galerkinformulation - IEEE mansactions on Electrical...

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IEEE mansactions on Electrical Insulation Vol. 27 No. 1, February 1002 135 A Galerkin Formulation of the Boundary Element Method for Two-dimensional and Axi-symmetric Problems in Electrostatics D. Beatovic, P. L. Levin, S. Sadovic' and R. Hutnak Department of Electrical Engineering, Worcester Polytechnic Institute, Worcester, MA ABSTRACT Charge simulation and boundary element techniques typical- ly solve for discretized charge densities on or within domain boundaries by satisfying, in general, the Cauchy condition for a discrete number of collocation points. No constraint is im- posed upon the approximation except at these locations, and the boundary conditions may not be met at other points along the boundary. We propose to process the Fkedholm integral equation relating potential to an unknown source density func- tion by the Galerkin weighted residual technique. In essence, this allows us to optimally satisfy the Dirichlet condition over the entire conductor surface. Solving the resulting equations requires evaluation of a second surface integration over weak- ly singular kernels, and the increased accuracy comes at some computational expense. The singularity issue is addressed ana- lytically for 2-D problems and semi-analytically for axi-symme- tric problems. We describe how the integrals are evaluated for both the standard and Galerkin Boundary element functions using zero, first, and second orde'r, interpolation functions. We demonstrate that the Galerkin solution is superior to the stan- dard collocation procedure for some canonical problems, in- cluding one in which analytical charge density becomes singu- lar. 1. INTRODUCTION applications, numerical techniques are generally the only viable approach. N order to calculate an electrostatic field distribution, Numerical techniques fall into two broad categories, I one has to solve Laplace's partial differential equation the finite element methods and finite difference methods, with boundary conditions prescribed over the boundary which solve the governing differential equations directly, I?. Analytical solutions are generally limited to relatively and the charge simulation methods, boundary element simple geometries. When more complicated geometries methods and surface charge simulation technique which are encountered, as is often the case in HV engineering rely upon reformulating the problem an integral equa- 0018-9367 $1.00 @ 1992 IEEE Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY KANPUR. Downloaded on August 15,2010 at 13:19:05 UTC from IEEE Xplore. Restrictions apply.
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136 Beatovic et al.: A Galerkin Formulation of the Boundary Element Method tion. The purpose of this work is to provide details of ex- tending the capabilities of boundary element techniques using the Galerkin method. 2. THE CHARGE SIMULATION METHOD HEN an electrostatic field is to be calculated by the W charge simulation method, the continuous chargk density on the surface of the conductor is replaced by fic- titious, discrete charges that are placed within the con- ductor [l]. Suppose that the number of such charges is
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galerkinformulation - IEEE mansactions on Electrical...

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