Computation of the magnetostatic interaction between linearly magnetized polyhedrons

• We develop a method to compute the magnetostatic interaction for finite elements.
• Hybrid analytical–numerical approach is used to resolve the singularity.
• Varying the order of numerical integration trades off performance and accuracy.
• Potential application: fast multipole computation of the demagnetising field.

In this paper we present a method to accurately compute the energy of the magnetostatic interaction between linearly (or uniformly, as a special case) magnetized polyhedrons. The method has applications in finite element micromagnetics, or more generally in computing the magnetostatic interaction when the magnetization is represented using the finite element method (FEM).The magnetostatic energy is described by a six-fold integral that is singular when the interaction regions overlap, making direct numerical evaluation problematic. To resolve the singularity, we evaluate four of the six iterated integrals analytically resulting in a 2d integral over the surface of a polyhedron, which is nonsingular and can be integrated numerically. This provides a more accurate and efficient way of computing the magnetostatic energy integral compared to existing approaches.The method was developed to facilitate the evaluation of the demagnetizing interaction between neighbouring elements in finite-element micromagnetics and provides a possibility to compute the demagnetizing field using efficient fast multipole or tree code algorithms.