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Global convergence of iterative solvers for problems of nonlinear magnetostatics

Herbert Egger, Felix Engertsberger, Bogdan Radu

Abstract

We consider the convergence of iterative solvers for problems of nonlinear magnetostatics. Using the equivalence to an underlying minimization problem, we can establish global linear convergence of a large class of methods, including the damped Newton-method, fixed-point iteration, and the Kacanov iteration, which can all be interpreted as generalized gradient descent methods. Armijo backtracking isconsidered for an adaptive choice of the stepsize. The general assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, as well as permanent magnets. The main results are proven on the continuous level, but they carry over almost verbatim to various approximation schemes, including finite elements and isogeometric analysis, leading to bounds on the iteration numbers, which are independent of the particular discretization. The theoretical results are illustrated by numerical tests for a typical benchmark problem.

Global convergence of iterative solvers for problems of nonlinear magnetostatics

Abstract

We consider the convergence of iterative solvers for problems of nonlinear magnetostatics. Using the equivalence to an underlying minimization problem, we can establish global linear convergence of a large class of methods, including the damped Newton-method, fixed-point iteration, and the Kacanov iteration, which can all be interpreted as generalized gradient descent methods. Armijo backtracking isconsidered for an adaptive choice of the stepsize. The general assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, as well as permanent magnets. The main results are proven on the continuous level, but they carry over almost verbatim to various approximation schemes, including finite elements and isogeometric analysis, leading to bounds on the iteration numbers, which are independent of the particular discretization. The theoretical results are illustrated by numerical tests for a typical benchmark problem.
Paper Structure (4 sections, 2 theorems, 17 equations, 1 figure, 1 table)

This paper contains 4 sections, 2 theorems, 17 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Numerical validation

Key Result

lemma 1

Let Assumption engertsberger:ass:1 be valid. Then the nonlinear variational problem engertsberger:eq:var has a unique solution which is also the unique minimizer of engertsberger:eq:min.

Figures (1)

  • Figure 1: Numerical solution of the magnetostatic problem obtained by the vector potential formulation using second-order finite elements. Left: geometric setup with iron (grey) and a coil (green) with the magnetic flux lines (red). Right: magnitude of the magnetic flux $\mathbf{b} = \operatorname{curl} \mathbf{a}$.

Theorems & Definitions (3)

  • lemma 1
  • proof
  • theorem 1