Micromagnetic Exchange Field Calculations for Unstructured Meshes

Abstract

Micromagnetic simulations are employed for predicting the behavior of magnetic materials from their microscopic properties, using the Landau-Lifshitz-Gilbert Equation dM / dt = - γ M × Heff - γ α (M × (M × Heff)) / Ms, with magnetization M, gyromagnetic ratio γ, dampening coefficient α and effective field Heff, Heff = Hexch + Hdem + Hext + Hanis. Here Hexch is the exchange field, Hdem the demagnetization field, Hext an applied field and Hanis the anisotropy field. By far the most time consuming part of these calculations consists of calculating the demagnetization field, Hdem. Standard approaches compute Hdem as the convolution of magnetization with the dipolar interaction kernel using fast-Fourier transforms[1][2][3]. The framework MagTense[4], by contrast, calculates the demagnetization field analytically from a mesh of computational elements, which can be prisms[5] or tetrahedrons[6], as shown in Fig. 1. Such unstructured grids present new challenges for computing the differential operator in the exchange field Hexch. Here we investigate differential operators on general unstructured polyhedral meshes[7] and apply them to the micromagnetics standard problems[8] using MagTense. Specifically, we employ a weighted least squares interpolation[9] and a potentially novel method for calculating the Laplacian directly. The results are analyzed in terms of accuracy and speed and compared to those obtained with prism meshes.