Abstract
With increasing complexity of EM problems, 1D and 2D axisymmetric approximations in p, z plane are sometimes necessary to quickly solve difficult symmetric problems using limited data storage and within shortest possible time. Inhomogeneous EM problems frequently occur in cases where two or more dielectric media, separated by an interface, exist and could pose challenges in complex EM problems. Simple, fast and efficient numerical techniques are constantly desired. This paper presents the application of simple and efficient Markov Chain Monte Carlo (MCMC) to EM inhomogeneous axisymmetric Laplace’s equations. Two cases are considered based on constant and mixed boundary potentials and MCMC solutions are found to be in close agreement with the finite difference solutions.
Highlights
Homogeneous and inhomogeneous Laplace’s equations with Dirichlet boundary conditions in Cartesian coordinates have been extensively studied using the Markov Chain Monte Carlo (MCMC) method [1]-[6]
Inhomogeneous EM problems frequently occur in cases where two or more different dielectric media separated by an interface exist and could pose challenges in complex EM problems
The Markov Chain Monte Carlo (MCMC) method involves no use of random number generator and not subject to randomness and the approach is potentially accurate [19]
Summary
Homogeneous and inhomogeneous Laplace’s equations with Dirichlet boundary conditions in Cartesian coordinates have been extensively studied using the MCMC method [1]-[6]. M. Ulam, various kinds of Monte Carlo methods such as fixed random walk, floating random walk and Exodus methods have evolved. Classical Monte Carlo methods like the fixed random walk, floating random walk, Exodus method have all been used successfully as numerical techniques for field computation in spite of their major limitation that they only allow single point calculations. The shrinking boundary and inscribed figure methods were proposed for whole-field calculation but they still offered no significant advantage over the conventional Monte Carlo techniques [16]-[17]. In this paper, we propose the application of simple and efficient MCMC method to the solution of axisymmetric inhomogeneous Laplace’s equations. To the best of the authors’ knowledge, solution of axisymmetric inhomogeneous Laplace’s equations with the MCMC is rare or not yet reported in the literature.
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