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RAIROMODLISATION MATHMATIQUE

ET ANALYSE NUMRIQUE

DANIELA CAPATINA-PAPAGHIUCNICOLAS RAYNAUDNumerical approximation of stiff transmission problemsby mixed finite element methodsRAIRO Modlisation mathmatique et analyse numrique, tome 32, no 5 (1998), p. 611-629.

SMAI, EDP Sciences, 1998, tous droits rservs.

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MATHEMATKAL MODELLING AND NUMERICAL ANALYSISMODLISATION MATHEMATIQUE ET ANALYSE NUMERIQUE

(Vol 32, n 5, 1998, p. 611 629)

NUMERICAL APPROXIMATION OF STIFF TRANSMISSION PROBLEMSBY MIXED FINITE ELEMENT METHODS (*)

Daniela CAPATINA-PAPAGHIUC and Nicolas RAYNAUD (l)

Abstract We are interested in the approximation of stiff transmission problems by the means of mixed finite element methods We provea resuit of uniform convergence in e, the small parameter, as the discretization 's step tends to 0, showing thus the robustness of the methodThe discrete problem is then numencally solved via hybnd methods, since the l-conditioning of the matrix makes the standard Uzawa'salgorithm impracticable The numencal results ascertain an optimal rate of convergence for both the stress tensor and the anti-planedisplacement So, contranly to primai methods, the mixed ones avoid the locking phenomenon over any regular triangulation Elsevier,Pans

Rsum On s'intresse l'approximation des problmes de transmission raide par mthodes d'lments finis mixtes On tablit unrsultat de convergence uniforme en , le petit paramtre quand le pas de discrtisation tend vers 0, prouvant ainsi la robustesse de lamthode Le problme discret est ensuite rsolu numriquement via une mthode hybride, car le mauvais conditionnement de la matrice rendimpraticable Valgorithme standard d'Uzawa Les rsultats numriques montrent une convergence optimale en O(h) pour le tenseur descontraintes et pour le dplacement anti-plane En conclusion, contrairement aux mthodes pnmales conformes, les mthodes mixteschappent au verrouillage numrique, et ceci indpendamment de triangulation Elsevier, Pans

1. INTRODUCTION

In this paper we study stiff transmission problems and their solution by Raviart-Thomas mixed finite elementmethods. A stiff transmission problem is a parameter-dependent problem which is best described by the followingphysical problem: an elastic body onto which is grafted a thin shell of thickness e and whose stiffness is anincreasing linear function of 1/e. One clearly sees that, when s tends towards zero, the thin shell acts like astiffener.

This kind of problem has already been solved using primai finite element methods, as in [4] and [21]; in ailinstances, locking has been exposed. Briefiy, the locking phenomenon can be described as a loss of convergencewhich stems trom the approximation scheme: although mathematical convergence is secured and computeraccuracy is adequate, the approximation does not square with the expected solution. A dfinition of locking isgiven in [4] and a criterion for avoiding locking is established in [11].

In previous cases, primai finite element methods have been employed. Mesh choice is then very important andconstraining altogether: the accuracy of the approximation for small values of the parameter s strongly dpendson the structure of the mesh, see [21]. Essentially, standard primai methods give place to numerical locking onarbitrary meshes.

Mixed methods weaken the continuity constraint at the internai edges of the triangulation by introducing anadditional field, the stress tensor. Actually, this is a side effect of the main purpose of mixed methods, which isto take into account the equilibrium constraint. As a resuit, the behavior of the computed solution is independentof the mesh structure. Moreover, we shall prove that the mixed methods are free of locking.

Let us note that the final System is never symmetrie definite positive since it corresponds to the solution of asaddle-point problem. For our model problem, the usual Uzawa's algorithm is impracticable because the resultingmatrix is ill-conditioned for small values of the parameter e.

(*) Manuscript received February 12, 1996, Revised May 15, 1997C1) Laboratoire de Mathmatiques Appliques, UPRES A 5033, Universit de Pau, 64000 Pau

E-mail daniela [email protected] fr

M2 AN Modlisation mathmatique et Analyse numrique 0764-583X/98/05Mathematical ModelLing and Numerical Analysis Elsevier, Pans

612 Daniela CAPATINA-PAPAGHIUC and Nicolas RAYNAUD

To counter this drawback, we shall make use of mixed hybrid methods, which dualize the coupling constraintsin order to obtain a symmetrie defmite positive System. The underlying idea of mixed hybrid methods is to finda field which satisfies the equilibrium conditions only at the element Ie vel (see [22]). For lo w-order methods orin particular instances, this field can be worked out by hand. However, such a field can be computed systematicallyby using the results established in [7].

