摘要
将基本解和径向基函数相结合反演一种逆热传导问题的初值和热源.由于方程的系数矩阵是病态的,所以文中用Tikhonov正则化方法求解线性方程组,通过L-曲线方法选择正则化参数.通过几个数值例子验证了方法的有效性和精确性.
An improved meshless method based on the fundamental solution and the radial basis functions is discussed for the simultaneous determination of a heat source and initial temperature.Since the coefficient matrix may be ill-conditioned,the Tikhonov regularization method is used to solve the resulted system of linear equations,and the L-curve technique is employed to choose a regularization parameter.The accuracy and efficiency of the proposed method is demonstrated by several numerical examples.
引文
[1] HAN H,INGHAM D B,YUAN Y.The boundary element method for the solution of the backward heat conduction equation[J].Comput Phys,1995,116(2):292.
[2] FARCAS A,LESNIC D.The boundary element method for the determination of a heat source dependent on one variable[J].Engrg Math,2006,54(4):375.
[3] LIJIMA K.Numerical solution of backward heat conduction problems by a high order lattice-free finite difference method[J].Chinese Inst Engrs,2004,27:611.
[4] MARI N L,LESNIC D.The method of fundamental solutions for the Cauchy problem associated with twodimensional Helmhotz-type equation[J].Comput Struct,2005,83(4/5):267.
[5] YAN L,YANG F L,FU C L.A meshless method for solving an inverse spacewise-dependent heat source problem[J].Comput Phys,2009,228(1):123.
[6] MERA N S.The method of fundamental solutions for the backward heat conduction problem[J].Inverse Probl Sci Eng,2005,13(1):65.
[7] HON Y C, WEI T.The method of fundamental solution for solving multidimensional inverse heat conduction problems[J].CMES Comput Model Eng Sci,2005,7(2):119.
[8] HON Y C,WEI T.A fundamental solutionmethod for inverse heat conduction problem[J].Eng Anal Boundary Elem,2004,28(5):489.
[9] BUHMANN M.Radial Basis Functions:Theory and Implementations[M].Cambridge:Cambridge University Press,2003.
[10] POWEL M J D.Radial Basis Functions for Multivariable Interpolation:A Review[M].Oxford:Oxford University Press,1987.
[11] WU Z M.Hermite-Birkhoff interpolation of scattered data by radial basis functions[J].Approx Theory Appl,1992,8(2):1.
[12] TOMAS J B, LESNIC D.A procedure for determining a spacewise dependent heat source and the initial temperature[J].Appl Anal,2008,87(3):265.
[13] WEN J,YAMAMOTO M,WEI T.Simultaneous determination of a time-dependent heat source and the initial temperature in an inverse heat conduction problem[J].Inverse Probl Sci Eng,2013,21(3):485.
[14] WEI T,WANG J C.Simultaneous determination for a space-dependent heat source and the initial data by the MFS[J].Eng Anal Bound Elem,2012,36(12):1848.
[15] AMIRFAKHRIAN M,ARGHAND M,KANSA E J.A new approximate method for an inverse timedependent heat source problem using fundamental solutions and RBFs[J].Eng Anal Bound Elem,2016,64:278.
[16] ARGHAND M,AMIRFAKHRIAN M.A meshless method based on the fundamental solution and radial basis function for solving an inverse heat conduction problem[J].Adv Math Phys,2015,2015:1.
[17] HANSEN P C.Rank-deficient and Discrete Illposed Problems:Numerical Aspects of Linear Inversion[M].Philadelphia:Society for Industrial and Applied Mathematics(SIAM)Press,1998.
[18] GOLUBGH, HEATHM, WAHBAG.Generalized cross-validation as a method for choosing agood ridge parameter[J].Technometrics,1979,21(2):215.
[19] HANSEN P C,O'LEARY D P.The use of the Lcurve in the regularization of discrete ill-posed problems[J].SIAM J Sci Comput,1993,14(6):1487.