Topology optimization

Problem

Let $D \subseteq \R^d$ be a bounded open design region with Lipschitz boundary, and let $0 < V < \abs{D}$ be a volume bound. Let $A$ be the elasticity tensor of a given isotropic elastic material, and let $f \in L^2(D; \R^d)$ be a load. For a measurable set $\omega \subseteq D$ let $\chi_\omega$ be its indicator function, and fix a small $\varepsilon > 0$. Let $u_\omega \in H^1(D; \R^d)$ be the displacement that solves the linear elasticity system $-\operatorname{div}\big((\chi_\omega + \varepsilon (1 - \chi_\omega)) A \, e(u)\big) = f$ in $D$, where $e(u) = \frac12 (\nabla u + \nabla u^{T})$. The boundary conditions are $u = 0$ on a part of $\partial D$ of positive surface measure and zero traction on the rest. The minimum compliance topology optimization problem is to find a set $\omega$ with $\abs{\omega} \le V$ that minimizes the compliance $\int_D f \cdot u_\omega \, dx$.