Shape optimization of bimaterial (micro)structures using XFEM and a level set description

This work focuses on the problem of finding the optimal microstructural design for a material to suit particular applications. For this purpose, an analysis method was developed and implemented to get the response of bimaterial structures. The analysis is based on the two following methods:
• The level set description is used to represent the geometries of the (micro)structures, which can be rather complex. It allows to handle conveniently large shape modifications, which are often encountered in optimization process.
• The extended finite element method (XFEM) allows circumventing the conform meshing of the (micro)structures. This is really advantageous for optimization applications, since a fixed mesh can be used throughout the simulations.
To perform shape optimization, a sensitivity analysis of the design variables is needed. Fol- lowing the work by Van Miegroet (2007), an analytical approach to the computation of the derivatives has been developed. The stiffness matrix derivative is computed analytically starting from its discretized expression. Then, the expression of the stiffness matrix derivative is used to evaluate the sensitivity of various objective functions such as the compliance (global) or the stresses (local).
The analytical sensitivity analysis is tested on simple examples, based on small scale struc- tures. Then, it is validated against other sensitivity approaches: a finite difference approach, a semi-analytical approach,... Finally, the analytical sensitivity analysis is illustrated by solving classic academic shape optimization problems.