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

This article extends recently developed finite variation sensitivity analysis (FVSA) approaches to an inverse homogenization problem. The design of metamaterials with prescribed mechanical properties is stated as a discrete density-based topology optimization problem, in which the design variables define the microstructure of the periodic base cell. The FVSA consists in estimating the finite variations of the objective and constraint functions after independently switching the state of each variable. It is used to properly linearize the functions of binary variables so the optimization problem can be solved through sequential integer linear programming. Novel sensitivity expressions were developed and it was shown that they are more accurate than the ones conventionally used in literature. Referred to as the conjugate gradient sensitivity (CGS) approach, the proposed strategy was quantitatively evaluated through numerical examples. In these examples, metamaterials with prescribed homogenized Poisson's ratios and minimal homogenized Young's moduli were obtained. A hexagonal base cell with dihedral $D_3$ symmetry was used to produce only metamaterials with isotropic properties. It was shown that, by using the CGS approach instead of the conventional sensitivity analysis, the sensitivity error was substantially reduced for the considered problem. The proposed developments effectively improved the stability and robustness of the discrete optimization procedures. In all the considered examples, when more accurate sensitivity analyses were performed, the parameters of the topology optimization method could be tuned more easily, yielding effective solutions even if the settings were not ideal.