Design Sensitivity Analysis for Shape Optimization of Nonlinear Magnetostatic Systems

This paper discusses the sensitivity analysis for the shape optimization of a nonlinear magnetostatic system, evaluated both by direct and adjoint approaches. The calculations rely on the Lie derivative concept of differential geometry, where the flow is the velocity field associated with the modification of a geometrical parameter in the model. The resulting sensitivity formulas can be expressed naturally in a finite-element setting through a volume integral in the layer of elements connected to the surface undergoing shape modification. The accuracy of the methodology is analyzed on a 2-D model of an interior permanent magnet motor and on a 3-D model of a permanent magnet system.

I. Introduction

The aim of shape optimization is to determine the layout of the structure that minimizes a performance measure, under some design constraints. Mathematical programming, which relies on the derivative of all the performance measures, is adopted to solve the problem. Thus, it is crucial to establish an efficient and accurate method for the calculation of the sensitivity of a response to shape variation.