Periodic micromagnetic finite element method

Abstract

Periodic micromagnetic finite element method (PM-FEM) is introduced to solve periodic unit cell problems using the Landau-Lifshitz-Gilbert equation. PM-FEM is applicable to general problems with 1D, 2D, and 3D periodicities. PM-FEM is based on a non-periodic FEM-based micromagnetic solver and extends it in several aspects to account for periodicities, including the computation of exchange and magnetostatic fields. For the exchange field, PM-FEM modifies the sparse matrix construction for computing the Laplace operator to include additional elements arising due to the periodicities. For the magnetostatic field, the periodic extensions include modifications in the local operators, such as gradient, divergence, and surface magnetic charges as well as the long-range superposition operator for computing the periodic scalar potential. The local operators are extended to account for the periodicities similar to handling the Laplace operator. For the long-range superposition operator, PM-FEM utilizes a periodic Green's function (PGF) and fast spatial convolutions. The PGF is computed rapidly via exponentially rapidly convergent sums. The spatial convolutions are accomplished via a modified fast Fourier transform based adaptive integral method that allows calculating spatial convolutions with non-uniform meshes in $O(N\log N)$ numerical operations. PM-FEM is implemented on CPU and GPU based computer architectures. PM-FEM allows efficiently handling cases of structures contained withing the periodic unit cell touching or not touching its boundaries as well as structures that protrude beyond the unit cell boundaries. PM-FEM is demonstrated to have about the same or even higher performance than its parent non-periodic code. The demonstrated numerical examples show the efficiency of PM-FEM for highly complex structures with 1D, 2D, and 3D periodicities.

Figures (7)

• Figure 1: Categories of 1D PBC unit cell (solid black) and its nearest images (shadowed), (a) non-touching, non-protruding case, (b) touching, non-protruding case, (c) non-touching, protruding case and (d) touching, protruding case.
• Figure 2: Illustration of (a) Protruding unit cell; (b) its geometry center (black dot) and its protruding parts in green and blue; (c) Regular unit cell after shifting the protruding parts.
• Figure 3: Hysteresis loop along $\hat{z}-$direction of infinite long periodic rod along $\hat{z}-$direction, the inset is the unit cell. The loop is of square shape and coercive field is around 3050 Oe, close to theoretical value 3085 Oe.
• Figure 4: Example of 2D T-NP-PBC case. (a) Equilibrium vortex state when no PBC is present and (b) the uniform state when the 2D T-NP-PBC is used. The yellow dashed line is the location of the line excitation. (c) Angles between wave vector of propagating spin wave and magnetization. (d) Simulated wavelength (circle marks) and theoretical prediction (blue dashed line).
• Figure 5: Example of 3D T-NP-PBC case. (a) Multi-grain structure (the unit cell is marked as a yellow cube); (b) multi-grain structure folded into the unit cell; (c) hysteresis loop for the periodic multi-grain structure. The unit cell size is $6\mu$m, the average grain size is $2.5\mu$m. The structure was meshed in a tetrahedral mesh with $70$ million elements and $12$ million nodes.