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1.
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2.
Lesnic, D. & Elliott, L., The decomposition approach to inverse heat conduction. Journal of Mathematical Analysis and Applications, 232, pp. 82–98, 1999. [Crossref]
3.
Liu, J., A stability analysis on Beck’s procedure for inverse heat conduction problem. Journal of Computational Physics, 123, pp. 65–73, 1996. [Crossref]
4.
Shen, S.Y., A numerical study of inverse heat conduction problems. Computers & Mathematics with Applications, 38, pp. 173–188, 1999. [Crossref]
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Hon, Y.C. & Wei, T., A fundamental solution method for inverse heat conduction problem. Engineering Analysis with Boundary Elements, 28, pp. 489–495, 2004. [Crossref]
6.
Guo, L. & Murio, D.A., A modified space-marching finite-difference algorithm for the two-dimensional inverse heat conduction problem with slab symmetry. Inverse Prob-lems, 7, pp. 247–259, 1991. [Crossref]
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Khalidy, N.A., A general space marching algorithm for the solution of two-dimensional boundary inverse heat conduction problems. Numerical Heat Transfer Part B, 34, pp. 339–360, 1998. [Crossref]
8.
Chantasiriwan, S., An algorithm for solving multidimensional inverse heat conduction problem. International Journal of Heat and Mass Transfer, 44, pp. 3823–3832, 2001. [Crossref]
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Hsu, T.R., Sun, N.S., Chen, G.G. & Gong, Z.L., Finite element formulation for two-dimensional inverse heat conduction analysis. ASME Journal of Heat Transfer, 114, pp. 553–557, 1992. [Crossref]
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Reinhardt, H.J., A numerical method for the solution of two-dimensional inverse heat conduction problems. International Journal for Numerical Methods in Engineering, 32, pp. 363–283, 1991. [Crossref]
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Hon, Y.C. & Wei, T., The method of fundamental solutions for solving multidimen-sional inverse heat conduction problems. Computer Modeling in Engineering & Sciences, 7, pp. 19–32, 2005.
13.
Yu, G.X., Sun, J., Wang, H.S., Wen, P.H. & Rose, J.W., Meshless inverse method to determine temperature and heat flux at boundaries for 2D steady-state heat conduction problems. Experimental Thermal and Fluid Science, 52, pp. 156–163, 2004. [Crossref]
14.
Atluri, S.N., The Meshless Method (MLPG) for Domain and BIE Discretizations, Tech Science Press Forsyth GA, 2004.
15.
Sladek, J., Sladek, V. & Hon, Y.C., Inverse heat conduction problems by meshless local Petrov–Galerkin method. Engineering Analysis with Boundary Elements, 30, pp. 650–661, 2006. [Crossref]
16.
Li, M., Hon, Y.C., Korakianitis, T. & Wen, P.H., Finite integration method for nonlocal elastic bar under static and dynamic loads. Engineering Analysis with Boundary Elements, 37(5), pp. 842–849, 2013. [Crossref]
17.
Wen, P.H., Hon, Y.C., Li, M. & Korakianitis, T., Finite integration method for partial differential equations. Applied Mathematical Modelling, 37, pp. 10092–10106, 2013. [Crossref]
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Li, M., Tian, Z.L., Hon, Y.C., Chen, C.S. & Wen, P.H., Improved finite integra-tion method for partial differential equations. Engineering Analysis with Boundary Elements, 64, pp. 230–236, 2016. [Crossref]
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Open Access
Research article

Heat Conduction in Anisotropic and Functionally Graded Media by Finite Integration Method

j. jin1,
j.l., zheng1,
t., huang1,
j.j., yang1,
h.s., wang2,
p.h., wen2,
j.m., li3
1
School of Communication and Transportation Engineering, Changsha University of Science and Technology, China
2
School of Engineering and Materials Science, Queen Mary University of London, United Kingdom
3
Department of Thermal Engineering, Tsinghua University, Beijing, China
International Journal of Computational Methods and Experimental Measurements
|
Volume 6, Issue 6, 2018
|
Pages 1149-1160
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
View Full Article|Download PDF

Abstract:

Based on the recently developed finite integration method (FIM) for solving one and two dimensional ordinary and partial differential equations, this paper extends FIM to both stationary and transient heat conduction inverse problems for anisotropic and functionally graded materials with high degree of accuracy. Lagrange series approximation is applied to determine the first order of integral and differential matrices, which are used to form the system equation matrix for two and three dimensional problems. Singular Value Decomposition (SVD) is applied to solve the ill-conditioned system of algebraic equations obtained from the integral equation, boundary conditions and scattered temperature measurements. The convergence and accuracy of this method are investigated with two examples for anisotropic media and functionally graded materials.

