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On nonlinear magnetic field solvers using local Quasi-Newton updates

Herbert Egger, Felix Engertsberger, Lukas Domenig, Klaus Roppert, Manfred Kaltenbacher

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.

On nonlinear magnetic field solvers using local Quasi-Newton updates

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.
Paper Structure (12 sections, 3 theorems, 40 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 12 sections, 3 theorems, 40 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discretized scalar potential formulation
  3. Globally convergent iterative solvers
  4. A method using local Quasi-Newton updates
  5. Quasi-Newton methods
  6. Local Quasi-Newton updates
  7. Numerical Results
  8. Nonlinear magnetostatics
  9. A vector hysteresis model
  10. Simulation of a typical load cycle
  11. Discussion
  12. Proof of Theorem \ref{['thm:2']}

Key Result

Theorem 1

Let $\mathcal{T}_h$ be a regular triangular resp. tetrahedral mesh of $\Omega \subset \mathbb{R}^d$, $d=2,3$. Further let $\mathbf{h}_s \in H(\operatorname{curl};\Omega)$ and $w_* : \Omega \times \mathbb{R}^d \to \mathbb{R}$ be piecewise continuous with respect to $\mathbf{x}$, and assume that for $ for all $\mathbf{h}_1,\mathbf{h}_2 \in \mathbb{R}^d$. Then the system eq:fem has a unique solution

Figures (2)

  • Figure 1: Left: Sketch of the geometry used in our computations with iron (grey), coil (green and red), air (cyan). Field evaluations at the point C6 are considered later on. Right: typical magnetic flux distribution $|\mathbf{b}| = |\partial_{\mathbf{h}} w_*(\mathbf{h})|$ in our tests.
  • Figure 2: Simulation results at the point C6 (see Figure \ref{['fig:1']}) for case 2 of TEAM problem 32. Left: Hysteresis loop for the hysteretic and anhysteretic material model. Right: Comparison of the hysteretic and anhysteretic model with the measurements teamproblem32.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Remark 8