A Coupled BEM-MLPG Technique for the ThermalAnalysis of Non-Homogeneous Media

This paper presents a technique that couples the boundary element method (BEM) with the meshless local Petrov-Galerkin (MLPG) method, formulated in the frequency domain. It is then used to study the transient heat diffusion through a two-dimensional unbounded medium containing confined subdomains where the material properties vary from point to point. To exploit the advantages of each method, the BEM is used for the homogeneous unbounded domain and the MLPG method is used for the non-homogeneous confined subdomains. The nodal points placed at the interface between the confined subdomains and the unbounded homogenous medium are used to couple the BEM and the MPLG method. The MLPG method is formulated using the moving leastsquares (MLS) approximation as the trial function and the Heaviside step function as the test function in local integral equations defined over small local sub-domains. The coupled BEM-MLPG approach is verified against the results provided by an analytical solution developed for a circular confined subdomain, in which the thermal diffusivity within the circular non-homogeneous region is assumed to vary in the radial direction. The proposed model is finally used to solve the case of a pair of non-homogeneous confined subdomains for which analytical solutions are not known. The analysis of time domain temperature responses is presented, which illustrates the applicability of the model.

[1]  M. N. Özişik,et al.  Finite Difference Methods in Heat Transfer , 2017 .

[2]  A. Luadsong,et al.  The MLPG with improved weight function for two-dimensional heat equation with non-local boundary condition , 2013 .

[3]  Linghui Wang,et al.  Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method , 2013, Appl. Math. Comput..

[4]  Satya N. Atluri,et al.  Applications of the MLPG Method in Engineering &Sciences: A Review , 2013 .

[5]  Xiaohua Zhang,et al.  An improved meshless method with almost interpolation property for isotropic heat conduction problems , 2013 .

[6]  J. Sládek,et al.  Physical decomposition of thin plate bending problems and their solution by mesh-free methods , 2013 .

[7]  Webe João Mansur,et al.  A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs , 2013 .

[8]  A. Tadeu,et al.  Closed Form Integration of Singular and Hypersingular Integrals in 3D BEM Formulations for Heat Conduction , 2012 .

[9]  S. Atluri,et al.  Development of 3D Trefftz Voronoi Cells with Ellipsoidal Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials , 2012 .

[10]  J. Sládek,et al.  Modified meshless local Petrov–Galerkin formulations for elastodynamics , 2012 .

[11]  A. Tadeu,et al.  Acoustic analysis of heterogeneous domains coupling the BEM with Kansa's method , 2012 .

[12]  J. Sládek,et al.  Inverse heat conduction problems in three-dimensional anisotropic functionally graded solids , 2012 .

[13]  Guang-Xiao Kou,et al.  Transient heat conduction analysis using the MLPG method and modified precise time step integration method , 2011, J. Comput. Phys..

[14]  Mehdi Dehghan,et al.  MLPG Method for Transient Heat Conduction Problem with MLS as Trial Approximation in Both Time and Space Domains , 2011 .

[15]  V. Sladek,et al.  Local integral equations implemented by MLS-approximation and analytical integrations , 2010 .

[16]  António Tadeu,et al.  Coupling BEM/TBEM and MFS for the simulation of transient conduction heat transfer , 2010 .

[17]  Ch. Zhang,et al.  On Increasing Computational Efficiency of Local Integral Equation Method Combined with Meshless Implementations , 2010 .

[18]  Delfim Soares,et al.  An Iterative Time-Domain Algorithm for Acoustic-Elastodynamic Coupled Analysis Considering Meshless Local Petrov-Galerkin Formulations , 2009 .

[19]  S. Atluri,et al.  Numerical Solution of Non-steady Flows, Around Surfaces in Spatially and Temporally Arbitrary Motions, by using the MLPG method , 2009 .

[20]  Y. Hon,et al.  The Inverse Problem of Determining Heat Transfer Coefficients by the Meshless Local Petrov-Galerkin Method , 2009 .

[21]  K. M. Liew,et al.  The reproducing kernel particle method for two-dimensional unsteady heat conduction problems , 2009 .

[22]  Zengjin Xu,et al.  Fracture analysis of cracked 2D planar and axisymmetric problems of magneto-electro-elastic materials by the MLPG coupled with FEM , 2009 .

[23]  Ch. Zhang,et al.  Local integral equation formulation for axially symmetric problems involving elastic FGM. , 2008 .

[24]  Y. Nie,et al.  A Study of Boundary Conditions in the Meshless Local Petrov-Galerkin (MLPG) Method for Electromagnetic Field Computations , 2008 .

