Finite variation sensitivity analysis in the design of isotropic metamaterials through discrete topology optimization

This paper extends recently developed Finite Variation Sensitivity Analysis (FVSA) approaches to an inverse homogenization problem. It is stated as a topology optimization problem, which is used to design metamaterials with prescribed mechanical properties. The objective function depends on the homogenized Poisson's ratio, and the homogenized Young's modulus is constrained. A hexagonal base cell, discretized by the finite element method, with dihedral D3 symmetry is used, which results in metamaterials with isotropic properties. Under these conditions, elemental stiffness matrices are not all identical, and each design variable is related to six different elements. The FVSA is suitable for discrete density topology optimization methods, it is used to properly linearize the functions of binary variables, then the optimization problem can be solved through sequential integer linear programming. In this work, novel sensitivity expressions, more accurate than the ones from the mainstream approach in literature, are developed. The obtained improvements are illustrated with numerical examples.