Solution of Axisymmetric Inhomogeneous Problems with the Markov Chain Monte Carlo
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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.
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References
V. F. Fusco and P. A. Linden, "A Markov Chain Approach for Static Field Analysis," Microwave and Optical Technology Letters, vol. 1, no. 6, 1988, pp. 216-220.
A. E. Shadare, M. N. O. Sadiku and S. M. Musa, "Analysis of Microstrip Line using Markov Chain Monte Carlo," International Conference on Scientific Computing, 2015, pp. 135-139.
A. E. Shadare, M. N. O. Sadiku and S. M. Musa, "Solution to Poisson's Equation in Rectangular Inhomogeneous Dielectric Media with the Markov Chain Monte Carlo," International Conference on Scientific Computing, 2017, pp. 10-15.
M. N. O. Sadiku, R. C. Garcia, S. M. Musa and S. R. Nelatury, "Markov Chain Monte Carlo Solution of Poisson's Equation," International Journal on Recent and Innovative Trends in Computing and Communication, vol. 3, Issue 1, January 2015.
K. Gu and M. N. O. Sadiku, "Absorbing Markov Chain Solution for Poisson's Equation," Proceedings of the IEEE Southeastcon, 2000, pp. 297-300.
R. C. Garcia, M. N. O. Sadiku and K. Gu, "Applying Absorbing Markov Chains to Solve Poisson's Equation in Inhomogeneous Regions," Proceedings of the IEEE Southeastcon, 2001, pp. 166-168.
P. R. P. Hoole and A. J. Pearmain, "A Review of the Finite-Difference Method for Multidielectric Electrostatic Field problems with Sharp-Edged Electrodes," Electric Power Systems Research, 24 (1992), pp. 19-30.
F. Sanchez-Quesada et al., "Monte-Carlo Method for Discrete Inhomogeneous Problems," Proceedings of the Institution of Electrical Engineers, 1978, vol. 125, Issue 12, pp. 1400-1402.
R. Schlott, "A Monte Carlo Method for the Dirichlet Problem of Dielectric Wedges," IEEE Transactions on Microwave Theory and Techniques, 1988, vol. 36, no. 4, pp. 724-730.
R. C. Garcia and M. N. O. Sadiku, "Monte Carlo Fixed-Radius Floating Random Walk Solution for Potential Problems," Proceedings of the IEEE Southeastcon, 1996, pp. 88-91.
J. H. Pickles, "Monte-Carlo Calculation of Electrically Induced Human-Body Currents" IEE Science Proceedings, Vol. 134, Pt. A, No. 9, pp. 705-711, November 1987.
M. N. O. Sadiku and R. C. Garcia, "Monte Carlo Floating Random Walk Solution of Poisson's Equation," Proceedings of Southeastcon, 1993.
K. Gu and M. N. O. Sadiku, "A triangular mesh random walk method for Dirichlet Problem," J. Franklin Inst., March, 1995, pp. 569-578.
M. N. O. Sadiku, S. O. Ajose, and Zhibao Fu, "Applying the Exodus Method to Solve Poisson's Equation," IEEE Trans. Microwave Theory and Techniques, vol. 42, no.4, April, 1994, pp. 661-666.
N. J. Jayant and L. L. C. Yannick, "An Improved Floating-Random-Walk Algorithm for Solving the Multi-Dielectric Dirichlet Problem," IEEE Transactions on Microwave Theory and Techniques, 1993, vol. 41, no. 2, pp. 325-329.
G. E. Zinsmeiter and J. A. Sawyerr, "A method for improving the efficiency of Monte Carlo Calculation of heat conduction problems," Transactions of the ASME, Journal of Heat Transfer, vol. 96, 1974, pp. 246-248.
G. E. Zinsmeiter and S. S. Pan, "A Modification of the Monte Carlo Method," International Journal for Numerical Methods in Engineering, vol. 10, 1976, pp. 1057-1064.
V. F. Fusco and P. A. Linden, "A Markov Chain Approach for Static Field Analysis," Microwave and Optical Technology Letter, vol. 1, no. 6, August 1988, pp. 216-220.
M. N. O. Sadiku and R. C. Garcia, "Whole Field Computation Using Monte Carlo Method," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 10, 1997, pp. 303-312.
