Topology optimization of 2-D mechanical metamaterials using a parametric level set method combined with a meshfree algorithm

Two-dimensional (2-D) micro-architectured mechanical metamaterials are designed using a topology optimization approach that integrates a parametric level set method (PLSM) with a meshfree method based on compactly supported radial basis functions (CS-RBF). The PLSM is employed as the optimization algorithm to achieve desired microstructures with targeted material properties. The effective elastic properties, including the bulk modulus, shear modulus and Poisson’s ratio, are predicted using a strain energy-based homogenization method and the CS-RBF meshfree algorithm. Two sets of optimizations are implemented: one is for the case of a single solid material, and the other is for the case of two solid materials, each involving an additional void phase. Three numerical examples are provided for each case with the same optimization objectives: maximizing the effective bulk modulus, maximizing the effective shear modulus, and minimizing the effective Poisson’s ratio under given volume fraction constraints. The numerical results reveal that the newly proposed approach can generate smooth topological boundaries and optimal microstructures. In particular, the current method can topologically optimize auxetic metamaterials with a negative Poisson’s ratio. It can also be extended to design other periodic metamaterials, including those with a negative coefficient of thermal expansion or frequency bandgaps.