh-Holland, Amsterdam, 1974.17. O. C. Zienkiewicz. R. L. Taylor and J. M, Too, 'Reduced integration technique in general analysis of plales and

shells', Inl. J. nlln!. Melh. Engng, 3 (1971).18. A. E. Taylor and D. C. Lay, Inlroduction 10 Fllnctional Analysis, 2nd edn, Wiley, N.Y., 1980.

finite element methods for constrained ...oden/dr._oden_reprints/...international journal for...

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