Efficient and Accurate Separable Models for Discretized Material Optimization: A Continuous Perspective Based on Topological Derivatives

Gangl P, Nees N, Stingl M (2024)

Publication Type: Journal article

Publication year: 2024

Journal

Journal of Geometric Analysis Springer Verlag (Germany)

Book Volume: 34

Article Number: 206

Journal Issue: 7

DOI: 10.1007/s12220-024-01663-0

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 stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.

Authors with CRIS profile

Nico Nees Lehrstuhl für Angewandte Mathematik (Kontinuierliche Optimierung) Michael Stingl Lehrstuhl für Angewandte Mathematik (Kontinuierliche Optimierung)

Involved external institutions

Johann Radon Institute AT Austria (AT)

How to cite

APA:

Gangl, P., Nees, N., & Stingl, M. (2024). Efficient and Accurate Separable Models for Discretized Material Optimization: A Continuous Perspective Based on Topological Derivatives. Journal of Geometric Analysis, 34(7). https://doi.org/10.1007/s12220-024-01663-0

MLA:

Gangl, Peter, Nico Nees, and Michael Stingl. "Efficient and Accurate Separable Models for Discretized Material Optimization: A Continuous Perspective Based on Topological Derivatives." Journal of Geometric Analysis 34.7 (2024).

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