Optimal parameters of method of fundamental solutions for Poisson problems in heat transfer by means of genetic algorithms

  • Jan A. Kołodziej Poznań University oj Technology
  • Tomasz Klekiel University of Zielona Góra

Abstract

This paper describes the application of the method of fundamental solutions to the solution of the boundary value problems of the two-dimensional steady heat transfer with heat sources. For interpolation of an inhomogeneous term in Poisson equation the radial basis functions are used. Three cases of boundary value problems are solved and five cases of radial basis functions are used. For comparison purposes the boundary value problems for which exact solution exists were chosen. Application of method of fundamental solutions with boundary collocation and radial basis function for solution of inhomogeneous boundary value problems introduces some number of parameters related with these tools. For optimal choosing of these parameters the genetic algorithm is used. The results of numerical experiences related to optimal parameters are presented .

Keywords

method of fundamental solutions; meshless method; radial basis functions; steady heat transfer, Poisson's equation; genetic algorithm; particular solution,

References

[1] K.E. Atkinson. The numerical evaluation of particular solutions for Poisson's equation. IMA Journal of Numerical Analysis, 5: 319- 38, 1985.
[2] K. Balakrishnan, P.A. Ramachandran. A particular solution Trefftz method for non-linear Poisson problems in heat and mass transfer. Journal of Computational Physics, 150: 239- 267, 1999.
[3] A. Bogomolny. Fundamental solutions method for elliptic boundary value problems. SIAM Journal of Numerical Analysis, 22: 644-69, 1985.
[4] G. Burges, E. Mahajerin. The fundamental collocation method applied to the non-linear Poisson equation in two dimensions. Computers and Structures, 27: 763- 767, 1987.
[5] R.E. Carlson, T.A. Foley. The parameter in multiquadric interpolation. Computers and Mathematics with Applications, 21: 29-42, 1991.
[6] A.P. Cisilino. Application of a simulated annealing algorithm in the optimal placement of source points in the method of the fundamental solutions . Computational Mechanics, 28: 129- 136,2002.
[7] A.H.D. Cheng, D.L. Young, C.C. Tsai. Solution of Poisson's equation by iterative DRBEM using compactly supported, positive definite radial basic function. Engineering Analysis with Boundary Elements, 24: 549-557, 2000.
[8] A.H.-D. Cheng, M.A. Golberg, E.J. Kansa, G. Zammito. Exponential convergence and H-c multiquadric collocation method for partial differential equations. Numerical Methods for Partial Differential Equations, 19: 571- 594, 2003.
[9] G. Faiweather, A. Karageorghis. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, 9: 69- 95, 1998.
[10] T.A. Foley. Interpolation and approximation of 3-D and 4-D scattered data. Math. Appl., 13: 711- 740, 1987.
[11] C. Franke. Scattered data interpolation: test of some methods. Mathematics and Computers, 38: 181- 200, 1982.
[12] M.A. Golberg. The method of fundamental solutions for Poisson's equation. Engineering Analysis with Boundary Elements, 18: 9- 17, 1995.
[13] M.A. Golberg, C.S. Chen, S. Karur. Improved multiquadric approximation for partial differential equations. Engineering Analysis with Boundary Elements, 18: 9- 17, 1996.
[14] R.L. Hardy. Multiquadric equation of topography and other irregular surfaces. Journal of Geophysical Research, 76: 1905- 1915, 1971.
[15] I. Herrera, F. Sabina. Connectivity as an alternative to boundary integral equations: Construction of bases. Proceedings of National Academy of Science USA, 75: 2059- 2063, 1978.
[16] F.J. Hickernell, Y.C. Hon. Radial basis function approximation of the surface wind field from scattered data. International Journal of Applied Scientific Computation, 4: 221- 247, 1998.
