Solving the Standard Micromagnetic Problems using Unstructured Meshes with MagTense

Abstract

Micromagnetics requires calculation of the demagnetization field, anisotropy field, applied field and exchange field. With the MagTense framework, demagnetization is calculated analytically [1] and both anisotropy and applied field are local, leaving only the exchange field in the form of a second order partial derivative. Standard micromagnetics are limited to finite element methods with large, extraneous simulation volumes or finite difference methods with homogeneous simulation grids, as the demagnetization is calculated using a fast Fourier transform [2], but with MagTense, no such limitation exists. However, here an optimal method of calculating second order partial derivatives on arbitrary meshes in micromagnetics must be determined, continuing previous research [3].We present solutions to the mumag standard micromagnetic problems (mumag) 3[4] and 4[5] using a direct second order partial derivate technique on four different meshes: Prismal, tetrahedral, grained prismal and grained tetrahedral. The grained meshes consist of voronoi generated and Lloyd iterated grain regions with lower resolution towards the center and higher towards the edges, along with high resolution intergrain regions.In fig. 1a) is shown an example of a grained prismal mesh with 9 grains and an intergrain region in gray. Of course, in the mumag standard problems considered, all regions have the same material properties, but it is easy to envision scenarios where they do not.In fig. 1b) is shown the exchange crossover length of mumag 3 as a function of resolution for three different meshes compared to published solutions. All mesh types converge to the correct value, with the tetrahedral meshes converging faster.In fig. 2 is shown the error compared to published solutions of mumag 4 as a function of resolution using regular-, 4 grained- and 9 grained prismal meshes. The unstructured meshes converge to the correct result in a way similar to a regular grid.In both cases it is apparent that while for these problems grained meshes offer few advantages, they are nonetheless correctly implemented and open up a slew of possibilities for accurately simulating complex geometries with far fewer elements in irregular meshes.  Fig. 1  Fig. 2