Regularization of shape optimization problems using FE-based parametrization

This paper introduces a general fully stabilized mesh based shape optimization strategy, which allows for shape optimization of mechanical problems on FE-based parametrization. The well-known mesh dependent results are avoided by application of filter methods and mesh regularization strategies. Filter methods are successfully applied to SIMP (Solid Isotropic Material with Penalization) based topology optimization for many years. The filter method presented here uses a specific formulation that is based on convolution integrals. It is shown that the filter methods ensure mesh independency of the optimal designs. Furthermore they provide an easy and robust tool to explore the whole design space with respect to optimal designs with similar mechanical properties. A successful application of optimization strategies with FE-based parametrization requires the combination of filter methods with mesh regularization strategies. The latter ones ensure reliable results of the finite element solutions that are crucial for the sensitivity analysis. This presentation introduces a new mesh regularization strategy that is based on the Updated Reference Strategy (URS). It is shown that the methods formulated on this mechanical basis result in fast and robust mesh regularization methods. The resulting grids show a minimum mesh distortion even for large movements of the mesh boundary. The performance of the proposed regularization methods is demonstrated by several illustrative examples.