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Higher order stray field computation on tensor product domains

Lukas Exl, Sebastian Schaffer

TL;DR

The paper tackles the computational bottleneck of nonlocal stray-field calculations in micromagnetism by introducing a higher-order, mesh-free framework. It represents magnetization with a functional Tucker tensor built from tensor-product B-spline bases and computes the magnetic field via a separable, Gaussian-sum approximated vector super-potential learned with a multilinear tensor-product extreme learning machine (ml-tp ELM). The approach achieves exponential convergence in rank and linear scaling with expansion size, while enabling continuous field evaluation and natural boundary handling. Numerical results across homogeneous cubes, flower/vortex states, thin films, and multi-domain setups demonstrate high accuracy and competitive performance against traditional grid-based methods, highlighting potential for dynamic simulations and multi-physics extensions. The methodology provides a versatile, physics-informed, mesh-free pathway for nonlocal micromagnetic computations and other nonlocal potentials.

Abstract

We present an extension of the tensor grid method for stray field computation on rectangular domains that incorporates higher-order basis functions. Both the magnetization and the resulting magnetic field are represented using higher-order B-spline bases, which allow for increased accuracy and smoothness. The method employs a super-potential formulation, which circumvents the need to convolve with a singular kernel. The field is represented with high accuracy as a functional Tucker tensor, leveraging separable expansions on the tensor product domain and trained via a multilinear extension of the extreme learning machine methodology. Unlike conventional grid-based methods, the proposed mesh-free approach allows for continuous field evaluation. Numerical experiments confirm the accuracy and efficiency of the proposed method, demonstrating exponential convergence of the energy and linear computational scaling with respect to the multilinear expansion rank.

Higher order stray field computation on tensor product domains

TL;DR

The paper tackles the computational bottleneck of nonlocal stray-field calculations in micromagnetism by introducing a higher-order, mesh-free framework. It represents magnetization with a functional Tucker tensor built from tensor-product B-spline bases and computes the magnetic field via a separable, Gaussian-sum approximated vector super-potential learned with a multilinear tensor-product extreme learning machine (ml-tp ELM). The approach achieves exponential convergence in rank and linear scaling with expansion size, while enabling continuous field evaluation and natural boundary handling. Numerical results across homogeneous cubes, flower/vortex states, thin films, and multi-domain setups demonstrate high accuracy and competitive performance against traditional grid-based methods, highlighting potential for dynamic simulations and multi-physics extensions. The methodology provides a versatile, physics-informed, mesh-free pathway for nonlocal micromagnetic computations and other nonlocal potentials.

Abstract

We present an extension of the tensor grid method for stray field computation on rectangular domains that incorporates higher-order basis functions. Both the magnetization and the resulting magnetic field are represented using higher-order B-spline bases, which allow for increased accuracy and smoothness. The method employs a super-potential formulation, which circumvents the need to convolve with a singular kernel. The field is represented with high accuracy as a functional Tucker tensor, leveraging separable expansions on the tensor product domain and trained via a multilinear extension of the extreme learning machine methodology. Unlike conventional grid-based methods, the proposed mesh-free approach allows for continuous field evaluation. Numerical experiments confirm the accuracy and efficiency of the proposed method, demonstrating exponential convergence of the energy and linear computational scaling with respect to the multilinear expansion rank.

Paper Structure

This paper contains 17 sections, 2 theorems, 43 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Method
  3. Fitting a multilinear tensor product ELM
  4. Derivatives
  5. Separable approximation of the super-potential
  6. Field and energy calculation
  7. Operator learner perspective and algorithm summary
  8. Storage and computational costs
  9. Numerical validation
  10. Fitting magnetization configurations
  11. Fitting the super-potential
  12. Field and energy computation
  13. Homogeneously magnetized cube
  14. Flower and vortex state
  15. Vortex state in a thin film
  16. ...and 2 more sections

Key Result

Lemma 1

Let $U_p \in \mathbb{R}^{n_p \times r_p},\, n_p \geq r_p,\,\, p=1,2,3,$ have full column rank, then the solution to eq:ml-tp-elm_fitting is given as

Figures (7)

  • Figure 1: Absolute errors of the GS approximation for $|x|$ with $S={\color{black}{50}}$ and $S=100$ terms.
  • Figure 2: Errors of the energy calculated with Alg. \ref{['alg:ho-stray-field']} for a homogeneously magnetized cube with $S=100$ and order $k=7$.
  • Figure 3: Errors of the energy for a homogeneously magnetized cube calculated with TG method, mumax$^3$ and Alg. \ref{['alg:ho-stray-field']} with $S=100$ for varying field rank $r'$ and order $k$ (super-potential method), and mesh size $n$ (TG and mumax$^3$).
  • Figure 4: Run time for the computation of the energy for a homogeneously magnetized cube calculated with TG method, mumax$^3$ and Alg. \ref{['alg:ho-stray-field']} with $S=100$ for varying field rank $r'$ and order $k=7$ (super-potential method), and mesh size $n$ (TG and mumax$^3$). Note that TG was computed on CPU, whereas super-potential method and mumax$^3$ utilize GPU.
  • Figure 5: Comparison of the Newton potential method versus the super-potential method for a homogeneously magnetized cube.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Remark 1