Fictitious minima of object functions, finite element meshes, and edge elements in electromagnetic device synthesis

The authors points out the nature of the discontinuities in the object function as a result of mesh error. That is, they are artificial and have no physical basis. These discontinuities, however accurate the mesh might be, persist. As such, the solution of inverse problems gets to be slow. The authors present three approaches to minimizing this error: adaptive meshes, edge elements, and crunched meshes. The latter is shown to be significantly faster for optimization, although the field solutions in the iterations have accuracy depending on the fineness of the initial mesh. Adaptive approaches on the other hand significantly slow down convergence. Edge elements improve flux-density-based object functions by making them C¹ continuous because no derivative of the potential is required, although multiple minima continue to exist; but the C¹ continuity makes it possible to utilize faster algorithms using the Hessian