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Shape derivatives in differential forms I: An intrinsic perspective

by R. Hiptmair and J. Li

(Report number 2011-42)

Abstract
We treat Zolesio’s velocity method of shape calculus using the formalism of differential forms, in particular, the notion of Lie derivative. This provides a unified and elegant approach to computing even higher order shape derivatives of domain and boundary integrals and skirts the tedious manipulations entailed by classical vector calculus. Hitherto unknown expressions for shape Hessians can be derived with little effort. The perspective of differential forms perfectly fits second-order boundary value problems.We illustrate its power by deriving the shape derivatives of solutions to second-order elliptic boundary value problems with Dirichlet, Neumann and Robin boundary conditions. A new dual mixed variational approach is employed in the case of Dirichlet boundary conditions.

Keywords: Differential forms, Lie derivative, shape derivative, Hadamard structure theorems, dual formulation

@Techreport{HL11_84,
  author = {R. Hiptmair and J. Li},
  title = {Shape derivatives in differential forms I: An intrinsic perspective},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-42},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-42.pdf },
  year = {2011}
}

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