Steady Heat Conduction Analysis of Functionally Graded Materials with Barycentric Lagrange Interpolation Collocation Method

doi:10.19596/j.cnki.1001-246x.8024

Abstract

Abstract: A new meshless method, named collocation method with barycentric Lagrange interpolation (BLIC), is presented for analyzing steady-state heat conduction in two-dimensional functionally graded materials(FGMs). Two-variable barycentric Lagrange interpolation function and its partial derivatives at Chebyshev nodes are derived. Then, based on a collocation method, it is directly introduced into controlling equations and boundary conditions of GEMs to obtain a linear discretijed equation. The method is a truly meshless method and takes advantages of both barycentric Lagrange interpolation and collocation method such as low computational cost, stable, accurate and easy to numerical implementation. Temperature fields of exponential, quadratic and trigonometrical FGMs models are simulated. It shows that the method is accurate and efficient, and is not sensitive to the change of material gradient parameters. It can be extended to solve transient heat conduction and thermal stress analysis of FGMs.

Key words: functionally graded material, heat conduction, meshless method, collocation method, barycentric Lagrange interpolation

CLC Number:

TQ038

WANG Leilei, JI Le, MA Wentao. Steady Heat Conduction Analysis of Functionally Graded Materials with Barycentric Lagrange Interpolation Collocation Method[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(2): 173-181.

