Abstract: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 (X-FEM). 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.

Abstract. This paper presents and describes an intermediate approach which takes its place between the two classical methods of shape and topology optimization. It is based on using the recent Level Set description of the geometry and the novel eXtended Finite Element Method (X-FEM). The method benefits from the fixed mesh work using X-FEM and from 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 remains small whereas various global and local constraints can be considered. A key problem which is investigated here is the sensitivity analysis and the way it can be carried out precisely and efficiently. Numerical applications revisit some classical 2D (academic) benchmarks from shape optimization and illustrate the great interest of using X-FEM and Level Set description together. The paper presents the results of stress constrained problems using the proposed X-FEM and Level Set based formulation. A central issue is the sensitivity analysis (related to the compliance and/or the stresses) and the way it can be carried out efficiently.A special attention is also paid to stress constrained problems which are often neglected in other Level Set Methods works.