Topology optimization of binary microstructures involving various non-volume constraints

In this paper, we use the Topology Optimization of Binary Structures (TOBS) method recently developed by Sivapuram and Picelli (2018) for microstructural optimization. This is the first work in topology optimization addressing various non-volume microstructural constraints with discrete (0/1) design variables. The objective and constraint functions are linearized at each iteration, and the obtained linear problem is solved through Integer Linear Programming (ILP) using sensitivities computed from asymptotic homogenization. A periodic filter is used to make the optimized solutions checkerboard-free and mesh-independent. Volume minimization problems subject to elastic and thermal constraints are considered. The examples consider different sets of constraints, including bulk and shear moduli, square/cubic symmetry, isotropy, thermal conductivity and a combination of them in two and three dimensions. The non-volume constraints are treated explicitly, i.e., without the use of Lagrange multiplier/penalty as used in conventional gradient-based binary topology optimization methods (Huang and Xie, 2010). The resulting microstructures are observed to be convergent in all the examples presented and in agreement with the Hashin–Shtrikman bounds.