Keywords: Finite integral method, Integration matrix, Inverse heat conduction, Lagrange series
References
1.
Jonas, P. & Louis, A.K., Approximate inverse for a one-dimensional inverse heat conduction problem. Inverse Problems, 16, pp. 175–185, 2000. [Crossref]
2.
Lesnic, D. & Elliott, L., The decomposition approach to inverse heat conduction. Journal of Mathematical Analysis and Applications, 232, pp. 82–98, 1999. [Crossref]
3.
Liu, J., A stability analysis on Beck’s procedure for inverse heat conduction problem. Journal of Computational Physics, 123, pp. 65–73, 1996. [Crossref]
4.
Shen, S.Y., A numerical study of inverse heat conduction problems. Computers & Mathematics with Applications, 38, pp. 173–188, 1999. [Crossref]
5.
Hon, Y.C. & Wei, T., A fundamental solution method for inverse heat conduction problem. Engineering Analysis with Boundary Elements, 28, pp. 489–495, 2004. [Crossref]
6.
Guo, L. & Murio, D.A., A modified space-marching finite-difference algorithm for the two-dimensional inverse heat conduction problem with slab symmetry. Inverse Prob-lems, 7, pp. 247–259, 1991. [Crossref]
7.
Khalidy, N.A., A general space marching algorithm for the solution of two-dimensional boundary inverse heat conduction problems. Numerical Heat Transfer Part B, 34, pp. 339–360, 1998. [Crossref]
8.
Chantasiriwan, S., An algorithm for solving multidimensional inverse heat conduction problem. International Journal of Heat and Mass Transfer, 44, pp. 3823–3832, 2001. [Crossref]
9.
Kurpisz, K. & Nowak, A.J., BEM approach to inverse heat conduction problems. Engineering Analysis with Boundary Elements, 10, pp. 291–729, 1992. [Crossref]
10.
Hsu, T.R., Sun, N.S., Chen, G.G. & Gong, Z.L., Finite element formulation for two-dimensional inverse heat conduction analysis. ASME Journal of Heat Transfer, 114, pp. 553–557, 1992. [Crossref]
11.
Reinhardt, H.J., A numerical method for the solution of two-dimensional inverse heat conduction problems. International Journal for Numerical Methods in Engineering, 32, pp. 363–283, 1991. [Crossref]
12.
Hon, Y.C. & Wei, T., The method of fundamental solutions for solving multidimen-sional inverse heat conduction problems. Computer Modeling in Engineering & Sciences, 7, pp. 19–32, 2005.
13.
Yu, G.X., Sun, J., Wang, H.S., Wen, P.H. & Rose, J.W., Meshless inverse method to determine temperature and heat flux at boundaries for 2D steady-state heat conduction problems. Experimental Thermal and Fluid Science, 52, pp. 156–163, 2004. [Crossref]
14.
Atluri, S.N., The Meshless Method (MLPG) for Domain and BIE Discretizations, Tech Science Press Forsyth GA, 2004.
15.
Sladek, J., Sladek, V. & Hon, Y.C., Inverse heat conduction problems by meshless local Petrov–Galerkin method. Engineering Analysis with Boundary Elements, 30, pp. 650–661, 2006. [Crossref]
16.
Li, M., Hon, Y.C., Korakianitis, T. & Wen, P.H., Finite integration method for nonlocal elastic bar under static and dynamic loads. Engineering Analysis with Boundary Elements, 37(5), pp. 842–849, 2013. [Crossref]
17.
Wen, P.H., Hon, Y.C., Li, M. & Korakianitis, T., Finite integration method for partial differential equations. Applied Mathematical Modelling, 37, pp. 10092–10106, 2013. [Crossref]
18.
Li, M., Tian, Z.L., Hon, Y.C., Chen, C.S. & Wen, P.H., Improved finite integra-tion method for partial differential equations. Engineering Analysis with Boundary Elements, 64, pp. 230–236, 2016. [Crossref]
Cite this:
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GB-T-7714-2015
Jin, J., Zheng, J., Huang, T., Yang, J., Wang, H., Wen, P., & Li, J. (2018). Heat Conduction in Anisotropic and Functionally Graded Media by Finite Integration Method. Int. J. Comput. Methods Exp. Meas., 6(6), 1149-1160. https://doi.org/10.2495/CMEM-V6-N6-1149-1160
J. Jin, J. Zheng, T. Huang, J. Yang, H. Wang, P. Wen, and J. Li, "Heat Conduction in Anisotropic and Functionally Graded Media by Finite Integration Method," Int. J. Comput. Methods Exp. Meas., vol. 6, no. 6, pp. 1149-1160, 2018. https://doi.org/10.2495/CMEM-V6-N6-1149-1160
@research-article{Jin2018HeatCI,
title={Heat Conduction in Anisotropic and Functionally Graded Media by Finite Integration Method},
author={J. Jin and J.L., Zheng and T., Huang and J.J., Yang and H.S., Wang and P.H., Wen and J.M., Li},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2018},
page={1149-1160},
doi={https://doi.org/10.2495/CMEM-V6-N6-1149-1160}
}
J. Jin, et al. "Heat Conduction in Anisotropic and Functionally Graded Media by Finite Integration Method." International Journal of Computational Methods and Experimental Measurements, v 6, pp 1149-1160. doi: https://doi.org/10.2495/CMEM-V6-N6-1149-1160
J. Jin, J.L., Zheng, T., Huang, J.J., Yang, H.S., Wang, P.H., Wen, J.M. and Li. "Heat Conduction in Anisotropic and Functionally Graded Media by Finite Integration Method." International Journal of Computational Methods and Experimental Measurements, 6, (2018): 1149-1160. doi: https://doi.org/10.2495/CMEM-V6-N6-1149-1160
Jin J.， Zheng J.， Huang T.， et al. Heat Conduction in Anisotropic and Functionally Graded Media by Finite Integration Method[J]. International Journal of Computational Methods and Experimental Measurements， 2018， 6(6)： 1149-1160. https://doi.org/10.2495/CMEM-V6-N6-1149-1160