The numerical approximation of the stiff transmission problem is surprising: it is bereft of locking, at least forthe lowest-order method. This is an important result since it means that we can apply mixed methods to solveefficiently locking-prone problems.

An outline of the paper is as follows: in Section 2, we introduc our model problem as well as the Ventcelproblem, formally obtained by letting e to go to 0, and we give a summary of theoretical results. This is essentiallya recall of some results of [19]. In Section 3 we give some numerical results, showing how locking occurs whenusing a primai finite element method on an arbitrary regular triangulation. The last section is devoted to the studyof mixed finite element methods for the stiff transmission problem. Firstly, a uniform convergence result isestablished, which shows that the Raviart-Thomas method is robust in the sense of [4], and this for any orderk e N. Secondly, the system obtained after discretization is hybridized. We obtain in this manner a linear systemwhose matrix is symmetrie de finite positive and whose unknowns are the multipliers introduced by the mixedhybrid method. Next, we recover by a local post-processing both the approximations of the displacement and ofthe stress tensor. Numerical results are also presented for the lowest-order method: they clearly show an optimalrate of convergence for any type of mesh structure.

2. STIFF TRANSMISSION PROBLEMS

2.1. Physical motivation

To fix the frame of our study, we consider a linear anti-plane displacement elasticity problem. Of course, thistype of problem is not the most mearringful in structural mechanics, However, the terminology linked to it willhelp us to illustrate the main features of the problem.

The gnerai situation of the elasticity problem studied hre is described as follows. Let Q* be an elastic body(an open set) whose boundary is denoted by dQ+; let us suppose that dQ+ = E \j T* where both E and F* areof non-vanishing measure. This elastic body has a thin shell, noted Q~ , grafted onto E\ let us prcise that e isa physical parameter which will tend to 0, characterizing the thickness of the thin shell. The boundary I* isclamped.

Figure 1. Elastic body Q + with a thin shell .

M2 AN Modlisation mathmatique et Analyse numriqueMathematical Modelling and Numerical Analysis

STTFF TRANSMISSION PROBLEMS 613

Both bodies Q+ and Q behave according to a linear and isotropic law which we suppose to be characterizedby their respective Lam coefficients , /u and /e, ju/e. It is thus obvious that the thin shell Q~ behaves itself asa tightener on 27, see figure 7. We dnote by Q the interior of

+ o> Q~ . For an anti-plane two-dimensionalproblem, the displacement is described by a scalar function u verifying:

(1)

-Au=f1 e

Aue

d u = i

= J

on Z

where v dnotes the unit normal to FE N outwardly directed with respect to Q and the unit normal to 27 outwardlydirected with respect to Q+. The superscripts + and - in the above transmission conditions indicate that the traceon 27 is worked out from the value of u in Q+ and Q~ respectively.

Problem (1) stems from various physical phenomena. It models, for example, the way heat spreads out in a bodyQ+ which is either partially or completely covered with a highly heat-conductive thin shell Q~ ; the quantityu is then the temprature. However, it is the study of an electric current scattering in a plate bordered by a highlyconductive rod which, in steady-state, constitutes the most typical example modeled by (1). This model is alsoa good approximation of wave scattering problems by a scatterer covered with a penetrable thin shell, providedwe limit ourselves to the main part of the operator of the partial differential quation set on Q+ (see [15] and [6]).

2.2. Variational formulation

We dnote by | , | the euclidean norm of the vector i; of IR2 and by ^ . T| the scalar product of the vectors ^ andr\ of U2 identified with column vectors. Problem (1) fits into the following setting. The thin shell Q~ is describedby:

Q~ = {x e [R2, x = m + n>(m) ; m G 27 and 0 < t < sh(m)} ,

fgiV = {xe IR2, x = m + eh( m ) v( m ) ; m e 27},

where h is a smooth real-valued