A local BIEM for analysis of transient heat conduction with nonlinear source terms in FGMs Jan Sladek a, * , Vladimir Sladek a , Ch. Zhang b a Institute of Construction and Architecture, Slovak Academy of Sciences, 84220 Bratislava, Slovakia b Department of Civil Engineering, University of Applied Sciences Zittau/Go ¨rlitz, D-02763 Zittau, Germany Received 6 February 2003; revised 13 May 2003; accepted 14 May 2003 Abstract The diffusion equation with nonlinear heat source intensity in functionally graded materials (FGMs) is considered. In FGMs the thermal material properties are dependent on spatial coordinates. For transient or steady-state heat problems in FGMs the conventional boundary integral equation method or boundary element method cannot be applied due to the lack of a fundamental solution. In this paper, a local boundary integral equation method is proposed to analyse a temperature distribution in a nonhomogeneous body under a microwave heating. To eliminate time variable in the heat equation, the Laplace transform technique is used. The boundary-domain integral formulation with a simple fundamental solution corresponding to the Laplace operator is related to all subdomains which cover the analysed domain. If such integral equations are considered on small subdomains with a simple geometry (circle), domain integrals can be easily evaluated. Physical fields (temperature, heat flux) on the local boundary and in the interior of the subdomain are approximated by the moving least-square. The method is completely element free. q 2003 Elsevier Ltd. All rights reserved. Keywords: Hot spots; Transient heat conduction analysis; Local boundary integral equations; Functionally graded materials; Laplace transform; Moving least- square approximation; Absorbtivity 1. Introduction Transient heat conduction equations with nonlinear source terms arise as governing equations in many different areas of mathematical physics. Examples in the literature include microwave heating process, mass transport in groundwater, and ignition. The use of microwave heating has attracted much attention in industrial processes due to its high efficiency. A crucial point in microwave heating is to analyse hot spots, regions where temperature is enormously running. This phenomenon is occurred due to dependence of material properties on temperature. Namely, thermal absorptivity increases with temperature. Hot spots out of control can damage the sample in an industrial process. The first attempt to analyse hot spots in a homogeneous body is given by Zhu et al. [1]. The dual reciprocity boundary element method (BEM) is proposed there. In the DRBEM, some interior nodes have to be supplemented to boundary nodes for an adequate spatial approximation of physical fields in transforming domain integrals into boundary integrals. Functionally graded materials (FGMs) are preferred and favoured in many engineering structures and components due to their excellent thermal properties [2,3]. In FGMs the volume fractions of composite constituents are varying continuously in space. Solution of boundary or initial boundary value problems for FGMs requires special numerical methods due to the high mathematical complex- ity caused by the material nonhomogeneity. Beside the well- established finite element method (FEM), the boundary integral equation method (BIEM) or BEM provides an efficient and popular alternative to the FEM for solving certain class of boundary or initial boundary value problems. However, the pure BIEM/BEM formulation can be efficiently applied only to problems where fundamental solutions or Green’s functions are available in simple forms. Unfortunately, for general FGMs the corresponding funda- mental solutions are either not available or they are mathematically too complicated. To avoid this difficulty, 0955-7997/04/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0955-7997(03)00093-6 Engineering Analysis with Boundary Elements 28 (2004) 1–11 www.elsevier.com/locate/enganabound * Corresponding author. Tel.: þ 421-7-54788662; fax: þ 421-7- 54772494. E-mail address: usarslad@savba.sk (J. Sladek).