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

Published online by Cambridge University Press:  03 June 2015

A. Tadeu ,
C. S. Chen ,
J. António  and
Nuno Simões
A. Tadeu*
Affiliation:
Department of Civil Engineering, Faculty of Sciences and Technology, University of Coimbra, Polo II-Pinhal de Marrocos, 3030-290 Coimbra, Portugal
C. S. Chen*
Affiliation:
Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA
J. António
Affiliation:
Department of Civil Engineering, Faculty of Sciences and Technology, University of Coimbra, Polo II-Pinhal de Marrocos, 3030-290 Coimbra, Portugal
Nuno Simões
Affiliation:
Department of Civil Engineering, Faculty of Sciences and Technology, University of Coimbra, Polo II-Pinhal de Marrocos, 3030-290 Coimbra, Portugal
*
URL:http://www.math.usm.edu/cschen/, Email: [email protected]
Corresponding author. Email: [email protected]
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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.

Keywords

65Y0435K05Transient heat transfermeshless methodsmethod of particular solutionsmethod of fundamental solutionsfrequency domainFourier transform
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 5 , October 2011 , pp. 572 - 585
Copyright
Copyright © Global-Science Press 2011

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References

[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. 1819–1843.3.0.CO;2-J>CrossRefGoogle Scholar
[2] Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Second edition, Oxford University Press, London, 1959.Google Scholar
[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. 1905–1918.CrossRefGoogle Scholar
[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. 195–201.CrossRefGoogle Scholar
[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. 469–486.CrossRefGoogle Scholar
[6] Chen, W. and Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl., 43(3-5) (2002), pp. 379–391.CrossRefGoogle Scholar
[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. 1–37.Google Scholar
[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. 85–110, 1976.Google Scholar
[9] Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9(1-2) (1998), pp. 69–95.CrossRefGoogle Scholar
[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. 757–763.CrossRefGoogle Scholar
[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. 103–176, 1999.Google Scholar
[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. 87–95.Google Scholar
[13] Kausel, E. and Roësset, J. M., Frequency domain analysis of undamped systems, J. Eng. Mech., 118(4) (1992), pp. 721–734.CrossRefGoogle Scholar
[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. 65–70.CrossRefGoogle Scholar
[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. 411–419.CrossRefGoogle Scholar
[16] Partridge, P. W., Brebbia, C. A. and Wrobel, L. C., The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, Boston, 1992.Google Scholar
[17] Shaw, R. P., Integral equation approach to diffusion, Int. J. Heat. Mass. Trans., 17(6) (1974), pp. 693–699.CrossRefGoogle Scholar
[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. 119–132.CrossRefGoogle Scholar
[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. 4511–4539.CrossRefGoogle Scholar
[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. 843–850.CrossRefGoogle Scholar
[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. 1–10.CrossRefGoogle Scholar