[Paper Review] Shape derivatives of boundary integral operators in electromagnetic scattering

Martin Costabel, Frédérique Le Louër|arXiv (Cornell University)|Feb 8, 2010

Numerical methods in inverse problems29 references10 citations

TL;DR

This paper establishes the shape differentiability of electromagnetic scattering solutions via boundary integral equations, proving that the solution is infinitely shape differentiable away from the obstacle boundary, with derivatives losing regularity on the boundary. It characterizes the first shape derivative as a solution to a new scattering problem, enabling gradient-based shape optimization in electromagnetics using integral equation methods.

ABSTRACT

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. Using Helmholtz decomposition, we can base their analysis on the study of scalar integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity and that the solutions of the scattering problem are infinitely shape differentiable away from the boundary of the obstacle, whereas their derivatives lose regularity on the boundary. We also give a characterization of the first shape derivative as a solution of a new electromagnetic scattering problem.

Motivation & Objective

• To analyze the shape differentiability of solutions to time-harmonic electromagnetic scattering by a penetrable obstacle.
• To establish the Gâteaux differentiability of electromagnetic boundary integral operators acting on tangential vector fields in the energy space $\boldsymbol{\mathsf{T}}\boldsymbol{\mathsf{H}}^{-\frac{1}{2}}(\operatorname{div}_{\Gamma},\Gamma)$.
• To characterize the first shape derivative as a solution to a new electromagnetic scattering problem with modified boundary conditions.
• To enable gradient-based shape optimization in applications such as radar and antenna design by providing explicit derivative representations.

Proposed method

• Use of Helmholtz decomposition to split the space $\boldsymbol{\mathsf{T}}\boldsymbol{\mathsf{H}}^{-\frac{1}{2}}(\operatorname{div}_{\Gamma},\Gamma)$ into gradient and curl components, reducing the analysis to scalar integral operators.
• Analysis of Gâteaux differentiability for pseudo-homogeneous kernels in standard Sobolev spaces, extending results from acoustic to electromagnetic settings.
• Study of surface differential operators (e.g., $\nabla_\Gamma$, $\operatorname{curl}_\Gamma$) under $\mathscr{C}^1$ deformations of the boundary.
• Derivation of explicit expressions for the first shape derivative of the electric and magnetic fields using commutator identities and surface vector calculus.
• Construction of a new boundary value problem whose solution equals the first shape derivative, using modified transmission conditions.
• Leverage integral representations and inverse operators to propagate differentiability through the solution map.

Experimental results

Research questions

• RQ1 Are electromagnetic boundary integral operators infinitely shape differentiable with respect to $\mathscr{C}^k$ deformations of the obstacle boundary?
• RQ2 Does the solution to the dielectric scattering problem retain infinite differentiability in the interior, or does regularity degrade near the boundary?
• RQ3 Can the first shape derivative of the electromagnetic fields be characterized as the solution to a new scattering problem with modified boundary conditions?
• RQ4 How do surface differential operators like $\nabla_\Gamma$ and $\operatorname{curl}_\Gamma$ behave under Gâteaux differentiation with respect to domain deformations?
• RQ5 What is the optimal regularity of the boundary deformation that preserves Gâteaux differentiability of the integral operators?

Key findings

• The electromagnetic boundary integral operators are infinitely shape differentiable without loss of regularity in the interior of the domain.
• The solution to the scattering problem is infinitely shape differentiable in the interior, but its derivatives lose regularity on the boundary $\Gamma$.
• The first shape derivative of the electromagnetic fields is characterized as the solution to a new scattering problem with modified transmission conditions involving surface derivatives of the deformation vector field.
• The shape derivative of the electric field satisfies a system of Maxwell’s equations with a source term involving $\nabla_\Gamma\xi$ and $\boldsymbol{\mathsf{n}}\times\partial_\mathbf{n}(\boldsymbol{\mathsf{E}}^i - \boldsymbol{\mathsf{E}}^s)$.
• The Gâteaux derivative of the boundary integral operators is well-defined for $\mathscr{C}^1$ deformations, even on Lipschitz domains, though full $\mathscr{C}^2$ regularity is required for higher-order derivatives.
• The Helmholtz decomposition allows reduction of the vector-valued problem to scalar operators, enabling a systematic analysis of differentiability in Sobolev spaces.