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Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

Published online by Cambridge University Press:  03 June 2015

Zhuo-Jia Fu ,
Wen Chen  and
Qing-Hua Qin
Zhuo-Jia Fu*
Affiliation:
Center for Numerical Simulation Software in Engineering and Sciences, Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu, P. R. China Research School of Engineering, Building 32, Australian National University, Canberra, ACT 0200, Australia
Wen Chen*
Affiliation:
Center for Numerical Simulation Software in Engineering and Sciences, Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu, P. R. China
Qing-Hua Qin*
Affiliation:
Research School of Engineering, Building 32, Australian National University, Canberra, ACT 0200, Australia
*
URL:http://em.hhu.edu.cn/chenwen/english.html, Email: paul212063@hhu.edu.cn
Corresponding author: Email: chenwen@hhu.edu.cn
Email: qinghua.qin@anu.edu.au
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Abstract

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.

Keywords

35k5541A3044A1065M7080M25Method of fundamental solutionboundary knot methodcollocation Trefftz methodKirchhoff transformationLaplace transformationmeshless method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 519 - 542
Copyright
Copyright © Global-Science Press 2012

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References

[1]Erdogan, F., Fracture mechanics of functionally graded materials, Compos. Eng., 5 (1995), pp. 753–770.Google Scholar
[2]Koike, Y., High-bandwidth graded-index polymer optical fibre, Polym., 32 (1991), pp. 1737–1745.Google Scholar
[3]Tani, J. and Liu, G. R., SH surface waves in functionally gradient piezoelectric plates, JSME Int. J., Ser. A, 36 (1993), pp. 152–155.Google Scholar
[4]Pompe, W., Worch, H., Epple, M., Friess, W., Gelinsky, M., Greil, P., Hempel, U., Scharnweber, D. and Schulte, K., Functionally graded materials for biomedical applications, Mater. Sci. Eng. A, 362 (2003), pp. 40–60.Google Scholar
[5]Kim, J.H. and Paulino, G. H., Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials, J. Appl. Mech., 69 (2002), pp. 502–514.Google Scholar
[6]Sutradhar, A. and Paulino, G. H., The simple boundary element method for transient heat conduction in functionally graded materials, Comput. Meth. Appl. Mech. Eng., 193 (2004), pp. 4511–4539.CrossRefGoogle Scholar
[7]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 (2002), pp. 119–132.Google Scholar
[8]Sladek, J., Sladek, V. and Zhang, C., Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method, Comput. Mater. Sci., 28 (2003), pp. 494–504.Google Scholar
[9]Sladek, V., Sladek, J., Tanaka, M. and Zhang, C., Transient heat conduction in anisotropic and functionally graded media by local integral equations, Eng. Anal. Bound. Elem., 29 (2005), pp. 1047–1065.Google Scholar
[10]Sladek, J., Sladek, V., Tan, C. L. and Atluri, S. N., Analysis of transient heat conduction in 3D anisotropic functionally graded solids by the MLPG, CMES-Comput. Model. Eng. Sci., 32 (2008), pp. 161–174.Google Scholar
[11]Wang, H., Qin, Q. H. and Kang, Y. L., A meshless model for transient heat conduction in functionally graded materials, Comput. Mech., 38 (2006), pp. 51–60.Google Scholar
[12]Marin, L. and Lesnic, D., The method of fundamental solutions for nonlinear functionally graded materials, Int. J. Solids Struct., 44 (2007), pp. 6878–6890.CrossRefGoogle Scholar
[13]Zheng, J. H., Soe, M. M., Zhang, C. and Hsu, T. W., Numerical wave flume with improved smoothed particle hydrodynamics, J. Hydro. B., 22 (2010), pp. 773–781.CrossRefGoogle Scholar
[14]Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. and Krysl, P., Meshless methods: An overview and recent developments, Comput. Meth. Appl. Mech. Eng., 139 (1996), pp. 3–47.Google Scholar
[15]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.Google Scholar
[16]Chen, C. S., Karageorghis, A. and Smyrlis, Y. S., The Method of Fundamental Solutions - A Meshless Method, Dynamic Publishers, (2008).Google Scholar
[17]Fairweather, G., Karageorghis, A. and Martin, P. A., The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem., 27 (2003), pp. 759–769.CrossRefGoogle Scholar
[18]Li, Z. C., Lu, T. T., Huang, H. T. and Cheng, A. H. D., Trefftz, collocation, and other boundary methods - a comparison, Numer. Meth. Part. D. E., 23 (2007), pp. 93–144.CrossRefGoogle Scholar
[19]Chen, W. and Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl., 43 (2002), pp. 379–391.Google Scholar
[20]Young, D. L., Tsai, C. C., Murugesan, K., Fan, C. M. and Chen, C. W., Time-dependent fundamental solutions for homogeneous diffusion problems, Eng. Anal. Bound. Elem., 28 (2004), pp. 1463–1473.Google Scholar
[21]Cao, L., Qin, Q. H. and Zhao, N., An RBF-MFS model for analysing thermal behaviour of skin tissues, Int. J. Heat Mass Tran., 53 (2010), pp. 1298–1307.Google Scholar
[22]Gaver, D. P., Observing stochastic processes and approximate transform inversion, Oper. Res., 14 (1966), pp. 444–459.Google Scholar
[23]Fu, Z. J., Qin, Q. H. and Chen, W., Hybrid-Trefftz finite element method for heat conduction in nonlinear functionally graded materials, Eng. Comput., 28 (2011), pp. 578–599.Google Scholar
[24]Fu, Z. J., Chen, W. and Qin, Q. H., Boundary Knot method for heat conduction in nonlinear functionally graded material, Eng. Anal. Bound. Elem., 35 (2011), pp. 729–734.Google Scholar
[25]Hansen, P., Regulazation tools: a Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), pp. 1–35.Google Scholar
[26]Karageorghis, A., A practical algorithm for determining the optimal pseudo-boundary in the method of fundamental solutions, Adv. Appl. Math. Mech., 1 (2009), pp. 510–528.Google Scholar
[27]Cisilino, A. P. and Sensale, B., Application of a simulated annealing algorithm in the optimal placement of the source points in the method of the fundamental solutions, Comput. Mech., 28 (2002), pp. 129–136.Google Scholar
[28]Nishimura, R., Nishimori, K. and Ishihara, N., Determining the arrangement of fictitious charges in charge simulation method using genetic algorithms, J. Electrostat., 49 (2000), pp. 95–105.Google Scholar
[29]Qin, Q. H., The Trefftz Finite and Boundary Element Method, WIT Press, Southampton (2000).Google Scholar