The generalized finite difference method for an inverse boundary value problem in three-dimensional thermo-elasticity

Abstract In this study, a new framework for the efficient and accurate solutions for an inverse problem associated with three-dimensional (3D) coupled thermo-elasticity equation is presented. The ill-conditioned problem is solved here with the generalized finite difference method (GFDM) together with the first-order Tikhonov regularization technique and the L-curve criterion. The GFDM uses the Taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables. The method is truly meshless that can be applied for solving problems merely defined over irregular clouds of points. In addition, for problems involving complex geometries, a new distance criterion for adaptive selection of nodes in the GFDM simulations is proposed. Preliminary numerical experiments show that the regularized GFDM proposed in this study are very promising for accurate and efficient inverse thermo-elasticity simulations, even with a comparatively large level of noise added into the input data.

[1]  H. Sadat,et al.  On the Solution Of Heterogeneous Heat Conduction Problems by a Diffuse Approximation Meshless Method , 2006 .

[2]  Luis Gavete,et al.  A note on the dynamic analysis using the generalized finite difference method , 2013, J. Comput. Appl. Math..

[3]  G. Payre.  Influence graphs and the generalized finite difference method , 2007 .

[4]  T. Liszka,et al.  The finite difference method at arbitrary irregular grids and its application in applied mechanics , 1980 .

[5]  Luis Gavete,et al.  A note on the application of the generalized finite difference method to seismic wave propagation in 2D , 2012, J. Comput. Appl. Math..

[6]  A. Cheng,et al.  Heritage and early history of the boundary element method , 2005 .

[7]  Daniel Lesnic,et al.  The method of fundamental solutions for an inverse boundary value problem in static thermo-elasticity , 2014 .

[8]  D. Lesnic,et al.  A survey of applications of the MFS to inverse problems , 2011 .

[9]  Wenzhen Qu,et al.  Solution of Two-Dimensional Stokes Flow Problems Using Improved Singular Boundary Method , 2015 .

[10]  Yan Gu,et al.  Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime , 2015, Comput. Math. Appl..

[11]  Luis Gavete,et al.  Generalized finite differences for solving 3D elliptic and parabolic equations , 2016 .

[12]  Wen Chen,et al.  A meshless singular boundary method for three‐dimensional elasticity problems , 2016 .

[13]  Yijun Liu,et al.  A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method , 2007 .

[14]  Luis Gavete,et al.  Solving second order non-linear elliptic partial differential equations using generalized finite difference method , 2017, J. Comput. Appl. Math..

[15]  Xiao-Wei Gao,et al.  An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals , 2010 .

[16]  Chuanzeng Zhang,et al.  A meshless singular boundary method for three-dimensional inverse heat conduction problems in general anisotropic media , 2015 .

[17]  P. Larsson,et al.  On the effect of substrate deformation at scratching of soft thin film composites , 2010 .

[18]  D. Ingham,et al.  An iterative boundary element algorithm for a singular Cauchy problem in linear elasticity , 2002 .

[19]  Luis Gavete,et al.  Influence of several factors in the generalized finite difference method , 2001 .

[20]  T. Liszka.  An interpolation method for an irregular net of nodes , 1984 .

[21]  B. Šarler,et al.  Application of the RBF collocation method to transient coupled thermoelasticity , 2017 .

[22]  Qingsong Hua,et al.  A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations , 2017 .

[23]  Xiaoqiao He,et al.  Application of the meshless generalized finite difference method to inverse heat source problems , 2017 .

[24]  Chuanzeng Zhang,et al.  A novel meshless local Petrov–Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading , 2015 .

[25]  Chia-Ming Fan,et al.  Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems , 2019, Appl. Math. Lett..

[26]  Chuanzeng Zhang,et al.  Singular boundary method for wave propagation analysis in periodic structures , 2018 .

[27]  Marc Duflot,et al.  Meshless methods: A review and computer implementation aspects , 2008, Math. Comput. Simul..

[28]  Luis Gavete,et al.  Solving third- and fourth-order partial differential equations using GFDM: application to solve problems of plates , 2012, Int. J. Comput. Math..

[29]  Uniqueness of the solution to an inverse thermoelasticity problem , 2009 .

[30]  Božidar Šarler,et al.  Dual reciprocity boundary element method for convective-diffusive solid-liquid phase change problems, Part 1. Formulation , 1998 .

[31]  Yan Li,et al.  On choosing the location of the sources in the MFS , 2015, Numerical Algorithms.

[32]  Toshiro Matsumoto,et al.  Application of boundary element method to 3-D problems of coupled thermoelasticity , 1995 .

[33]  Daniel Lesnic,et al.  Regularized MFS solution of inverse boundary value problems in three-dimensional steady-state linear thermoelasticity , 2016 .

[34]  L. Gavete,et al.  Solving parabolic and hyperbolic equations by the generalized finite difference method , 2007 .

[35]  A. Karageorghis,et al.  The MFS for the Cauchy problem in two-dimensional steady-state linear thermoelasticity , 2013 .

[36]  Wen Chen,et al.  A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations , 2018, Comput. Math. Appl..

[37]  Guangyao Li,et al.  A nodal integration technique for meshfree radial point interpolation method (NI-RPIM) , 2007 .

[38]  Luis Gavete,et al.  Improvements of generalized finite difference method and comparison with other meshless method , 2003 .

[39]  Chuanzeng Zhang,et al.  A general algorithm for evaluating nearly singular integrals in anisotropic three-dimensional boundary element analysis , 2016 .

[40]  Timon Rabczuk,et al.  Goal-oriented error estimation and adaptive mesh refinement in dynamic coupled thermoelasticity , 2016 .

[41]  Wen Chen,et al.  Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method , 2017, Comput. Math. Appl..

[42]  F. Darwish,et al.  Thermoelastic response of the solid boundaries of a semi-infinite rectangular micro-channel flow , 2016 .

[43]  Aixia Zhang,et al.  A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics , 2018 .

[44]  Daniel Lesnic,et al.  A moving pseudo-boundary MFS for void detection in two-dimensional thermoelasticity , 2014 .

[45]  Robert Vertnik,et al.  Meshfree explicit local radial basis function collocation method for diffusion problems , 2006, Comput. Math. Appl..