Non-uniform FFT for the finite element computation of the micromagnetic scalar potential

Abstract

We present a quasi-linearly scaling, first order polynomial finite element method for the solution of the magnetostatic open boundary problem by splitting the magnetic scalar potential. The potential is determined by solving a Dirichlet problem and evaluation of the single layer potential by a fast approximation technique based on Fourier approximation of the kernel function. The latter approximation leads to a generalization of the well-known convolution theorem used in finite difference methods. We address it by a non-uniform FFT approach. Overall, our method scales O(M + N + N log N) for N nodes and M surface triangles. We confirm our approach by several numerical tests.

A magnetostatic energy formula arising from the L2-orthogonal decomposition of the stray field

“…(1). This approach is common in micromagnetics [6,7] and requires the solution of two Dirichlet problems, the one in Eqn. (12) and a second one with zero right hand side and Dirichlet data obtained from the evaluation of the single layer potential on the boundary nodes.…”

“…Note, that neither evaluations of a nonlocal boundary integral inside the domain Ω nor the solution of an equivalent Dirichlet problem is required to obtain the energy from (2). This is an interesting observation, since most numerical methods in micromagnetics implement the magnetic self-energy in the form (1) with first computing the nonlocal field via the convolution with the Green's kernel G(x) = 1 4π 1 |x| [1,11] h s (x) = −∇u(x) = Ω ∇G(x − x ) ∇ · m dx − ∂Ω ∇G(x − x ) m · n ds x (5) or the solution of the PDE ∆u = ∇ · m in R 3 (6) with the help of boundary integral operators, which account for the contribution of the field in the external region R 3 \ Ω. In any of these cases, the computation of the energy requires the evaluation of a part of the field by nonlocal convolutions evaluated on the boundary and inside the magnet or an additional Dirichlet problem.…”

Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation

“…This grants us the possibility to restrict the local correction in the interval [0, δ]. We mention that this approach has already been proven effective by Exl and Schrefl [19] in the context of computational micromagnetics. Plugging the finite Fourier series expansion into the Gaussian convolution, the evaluation boils down to four Fourier transforms (forward and backward pair counted as two).…”

MagTense: A micromagnetic framework using the analytical demagnetization tensor

“…All methods have inherent advantages and shortcomings. The finite difference Fourier-based approaches are extremely fast but unfortunately require that the magnetic moments lie on a regular grid Vansteenkiste and Van de Wiele [2011]; Vansteenkiste et al [2014]; Ferrero and Manzin [2020], although methods for non-uniform grids Livshitz et al [2009]; Exl and Schrefl [2014] and multi-layer materials Lepadatu [2019] have also been proposed. The tensor-grid method Exl et al [2012] is still under investigation for stability and correctness Abert et al [2013].…”