Laurent Van Miegroet

Generalized shape optimization using X-FEM and Level Set methods

This paper presents an intermediate approach between parametric shape optimization and topology optimization. It is based on using the recent Level Set description of the geometry and the novel eXtended Finite Element Method (XFEM). The method takes benefit of the fixed mesh work using X-FEM and of the curves smoothness of the Level Set description. Design variables are shape parameters of basic geometric features. The number of design variables of this formulation is small whereas various global and local constraints can be considered. The Level Set description allows to modify the connectivity of the structure as geometric features can merge or separate from each other. However no new entity can be introduced. A central problem that is investigated here is the sensitivity analysis and the way it can be carried out efficiently. Numerical applications revisit the classical elliptical hole benchmark from shape optimization.

Stress constrained optimization using X-FEM and Level-Set description

Topology optimization has experienced an incredible soar since 1988 and is now available within several commercial finite element (FE) codes. Meanwhile, parametric shape optimization has found few industrial applications. This is may be due to its inherent difficulties to deal with mesh management with boundary modifications. Recently the extended finite element method (X-FEM) has been proposed (see [1] for a review) as an alternative to remeshing methods. The X-FEM method is naturally associated with the Level Set description of the geometry to provide an efficient treatment of problems involving discontinuities and propagations. Up to now the X-FEM method has been mostly developed for crack propagation problems, but the potential interest of the X-FEM method and the Level Set description for other problems like shape and topology optimization was identified very early (see [2]). In this paper, the authors present an intermediate approach between parametric shape and topology optimization by using the X-FEM and Level Set Description. The method benefits from fixed mesh work using X-FEM and from smooth curves representation of the Level Set description. One major characteristic of the approach is to be able to model exactly void and solid structures.