Finite element iterative solution of skin effect problems in open boundaries

Abstract

The paper proposes an iterative procedure, called current iteration, for the finite element solution of two-dimensional steady-state skin effect problems in open boundaries. In the procedure a fictitious boundary is introduced enclosing all the conductors. On it, the magnetic vector potential is first guessed and then iteratively updated according to the current density computed in the conductors. Conditions are obtained implying convergence to the exact solution of the unbounded problem whatever the initial guess. The choice of the fictitious boundary and the selection of the relaxation parameter in such a way that computational efficiency is obtained are discussed. The greatest advantage of the procedure is its ease of implementation in a pre-existing finite element code for bounded problems, An axisymmetric version of the procedure is also described since implementation only involves minor changes as compared with the 2-D one. Examples are provided in order to clarify and validate the procedure and compare it with other techniques.

Improving the integral equation in the hybrid FEM‐RBCI method for scalar electromagnetic scattering problems

“…Note that a similar approach has been successfully applied to the solution of electrostatic (Aiello et al, 1994(Aiello et al, , 1996a, and time-harmonic skin effect problems (Aiello et al, 1996b) in open boundaries, in which a Dirichlet boundary condition is used on the fictitious boundary G F .…”

Comparing FEM-BEM and FEM-DBCI for open-boundary electrostatic field problems

“…First, in the FEM-DBCI integral equation (18) only one dense matrix N is present, whereas in the FEM-BEM integral equation (8) two dense matrices H and G appear. Since the number of entries of N is (in general) approximately equal to the sum of those of H and G, the two methods exhibit comparable memory requirements for the integral equations.…”

“…In the literature several methods have been devised to enable the Finite Element Method (FEM) [1] to solve static and quasi-static electromagnetic field problems in unbounded domains, such as infinite elements [2], coordinate transformations [3], the hybrid FEM/BEM (Boundary Element Method) method [4,5], and the hybrid FEM-DBCI (Dirichlet Boundary Condition Iteration) method proposed by the authors to solve electrostatic [6,7], timeharmonic skin effect [8,9] and eddy current [10] problems.…”

Finite-element mesh adaptation of 2-D time-harmonic skin effect problems

“…1). By selecting a Cartesian reference frame having the -axis parallel to the conductors, the magnetic vector potential is -directed and in the time-harmonic steady-state case satisfies the following integro-differential equation in the -th conductor [6], [7]:…”