Solution of Axisymmetric Potential Problem in Oblate Spheroid Using the Exodus Method

This paper presents the use of Exodus method for computing potential distribution within a conducting oblate spheroidal system. An explicit finite difference method for solving Laplace’s equation in oblate spheroidal coordinate systems for an axially symmetric geometry was developed. This was used to determine the transition probabilities for the Exodus method. A strategy was developed to overcome the singularity problems encountered in the oblate spheroid pole regions. The potential computation results obtained correlate with those obtained by exact solution and explicit finite difference methods. 1. Introduction An oblate spheroid is the surface generated by the rotation of an ellipse about its minor axis, and depending upon the ellipse’s eccentricity, the spheroid will be flattened about the minor axis [1]. An oblate spheroidal shell, for instance, is considered as a continuous system constructed from two spherical shell caps by matching the continuous boundary conditions [2]. Oblate and prolate spheroidal coordinates are widely used in many fields of science and engineering, such as potential theory, fluid mechanics, heat and mass transfer, thermal stress, and elastic inclusions. For example, oblate and prolate spheroids being surfaces of revolution can be more easily conformed to most districts of human body (e.g., extremities) which is of interest for dedicated MRI systems [3]. Oblate spheroidal coordinates are the natural choice for the translation of any ellipsoid parallel to a principal axis [4]. There is a more recent improvement in the lightning ground tracking systems based on the time-of-arrival (TOA) technique because of the refinement in the mathematics to more accurately accommodate the oblate shape of the earth spheroid. Approximating the earth as a perfect sphere affects not only the accuracy of time clock offset calculations, but also the accuracy of stroke coordinate computation given receiver time differences. Oblate solution mathematics can provide a substantial systematic error reduction of up to 50% percent [5]. In this paper, Exodus method is used to compute potential distribution inside conducting oblate spheroidal shells maintained at two potentials. This work is a continuation of our previous work in which a fixed random walk Monte Carlo method (MCM) was used for numerical computation of potential distribution with two conducting oblate spheroidal shells maintained at two potentials [6]. An explicit Neumann boundary condition was imposed at the pole regions of the oblate spheroid to treat the presence of singularities in