^{1}

^{1}

^{1}

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.

Most real-world EM problems are difficult to solve using analytical methods and in most cases, analytical solutions are outright intractable [

Several methods such as the Method of Lines [

However, to the best of the authors’ knowledge, the solutions of Laplace’s equation in axisymmetric homogeneous domains with MCMC with inhomogeneous Dirichlet boundary condition are yet to be reported in the literature. Thus, this paper presents the MCMC solution of Laplace’s equation in axisymmetric homogeneous region.

When it is necessary and convenient, electromagnetic problems in cylindrical coordinates may be approximated to axisymmetric solution region. Suppose a cylindrical coordinate system is as shown in

The Laplace’s equation in the axisymmetric region R depicted in

The corresponding finite difference equivalence of Equation (1) for

The transition probabilities in the Equation (2) are given as [

Similarly, the finite difference equivalence of Equation (1) for

The corresponding transition probabilities in the Equation (3) are given as [

A Markov chain is a mathematical model that represents a sequence of random variables

The transition probability P is defined as

The sum of each row elements in P matrix is 1. This shows that the matrix P is stochastic as described in Equation (5).

The size of the transition matrix P is

From the Equation (6),

The transition matrix P in which absorbing nodes and the non-absorbing nodes are numbered first and last respectively is given by [

where

R is a

Q is a

I is a

0 is a

To solve Laplace’s equation in the region R, the elements of matrix Q for nodes in the

Equation (8) describes the probabilities of moving from one node to another in the

Similarly, the elements of matrix Q for nodes at the line of symmetry,

The same Equation (8) and Equation (9) apply to except that j is a fixed node.

For any absorbing Markov chains, the fundamental matrix,

The absorption probability matrix B which describes the probabilities that a randomly walking particle starting at free node i will be absorbed at a fixed node j is given as

The size of matrix B is

where

Finally, the potential at any free node i is given as

where

In this section, simulation results are presented for the two cases of homogeneous axisymmetric problems considered. The inhomogeneous Dirichlet boundary condition with different levels of complexity was enforced for the two cases presented. At the line of symmetry, the Neumann boundary condition was imposed. Simulation results are reported for both cases.

Case I: Axisymmetric homogeneous Problem with Inhomogeneous Dirichlet Boundary Condition

Suppose the axisymmetric cylinder given in

and

In order to demonstrate the effectiveness of the MCMC method for axisymmetric homogeneous problems, the following simulation is carried out. The parameters used for the simulations are given in

From the given parameters, the elements of matrix Q with the size 3160 × 3160 are formed based on the Equation (8) and Equation (9). Similarly, the elements of matrix R with size 3160 × 159 are formed from the Equation (8) and Equation (9) except that j is a fixed node. Then the identity matrix I with size 3160 × 3160 is determined.

Based on the foregoing, the fundamental matrix N with size 3160 × 3160 and the absorption matrix B with size

Parameter | Value |
---|---|

V_{0} | 100 V |

a | 1 m |

L | 2 m |

where

A solution to the present problem is also obtained with the finite difference method and comparison was made with the MCMC and analytical solutions. Simulation results are presented in

Case II: Axisymmetric Homogeneous Problem with Inhomogeneous Dirichlet Boundary Condition.

In this section, another case of Laplace’s equation with inhomogeneous Dirichlet boundary conditions is presented. Suppose the cylinder given in

Coordinate ( | Analytical (V) | FDM (V) | FDM (V) | MCMC (V) | MCMC (V) |
---|---|---|---|---|---|

(0.25, 0.3) | 10.8381 | 10.8427 | 10.8387 | 10.7346 | 10.8394 |

(0.35, 1.5) | 17.6737 | 17.6838 | 17.6756 | 17.3850 | 17.6765 |

(0.5, 1.05) | 27.9346 | 27.9713 | 27.9431 | 27.2701 | 27.9441 |

(0.6, 1.6) | 16.2614 | 16.2642 | 16.2616 | 16.0837 | 16.2621 |

(0.8, 0.6) | 26.3891 | 26.3983 | 26.3960 | 26.2916 | 26.3963 |

where a and L are given in

Similarly, recall that the analytical solution for region II depicted in

as:

where

With the step size of 0.025 m, the MCMC results are reported in the

Coordinate ( | Analytical (V) | FDM (V) | FDM (V) | MCMC (V) | MCMC (V) |
---|---|---|---|---|---|

(0.25, 0.3) | 12.7037 | 12.5030 | 12.7054 | 12.5798 | 12.7065 |

(0.35, 1.5) | 55.6216 | 55.3465 | 55.6180 | 55.2416 | 55.6195 |

(0.5, 1.05) | 38.8788 | 38.6227 | 38.8909 | 38.0625 | 38.8926 |

(0.6, 1.6) | 53.0784 | 52.9373 | 53.0772 | 52.8529 | 53.0780 |

(0.8, 0.6) | 27.8036 | 27.7157 | 27.8114 | 27.6848 | 27.8119 |

This paper presents a comprehensive application of MCMC method to Laplace’s equation in homogeneous axisymmetric domains. Two broad cases of homogeneous axisymmetric problems were investigated. For Case I, the MCMC method was used to solve Laplace’s equation with inhomogeneous Dirichlet boundary condition in which the top and bottom boundaries were characterized by the same potential. For case II, the MCMC method was investigated with the Laplace’s equation using inhomogeneous Dirichlet boundary condition in which the top and bottom boundaries (prescribed potentials) were at different potentials. Also, Neumann boundary condition was imposed at the line of symmetry. Several plots were reported. The MCMC results were compared with the analytical and finite difference solutions. The proposed MCMC method agreed with the analytical and finite difference solutions for all reported cases. However, the difference in computation time between MCMC and FDM is in the order of seconds for the problems considered and thus cannot be used as a basis for comparison.

The authors declare no conflicts of interest regarding the publication of this paper.

Shadare, A.E., Sadiku, M.N.O. and Musa, S.M. (2019) Markov Chain Monte Carlo Solution of Laplace’s Equation in Axisymmetric Homogeneous Domain. Open Journal of Modelling and Simulation, 7, 203-216. https://doi.org/10.4236/ojmsi.2019.74012

MCMC: Markov Chain Monte Carlo;

1D: One dimension;

2D: Two dimension;

3D: Three dimension;

EM: Electromagnetics;

FDM: Finite difference method;

P: Transition probability;

N: Fundamental matrix;

B: Absorption probability matrix.