On nonlinear magnetic field solvers using local Quasi-Newton updates

TL;DR

This paper introduces a local Quasi-Newton method for nonlinear magnetic field problems that converges quickly like Newton's method but without requiring derivative information, making it suitable for nonsmooth and hysteretic materials.

Contribution

It proposes a novel local Quasi-Newton scheme that can be integrated into standard finite-element codes, offering a derivative-free alternative with proven convergence.

Findings

The method achieves convergence rates comparable to Newton's method.

It effectively handles nonsmooth and hysteretic material behaviors.

Theoretical analysis confirms global mesh-independent convergence.

Abstract

Fixed-point or Newton-methods are typically employed for the numerical solution of nonlinear systems arising from discretization of nonlinear magnetic field problems. We here discuss an alternative strategy which uses local Quasi-Newton updates to construct appropriate linearizations of the material behavior during the nonlinear iteration. The resulting scheme shows similar fast convergence as the Newton-method but, like the fixed-point methods, does not require derivative information of the underlying material law. As a consequence, the method can be used for the efficient solution of models with hysteresis which involve nonsmooth material behavior. The implementation of the proposed scheme can be realized in standard finite-element codes in parallel to the fixed-point and the Newton method. A full convergence analysis of all three methods is established proving global mesh-independent convergence. The theoretical results and the performance of the nonlinear iterative schemes are evaluated by computational tests for a typical benchmark problem.