Topology optimization of auxetic microstructures with isotropic and orthotropic multiple materials based on element-free Galerkin method

The topology optimization (TO) framework for isotropic and orthotropic multi -material periodic microstructures with auxetic performance is proposed based on element -free Galerkin method (EFGM). Negative Poisson's ratio is chosen as the objective function while satisfying specified volume constraints, and the effective elastic properties of the multi -material microstructures are computed by the energy -based homogenization method under periodic boundary conditions. The alternating active -phase algorithm is integrated into meshless solid isotropic microstructure with penalization (SIMP) scheme, and the optimal topological configuration of the multiphase composite is obtained by solving the subproblem of each two-phase materials' distribution. Several topological optimization examples are presented to evaluate the effects of different material design parameters on the topological configurations, minimum objective functions, and negative Poisson's ratios (NPRs) of microstructures. The results demonstrate that the optimum microstructures with isotropic and orthotropic multiple materials can generate more diversified patterns of re-entrant and chiral topological configurations and wider ranges of NPRs in comparison to those with a single material. The auxetic performance of microstructures will be effectively improved by reasonably adjusting the material design parameters and the obtained NPRs will be easy to get close to or break through -1.