Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency

TL;DR

This paper extends classical energy minimization principles to lossy inhomogeneous bodies in acoustics, elastodynamics, and electromagnetism at fixed frequency, enabling better analysis and potential tomography applications.

Contribution

It develops new variational principles for lossy inhomogeneous media, linking boundary measurements to internal properties, and generalizes existing saddle point principles for these fields.

Findings

Primary electromagnetic principles relate to power dissipation.

Boundary data constrains internal moduli.

Derived saddle point principles generalize prior theories.

Abstract

The classical energy minimization principles of Dirichlet and Thompson are extended as minimization principles to acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency. This is done by building upon ideas of Cherkaev and Gibiansky, who derived minimization variational principles for quasistatics. In the absence of free current the primary electromagnetic minimization variational principles have a minimum which is the time-averaged electrical power dissipated in the body. The variational principles provide constraints on the boundary values of the fields when the moduli are known. Conversely, when the boundary values of the fields have been measured, then they provide information about the values of the moduli within the body. This should have application to electromagnetic tomography. We also derive saddle point variational principles which correspond to variational principles of Gurtin, Willis, and Borcea.