Abstract
A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter. First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem. Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively. Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined. Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients. The accuracy of the proposed method is assessed through two numerical examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ingber MS (1994) A triple reciprocity boundary element method for transient heat conduction analysis. In Boundary Element Technology, Brebbia CA, Kassab AJ (eds), vol. IX. Elsevier Applied Science Amsterdam, 41–49
Jirousek J, Qin QH (1996) Application of Hybrid-Trefftz element approach to transient heat conduction analysis, Compu Struc 58:195–201
Nardini D, Brebbia CA (1982) A new approach to free vibration analysis using boundary elements. In Boundary Element Methods in Engineering Proceedings. 4th International Seminar, Southampton. ed. C. A. Brebbia, Springer-Verlag, Berlin and New York, 312–326
Bulgakov V, Sarler B, Kuhn G (1998) Interative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation. Int J Numer Meth Eng 43:713–732
Blobner J, Bialecki RA, Kuhn G (1999) Transient non-linear heat conduction-radiation problems-a dual reciprocity formulation. Int J Numer Meth Eng 49:1865–1882
Singh KM, Tanaka M (1999) Dual reciprocity boundary element analysis of nonlinear diffusion-temporal discretization. Eng Anal Bound Elem 23:419–433
Bialecki RA, Jurgas P, Kuhn G (2002) Dual reciprocity BEM without matrix inversion for transient heat conduction. Eng Anal Bound Elem 26:227–236
Nowak AJ, Brebbia CA (1989) The multiple-reciprocity method. A new approach for transforming BEM domain integrals to the boundary. Eng Anal Bound Elem 6:164–167
Zhu T, Zhang JD, Atluri SN (1998) A local boundary integral equations (LBIE) method in computational mechanics, and a meshless discretization approach. Comput Mech 21:223–235
Sladek J, Sladek V, Zhang Ch (2003) Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method. Comput Mater Sci 28:494–504
Chen W (2002) High-order fundamental and general solutions of convection-diffusion equation and their applications with boundary particle method. Eng Anal Bound Elem 26:571–575
Chen W (2002) Meshfree boundary particle method applied to Helmholtz problems. Eng Anal Bound Elem 26:577–581
Kupradze VD (1965) Potential methods in the theory of elasticity. Israel Program for Scientific Translations. Jerusalem
Sun HC, Zhang LZ, Xu Q, Zhang YM (1999) Nonsingularity Boundary Element Methods, Dalian: Dalian University of Technology Press (in Chinese)
Golberg MA, Chen CS (1996) Discrete Projection Methods for Integral Equations. Computational Mechanics. Southampton
Ingber MS, Chen CS, Tanski JA (2004) A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. Int J Numer Meth Eng 60:2183–2201
Schaback R (1995) Error estimates and condition numbers for radial basis function interpolation. Adv Comput Math 3:251–264
Katsikadelis JT (1994) The analog equation method - a powerful BEM - based solution technique for solving linear and nonlinear engineering problems. In: Brebbia CA, (ed) Boundary Element Method XVI, CLM Publications Southampton, p 167
Golberg MA, Chen CS (1999) The method of fundamental solutions for potential, Helmholtz and diffusion problems. Boundary Integral Methods: Numerical and Mathematical Aspects, In: Golberg MA(ed) Computational Mechanics Publications Southampton, 103–176
Mitic P, Rashed YF (2004) Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources. Eng Anal Bound Elem 28:143–153
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, H., Qin, QH. & Kang, YL. A meshless model for transient heat conduction in functionally graded materials. Comput Mech 38, 51–60 (2006). https://doi.org/10.1007/s00466-005-0720-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-005-0720-3