Holding the fields constant: A shape-calculus approach to electromagnetic forces

TL;DR

This paper uses shape calculus to rigorously analyze electromagnetic force computation, clarifying when passive field advection suffices and when additional fields are needed, especially in nonlinear materials.

Contribution

It provides a mathematical justification for the virtual work principle in electromagnetic forces and clarifies the conditions under which passive field advection is valid.

Findings

Passive advection suffices for linear materials.

Adjoint fields must be considered in nonlinear cases.

Shape calculus offers a rigorous framework for electromagnetic force analysis.

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.