A generalised Maxwell Stress Tensor for semi-analytic force and torque between permanent magnets, coils, and soft iron

The Maxwell Stress Tensor is a computationally efficient method for calculating the force and torque between two arbitrary collections of rigidly-connected permanent magnets, coils, and/or iron (soft magnet) segments, when using exact analytic magnetic field solutions. However, use of the tensor exacerbates numerical errors present in the closed-surface free space mesh of a region, whether that be from an approximate field solution such as a finite sum, or discretisation errors that create a numeric non-zero divergence. Using a specialised identity of the divergence theorem, this article derives a generalised Maxwell Stress Tensor, which is interchangeable with the standard form and significantly reduces or removes numerical error sources from the meshing. The application focus of this work is modelling of non-periodic permanent magnet machines without geometrical assumptions through superposition of analytic magnetic field solutions (B and H) from a large number of elements. The influence of relative permeability can be included in these elements through varying the volumetric magnetic charge or current densities. Case studies with analytic or finite element force solutions are used to verify the result and compare the accuracy and computational efficiency with traditional semi-analytic methods. The proposed tensor enables parametric studies with accuracy not previously possible using an elemental modelling method, and can be applied to existing multiphysics models.