Efficient and accurate separable models for discrete material optimization: A continuous perspective

TL;DR

This paper introduces new separable approximation models for multi-material design optimization that are both accurate and computationally efficient, leveraging advanced mathematical concepts like the Sherman-Morrison-Woodbury identity and topological derivatives.

Contribution

The paper presents novel separable models based on convex approximations, demonstrating their accuracy and efficiency in discrete material optimization problems.

Findings

Two models show high accuracy in numerical experiments.

Evaluation can be efficient after offline precomputation.

Suboptimal decisions are avoided with the most accurate models.

Abstract

Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two -- at a first sight -- very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline phase. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.