# Analysis of the Minimal Compliance Problem and Related Filters

## Andreas Thalhammer

Nov. 25, 2014, 3:30 p.m. S2 059

In this talk we consider the topology optimization of elastic continua. For this, we derive the so-called \textit{Minimal Compliance Problem:}

\begin{align*}
\ell(\mathbf{u}(\rho)) &\to \min_{\rho\in L_\infty(\Omega)}\\
\text{subject to } \quad \int_\Omega \rho ( \mathbf{x}) d \mathbf{x} &\leq m_0, \\
\rho_{\min} \leq \rho(\mathbf{x}) &\leq 1,\qquad\; \dot{\forall} \mathbf{x} \in \Omega.
\end{align*}

Whereas we are able to show existence of solutions for this problem, a disadvantage of this formulation is that we cannot guarantee a $0-1$-structure for the material-void problem, since we allow intermediate values of the density $\rho$.

In order to force a \textit{0-1}-structure, we are discussing material interpolation methods such as SIMP and RAMP. Although the sharp contrast in the numerical output is forced by these methods, existence of solutions of the modified problem - in contrast to the original formulation - cannot be proved directly.

As a possible remedy, the RIDC method is presented, whose main idea is to add an additional constraint $P_S(\rho) \leq \varepsilon_P$ to the Minimal Compliance Problem. For this modified problem, it is possible to show existence of solutions if the integral operators $P$ and $S$ satisfy specific assumptions.