The existence of solutions to topology optimization problems

Abstract

Topology optimization is to determine a shape or topology, having minimum cost. We are devoted entirely to minimum compliance (maximum stiffness) as minimum cost. An optimal shape $\Omega$ is realized as a distribution of material on a reference domain $D$, strictly larger than $\Omega$ in general. The optimal shape $\Omega$ and an equilibrium $u(\Omega)$ on $\Omega$ are approximated by material distributions on the domain $D$ and equilibriums also on $D$, respectively. This note gives a sufficient setting to the existence of an optimal material distribution.