Sensitivity analysis of discrete variable topology optimization

This paper studies sensitivity analysis for discrete variable topology optimization. Minimum compliance of plane stress structures is considered. The element thickness is the design variable and is named as element density, whose value is 0 or 1. According to the concerned element density and its surrounding density distribution, all the design elements are classified into three types: white interface elements, black elements, and white isolated elements. Their sensitivities are studied by shape sensitivity analysis, topological and configuration derivative, respectively. The white interface element sensitivity obtained by shape sensitivity justifies the sensitivity filter. Based on theoretical deduction and inspired by the analytical, topological derivative formula, the black element sensitivity for inserting the square hole that is consistent with the background finite element mesh is a linear combination of three quadratic forms of stress components. The combination coefficients are dependent on material constants and irrelevant to the stress and strain state, which can be determined by parameter fitting through special load conditions. The white isolated element sensitivity can also be determined by parameter fitting inspired by the configuration derivative. The obtained formula resolves the paradox of the white isolated element sensitivity. The present can further solidify the theoretical foundation for the discrete variable topology optimization methods via Sequential Approximate Integer Programming (SAIP).