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

Abstract

SUMMARYThe 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 parametrization 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.

Development of a benchmark problem and a general optimisation package with Powell's method to develop the benchmark

“…The influence of this leakage flux requires significant correction in the shape of the pole face close to the left edge in order to achieve the desired constant flux density in the air gap. As shown by Weeber and Hoole [4], the constraints can be based on specification of minimum percentage of leakage flux or as shown by Subramanaiam et al [6], the jaggedness of the resulting geometry. This example is frequently used in the demonstration of electromagnetic optimisation methods and it can be considered as a standard demonstration example [7,8].…”

“…1). It is one of the oldest problems studied as electromagnetic devices were optimised [3] and has been developed since [4][5][6][7]. Using the package specially developed for this purpose and described below, we study and define in detail many relevant features of the pole-face problem.…”

A common algorithm for various parametric geometric changes in finite element design sensitivity computation

“…With these parameterized mesh generators, points are introduced on a scaled basis on lines defining boundaries but the mesh is recreated from these boundary points. As a result, C 1 continuity is lost; to preserve it, the connections between nodes must not be altered as the shapes change [3].…”

“…This method cyclically solves first a structural problem to find the displacement of nodes and then the desired electromagnetic problem. However, the process is time-consuming, involves repeatedly solving a larger structural problem and then the electromagnetic field problem at hand [3]; numerical errors in solving the structural problems accumulate to the original electromagnetic problem.…”

Optimizing defibrillation electrodes: automating the search for better configurations

“…Optimization algorithms have been used to solve problems in many disciplines, including structural engineering [5] and electromagnetics [6]. Most define a "cost" or "objective" function, which measures how closely a particular combination of input parameters meets a desired goal.…”