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Holding the Fields Constant: A Shape-Calulus Approach to Electromagnetic Forces

by P. Panchal and R. Hiptmair and S. Kurz

(Report number 2024-19)

Abstract
Using the mathematical theory of shape calculus as tool, we perform a rigorous investigation of the virtual work principle for the computation of electromagnetic forces in static settings. The main goal is to shed light on the widely held belief that the virtual work principle entails passively advecting the fields with the virtual displacement when computing deformation-dependent field energies gradients. The adjoint approach to differentiation of functionals under variational constraints provides a mathematical justification for this belief. However, it also shows that passively advecting the actual electromagnetic fields is sufficient only in the case of linear materials. In the general case, also fields arising as solutions of adjoint variational problems have to be taken into account.

Keywords: shape calculus, electromagnetic forces, virtual work principle, adjoint variational problems

@Techreport{PHK24_1101,
  author = {P. Panchal and R. Hiptmair and S. Kurz},
  title = {Holding the Fields Constant: A Shape-Calulus Approach to Electromagnetic Forces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-19.pdf },
  year = {2024}
}

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