Multi-material continuum topology optimization with arbitrary volume and mass constraints

A framework is presented for multi-material compliance minimization in the context of continuum based topology optimization. We adopt the common approach of finding an optimal shape by solving a series of explicit convex (linear) approximations to the volume constrained compliance minimization problem. The dual objective associated with the linearized subproblems is a separable function of the Lagrange multipliers and thus, the update of each design variable is dependent only on the Lagrange multiplier of its associated volume constraint. By tailoring the ZPR design variable update scheme to the continuum setting, each volume constraint is updated independently. This formulation leads to a setting in which sufficiently general volume/mass constraints can be specified, i.e., each volume/mass constraint can control either all or a subset of the candidate materials and can control either the entire domain (global constraints) or a sub-region of the domain (local constraints). Material interpolation schemes are investigated and coupled with the presented approach. The key ideas presented herein are demonstrated through representative examples in 2D and 3D.