A structural mapping technique for geometric parametrization in the optimization of magnetic devices

Abstract

Abstract The continuity and differentiability of object functions is a basic prerequisite for the application of gradient methods in optimization. However, for parameters defining the shape of an electromagnetic device, the finite element discretization in the field analysis introduces discontinuities into the object function which slow down the convergence rate. Additionally, depending on the geometric parametrizaiion employed, the optimization frequently yields shape contours that are impracticable for manufacturing purposes. This paper investigates the problems inherent in geometric parametrization and shows that the discontinuities in the object function are caused by changes in mesh topology as the geometric parameters vary; these changes inevitably follow from the use of free meshing algorithms. As a solution to these shortcomings a structural mapping technique is outlined that maps surface displacements onto the parameters of the finite element mesh and obtains the parameter dependent geometric variations without a change in mesh topology. This resulting geometric parametrization yields continuous object functions without artificial local minima and results in smooth surface contours of the optimized device. Using this new parametrization technique, design sensitivity analysis, is shown to be a reliable and essential part in the efficient application of gradient methods for shape optimization.