Natural element method for radiative heat transfer in two-dimensional semitransparent medium Yong Zhang, Hong-Liang Yi ⇑ , He-Ping Tan ⇑ School of Energy Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Harbin 150001, PR China article info Article history: Received 11 June 2012 Received in revised form 22 September 2012 Accepted 24 September 2012 Available online 27 October 2012 Keywords: Radiative heat transfer Meshless method Natural element method Voronoi diagram abstract The natural element method (NEM) is employed for solving radiative heat transfer problem in a two- dimensional enclosure containing an absorbing, emitting and isotropically scattering medium. Medium boundaries are considered to be opaque, diffuse as well as gray. The NEM referred to as natural neighbor Galerkin method is a new technique in the field of computational mechanics and can be considered as a meshless numerical method. Unlike most of other meshless methods such as element-free Galerkin method or those based on radial basis functions, the shape functions used in NEM, which are based on the Voronoi tesselation of a set of nodes, are strictly interpolant and the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations. Three types of radiative heat transfer problems are addressed. For pure radiation, studies are made for three represen- tative benchmark problems dealing with radiative equilibrium and non-radiative equilibrium. For cou- pled heat transfer, transient conduction and radiation problem in a rectangular geometry, steady conduction and radiation problem in a gray cylindrical ring are solved. For these problems considered, tests are presented for various parameters, such as the aspect ratio, conduction-radiation parameter, optical thickness, scattering albedo and surface emissivity. Results for NEM are compared with those for finite element method (FEM) generated by the authors and those reported in the literatures. By com- parison, it is shown that NEM is efficient, accurate and stable, and can be used for solving radiative heat transfer problem in 2-D semitransparent media. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Many numerical methods have been developed to solve the problem of radiative heat transfer in semitransparent media. Well known examples are the Monte Carlo method, the zonal method, the discrete ordinates method (DOM), the finite volume method (FVM) [1,2] and the finite element method (FEM) [3,4]. As to cou- pled problem with radiation transfer, The spatial distributions of radiative information as source terms are first solved, and then en- ergy equation can be solved by finite difference method (FDM) [5], FVM and FEM [6]. Combination of the methods mentioned above can be used to deal with the coupled radiation and conduction in multi-dimensional complicated geometry. However, in the discretization of the domain of the problem, those traditional methods, especially the FEM, depend on the qual- ity of pre-defined mesh severely, and they are difficult to deal with the problems associated with severe mesh distortion. To avoid the drawbacks of FEM, considerable effort has been devoted to the development of the so-called meshless method. More recently meshless methods have emerged and have been used successfully in fluid flow and related heat transfer problems [7–9]. The common meshless methods include such as the smooth particle hydrody- namics (SPH) [10], the reproducing kernel particle method (RKPM) [11], the diffuse element method (DEM) [12], the element-free Galerkin method (EFG) [13], the partition of unity finite element method (PUFE) [14], the meshless local Petrov–Galerkin method (MLPG) [15] and so on. A common feature of the meshless methods is that the approximation in terms of discrete nodes is constructed and no predefined nodal connectivity is required. With this advan- tage, these methods can cope with physical problems associated with large transformation of the domain considered without being restricted by mesh distortion, as opposed to the traditional mesh methods such as FVM and FEM. Recently, meshless methods were introduced into the commu- nity of radiative heat transfer. Liu and his coworkers [16] solved the multi-dimensional radiative transfer equation and coupled radiative and conductive heat transfer using a MLPG method. Liu and Tan [17] employed an least-squares collocation meshless method (LSCM) proposed by Zhang et al. [18] for solving the radi- ative heat transfer in semitransparent media. As it was overviewed by Belytschko et al. [19] and Duarte and Oden [20], the moving least square technique (MLS) plays a crucial 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.09.051 ⇑ Corresponding authors. Tel.: +86 451 86412674; fax: +86 451 86221048. E-mail addresses: yihongliang@hit.edu.cn (H.-L. Yi), tanheping@hit.edu.cn (H.-P. Tan). International Journal of Heat and Mass Transfer 56 (2013) 411–423 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt