Monte Carlo Increased-Radius Floating Random Walk Solution For Potential Problems

Abstract

In this paper, a new Monte Carlo walk method is introduced. The increased radius floating random walk combines of the two classical Monte Carlo methods and derived from fixed-radius floating walk method. In this paper, the method is used to solve typical Laplace’s equations in rectangular region. Also, this method is easily applied to Poisson equations. Lower walk number and hence lower computation time are obtained from new method compared with the fixed random walk, floating random walk and fixed-radius random walk methods. Analyzes were performed on an average computer and the solution time was reduced by 80%. The results are also compared with Finite Element Method. Increased radius walk method’s results are good agreement with other methods.

Keywords

• Floating random walk
• Laplace equations
• Monte Carlo method
• numerical methods
• potential problems

References

1. [1] J. H. Pickles, “Monte Carlo field calculations,” Proceedings of the Institution of Electrical Engineers, vol. 124, no. 12, p. 1271, 1977, doi: 10.1049/piee.1977.0268.
2. [2] W. Yu, K. Zhai, H. Zhuang, and J. Chen, “Accelerated floating random walk algorithm for the electrostatic computation with 3-D rectilinear-shaped conductors,” Simulation modelling practice and theory, vol. 34, pp. 20–36, May 2013, doi: 10.1016/j.simpat.2013.01.003.
3. [3] K. Chatterjee and J. Poggie, “A parallelized 3D floating random-walk algorithm for the solution of the nonlinear Poisson-Boltzmann equation,” Electromagnetic waves, vol. 57, pp. 237–252, Jan. 2006, doi: 10.2528/pier05072802.
4. [4] R. Sotner, A. Lahiri, A. Kartci, N. Herencsar, J. Jerabek, and K. Vrba, “Design of novel precise quadrature oscillators employing ECCIIs with electronic control,” Advances in electrical and computer engineering, vol. 13, no. 2, pp. 65–72, Jan. 2013, doi: 10.4316/aece.2013.02011.
5. [5] O. A. Mousavi, M. S. Farashbashi-Astaneh, and G. B. Gharehpetian, “Improving power system risk evaluation method using Monte Carlo simulation and Gaussian mixture method,” Advances in electrical and computer engineering, vol. 9, no. 2, pp. 38–44, 2009, doi: 10.4316/aece.2009.02007.
6. [6] M. N. O. Sadiku, “Monte Carlo methods in an introductory electromagnetic course,” IEEE transactions on education, vol. 33, no. 1, pp. 73–80, Jan. 1990, doi: 10.1109/13.53630.
7. [7] M. N. O Sadiku, C. M. Akujuobi, and S. M. Musa, “Monte carlo analysis of time-dependent problems,” in Southeastcon’ 06 Proceedings of the IEEE, pp. 7-10, 2006.
8. [8] R. C. Garcia, and M. N. O. Sadiku, “Monte carlo fixed-radius floating random walk solution for potential problems,” in Southeastcon ’96. Bringing Together Education, Science and Technology Proceedings of the IEEE, 1996, pp. 88-91.

9. [9] G. E. Zinsmeister, and J. A. Sawyerr, “A method for improving the efficiency of monte carlo calculation of heat conduction problems,” Journal of Heat Transfer, vol. 96, pp. 246–248, 1974.
10. [10] G. E. Zinsmeister, and S. S. Pan, “A modification of the monte carlo method,” International Journal for Numerical Methods Engineering, vol. 10, pp. 1057–1064, 1976.
11. [11] M. N. O. Sadiku, Monte Carlo Methods for Electromagnetics. CRC Press, 1st Ed., 2009.
12. [12] M. N. O. Sadiku, Numerical Techniques in Electromagnetics. CRC Press, 2nd Ed., 2001.
13. [13] M. N. O. Sadiku, and R. C. Garcia, “Whole field computation using monte carlo method,” International Journal of Numerical Modelling,” vol. 10, pp. 303–312, 1997.
14. [14] D. Meeker, FEMM (Finite Element Method Magnetics), www.femm.info
15. [15] B. Bowerman, and R. T. O’Connell, Applied Statistics. The McGraw-Hill Companies, Inc., 1997.