M. N. O. Sadiku and K. Gu, "Floating random walk method on axisymmetric potential problems," 1994 International Symposium on Electromagnetic Compatibility, Sendia, Japan, May, 1994, pp. 659-662.
M. N. O. Sadiku, "Monte Carlo Solution of Axisymmetric Potential Problems," IEEE Transactions on Industry Applications, vol. 29, no. 6, 1993, pp. 1042 - 1046.
F. Z. Louai, N. N. Said and S. Drid, "Numerical Analysis of Electromagnetic Axisymmetric Problems Using Element Free Galerkin Method," Journal of Electrical Engineering, vol. 57, No. 2, 2006, pp. 99-104.
A. N. Hayati, M. M. Ahmadi, and S. A. Sadrnejad, "Analysis of Axisymmetric Problems By Element-Free Galerkin Method," International Journal of Modeling and Optimization, vol. 2, No. 6, December 2012.
R. D. Soares, F. J. S. Moreira, R. C. Mesquita, D. A. Lowther and N. Z. Lima, "A Modified Meshless Local Petrov-Galerkin Applied to Electromagnetic Axisymmetric Problems," IEEE Transactions on Magnetics, vol. 50, No. 2, February 2014.
J. B. M. Melissen and J. Simkin, "A New Coordinate Transform For The Finite Element Solution of Axisymmetric Problems In Magnetostatics," IEEE Transactions on Magnetics, vol. 26, No. 2, March 1990.
A. Boglietti, M. Chiampi, D. Chiarabaglio and M. Tartaglia, "Finite Element Approximation In Axisymmetrical Domains," IEEE Transactions on Magnetics, vol. 26, No. 2, March 1990
A. Yoneta, M. Tsuchimoto and T. Honma, "An Analysis of Axisymmetric Modified Helmholtz Equation By Using Boundary Element Method," IEEE Transactions on Magnetics, vol. 26, no. 2, March 1990.
M. N. O. Sadiku and Raymond C. Garcia, "Method of Lines Solution of Axisymmetric Problems," Proceedings of the IEEE Southeastcon, 2000, pp. 527-530.
S. Gratkowski, T. Todaka, M. Enokizono, and R. Sikora, "Asymptotic Boundary Conditions for the Finite Element Modeling of Axisymmetric Electrical Field Problems," IEEE Transactions on Magnetics, vol. 36, No. 4, pp. 717-721, July 2000.
Q. Chen, A. Konrad and S.Baronijan, "Asymptotic Boundary Conditions for Axisymmetric Finite Element Electrostatic Analysis,' " IEEE Transactions on Magnetics, vol. 30, No. 6, pp. 4335- 4337, November 1994.
M. Ehrich, J. Kuhlmann and D. Netzler, "High Accuracy Integration of Boundary Integral Equations Describing Axisymmetric Field Problems," Asia Pacific Microwave Conference, 1997, pp. 462-464.
L. Krahenbuhl and A. Nicolas, "Axisymmetric Formulation for Boundary Integral Equation Methods in Scalar Potential Problems," IEEE Transactions on Magnetics, vol. MAG-19, NO. 6, pp. 2364- 2366, November 1983.
E. Godoy, V. Boccardo and M. Durán, "A Dirichlet-to-Neumann Finite Element Method for Axisymmetric Elastostatics in a Semi-Infinite Domain," Journal of Computational Physics, 2017, pp. 1-26.
M. N. O. Sakidu, Computational Electromagnetics with MATLAB. Boca Raton, FI, Boca Raton, CRC Press, Fourth Edition, 2019.
M. N. O. Sakidu, Monte Carlo Methods for Electromagnetics. CRC Press, Boca Raton, FI, 2009.
M. N. O. Sadiku, "Monte Carlo Methods in an Introductory Electromagnetic Course," IEEE Transactions on Education, 33, 1, February 1990, pp. 73-80.
M. A. Kolbehdari and M. N. O. Sadiku, "Finite & Infinite Element Analysis of Coupled Cylindrical Microstrip Line in a Nonhomogeneous Dielectric Media," Proceedings of IEEE Southeastcon, 1995. pp. 269-273.
M. N. O. Sadiku and D. T. Hunt, "Solution of Dirichlet Problems by the Exodus Method," IEEE Transactions on Microwave Theory and Techniques, 40, 1, January 1992, pp. 89-95.