Direct Exchange Calculation for Unstructured Micromagnetic Meshes

In micromagnetic simulations it is necessary to compute the effective field associated to the exchange interaction, which corresponds to applying the Laplace operator to the magnetization field. The most widely used approach for computing the exchange field is to apply finite-difference schemes to homogeneous Cartesian meshes. In many situations it would be computationally advantageous to perform micromagnetic simulation on inhomogeneous meshes, but doing so presents a challenge with regard to the calculation of the exchange field. Here we present the construction of an exchange field calculation method for inhomogeneous meshes. A novel second order differential operator method is introduced and applied to a series of analytical functions on irregular meshes and compared to existing methods, finding an accuracy improvement of up to 81% relative to the second order finite difference method. The method is then used in combination with the MagTense framework and tested on the μmag standard micromagnetics problems 3 and 4. Here we find that especially for tetrahedral meshes, the proposed method results in much faster convergence as function of mesh size. We demonstrate that the method converges stably even for meshes where some mesh elements have a volume as small as 1/8 compared to the base resolution. Finally, we show that the method works equally well for meshes with spatially varying material parameters.