Boundary Knot Method for 2D Transient Heat Conduction Problems

SHI Jin-hong; FU Zhuo-jia; CHEN Wen

doi:10.3879/j.issn.1000-0887.2014.02.001

Volume 35 Issue 2

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2014 > 35(2): 111-120

SHI Jin-hong, FU Zhuo-jia, CHEN Wen. Boundary Knot Method for 2D Transient Heat Conduction Problems[J]. Applied Mathematics and Mechanics, 2014, 35(2): 111-120. doi: 10.3879/j.issn.1000-0887.2014.02.001

Citation:

PDF( 1170 KB)

Boundary Knot Method for 2D Transient Heat Conduction Problems

doi: 10.3879/j.issn.1000-0887.2014.02.001

¹College of Mechanics and Materials, Hohai University, Nanjing 210098, P.R.China;²State Key Laboratory of HydrologyWater Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, P.R.China

Funds: The National Basic Research Program of China (973 Program)（2010CB832702）; The National Science Fund for Distinguished Young Scholars of China（11125208）; The National Natural Science Foundation of China（11372097;11302069）

Received Date: 2013-10-07
Rev Recd Date: 2013-11-19
Publish Date: 2014-02-15

Abstract

Abstract

The boundary knot method (BKM) in conjunction with the dual reciprocity method (DRM) was introduced to solve 2D transient heat conduction problems. With the finite difference scheme applied to deal with the time derivative term, the transient heat conduction equation was converted to a set of nonhomogeneous modified Helmholtz equations. Then the numerical solution to the nonhomogeneous problems was divided into two parts: the particular solution and the homogeneous solution. The DRM with few inner interpolation nodes was employed to get the particular solution, and the BKM with boundaryonly nodes used to obtain the homogeneous solution. Numerical results show that the present combined method has the merits of high accuracy, wide applicability, good stability and rapid convergence, which were appealing to solving transient heat conduction problems.
- transient heat conduction,
- boundary knot method,
- dual reciprocity method,
- difference scheme,
- modified Helmholtz equation

FullText(HTML)

References(24)

References

[1]	Burris K W, Beardsley M B, Chuzhoy L. Component having a functionally graded material coating for improved performance[P]. US Patent: 6087022, 2000-7-11.
[2]	Renner E. Thermal engine[P]. US Patent: 3937019, 1976-2-10.
[3]	Bruch J C, Zyvoloski G. Transient two-dimensional heat conduction problems solved by the finite element method[J]. International Journal for Numerical Methods in Engineering,1974, 8(3): 481-494.
[4]	Blobner J, Biaecki R A, Kuhn G. Transient non-linear heat conduction-radiation problems—a boundary element formulation[J]. International Journal for Numerical Methods in Engineering,1999, 46(11): 1865-1882.
[5]	Li Q H, Chen S S, Kou G X. Transient heat conduction analysis using the MLPG method and modified precise time step integration method[J]. Journal of Computational Physics,2011, 230(7): 2736-2750.
[6]	Valtchev S S, Roberty N C. A time-marching MFS scheme for heat conduction problems[J]. Engineering Analysis With Boundary Elements,2008, 32(6): 480-493.
[7]	Jirousek J, Qin Q H. Application of hybrid-Trefftz element approach to transient heat conduction analysis[J]. Computers & Structures,1996, 58(1): 195-201.
[8]	Brebbia C A, Telles J C F, Wrobel L C. Boundary Element Techniques: Theory and Applications in Engineering [M]. Berlin: Springer-Verlag, 1984.
[9]	欧阳华江. 广义热传导方程有限元算法的计算准则[J]. 应用数学和力学, 1992, 13(6): 563-571.(OUYANG Hua-jiang. Criteria for finite element algorithm of generalized heat conduction equation[J]. Applied Mathematics and Mechanics,1992, 13(6): 563-571.(in Chinese))
[10]	张雄, 刘岩. 无网格法[M]. 北京: 清华大学出版社, 2004.(ZHANG Xiong, LIU Yan. Meshless Methods [M]. Beijing: Tsinghua University Press, 2004.(in Chinese))
[11]	Wang H, Qin Q, Kang Y. A meshless model for transient heat conduction in functionally graded materials[J]. Computational Mechanics,2006, 38(1): 51-60.
[12]	Fu Z J, Chen W, Qin Q H. Three boundary meshless methods for heat conduction analysis in nonlinear FGMs with Kirchhoff and Laplace transformation[J]. Advances in Applied Mathematics and Mechanics,2012, 4(5): 519-542.
[13]	Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems[J]. Advances in Computational Mathematics,1998, 9(1/2): 69-95.
[14]	Chen W, Tanaka M. A meshless, integration-free, and boundary-only RBF technique[J].Computers & Mathematics With Applications,2002, 43(3): 379-391.
[15]	Fu Z J, Chen W, Qin Q H. Boundary knot method for heat conduction in nonlinear functionally graded material[J]. Engineering Analysis With Boundary Elements,2011, 35(5): 729-734.
[16]	Chen W, Hon Y C. Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems[J]. Computer Methods in Applied Mechanics and Engineering,2003, 192(15): 1859-1875.
[17]	Hon Y C, Chen W. Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry[J]. International Journal for Numerical Methods in Engineering,2003, 56(13): 1931-1948.
[18]	Jin B, Zheng Y. Boundary knot method for some inverse problems associated with the Helmholtz equation[J]. International Journal for Numerical Methods in Engineering,2005, 62(12): 1636-1651.
[19]	Partridge P W, Brebbia C A, Wrobel L C. The Dual Reciprocity Boundary Element Method [M]. Southampton: Computational Mechanics Publications, 1992.
[20]	Biaecki R A, Jurgas＇P, Kuhn G. Dual reciprocity BEM without matrix inversion for transient heat conduction[J]. Engineering Analysis With Boundary Elements,2002, 26(3): 227-236.
[21]	Bulgakov V, Arler B, Kuhn G. Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation[J]. International Journal for Numerical Methods in Engineering,1998, 43(4): 713-732.
[22]	Chapko R, Kress R. Rothe’s method for the heat equation and boundary integral equations[J]. Journal of Integral Equations and Applications,1997, 9(1): 47-69.
[23]	Golberg M. Recent developments in the numerical evaluation of particular solutions in the boundary element method[J]. Applied Mathematics and Computation,1996, 75(1): 91-101.
[24]	Chen C, Rashed Y F. Evaluation of thin plate spline based particular solutions for Helmholtz-type operators for the DRM[J]. Mechanics Research Communications,1998, 25(2): 195-201.