[25]  G. Juncu.  Unsteady conjugate forced convection heat/mass transfer from a finite flat plate , 2008 .

[26]  Wen-Quan Tao,et al.  Meshless method based on the local weak-forms for steady-state heat conduction problems , 2008 .

[27]  K. Liew,et al.  Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems , 2008 .

[28]  V. Vavourakis,et al.  A MLPG4 (LBIE) Formulation in Elastostatics , 2007 .

[29]  Vladimir Sladek,et al.  Meshless Local Petrov-Galerkin Method for Plane Piezoelectricity , 2006 .

[30]  Vladimir Sladek,et al.  Inverse heat conduction problems by meshless local Petrov–Galerkin method , 2006 .

[31]  Satya N. Atluri,et al.  The Applications of Meshless Local Petrov-Galerkin (MLPG) Approaches in High-Speed Impact, Penetration and Perforation Problems , 2006 .

[32]  Vladimir Sladek,et al.  Meshless Local Petrov-Galerkin (MLPG) Method for Shear Deformable Shells Analysis , 2006 .

[33]  Y. Smyrlis,et al.  The Method of Fundamental Solutions: A Weighted Least-Squares Approach , 2006 .

[34]  J. Eberhardsteiner,et al.  Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov–Galerkin Method , 2006 .

[35]  Vladimir Sladek,et al.  Transient heat conduction in anisotropic and functionally graded media by local integral equations , 2005 .

[36]  Vladimir Sladek,et al.  Meshless Local Petrov-Galerkin Method for Stress and Crack Analysis in 3-D Axisymmetric FGM Bodies , 2005 .

[37]  Satya N. Atluri,et al.  Meshless Local Petrov-Galerkin (MLPG) Formulation for Analysis of Thick Plates , 2004 .

[38]  Satya N. Atluri,et al.  Meshless Local Petrov-Galerkin Method for Heat Conduction Problem in an Anisotropic Medium , 2004 .

[39]  Vladimir Sladek,et al.  Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method , 2003 .

[40]  Tzi-Kang Chen,et al.  A coupled finite element and meshless local Petrov–Galerkin method for two-dimensional potential problems , 2003 .

[41]  Ch. Zhang,et al.  Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids , 2003 .

[42]  Graeme Fairweather,et al.  The method of fundamental solutions for scattering and radiation problems , 2003 .

[43]  J. Sládek,et al.  Meshless local boundary integral equation method for 2D elastodynamic problems , 2003 .

[44]  Wilfried B. Krätzig,et al.  Automatic adaptive generation of a coupled finite element/element-free Galerkin discretization , 2002 .

[45]  S. Atluri,et al.  The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods , 2002 .

[46]  Y. Ochiai.  Steady Heat Conduction Analysis in Orthotropic Bodies by Triple-reciprocity BEM , 2001 .

[47]  Romesh C. Batra,et al.  DETERMINATION OF CRACK TIP FIELDS IN LINEAR ELASTOSTATICS BY THE MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD , 2001 .

[48]  YuanTong Gu,et al.  Coupling of element free Galerkin and hybrid boundary element methods using modified variational formulation , 2000 .

[49]  Satya N. Atluri,et al.  The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity , 2000 .

[50]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[51]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[52]  D Dries Hegen,et al.  Element-free Galerkin methods in combination with finite element approaches , 1996 .

[53]  Zhiqiang Cai,et al.  The finite volume element method for diffusion equations on general triangulations , 1991 .

[54]  Satya N. Atluri,et al.  SGBEM Voronoi Cells (SVCs), with EmbeddedArbitrary-Shaped Inclusions, Voids, and/or Cracks, forMicromechanical Modeling of Heterogeneous Materials , 2013 .

[55]  Vladimir Sladek,et al.  A local integral equation formulation to solve coupled nonlinear reaction–diffusion equations by using moving least square approximation , 2013 .

[56]  G. Liu.  Meshless Methods Coupled with Other Numerical Methods , 2011 .

[57]  G. Liu.  Meshless Local Petrov-Galerkin ( MLPG ) method in combination with finite element and boundary element approaches , 2008 .

[58]  V. Vavourakis,et al.  A new singular/hypersingular MLPG (LBIE) method for 2D elastostatics , 2005 .

[59]  S. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) method , 2002 .

[60]  S. Mukherjee,et al.  Boundary element techniques: Theory and applications in engineering , 1984 .

[61]  Edward L. Wilson,et al.  Numerical methods in finite element analysis , 1976 .

[62]  L. Rosenhead.  Conduction of Heat in Solids , 1947, Nature.