A QR accelerated volume-to-surface boundary condition for finite element solution of eddy current problems Page: 3 of 19
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A QR Accelerated Volume-to-Surface Boundary
Condition for the Finite Element Solution of Eddy
Current Problems
Daniel A. White, Benjamin J. Fasenfest, Robert N. Rieben, and Mark L. Stowell
Abstract
We are concerned with the solution of time-dependent electromagnetic eddy current problems using a finite element formulation
on three-dimensional unstructured meshes. We allow for multiple conducting regions, and our goal is to develop an efficient compu-
tational method that does not require a computational mesh of the air/vacuum regions. This requires a sophisticated global boundary
condition specifying the total fields on the conductor boundaries. We propose a Biot-Savart law based volume-to-surface boundary
condition to meet this requirement. This Biot-Savart approach is demonstrated to be very accurate. In addition, this approach can be
accelerated via a low-rank QR approximation of the discretized Biot-Savart law.
Index Terms
Maxwell's equations, computational electromagnetics, low-rank approximation, Biot-Savart law, eddy currents, electromagnetic
diffusion, parallel processing.
I. INTRODUCTION
In this paper, we present a finite element method for solving the multiply connected eddy current problem. While much of this
presentation is applicable towards frequency-domain analysis, our emphasis here is on transient simulation. Various formulations
for the eddy current equations exist and have been extensively reviewed and studied in the literature. These include formulations
which solve for the electric field (the E field formulation) [1], [2], [3], the magnetic field (the H field formulation) [4], [5] or
for the potential field (the A-0 potential formulation) [6], [7], [8], [9]. Each formulation has its advantages and disadvantages
for problems in computational electromagnetics. However, it has been shown that when using H(Curl) and H(Div) conforming
finite element methods there is no difference in accuracy for these three formulations, even for secondary quantities such as B
and 1 [10]. The difference between the three formulations, which use primary field variables E,H, and A, respectively, is in the
boundary conditions and the source terms, and is therefore simply a matter of which formulation is most convenient for a given
electromagnetics problem.
The most difficult electromagnetic diffusion problems encountered in practice are those that involve multiple conductors sep-
arated by a non-conducting region, the so-called multiply connected eddy current problem. While the currents are zero in the
non-conducting region clearly the fields are not, and some method must be used to account for these fields. One approach is to
simply mesh the non-conducting region and use a small value of conductivity in this region. While seemingly a crude approach,
it works well in practice for many problems, for example using a conductivity at least 103 times smaller than the metal results in
fields correct to within the discretization error [7] [11]. The difficulty with this is twofold; one is the large number of unknowns
and the second is matrix ill-conditioning. More sophisticated approaches include forming a magnetostatic problem in the non-
conducing region using either the vector or scalar magnetic potential and coupling the two finite element solutions [6] [12], or
employing a surface integral equation to correctly model the global boundary condition [1] [2] [13].
In the context of Galerkin approximations of electromagnetics PDE's, the choice of the finite element space plays a crucial
role in the stability and convergence of the discretization. For instance, in numerical approximations of the magnetic and electric
field intensities, H(Curl) conforming finite element spaces (or edge elements) are preferred over traditional nodal vector spaces
since they eliminate spurious modes in eigenvalue computations and they prevent fictitious charge build-up in time-dependent
computations. The lowest order H(Curl) conforming basis functions were developed by Whitney [14] before the advent of
finite element programs. Arbitrary order versions were introduced by N6d6lec [15], [16] as a generalization of the mixed finite
element spaces introduced by P.A. Raviart and J.M. Thomas [17] for H(Div) conforming methods. Application of these H(Curl)
and H(Div) basis functions toward electromagnetics is becoming quite popular and applications can be found in several recent
textbooks [18], [5], [19]
In this paper we focus on the A field formulation for the eddy current problem using H(Curl) basis functions. We assume
a given initial condition (which may be zero) and the problem is driven by either a time-varying voltage or a prescribed time-
varying current. We allow for multiple conducting regions separated by air. Each individual conducting region is assumed to be
This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under
contract No. W-7405-Eng-48
Defense Sciences Engineering Division, Lawrence Livermore National Laboratory, whit e 37c@llnl . govI
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White, D; Fasenfest, B; Rieben, R & Stowell, M. A QR accelerated volume-to-surface boundary condition for finite element solution of eddy current problems, article, September 8, 2006; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc893391/m1/3/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.