Abstract
With increasing complexity of today’s electromagnetic problems, the need and opportunity to reduce domain sizes, memory requirement, computational time and possibility of errors abound for symmetric domains. With several competing computational methods in recent times, methods with little or no iterations are generally preferred as they tend to consume less computer memory resources and time. This paper presents the application of simple and efficient Markov Chain Monte Carlo (MCMC) method to the Laplace’s equation in axisymmetric homogeneous domains. Two cases of axisymmetric homogeneous problems are considered. Simulation results for analytical, finite difference and MCMC solutions are reported. The results obtained from the MCMC method agree with analytical and finite difference solutions. However, the MCMC method has the advantage that its implementation is simple and fast.
Highlights
Most real-world EM problems are difficult to solve using analytical methods and in most cases, analytical solutions are outright intractable [1]
This paper presents the application of simple and efficient Markov Chain Monte Carlo (MCMC) method to the Laplace’s equation in axisymmetric homogeneous domains
The inhomogeneous Dirichlet boundary condition with different levels of complexity was enforced for the two cases presented
Summary
Most real-world EM problems are difficult to solve using analytical methods and in most cases, analytical solutions are outright intractable [1]. The resulting two-dimensional approximation can be treated like a Cartesian coordinate problem with either z or ρ constant interface Several methods such as the Method of Lines [11], the Finite Element method [12] [13], Finite difference method [14] and the Boundary Integral Equation methods [15] [16] have all been applied in the modeling and analysis of axisymmetric problems. The shrinking boundary and the inscribed figure methods later proposed for whole-field calculations are not significantly superior to the classical Monte Carlo methods [27] [28] To address this gap, Markov Chains for whole-field computations was proposed by Andrey Markov [29] [30]. This paper presents the MCMC solution of Laplace’s equation in axisymmetric homogeneous region
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