Skip to main content Accessibility help
×
Home

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

  • A. Tadeu (a1), C. S. Chen (a2), J. António (a1) and Nuno Simões (a1)

Abstract

Fourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

Copyright

Corresponding author

URL: http://www.math.usm.edu/cschen/ , Email: tadeu@dec.uc.pt
Corresponding author. Email: cs.chen@usm.edu

References

Hide All
[1] Carini, A., Diligenti, M. and Salvadori, A., Implementation of a symmetric boundary element method in transient heat conduction with semi-analytical integrations, Int. J. Numer. Methods. Eng., 46 (1999), pp. 18191843.
[2] Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Second edition, Oxford University Press, London, 1959.
[3] Chang, Y. P., Kang, C. S and Chen, D. J., Use of fundamental Green’s functions for solution of problems of heat-conduction in anisotropic media, Int. J. Heat. Mass. Trans., 16 (1973), pp. 19051918.
[4] Chen, C. S. and Rashed, Y. F., Evaluation of thin plate spline based particular solutions for Helmholtz-type operators for the DRM, Mech. Res. Commun., 25(2) (1998), pp. 195201.
[5] Chen, C. S., Golberg, M. A. and Rashed, Y. F., A mesh-free method for linear diffusion equations, Numer. Heat. Trans. B. Fundament., 33 (1998), pp. 469486.
[6] Chen, W. and Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl., 43(3-5) (2002), pp. 379391.
[7] Cho, H., Golberg, M. A., Muleshkov, A. S. and Li, X., Trefftz methods for time dependent partial differential equations, CMC., 1(1) (2004), pp. 137.
[8] Duchon, J., Splines minimizing rotation invariant semi-norms in Sobolev spaces, in: Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571, ed. Schempp, W. and Zeller, K., Springer-Verlag, Berlin, pp. 85110, 1976.
[9] Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9(1-2) (1998), pp. 6995.
[10] Fu, Z. J., Chen, W. and Yang, W., Winkler plate bending problems by a truly boundary-only boundary particle method, Comput. Mech., 44(6) (2009), pp. 757763.
[11] Golberg, M. A. and Chen, C. S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, in Boundary Integral Methods: Numerical and Mathematical Aspects, ed. Golberg, M. A., Computational Mechanics Publications, pp. 103176, 1999.
[12] Golberg, M. A. and Chen, C. S., An efficient mesh-free method for nonlinear reaction-diffusion equations, Comput. Model. Eng. Sci., 2 (2001), pp. 8795.
[13] Kausel, E. and Roësset, J. M., Frequency domain analysis of undamped systems, J. Eng. Mech., 118(4) (1992), pp. 721734.
[14] Lesnic, D., Elliot, L. and Ingham, D. B., Treatment of singularities in time-dependent problems using the boundary element method, Eng. Anal. Bound. Elem., 16 (1995), pp. 6570.
[15] Muleshkov, A. S., Golberg, M. A. and Chen, C. S., Particular solutions of Helmholtz-type operators using higher order polyharmonic splines, Comput. Mech., 23 (1999), pp. 411419.
[16] Partridge, P. W., Brebbia, C. A. and Wrobel, L. C., The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, Boston, 1992.
[17] Shaw, R. P., Integral equation approach to diffusion, Int. J. Heat. Mass. Trans., 17(6) (1974), pp. 693699.
[18] Sutradhar, A., Paulino, G. H. and Gray, L. J., Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method, Eng. Anal. Bound. Elem., 26(2) (2002), pp. 119132.
[19] Sutradhar, A. and Paulino, G. H., The simple boundary element method for transient heat conduction in functionally graded materials, Comput. Methods. Appl. Mech. Eng., 193(42-44) (2004), pp. 45114539.
[20] Wrobel, L. C. and Brebbia, C. A., A formulation of the boundary element method for ax-isymmetric transient heat conduction, Int. J. Heat. Mass. Trans., 24 (1981), pp. 843850.
[21] Zhu, S., Satravaha, P. and Lu, X., Solving linear diffusion equations with the dual reciprocity method in Laplace space, Eng. Anal. Bound. Elem., 13 (1994), pp. 110.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed