Tamtam Proceedings - lamsin

302 Ben Dhia1. IntroductionZooming numerically with great flexibility a globally defined numerical model whilesaving human and machine resources is essential in the designing and analyzing of complexproblems such as the engineering ones. Nnumerical methods have been developedduring the last decade to address this issue (see [1], for references on this subject).The Arlequin method [2, 3] is one of these approaches which proved to be practicallyrelevant [4, 5, 6, 1]. It creates a multimodel framework in which the models are not addedbut crossed and glued partially to each others. More precisely, it consists in1) a superposition of mechanical states in a subzone, denoted S of the whole domainΩ occupied by the studied mechanical system ;2) an energy distribution between the mechanical states in S, by using weight functionsbuilding this way a partition of models ;3) a gluing of these states in a subzone of S called the gluing zone.Since based on superposition of models, the Arlequin method may recall the overset gridmethods (also known as Chimera methods) introduced by the computational fluid community(eg. [7]). These methods are closely related to the overlapping Schwarz methodsstemming from the classical alternating Schwarz algorithm. However, the overlappingSchwarz method may not seem to be in essence the most appropriate tool to address themultimodel or multiscale issues.In the Arlequin framework, the crossing and gluing processes [3] leads to some relevantmultiscale models. The gluing needs however to be addressed with sufficient care. Manycoupling operators have been suggested in the first papers by Ben Dhia [2, 3]. Some ofthese operators have been theoretically proved in [8, 5] to be particularly well-suited, inthe continuous and discrete ranges. This mathematical aspect of the approach is hereindeveloped further.By construction, the Arlequin framework allows the “cohabitation” of incompatible models,sharing energies of the system in the superposition zones. These energ distributionsrequire the definition of a kind of “partitions of the unity” that need also a special careand constitute a second issue which is here addressed from a mathematical point of view.In the next section, we recall the continuous and the discrete mixed Arlequin equationsfor a model elasticity problem by using either a penalty coupling operator or lagrangianones. These problems are then analyzed mathematically and under some hypotheses andchoices of the Arlequin bricks, existence and uniqueness results are given. Section 4 isdevoted to the mathematical analysis of the behaviour of the Arlequin solutions whendifferent models are superposed and when, by the sharing of energy mechanism, one ofthe model is much more stressed than the other. The effectiveness of the choices of someTAMTAM –Tunis– 2005