Meshless method for solving coupled radiative and conductive heat transfer in complex multi-dimensional geometries Hamou Sadat ⇑ , Cheng-An Wang, Vital Le Dez Institut PPRIME, CNRS, Université de Poitiers, ENSMA UPR 3346, Département Fluides, Thermique, Combustion, ESIP, Campus Sud 40, avenue du recteur Pineau, 86022 Poitiers Cedex, France article info Keywords: Meshless method Coupled radiative–conductive transfer Discrete ordinates Even parity Complex geometries abstract A meshless method DAM is employed to solve the coupled radiative and conductive heat transfer problem in a semi-transparent medium enclosed in complex 2D and 3D geome- tries. The meshless method for radiative transfer is based on the even parity formulation of the discrete ordinates method. Cases of combined conduction–radiation are presented and the results are compared with other benchmark approximate solutions. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The coupled problem of conduction and radiation in participating media at high temperature arises in many engineering applications. When the geometry is complex, finite element [2–7] and control volume based finite element methods [8] have been generally employed for the spatial discretization. More recently meshless methods have emerged and have been used successfully in fluid flow [9–17] and related heat transfer problems [18–21]. These methods which do not require the use of a finite element mesh have been also applied in the numerical solution of radiative transfer problems. Liu et al. [22] have proposed a meshless local Petrov–Galerkin (MLPG) approach for solving the coupled radiative and conductive heat transfer in a one-dimensional slab with graded index media. Tan et al. [23] used a least-squares collocation meshless method to solve the coupled radiative and conductive heat transfer in 2D rectangular and cylindrical enclosures. A meshless local Petrov– Galerkin approach with upwind scheme for radiative transfer based on the discrete ordinate equations has been presented in Liu and Tan [24]. Sadat [25] used a moving least squares collocation meshless method with the even parity formulation of the RTE. Another second order method which differs from the even parity formulation was proposed [26] and its comparison with first order formulation was presented in [27]. Like in [25], it was found that the second order approach is more accurate. Wang et al. [28] used the meshless method presented in [25] to solve several purely radiative transfer problems in 2D and 3D geometries. In this paper we extend the method to solve coupled radiative and conductive problems. The details of the the- oretical and numerical calculation procedures are given first. Then several 2D and 3D test cases in irregular geometries are considered the results of which are compared with the available literature results. 2. The discrete ordinates method The discrete ordinates method (DOM) is based on the use of numerical quadratures to approximate the integrals appear- ing in the calculation of the incident radiation and radiative fluxes. It uses a discretization of the angular space by a finite 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.101 ⇑ Corresponding author. E-mail address: hamou.sadat@univ-poitiers.fr (H. Sadat). Applied Mathematics and Computation 218 (2012) 10211–10225 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc