Quasi-Newton Methods for Topology Optimization Using a Level-Set Method

TL;DR

The paper addresses PDE-constrained topology optimization using a level-set representation, introducing a gradient-descent perspective on level-set evolution and a limited-memory BFGS quasi-Newton method for updating the level-set function $\psi$ in the presence of the generalized topological derivative $\mathcal{D}J(\Omega)$. By casting the optimization in an infinite-dimensional LBFGS framework, the authors achieve substantial reductions in iteration counts across inverse linear and semilinear Poisson problems, elasticity compliance minimization, and Navier–Stokes flow design, while maintaining only modest increases in per-iteration cost. They demonstrate mesh-independent performance and provide extensive numerical results showing that LBFGS typically outperforms the Amstutz–Andrä sphere/convex combination strategies, with elasticity as an exception where all methods are comparably effective. The work contributes a practical, scalable solver for PDE-constrained topology optimization and offers open-source replication through the cashocs package.

Abstract

The ability to efficiently solve topology optimization problems is of great importance for many practical applications. Hence, there is a demand for efficient solution algorithms. In this paper, we propose novel quasi-Newton methods for solving PDE-constrained topology optimization problems. Our approach is based on and extends the popular solution algorithm of Amstutz and Andrä (A new algorithm for topology optimization using a level-set method, Journal of Computational Physics, 216, 2006). To do so, we introduce a new perspective on the commonly used evolution equation for the level-set method, which allows us to derive our quasi-Newton methods for topology optimization. We investigate the performance of the proposed methods numerically for the following examples: Inverse topology optimization problems constrained by linear and semilinear elliptic Poisson problems, compliance minimization in linear elasticity, and the optimization of fluids in Navier-Stokes flow, where we compare them to current state-of-the-art methods. Our results show that the proposed solution algorithms significantly outperform the other considered methods: They require substantially less iterations to find a optimizer while demanding only slightly more resources per iteration. This shows that our proposed methods are highly attractive solution methods in the field of topology optimization.

Figures (18)

• Figure 1: Evolution of the optimization for the linear Poisson problem \ref{['eq:state_system_poisson']}.
• Figure 2: Plots of the topological derivative at two iterations of the sphere combination algorithm for problem \ref{['eq:state_system_poisson']}.
• Figure 3: Evolution of the cost functional for the linear Poisson problem \ref{['eq:state_system_poisson']} for several discretization levels.
• Figure 4: Evolution of the optimization for the semilinear Poisson problem \ref{['eq:semilinear_poisson']}.
• Figure 5: Schematic of the cantilever problem.

Theorems & Definitions (6)

• Definition 2.1
• Definition 2.2
• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2