The generalized finite difference method for long-time transient heat conduction in 3D anisotropic composite materials

Abstract In this paper we investigate the application of the generalized finite difference method (GFDM) to three-dimensional (3D) transient heat conduction in anisotropic composite (layered) materials. In our computations, the Krylov deferred correction (KDC) method, a pseudo-spectral type time-marching technique, is introduced to perform temporal discretization in time-domain. The KDC method allows discretizing the temporal direction using relatively large time-steps, making the method very promising for dynamic simulations, particularly when high precision is desired. A multi-domain GFDM scheme is also employed where the composite material considered is decomposed into several sub-domains and, in each sub-domain, the solution is approximated by using the GFDM expansion. On the sub-domain interface, compatibility of temperatures and normal heat fluxes is imposed. The method is tested on several benchmark numerical examples and its relative merits and disadvantages are discussed.

[1]  A. Karageorghis,et al.  THE METHOD OF FUNDAMENTAL SOLUTIONS FOR HEAT CONDUCTION IN LAYERED MATERIALS , 1999 .

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

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

[4]  Yijun Liu,et al.  Interfacial stress analysis for multi-coating systems using an advanced boundary element method , 2000 .

[5]  Jingfang Huang,et al.  Krylov deferred correction accelerated method of lines transpose for parabolic problems , 2008, J. Comput. Phys..

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

[7]  Daniel Lesnic,et al.  The method of fundamental solutions for nonlinear functionally graded materials , 2007 .

[8]  Wen Chen,et al.  Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method , 2016, Comput. Math. Appl..

[9]  Chuanzeng Zhang,et al.  Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method , 2014 .

[10]  Wenzhen Qu,et al.  A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems , 2018, International Journal for Numerical Methods in Engineering.

[11]  Xiao-Wei Gao,et al.  Interface integral BEM for solving multi-medium elasticity problems , 2009 .

[12]  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..

[13]  Daniel Lesnic,et al.  Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions , 2008 .

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

[15]  Chuanzeng Zhang,et al.  Stress analysis for thin multilayered coating systems using a sinh transformed boundary element method , 2013 .

[16]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

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

[18]  Chia-Ming Fan,et al.  Application of the Generalized Finite-Difference Method to Inverse Biharmonic Boundary-Value Problems , 2014 .

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

[20]  W. Yeih,et al.  Generalized finite difference method for solving two-dimensional inverse Cauchy problems , 2015 .

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

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

[23]  Božidar Šarler,et al.  Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions , 2009 .

[24]  Jingfang Huang,et al.  Accelerating the convergence of spectral deferred correction methods , 2006, J. Comput. Phys..

[25]  B. Tomas Johansson,et al.  A method of fundamental solutions for transient heat conduction in layered materials , 2009 .

[26]  Jingfang Huang,et al.  A Numerical Framework for Integrating Deferred Correction Methods to Solve High Order Collocation Formulations of ODEs , 2015, Journal of Scientific Computing.

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

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

[29]  YuanTong Gu,et al.  A meshfree method: meshfree weak–strong (MWS) form method, for 2-D solids , 2003 .

[30]  Yan Liu,et al.  Material point method with enriched shape function for crack problems , 2017 .

[31]  Xiang Chen,et al.  A robust and efficient polyhedron subdivision and intersection algorithm for three-dimensional MMALE remapping , 2017, J. Comput. Phys..

[32]  Wen Chen,et al.  Domain-decomposition generalized finite difference method for stress analysis in multi-layered elastic materials , 2018, Engineering Analysis with Boundary Elements.

[33]  Jingfang Huang,et al.  Arbitrary order Krylov deferred correction methods for differential algebraic equations , 2007, J. Comput. Phys..

[34]  Ji Lin,et al.  Simulation of Seismic Wave Scattering by Embedded Cavities in an Elastic Half-Plane Using the Novel Singular Boundary Method , 2018, Advances in Applied Mathematics and Mechanics.