Numerical calculation of total force upon permanent magnets using equivalent source methods

Abstract

The application of the equivalent source methods for the numerical calculation of the total magnetic force acting upon a permanent magnet is proposed. These methods are formulated in terms of the external field, which allows the complete avoidance of the numerical inaccuracies affecting force computation due to the singularity of the self‐field of the magnet on its edges. It is shown, with the help of some 2D and 3D test cases, that the proposed formulae provide reliable and stable results, even when the FEM mesh is not refined. Such results have also been compared with those derived from more traditional methods, such as the surface integration of the Maxwell’s stress tensor and the virtual work method, exhibiting better precision and lower computational costs.

Magnetic Flux Density and Vector Potential of Uniform Polyhedral Sources

“…As expected, the magnetic flux density at the vertex is nearly singular and only the regularization (19) limits its value; lowering the value of sharpens the profile of the computed magnetic flux density at the vertex. The error with respect to the exact solution is relevant near the edges (up to 30%), but, inside and outside, decreases logarithmically, as it is well known [16], [25].…”

“…This is a well-known aspect of the uniform magnetization assumption, which anyway does not prevent its use for the analysis of large iron-dominated structures and in geophysics [9], [13]. A detailed analysis of the singularity of the self-field of a uniformly magnetized polyhedron at its edges can be found in [25]. …”

“…The main difficulty with (25) is that it is nearly singular at the edges, also when (19) is utilized, since in this case the singularity is not removed by the multiplication times a zeroing coefficient. The gradient (25) is required only in (12) for the evaluation of magnetic flux density produced by a uniformly magnetized body. When inner edges of the polyhedral mesh are concerned, it is seen from (12) that the singularity cancels out since adjacent elements with the same magnetization share the same face with opposite orientations.…”

“…This leads to (23) The first integral is simply reduced using an alternate form of the Stokes' theorem [20] while the second is recognized as (24) Denoting with the unit vector along the e-th edge of the face, and taking into account definition (16), we obtain (25) As for the function evaluated in (17), also its gradient is related only to the elementary functions and defined in (18) and (21). The main difficulty with (25) is that it is nearly singular at the edges, also when (19) is utilized, since in this case the singularity is not removed by the multiplication times a zeroing coefficient. The gradient (25) is required only in (12) for the evaluation of magnetic flux density produced by a uniformly magnetized body.…”

Magnets in motion

“…These methods are mainly used for static analysis of magneto-mechanical systems within engineering tasks. Comparative studies of these methods can be found in [Delfino et al 2001;de Medeiros et al 1998]. Most of these methods resort in some way to finite element (FE) computations -at least for the computation of the external magnetic field.…”