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Numerical eigenvalue optimization by shape variations for Maxwell's eigenvalue problem
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DOI：10.1080/10556788.2026.2651158.png)
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Abstract
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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 utilize the mixed variational formulation of the Maxwell eigenvalue problem as proposed by Kikuchi (1987) in function spaces of $ H(\operatorname {curl}; \Omega ) $ H(curl;Omega) and $ H<^>1(\Omega ) $ H1(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 ) $ H(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 eigenfunctions. This allows us to derive adjoint formulas for 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 problem is discretized by mixed finite elements and a numerical example shows the efficiency of the proposed approach.
Keywords:
Shape optimization
Maxwell eigenvalue
adjoint calculus
damped inverse BFGS method
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1.4
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