Volume 4, Issue 5
Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

Zhuo-Jia Fu, Wen Chen & Qing-Hua Qin

Adv. Appl. Math. Mech., 4 (2012), pp. 519-542.

Published online: 2012-04

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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

Method of fundamental solution boundary knot method collocation Trefftz method Kirchhoff transformation Laplace transformation meshless method

  • AMS Subject Headings

35k55 41A30 44A10 65M70 80M25

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COPYRIGHT: © Global Science Press

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@Article{AAMM-4-519, author = {Zhuo-Jia Fu, Wen Chen and Qing-Hua Qin}, title = {Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {5}, pages = {519--542}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m1170}, url = {http://global-sci.org/intro/article_detail/aamm/134.html} }
TY - JOUR T1 - Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation AU - Zhuo-Jia Fu, Wen Chen & Qing-Hua Qin JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 519 EP - 542 PY - 2012 DA - 2012/04 SN - 4 DO - http://dor.org/10.4208/aamm.10-m1170 UR - https://global-sci.org/intro/aamm/134.html KW - Method of fundamental solution KW - boundary knot method KW - collocation Trefftz method KW - Kirchhoff transformation KW - Laplace transformation KW - meshless method AB -

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.

Zhuo-Jia Fu, Wen Chen & Qing-Hua Qin. (1970). Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation. Advances in Applied Mathematics and Mechanics. 4 (5). 519-542. doi:10.4208/aamm.10-m1170
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