References

[1] MIYANMOTO Y, KAYSSER W A, RABIN B H, et al. Functionally graded materials design, processing and applications[M]. New York:Springer, Science Business Media, 1999.
[2] XU Yangjian, TU Daihui. Transient temperature field analysis of functionally gradient material plate with temperature-dependent material properties under convective heat transfer boundary by finite element method[J]. Acta Material Compositae Sinica, 2003, 20(2):94-99.
[3] SLADEK J, SLADEK V, ZHANG C. Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method[J]. Comput Mater Sci, 2003, 28:494-504.
[4] SUTRADHAR A, PAULINO G H. The simple boundary element method for transient heat conduction in functionally graded materials[J]. Comput Method Appl Mech, 2004, 193:4511-4539.
[5] WANG H, QIN Q H, KANG Y L. A meshless model for transient heat conduction in functionally graded materials[J]. Comput Mech, 2006, 38(1):51-60.
[6] CAO L L, QIN Q H, ZHAO N. Hybrid graded element model for transient heat conduction in functionally graded materials[J]. Acta Mechanica Sinica, 2012, 28(1):128-139.
[7] CAO L L, WANG H, QIN Q H. Fundamental solution based graded element model for steady-state heat transfer in FGM[J]. Acta Mechanica Solida Sinica, 2012, 25(04):377-392.
[8] MIRZAEI D, DEHGHAN M. New implementation of MLBIE method for heat conduction analysis in functionally graded materials[J]. Eng Anal Bound Elem, 2012, 36(4):511-519.
[9] LIU Junqiao, MIAO Fusheng, LI Xing. Boundary element analysis of the two-dimensional heat conduction equation for anisotropic functional graded materials[J]. Journal of Xi'an Jiaotong University, 2013, 47(5):77-81.
[10] ZHANG Hengliang, KAN Weimin, HU Xuejiao. Green's function approach to the nonlinear transient heat transfer analysis of functionally graded materials[J]. International Journal of Thermal Sciences, 2013, 71:292-301.
[11] LI Qinghua, CHEN Shenshen, ZENG Jihui. A meshless model for transient heat conduction analysis of 3D axisymmetric functionally graded solids[J]. Chinese Phys B, 2013, 22(12):51-57.
[12] LI M, CHEN C S, CHU C C, et al. Transient 3D heat conduction in functionally graded materials by the method of fundamental solutions[J]. Eng Anal Bound Elem, 2014, 45:62-67.
[13] LI G Y, GUO S P, ZHANG J M, et al. Transient heat conduction analysis of functionally graded materials by a multiple reciprocity boundary face method[J]. Eng Anal Bound Elem, 2015, 60:81-88.
[14] YU B, ZHOU H L, YAN J, et al. A differential transformation boundary element method for solving transient heat conduction problems in functionally graded materials[J]. Numerical Heat Transfer Part A:Applications, 2016, 70(3):293-309.
[15] MIAO Yu, SUN Tiancan, ZHENG Junjie. Application of new fast multipole hybrid boundary point method in composite heat conduction[J]. Chinese Journal of Computational Physics, 2014, 31(3):335-342.
[16] HU Guodong, ZHANG Huihua, TAN Yuxin. Numerical manifold study of steady heat conduction problems in functionally graded materials[J]. Chinese Journal of Applied Mechanics, 2017, 34(2):311-317.
[17] ZHANG H H, HAN S Y, FAN L F, et al. The numerical manifold method for 2D transient heat conduction problems in functionally graded materials[J]. Eng Anal Bound Elem, 2018, 88:145-155.
[18] LIU Yingkai, CHENG Zhangqi. Transient heat conduction model for functionally graded materials based on peridynamics[J]. Chinese Quarterly of Mechanics, 2018, 39(1):82-89.
[19] LIU Shuo, FANG Guodong, WANG Bing, et al. Study of thermal conduction problem using coupling peridynamics and finite element method[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2):339-348.
[20] FU Z J, QIANG X, CHEN W, et al. A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations[J]. Computers and Mathematics with Applications, 2018, 76:760-773.
[21] BERRUT J P, TREFETHEN L N. Barycentric Lagrange interpolation[J]. SIAM Review, 2004, 46(3):501-517.
[22] WANG Z Q, LI S C, YANG P, et al. A highly accurate regular domain collocation method for solving potential problems in the irregular doubly connected domains[J]. Mathematical Problems in Engineering, 2014(4):1-9.
[23] WANG Zhaoqing, ZHUANG Meiling, JIANG Jian. Nonlinear mems microbeam analysis by barycentric rational interpolation iteration collocation method[J]. Chinese Journal of Solid Mechanics, 2015, 36(5):435-459.
[24] LI Shuchen, WANG Zhaoqing, YUAN Chao. Barycentric interpolation collocation method for solving elastic problems[J]. Journal of Central South University (Science and Technology), 2013, 44(5):2031-2040.
[25] WANG Zhaoqing, XU Zikang. Mixed displacement-pressure collocation method for plane elastic problems[J]. Chinese Journal of Computational Physics, 2018, 35(1):77-86.
[26] WANG Zhaoqing, JI Siyuan, XU Zikang, et al. A regular domain collocation method based on barycentric interpolation for solving plane elastic problems in irregular domains[J]. Chinese Journal of Computational Mechanics, 2018, 35(2):195-201.
[27] MA Wentao, ZHANG Baowen, MA Hailong. A meshless collocation approach with barycentric rational interpolation for two-dimensional hyperbolic telegraph equation[J]. Applied Mathematics and Computation, 2016, 279:236-248.
[28] LIU Ting, MA Wentao. Barycentric Lagrange interpolation collocation method for two-dimensional hyperbolic telegraph equation[J]. Chinese Journal of Computational Physics, 2016, 33(3):341-348.
[29] LIU Hongyan, HUANG Jin, PAN Yubin, et al. Barycentric interpolation collocation methods for solving linear and nonlinear high-dimensional Fredholm integral equations[J]. Journal of Computational and Applied Mathematics, 2018, 327:141-154.
[30] BELYTSCHKO T, LU Y Y, GU L. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering, 1994, 37:229-256.
[31] MA Wentao. A smoothed meshfree Galerkin elasticity problem[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(5):1115-1124.
[32] SHI Baojun, YUAN Mingwu, SHU Dongwei. Solving Helmholtz equation by least-square collocation method based on reproducing kernel particle method[J]. Chinese Journal of Theoretical and Applied Mechanics, 2006, 38(1):125-129.
[33] LIU Guojun, MA Wentao, MA Hailong, et al. A multiple-scale higher order polynomial collocation method for 2D and 3D elliptic partial differential equations with variable coefficients[J]. Applied Mathematics and Computation, 2018, 331:430-444.
[34] SKOURAS E D, BOURANTAS G C, LOUKOPOULOS V C, et al. Truly meshless localized type techniques for the steady-state heat conduction problems for isotropic and functionally graded materials[J]. Eng Anal Bound Elem, 2011, 35(3):452-464.