逆热传导问题的一种新型无网格方法
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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.
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
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[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.
[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.