Sensitivity analysis with the extended finite element method on bimaterial structures

Material tailoring can be formulated as a structural optimization problem. To design com- posite microstructures, which can exhibit very complex geometries, a level set description is used to efficiently represent the microstructures. Fixed mesh methods and non conforming finite el- ement approximations, such as the extended finite element method (XFEM), presents several advantages to solve these problems of optimal material design. The key issue to solve an optimization problem is to perform an accurate sensitivity analysis. Recently, van Miegroet et Duysinx (2007) developed a semi-analytical procedure to achieve shape optimization using XFEM on void-material structures. The shape is represented by a level set function, whose parameters are considered as design variables. The optimization problem is solved by applying mathematical programming algorithms. To realize material tailoring, we extend this work to be able to deal with bimaterial structures. Working on void-material structures, the finite element approximation on a partially or on a fully filled element remains the same and the number of degrees of freedom does not change either. These characteristics of the approximation slightly ease the derivation procedure. A valid sensitivity analysis can be performed implementing a semi-analytical method based on a finite element computation of the stiffness matrix derivative. Dealing with material-material interfaces, the approximations and the number of degrees of freedom associated to an element filled with only one material or an element filled with two different materials are different. A new approach, adapted to the bimaterial framework, to achieve the sensitivity analysis is needed. To circumvent the problems linked to a finite difference computation, an analytical method is developed to perform the sensitivity analysis. Several other method, just like the global finite dif- ference, the semi-analytical computation, ... are also implemented to evaluate the performance and the validity of the analytical method. The developments are illustrated on academic test cases. Those are designed so that all the possible pathologic cases arise. The performance osf each method can then be easily evaluated. Finally, the analytical method is used in a simple optimization problem, where the shape of a solid inclusion is optimized to reach minimal compliance