Topological derivative-based topology optimization of incompressible structures using mixed formulations

Abstract

In this work an algorithm for topological optimization, based on the topological derivative concept, is proposed for both nearly and fully incompressible materials. In order to deal with such materials, a new decomposition of the Polarization tensor is proposed in terms of its deviatoric and volumetric components. Mixed formulations applied in the context of linear elasticity do not only allow to deal with incompressible material behavior but also to obtain a higher accuracy in the computation of stresses. The system is stabilized by means of the Variational Multiscale method based on the decomposition of the unknowns into resolvable and subgrid scales in order to prevent fluctuations. Several numerical examples are presented and discussed to assess the robustness of the proposed formulation and its applicability to Topology Optimization problems for incompressible elastic solids.