Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov–Galerkin Method

The meshless local Petrov–Galerkin method is used to analyze transient heat conduction in 3-D axisymmetric solids with continuously inhomogeneous and anisotropic material properties. A 3-D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem. The cross section is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen subdomains, called local boundary integral equations. These integral formulations are either based on the Laplace transform technique or the time difference approach. The local integral equations are nonsingular and take a very simple form, despite of inhomogeneous and anisotropic material behavior across the analyzed structure. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.

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

[2]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[3]  N. Noda,et al.  A crack in functionally gradient materials under thermal shock , 1994, Archive of Applied Mechanics.

[4]  Ch. Zhang,et al.  Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients , 2005 .

[5]  Naotake Noda,et al.  Thermal stress intensity factors for a crack in a strip of a functionally gradient material , 1993 .

[6]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[7]  Yoshinari Miyamoto,et al.  Functionally Graded Materials. , 1995 .

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

[9]  Solution of parabolic differential equations by the boundary element method using discretisation in time , 1980 .

[10]  K. Bathe,et al.  The method of finite spheres , 2000 .

[11]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

[12]  Alok Sutradhar,et al.  Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method , 2002 .

[13]  S. Suresh,et al.  Fundamentals of functionally graded materials , 1998 .

[14]  B. Davies,et al.  Numerical Inversion of the Laplace Transform: A Survey and Comparison of Methods , 1979 .

[15]  G. Paulino,et al.  ISOPARAMETRIC GRADED FINITE ELEMENTS FOR NONHOMOGENEOUS ISOTROPIC AND ORTHOTROPIC MATERIALS , 2002 .

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

[17]  A. Kawasaki,et al.  Functionally graded materials : design, processing and applications , 1999 .

[18]  S. Atluri.  The meshless method (MLPG) for domain & BIE discretizations , 2004 .

[19]  J. Sládek,et al.  A meshless local boundary integral equation method for heat conduction analysis in nonhomogeneous solids , 2004 .

[20]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[21]  Masataka Tanaka,et al.  Transient heat conduction problems in inhomogeneous media discretized by means of boundary-volume element , 1980 .

[22]  N. Noda,et al.  An internal crack parallel to the boundary of a nonhomogeneous half plane under thermal loading , 1993 .

[23]  S. Atluri,et al.  A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach , 1998 .

[24]  F. Erdogan,et al.  CRACK PROBLEMS IN FGM LAYERS UNDER THERMAL STRESSES , 1996 .

[25]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[26]  Sergey E. Mikhailov.  Localized boundary-domain integral formulations for problems with variable coefficients , 2002 .

[27]  R. Batra,et al.  STRESS INTENSITY RELAXATION AT THE TIP OF AN EDGE CRACK IN A FUNCTIONALLY GRADED MATERIAL SUBJECTED TO A THERMAL SHOCK , 1996 .

[28]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[29]  Carlos Alberto Brebbia,et al.  Recent advances in boundary element methods , 1978 .

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

[31]  Glaucio H. Paulino,et al.  Transient thermal stress analysis of an edge crack in a functionally graded material , 2001 .

[32]  Z.-H. Jin,et al.  AN ASYMPTOTIC SOLUTION OF TEMPERATURE FIELD IN A STRIP A FUNCTIONALLY GRADED MATERIAL , 2002 .