Numerical Eigenvalue Optimization by Shape-Variations for Maxwell's Eigenvalue Problem

TL;DR

This paper develops a shape-variation based optimization method for Maxwell eigenvalues, utilizing a mixed variational formulation, domain transformations, and a damped inverse BFGS algorithm, demonstrated through numerical experiments.

Contribution

It introduces a novel shape-optimization framework for Maxwell eigenvalues using mixed variational formulations and a specialized BFGS method.

Findings

Efficient numerical approach for electromagnetic eigenvalue optimization.

Successful application of domain transformations and mixed finite elements.

Numerical example demonstrates the method's effectiveness.

Abstract

In this paper we consider the free-form optimization of eigenvalues in electromagnetic systems by means of shape-variations with respect to small deformations. The objective is to optimize a particular eigenvalue to a target value. We introduce the mixed variational formulation of the Maxwell eigenvalue problem introduced by Kikuchi (1987) in function spaces of (H(\operatorname{curl}; \Omega)) and (H^1(\Omega)). To handle this formulation, suitable transformations of these spaces are utilized, e.g., of Piola-type for the space of (H(\operatorname{curl}; \Omega)). This allows for a formulation of the problem on a fixed reference domain together with a domain mapping. Local uniqueness of the solution is obtained by a normalization of the the eigenfunctions. This allows us to derive adjoint formulas for the derivatives of the eigenvalues with respect to domain variations. For the solution of the resulting optimization problem, we develop a particular damped inverse BFGS method that allows for an easy line search procedure while retaining positive definiteness of the inverse Hessian approximation. The infinite dimensional problem is discretized by mixed finite elements and a numerical example shows the efficiency of the proposed approach.