Analysis of a combined Filtered/phase-field approach to topology optimization in elasticit

Abstract

We advance a combined filtered/phase-field approach to topology optimization in the setting of linearized elasticity. Existence of minimizers is proved and rigorous parameter asymptotics are discussed by means of variational convergence techniques. Moreover, we investigate an abstract space discretization in the spirit of conformal finite elements. Eventually, stationarity is equivalently reformulated in terms of a Lagrangian.

Figures (1)

• Figure 1: Cantilever beam topology optimization: optimal field $(\alpha \phi + \beta K \phi)$ obtained for $\alpha=1-\beta=0$ (F method), $\alpha=\beta=0.5$ (combined F/PF) and $\alpha=1-\beta=1$ (PF method) employing different values of the filter radius $r_f$ (for F and F/PF, left) and of the phase-field parameter $\gamma$ (for F/PF and PF, right). Simulation parameters (if not differently specified): $\eta =1$ N/m, $r_f=0.1$ m, and $\gamma=0.01$ m.

Theorems & Definitions (9)

• Theorem 2.1: Existence
• proof : Proof of Theorem \ref{['thm:ex']}
• Theorem 3.1: $\Gamma$-convergence
• proof
• Corollary 3.2: Parameter asymptotics
• Theorem 4.1: Space discretization
• proof
• Theorem 5.1: Lagrangian formulation
• proof