Shape optimization of microstructural designs subject to local stress constraints within an XFEM-level set framework

The present paper investigates the tailoring of bimaterial microstructures minimizing their local stress field exploiting shape optimization. The problem formulation relies on the extended finite element method (XFEM) combined with a level set representation of the geometry, to deal with complex microstructures and handle large shape modifications while working on fixed meshes. The homogenization theory, allowing extracting the behavior of periodic materials built from the repetition of a representative volume element (RVE), is applied to impose macroscopic strain fields and periodic boundary conditions to the RVE. Classical numerical homogenization techniques are adapted to the selected XFEM-level set framework. Following previous works on analytical sensitivity analysis [31], the scope of the developed approach is extended to tackle the problem of stress objective or constraint functions. Finally, the method is illustrated by revisiting 2D classical shape optimization examples: finding the optimal shapes of single or multiple inclusions in a microstructure while minimizing its local stress field.