Copyright c  2008 Tech Science Press CMES, vol.32, no.3, pp.161-174, 2008 Analysis of Transient Heat Conduction in 3D Anisotropic Functionally Graded Solids, by the MLPG Method J. Sladek 1 , V. Sladek 1 , C.L. Tan 2 and S.N. Atluri 3 Abstract: A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of steady-state and transient heat conduction problems in a continuously non- homogeneous anisotropic medium. The Laplace transform is used to treat the time dependence of the variables for transient problems. The an- alyzed domain is covered by small subdomains with a simple geometry. A weak formulation for the set of governing equations is transformed into local integral equations on local subdomains by using a unit test function. Nodal points are ran- domly distributed in the 3D analyzed domain and each node is surrounded by a spherical subdo- main to which a local integral equation is applied. The meshless approximation based on the Mov- ing Least-Squares (MLS) method is employed for the implementation. Several example problems with Dirichlet, mixed, and/or convection bound- ary conditions, are presented to demonstrate the veracity and effectiveness of the numerical ap- proach. Keyword: meshless method, local weak form, Heaviside step function, moving least squares in- terpolation, Laplace transform 1 Introduction Functionally graded materials are multi-phase materials with the phase volume fractions vary- ing gradually in space, in a pre-determined pro- file. This results in continuously graded thermo- mechanical properties at the (macroscopic) struc- 1 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia 2 Department of Mechanical & Aerospace Engineering, Carleton University, Ottawa, Canada K1S 5B6 3 Center of Aerospace Research & Education, University of California at Irvine, Irvine, CA 92697-3975, USA tural scale. Often, these spatial gradients in mate- rial behaviour render FGMs as superior to con- ventional composites. FGMs possess some ad- vantages over conventional composites because of their continuously graded structures and prop- erties [Suresh and Mortensen (1998); Miyamoto et al. (1999)]. FGMs may exhibit isotropic or anisotropic material properties, depending on the processing technique and the practical engineer- ing requirements. Recent progress in the devel- opment and research of FGMs has also enhanced interests in the development of numerical meth- ods for the solution of heat conduction problems in continuously non-homogeneous solids. The lit- erature on heat conduction problems in FGM ma- terials has focused mainly on problems with ex- ponential variations of thermal properties, formu- lated in Cartesian coordinates and under steady- state boundary conditions [Noda and Jin (1993); Erdogan and Wu (1996); Jin and Noda (1993)]. Transient heat transfer in FGMs with the expo- nential spatial variation has also been examined, but to a lesser extent [Jin and Batra, 1996; Noda and Jin (1994); Jin and Paulino (2001); Sutradhar et al. (2002); Jin (2002)]. Due to the high mathematical complexity of the initial-boundary value problems, analytical ap- proaches for the thermo-mechanics of FGMs are restricted to simple geometries and boundary con- ditions. Transient heat conduction analysis in FGM demands accurate and efficient numerical methods. The finite element method (FEM) can be successfully applied to problems with an arbi- trary variation of material properties by using spe- cial graded elements [Kim and Paulino (2002)]. In commercial computer codes, however, mate- rial properties are considered to be uniform on each element. The boundary element method (BEM) is also a suitable numerical tool for this