The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

In this paper, we present a framework for automated shape differentiation in the finite element software . Our approach combines the mathematical Lagrangian approach for differentiating PDE-constrained shape functions with the automated differentiation capabilities of . The user can decide which degree of automatisation is required, thus allowing for either a more custom-like or black-box–like behaviour of the software. We discuss the automatic generation of first- and second-order shape derivatives for unconstrained model problems as well as for more realistic problems that are constrained by different types of partial differential equations. We consider linear as well as nonlinear problems and also problems which are posed on surfaces. In numerical experiments, we verify the accuracy of the computed derivatives via a Taylor test. Finally, we present first- and second-order shape optimisation algorithms and illustrate them for several numerical optimisation examples ranging from nonlinear elasticity to Maxwell’s equations.

We aim at finding an optimal design for an interior permanent magnet electric motor by means of a sensitivity-based topology optimization method. The gradient-based ON/OFF method has been successfully applied to optimization problems of this form. We show that this method can be improved by considering the mathematical concept of topological derivatives (TDs). TDs for optimization problems constrained by linear partial differential equations (PDEs) are well understood, whereas little is known about TDs in combination with nonlinear PDE constraints. We derive the TD for an optimization problem constrained by the equation of nonlinear 2-D magnetostatics, illustrate its advantages over the sensitivities used in the ON/OFF method, and show numerical results for the optimization of an interior permanent magnet electric motor obtained by a level-set algorithm, which is based on the TD.

This work deals with shape optimization of electric machines using isogeometric analysis. Isogeometric analysis is particularly well suited for shape optimization as it allows to easily modify the geometry without remeshing the domain. A 6-pole interior permanent magnet synchronous machine (IPMSM) is modeled using a multipatch isogeometric approach and rotation of the machine is realized by modeling the stator and rotor domain separately and coupling them at the interface using harmonic basis functions. Shape sensitivity analysis is used to find the shape derivative of the optimization goal functional, i.e., the total harmonic distortion of the electromotive force. This method allows for a freeform shape optimization which is not restricted by a choice of a set of optimization parameters. This freeform shape optimization is applied for the first time to an isogeometric model of an IPMSM minimizing the total harmonic distortion of the electromotive force as a goal functional, where a reduction of 75 % is achieved.

In this paper we perform the rigorous derivation of the topological derivative for optimization problems constrained by a class of quasi-linear elliptic transmission problems. In the case of quasi-linear constraints, techniques using fundamental solutions of the differential operators cannot be applied to show convergence of the variation of the states. Some authors succeeded showing this convergence with the help of technical computations under additional requirements on the problem. Our main objective is to simplify and extend these previous results by using a Lagrangian framework and a projection trick. Besides these generalisations the purpose of this manuscript is to present a systematic derivation approach for topological derivatives.

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.