Adjoint variable method for time‐harmonic Maxwell equations

Abstract

Purpose

The purpose of this paper is to study the optimization problem of low‐frequency magnetic shielding using the adjoint variable method (AVM). This method is compared with conventional methods to calculate the gradient.

Design/methodology/approach

The equation for the vector potential (eddy currents model) in appropriate Sobolev spaces is studied to obtain well‐posedness. The optimization problem is formulated in terms of a cost functional which depends on the vector potential and its rotation. Convergence of a steepest descent algorithm to a stationary point of this functional is proved. Finally, some numerical results for an axisymmetric induction heater are presented.

Findings

Using Friedrichs' inequality, the existence and uniqueness of the vector potential, its gradient and the corresponding adjoint variable can be proved. From the numerical results, it is concluded that the AVM is advantageous if the number of parameters to optimize is larger than two.

Research limitations/implications

The AVM is only faster than conventional methods if the gradients can be calculated with sufficient accuracy.

Originality/value

Theoretical results for eddy currents model are often based on a non‐vanishing conductivity. The theoretical value of this paper is the presence of non‐conducting materials in the domain. From a practical viewpoint, it has been demonstrated that the AVM can yield a significant reduction of computational time for advanced optimization problems.

Citation

Durand, S., Cimrák, I. and Sergeant, P. (2009), "Adjoint variable method for time‐harmonic Maxwell equations", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 28 No. 5, pp. 1202-1215. https://doi.org/10.1108/03321640910969458