[17] E.J. Kansa, Y.C. Hon. Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Computers and Mathematics with Applications, 39: 123-137,2000.
[18] A. Karageorghis, G. Fairweather. The Almansi of fundamental solutions for solving biharmonic problems. International Journal for Numerical Methods in Engineering, 26: 1668- 1682, 1988.
[19] A. Karageorghis, G. Fairweather. The method of fundamental solutions for numerical solution of the biharmonic equation. Journal of Computational Physics, 69: 434- 459, 1987.
[20] A. Karageorghis, G. Fairweather. The simple layer potential method of fundamental solutions for certain biharmonic equation. International Journal for Numerical Methods in Fluids, 9: 1221- 1234, 1989.
[21] J.A. Kołodziej. Review of application of boundary collocation methods in mechanics of continuous media. Solid Mechanics Archives, 12: 187- 231, 1987.
[22] C.K. Lee, X. Liu, C.S. Fan. Local multiquadric approximation for solving boundary value problems. Computational Mechanics, 30: 396-409, 2003.
[23] J. Li. Mathematical justification for RBS-MFS. Engineering Analysis with Boundary Elements, 25: 897- 901 , 2001.
[24] J. Li, Y.C. Hon, C.S. Chen. Numerical comparison of two meshless methods using radial basis functions. Engineering Analysis with Boundary Elements, 26: 205- 225, 2002.
[25] Z. Michalewicz. Genetic Algorithm + Data Structures = Evolutionary Programs. Springer-Verlag, AI Series, New York,1992.
[26] P. Mitic, Y.F. Rashed. Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources. Engineering Analysis with Boundary Elements, 28: 143- 153,2004.
[27] R. Nishimura, K. Nishimori, N. Ishihara. Determining the arrangement of fictitious charges in charge simulation method using genetic algorithms. Journal of Electrostatics, 49: 95- 105, 2000.
[28] R. Nishimura, K. Nishimori, N. Ishihara. Automatic arrangement of fictitious charges and contour points in charge simulation method for polar coordinate system. Journal of Electrostatics, 51-52: 618- 624, 2001.
[29] R. Nishimura, M. Nishihara, K. Nishimori, N. Ishihara. Automatic arrangement of fictitious charges and contour points in charge simulation method for two spherical electrodes. Journal of Electrostatics, 57: 337- 346, 2003.
[30] H. Power, J .J. Rodriguez. An adaptative dual reciprocity scheme for the numerical solution of then Poisson equation. Engineering Analysis with Boundary Elements, 26: 283-300, 2002. .
[31] A. Poullikkas, A. Karageorghis, G. Georgiou. The method of fundamental solutions for inhomogeneous elliptic problems. Computational Mechanics, 22: 100- 107, 1998.
[32] S. Rippa. An algorithm for selecting a good parameter c in radial basis function. Advances in Computational Mathematics, 11: 193-210, 1999.
[33] R. Schaback. Error estimates and condition numbers for radial basis function interpolation. Advances in Computational Mathematics, 3: 251- 264, 1995.
[34] C.J. Trahan, R.E. Wyatt. Radial basis function interpolation in the quantum trajectory method: optimization of the multi-quadric shape parameter. Journal of Computational Physics, 185: 27-49, 2003.
[35] A. Uściłowska-Gajda, J.A. Kołodziej, M. Ciałkowski, A. Frąckowiak. Comparison of two types of Trefftz method for the solution of inhomogeneous elliptic problems. Computer Assisted Mechanics and Engineering Sciences, 10: 375- 389, 2003.
Published
Jul 22, 2022
How to Cite
KOŁODZIEJ, Jan A.; KLEKIEL, Tomasz. Optimal parameters of method of fundamental solutions for Poisson problems in heat transfer by means of genetic algorithms. Computer Assisted Methods in Engineering and Science, [S.l.], v. 15, n. 2, p. 99-112, july 2022. ISSN 2956-5839. Available at: <https://cames.ippt.gov.pl/index.php/cames/article/view/750>. Date accessed: 23 dec. 2024.
Citation Formats
Issue
Vol 15 No 2 (2008)
Section
Articles