Multi-material topology optimization considering isotropic and anisotropic materials combination

Multi-material topology optimization that is based on homogenization schemes has evolved from the conventional optimization methods to the general multi-material solution methodology. While the former interpolation schemes restrict the solution to the isotropic material with constant Poisson's ratio, the latter generalizes the application to both isotropic and anisotropic materials, as well as a combination of them. Within density-based multi-material topology optimization, the discrete material optimization scheme is a well-known tool to solve the mixture of isotropic and anisotropic materials; nevertheless, most of its applications are based on open-source finite element codes. An alternative to the discrete material optimization scheme is the element duplication method that relies on the idea of stacking multiple elements and assigning one candidate material to each stacked element, thus avoiding the need-to-know important information from open-source finite element engines to compute the first-order sensitivities. Besides its simple implementation along with commercial finite element solvers, the element duplication procedure diminishes computational efficiency due to the element stacking process. In this paper, a solution for this process is proposed for the multi-material topology optimization problem by considering the mixture of isotropic and anisotropic materials without the need for stacking elements in commercial finite element engines, improving the numerical efficiency of element duplication methods as well as being an alternative to compute the sensitivities in the discrete material optimization scheme. Traditional topology optimization response sensitivities are thoroughly discussed, and several numerical examples are presented demonstrating the effectiveness of the proposed approach. In addition, a new approach to compute displacement sensitivities is presented.