Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids J. Sladek, V. Sladek, J. Krivacek, Ch. Zhang Abstract An advanced computational method for tran- sient heat conduction analysis in 3-D axisymmetric con- tinuously nonhomogeneous functionally graded materials (FGM) is proposed. The analysed domain is covered by small circular subdomains. On each subdomain local boundary integral equations for the transient heat conduction problem are derived in the Laplace transform domain. The meshless approximation based on the mov- ing least-squares method is employed for the numerical implementation. The Stehfest algorithm is applied for the numerical Laplace inversion to obtain the temporal vari- ation. Numerical results are presented for finite full and hollow cylinders with an exponential variation of material parameters with spatial coordinates. Keywords Local boundary integral equation, Axisym- metric, Functionally graded materials, Laplace transform, Stehfest algorithm, Subdomains, Meshless approximation 1 Introduction Functionally graded materials are multi-phase materials with the volume fractions of the constituents varying gradually in a pre-determined profile, thus giving a non- uniform structure in the materials with continuously graded macro-thermomechanical properties. FGMs posses some advantages over conventional composites because of their continuously graded structures and properties (Suresh and Mortensen, 1998; Miyamoto et al., 1999). Since FGM are frequently used for structures under thermal load, it is needed to analyze the temperature distribution in such materials. The optimum design of FGMs to improve their thermal stress resistance requires the thermal solutions for FGMs with arbitrarily graded material properties. In literature one can find only few limited number of papers where heat conduction problems in FGM materials are analyzed. They are mostly focused on problems with exponential variations of thermal properties with Cartesian coordinates under stationary boundary conditions (Noda and Jin, 1993; Erdogan and Wu, 1996; Jin and Noda, 1993). The transient heat transfer in FGMs with the exponential spatial variation has been examined by several authors (Jin and Batra, 1996; Noda and Jin, 1994; Jin and Paulino, 2001; Sutradhar et al., 2002). Due to the high mathematical complexity of the initial- boundary value problem, analytical approaches for FGM are restricted to simple geometry and boundary conditions (Jin, 2002). Thus, the transient heat conduction analysis in FGM demands accurate and efficient numerical methods. The finite element method can be successfully applied to problems with an arbitrary variation of material properties by using special graded elements (Kim and Paulino, 2002). The boundary element method (BEM) is a suitable numerical tool for this purpose too. However, to the knowledge of the authors, only Sutradhar et al. (2002) ap- plied the BEM to 3-D transient heat conduction analysis in FGM, where the Green’s function approach and an expo- nential variation of material parameters were used. Up to date no special BEM has been presented in literature to analyze 3-D axisymmetric functionally graded bodies. A pure BEM formulation can be applied only to problems, where fundamental solutions are available. For a general nonhomogeneous body the fundamental solution for transient heat conduction problem is yet not known in literature. One possibility to obtain a BEM formulation is based on the use of fundamental solutions for a fictitious homogeneous medium, as has been suggested for the first time by Butterfield (1978) for potential flow problems. This approach, which is the basis of the global BEM, however, leads to a boundary-domain integral formulation with additional domain integrals for the gradients of primary fields to obtain a unique formulation. This approach has been applied to the heat conduction analysis in anisotropic nonhomogeneous media by Tanaka and Tanaka (1980). Axial symmetry of the geometry and boundary condi- tions for a 3-D body reduces the dimensionality of an initial-boundary value problem into a 2-D problem. The transient heat conduction problem in 3-D axisymmetric Computational Mechanics 32 (2003) 169–176 Ó Springer-Verlag 2003 DOI 10.1007/s00466-003-0470-z Received: 20 November 2002 / Accepted: 10 June 2003 J. Sladek (&), V. Sladek, J. Krivacek Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia E-mail: usarslad@savba.sk Ch. Zhang Department of Civil Engineering, University of Applied Sciences Zittau/Go ¨rlitz, 02763 Zittau, Germany The authors acknowledge the support by the Slovak Science and Technology Assistance Agency registered under number APVT- 51-003702, and the Project for Bilateral Cooperation in Science and Technology supported jointly by the International Bureau of the German BMBF and the Ministry of Education of Slovak Republic under the project number SVK 01/020. 169