insight - Computational Complexity - # Nonlinear Magnetic Field Solvers with Local Quasi-Newton Updates

Core Concepts

The proposed iterative scheme combines the advantages of fixed-point and Newton-type methods, achieving fast convergence without requiring derivative information of the underlying material law.

Abstract

The content discusses an efficient numerical solution strategy for nonlinear magnetic field problems arising in high-power low-frequency applications like electric machines and transformers.

The key idea is to employ local Quasi-Newton updates to construct appropriate linearizations of the nonlinear material behavior during the iterative solution process. This combines the advantages of fixed-point methods, which do not require derivative information, and Newton-type methods, which exhibit fast convergence.

The authors provide a detailed convergence analysis, proving global mesh-independent r-linear convergence of the proposed scheme. The method can handle both smooth and non-smooth material laws, including models with hysteresis.

Numerical experiments demonstrate the performance of the new approach and compare it to the classical fixed-point and Newton-type methods. The results show that the Quasi-Newton based scheme achieves convergence rates similar to the Newton method, while only requiring evaluations of the material law itself, and not its derivatives.

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arxiv.org

Stats

The material law is represented by the co-energy density w*(h) = μ0|h|^2/2 + U*(h), where U*(h) is a nonlinear function.
The bounds on the local permeability tensors μ^n are given by μ1 ≤ ξ^T μ^n_T ξ ≤ μ2|ξ|^2 for all ξ ∈ ℝ^d and T ∈ Th, with constants μ1, μ2 > 0.

Quotes

"The key idea of the scheme is to employ the low-rank update formulas provided by Quasi-Newton methods, in order to extract significant information about the linearized material behavior, i.e., the Jacobian ∂_hb(h) or generalization of it, from previous values of h and b(h) which are accessible during simulation."
"Together with appropriate strategies for choosing the step size in the underlying iterative method, this allows us to establish a complete convergence analysis of the method proving global r-linear convergence with iteration numbers that are independent of the finite-element mesh."

Key Insights Distilled From

by Herbert Egge... at **arxiv.org** 09-11-2024

Deeper Inquiries

The proposed Quasi-Newton based scheme can be extended to handle time-dependent or coupled magnetoelectric problems by incorporating a time discretization approach alongside the existing iterative framework. In time-dependent scenarios, the governing equations can be formulated using a time-stepping method, such as the implicit or explicit Euler method, or higher-order Runge-Kutta methods. Each time step would involve solving a nonlinear system similar to the one described in the context, where the local Quasi-Newton updates can be applied to approximate the local permeabilities at each time step.
For coupled magnetoelectric problems, the scheme can be adapted by simultaneously solving the magnetic and electric field equations. This involves defining a coupled system of equations that includes both the magnetic field intensity and the electric field, potentially represented through a vector potential formulation. The Quasi-Newton updates can be utilized to derive local permeabilities that account for the interaction between the magnetic and electric fields, ensuring that the material behavior is accurately captured. Additionally, the iterative solver can be modified to handle the coupling by updating both fields in a staggered or simultaneous manner, depending on the specific problem requirements.

The potential limitations of the Quasi-Newton based method primarily arise from the assumptions made regarding the material models and the geometrical configurations of the problem. One significant limitation is the reliance on the smoothness of the material behavior. While the method is designed to handle nonsmooth material laws, such as those exhibiting hysteresis, extreme cases of non-differentiability or highly complex material responses may lead to convergence issues or require additional modifications to the algorithm.
In terms of problem geometries, the method may face challenges when applied to highly irregular or complex geometries where the finite-element mesh becomes excessively refined. The local Quasi-Newton updates, while efficient, may struggle to maintain the required conditions for convergence if the mesh is not adequately designed to capture the material behavior across all elements. Furthermore, the performance of the method could degrade in scenarios involving significant geometric nonlinearity or when the magnetic field exhibits strong spatial variations, necessitating careful mesh design and possibly more sophisticated numerical techniques.

Yes, the local Quasi-Newton updates can be combined with other iterative solvers beyond the fixed-point and Newton-type methods. For instance, they can be integrated with optimization-based iterative methods, such as the Gauss-Seidel or Jacobi methods, which are commonly used for solving linear systems. By incorporating local Quasi-Newton updates into these frameworks, one can enhance the convergence properties and efficiency of the solvers, particularly in cases where the underlying problem exhibits strong nonlinearity.
Additionally, the local Quasi-Newton updates can be utilized in conjunction with advanced techniques such as the Broyden's method or other quasi-Newton methods that focus on updating the Jacobian approximation in a more generalized manner. This flexibility allows for the development of hybrid iterative schemes that leverage the strengths of multiple approaches, potentially leading to improved convergence rates and robustness in solving complex nonlinear problems.
Moreover, the integration of local Quasi-Newton updates with multigrid methods could also be explored, where the updates are applied at different levels of mesh refinement to accelerate convergence across scales. This combination could be particularly beneficial in large-scale simulations where computational efficiency is critical.

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