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

Christine Herter, Sebastian Schöps, Winnifried Wollner

TL;DR

This work addresses free-form shape optimization of Maxwell's eigenvalues by deforming the domain to match a target eigenvalue $\lambda_*$, leveraging a Kikuchi mixed formulation on a fixed reference domain via a deformation map $F_q$ with Jacobian $J_q$. An adjoint framework is developed to obtain derivatives with respect to domain variations, and a damped inverse BFGS method is employed to solve the resulting infinite-dimensional optimization while maintaining positive definiteness and enabling a line search. The method is demonstrated on a planar 5-cell cavity, achieving rapid convergence with modest mesh sizes and only small domain deformation, illustrating its practicality for accelerator cavity design. These results establish a robust workflow for reliable eigenvalue tuning in electromagnetic devices using shape optimization techniques.

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}; Ω)) and (H^1(Ω)). To handle this formulation, suitable transformations of these spaces are utilized, e.g., of Piola-type for the space of (H(\operatorname{curl}; Ω)). 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.

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

TL;DR

This work addresses free-form shape optimization of Maxwell's eigenvalues by deforming the domain to match a target eigenvalue , leveraging a Kikuchi mixed formulation on a fixed reference domain via a deformation map with Jacobian . An adjoint framework is developed to obtain derivatives with respect to domain variations, and a damped inverse BFGS method is employed to solve the resulting infinite-dimensional optimization while maintaining positive definiteness and enabling a line search. The method is demonstrated on a planar 5-cell cavity, achieving rapid convergence with modest mesh sizes and only small domain deformation, illustrating its practicality for accelerator cavity design. These results establish a robust workflow for reliable eigenvalue tuning in electromagnetic devices using shape optimization techniques.

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}; Ω)) and (H^1(Ω)). To handle this formulation, suitable transformations of these spaces are utilized, e.g., of Piola-type for the space of (H(\operatorname{curl}; Ω)). 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.
Paper Structure (10 sections, 1 theorem, 43 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 1 theorem, 43 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Modeling the Optimization Problem of Maxwell's Eigenvalue Problem
  3. Mapping
  4. Shape Optimization Problem
  5. Adjoint Calculus for a Generalized Eigenvalue Optimization Problem
  6. An Adjoint Representation for the Derivatives
  7. Derivative Formulas for the Domain Transformation
  8. A Damped Inverse BFGS Method
  9. Numerical Examples on a 5-cell Cavity
  10. Conclusion

Key Result

Lemma 1

Let $\widetilde{d}^k,y^k$ be given and assume that $B_0\in \mathcal{L}(Q,Q)$ is a symmetric, positive definite operator. Then the damping step in Algorithm BFGSInverseAlgorithm ensures that $\langle y^k, \widetilde{d}^k \rangle > 0$. By induction $B_{k}$, and thus $B_{k+1}$, remains positive definit

Figures (4)

  • Figure 1: Electric field of the $\mathrm{TM}_{010}$, also called $\pi$-mode
  • Figure 2: Geometry of a Cavity.
  • Figure 3: Mesh of the 5-cell Cavity
  • Figure 4: Deformation of a 5-cell cavity after optimization of the first eigenvalue to target value $\lambda_{*} = 6017$ using the BFGS method from \ref{['BFGSInverseAlgorithm']} with regularization parameters $\alpha = 100$ and $\beta = 10^{-6}$. The chosen refinement level is 2, the number of DoFs is 86723. Lagrange elements of order 1 and lowest order Nédélec elements are used.

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof