Topological derivative for the nonlinear magnetostatic problem

Samuel  Amstutz; Peter  Gangl

doi:doi:10.1553/etna_vol51s169

Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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Topological derivative for the nonlinear magnetostatic problem

Samuel Amstutz,

Peter Gangl

ETNA - Electronic Transactions on Numerical Analysis, pp. 169-218, 2019/07/24

doi: 10.1553/etna_vol51s169

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doi:10.1553/etna_vol51s169

Abstract

The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical engineering, we derive the topological derivative for an optimization problem which is constrained by the quasilinear equation of two-dimensional magnetostatics. Here, the main ingredient is to establish a sufficiently fast decay of the variation of the direct state at scale 1 as |x|→∞. In order to apply the method in a bi-directional topology optimization algorithm, we derive both the sensitivity for introducing air inside ferromagnetic material and the sensitivity for introducing material inside an air region. We explicitly compute the arising polarization matrices and introduce a way to efficiently evaluate the obtained formulas. Finally, we employ the derived formulas in a level-set based topology optimization algorithm and apply it to the design optimization of an electric motor.

Keywords: topological derivative, nonlinear magnetostatics, topology optimization, electrical machine

ETNA

Published Online: 2019/07/24 09:29:05

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Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

Topological derivative for the nonlinear magnetostatic problem

Abstract