On the Equivalence of Finite Element and Finite Integration Formulations

Abstract

The paper offers a comparative study of numerical methods of analysis of electromagnetic fields. The focus is on the finite element method (FEM) and finite integration technique (FIT), but with the cell and equivalent network approaches also considered. It is shown how the approximate integrals describing coefficients of the FEM need to be derived for a mesh with parallelepiped elements to achieve consistency with FIT equations. The equivalence of FEM and FIT formulations for a triangular mesh in 2D is highlighted. The TEAM Workshops Problem No. 7 is used as an example for numerical comparisons. Two formulations have been considered: 1) using the edge values of the magnetic vector potential and the nodal values of the electric scalar potential ; and 2) expressed in terms of the edge values of both magnetic and electric -0 vector potentials.Index Terms-Eddy currents, electrical engineering education, finite element method (FEM), finite integration technique (FIT), magnetic circuits.

On the equivalence of finite difference and edge element formulations in magnetic field analysis using vector potential

“…A comparison between calculated results using accurate and approximate representation of reluctances R mvi for selected examples was previously reported in Demenko et al (1998Demenko et al ( , 2010. The results of this evaluation are also relevant to the comparison between EEM and FDM as in the approximate description the relationship (17) was used.…”

Analogies Between Finite-Difference and Finite-Element Methods for Scalar and Vector Potential Formulations in Magnetic Field Calculations

“…Curvilinear cuboidal elements were used in polar coordinates, which in the slot region were supplemented by triangular curvilinear prisms of 3 edges parallel to the machine shaft axis (for better representation of the slot shape), thus the types of elements unusual for the FDM. The model for such a type of prisms was introduced in [5]. The FEM parameters were then established directly using accurate expressions (1), whereas in the case of FDM formula (6) was used.…”

“…The authors of this paper have previously shown how suitable assumptions and approximations allowed the FEM equations to be identical to those obtained from the Finite Integration Technique (FIT) or equivalent reluctance networks [4], [5]. Analogies between edge formulations (EEM) in FEM and FDM were established when a magnetic vector potential A was employed [6].…”

Steady-state time-periodic finite element analysis of a brushless DC motor drive considering motion

“…The square brackets denote block matrices and block vectors that result from the approximation of a stator stack skew by the axially distributed mesh slices, and R and n S are the machine length and number of mesh slices, respectively. There are a few ways to derive the matrices in Equation (1) depending on the discretization method used [20], that will not be discussed here. The above system of equations describes the machine in transient conditions.…”