Analysis of the convergence of Newton method by finite element simulation with vector hysteresis stop model

Abstract

Purpose

The goal of this research is to investigate the convergence behavior of the Newton iteration, when solving the nonlinear problem with consideration of hysteresis effects. Incorporating the vector hysteresis model in the magnetic vector potential formulation has encountered difficulties. One of the reasons is that the Newton method is very sensitive regarding the starting point and states distinct requirements for the nonlinear function in terms of monotony and smoothness. The other reason is that the differential reluctivity tensor of the material model is discontinuous due to the properties of the stop operators. In this work, line search methods to overcome these difficulties are discussed.

Design/methodology/approach

To stabilize the Newton iteration, line search methods are studied. The first method computes an error-oriented search direction. The second method is based on the Wolfe-Powell rule using the Armijo condition and curvature condition.

Findings

In this paper, the differentiation of the vector stop model, used to evaluate the Jacobian matrix, is studied. Different methods are applied for this nonlinear problem to ensure reliable and stable finite element simulations with consideration of vector hysteresis effects.

Originality/value

In this paper, two different line search Newton methods are applied to solve the magnetic field problems with consideration of vector hysteresis effects and ensure a stable convergence successfully. A comparison of these two methods in terms of robustness and efficiency is presented.

Citation

Xiao, X., Müller, F., Nell, M.M. and Hameyer, K. (2023), "Analysis of the convergence of Newton method by finite element simulation with vector hysteresis stop model", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 42 No. 4, pp. 893-902. https://doi.org/10.1108/COMPEL-09-2022-0328