AD-01: Solving periodic micromagnetic problems using finite element method

Periodic problems often arise in micromagnetic modeling, e.g., when considering an infinite space, film, or wire as well as dispersion diagrams of periodically modulated structures. Finite element method-based solvers typically require calculating the static magnetic periodic scalar potential (PSP), which, in principle, can be done via static scalar periodic Green’s function (PGF). However, the scalar PGF diverges even for the 1D periodicity case. Here, we present an efficient technique for computing the PSP via a convolution with the PGF in 1D, 2D, and 3D periodicity cases. We represent the PGF as the near-zone term over a small number of nearby periodic images and far-zone term over the rest of the infinite periodic sum. PSP for the near-zone term is evaluated via a modified non-uniform Fast Fourier Transform method [1]. For the far-zone PSP term, we prove that while the infinite PGF sum diverges, the corresponding PSP converges for the micromagnetic case of zero total magnetic charge. We use a rapidly convergent spectral representation combined with interpolations for the infinite sum [2]. The accuracy can be controlled by a proper selection of the number of the near-zone images, interpolation order, and the interpolation grid density. Figure 1a illustrates the error control for the case of 1D periodicity. Figure 1b shows the computational time versus the number of randomly distributed charges in the periodic unit cell. Similar results are obtained for 2D and 3D periodicity cases and the technique can account for phase shifts needed for periodic dispersion diagrams.References: [1] Y. Brick, V. Lomakin and A. Boag, IEEE Transactions on Antennas and Propagation, vol. 62, no. 8, pp. 4314-4324 (2014) [2] S. Li, D. A. Van Orden, and V. Lomakin, IEEE Transactions on Antennas and Propagation, vol. 58, no. 12, pp. 4005–4014 (2010)