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Arlequin method 307refinements of the superposed models. An absolute limit situation consists in superposing(locally) a rigid model to a deformable one. In this situation there is no need for thedistribution of the internal energies. Notice that in these very particular situations, onecan establish a bridge between the fictitious domain method with a distributed Lagrangemultiplier [12] and the second mixed Arlequin method.3.2.2. A limit behaviourWhen considering deformable bodies, the stability analysis of the Arlequin problemsrequires that each α i must be strictly positive. But one can ask the question of existenceof a limit behaviour of the Arlequin solutions whenever either α 1 or α 2 tends to theunity (the other tending to zero) in situations where, in the unglued part of S, the twomodels are quite different. For this, let us for instance assume that in the unglued partof S, one model is fractured and the other is not. Moreover, we assume that the crackis strictly embedded in the interior of the unglued part of the fractured structure. Let usthen define two global monomodel problems, we denote by M 1 and M 2 , respectively. Thefirst problem is associated to the fractured domain, while the second is associated to the“same” but sound domain. We denote by u M1 and u M2 the respective solutions. Now, ifin the Arlequin framework α i is associated with the Arlequin model part of M i , i = 1, 2then we can prove the following result (important in practice) :Theorem 5- Under ad hoc hypotheses, the Arlequin solutions tend to u Mi when α i tendsto 1 and when β i has the same order as α i , i = 1, 2This result is based in the following lemma :Lemma- Under ad hoc hypotheses, the restriction of the displacement field u 1 , defined inthe second mixed Arlequin problem, is bounded by the norm of u 2 , independently of α 1and β 1 and vice versa.4. Numerical resultsEach point developed here will be exemplified by a numerical result, obtained by G.Rateau during his PhD thesis (collaboration with EdF). A zoomed deformed tyre is givenhere : a 2D-Model is superposed to a thin global curved beam.RemerciementsThe support of Électricité de France is greatfully acknowledged.5. Bibliographie[1] BEN DHIA H., RATEAU G., « The Arlequin method as a flexible engineering design tool »,International Journal for Numerical Methods in Engineering, accepted in 2004, to appear inTAMTAM –Tunis– 2005