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Shape Derivatives in Differential Forms II: Shape Derivatives for Scattering Problems

by R. Hiptmair and J.-Z. Li

(Report number 2017-24)

Abstract
In this paper we study shape derivatives of solutions of acoustic and electromagnetic scattering problems in frequency domain from the perspective of differential forms following $[$Ralf Hiptmair and Jingzhi Li, Shape derivatives in differential forms I: An intrinsic perspective, Annali di Matematica Pura ed Applicata, 192 (2013), pp. 1077-1098$]$. Relying on variational formulations, we present a unified framework for the derivation of strong and weak forms of derivatives with respect to variations of the shape of an impenetrable (resp. penetrable) scatterer, when we impose Dirichlet, Neumann, or impedance (resp. transmission) conditions on its boundary (resp. interface). In 3D for degrees $l=0$ and $l=1$ of the forms we obtain known and new formulas for shape derivatives of solutions of Helmholtz and Maxwell equations. They can form the foundation for numerical approximation with finite elements or boundary elements.

Keywords: Shape derivative, shape calculus, differential forms, acoustic scattering, electromagnetic scattering

@Techreport{HL17_720,
  author = {R. Hiptmair and J.-Z. Li},
  title = {Shape Derivatives in Differential Forms II: Shape Derivatives for Scattering Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-24.pdf },
  year = {2017}
}

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