Quasi-Newton methods for topology optimization using a level-set method

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 (J Comput Phys 216: 573–588, 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.

Topology optimization for uniform flow distribution in electrolysis cells

“…The method was introduced by Amstutz and Andrä [4]. For the sake of brevity, we only give a short presentation here and refer the reader to [10] for an overview over established topology optimization algorithms as well as novel quasi-Newton methods for topology optimization. First, we introduce a continuous level-set function 𝜓 ∶ 𝐷 → ℝ to characterize the fluid as well as the solid part of the hold-all domain 𝐷 by 𝜓(𝑧)…”

“…The method was introduced by Amstutz and Andrä [4]. For the sake of brevity, we only give a short presentation here and refer the reader to [10] for an overview over established topology optimization algorithms as well as novel quasi‐Newton methods for topology optimization. First, we introduce a continuous level‐set function $$\psi :D\rightarrow \mathbb {R}$$ to characterize the fluid as well as the solid part of the hold‐all domain D by $$$$\begin{equation*} \psi (z){\begin{cases} <0, & z\in \Omega ,\\ =0, & z\in \partial \Omega ,\\ >0.$$…”