Use of geometry in finite element thermal radiation combined with ray tracing (Q984972)

The authors consider a steady-state 3D heat transfer equation \(\text{div}(k \overrightarrow{\text{grad}T})+Q=0\) in the volume \(\mathcal{V}\) where \(Q\) is the volume source. The volume \(\mathcal{V}\) is limited by the surface \( \mathcal{S}\) which can be decomposed in two distinct pieces \(\Gamma _{T}\) and \(\Gamma _{q}\). On \(\Gamma _{T}\) (resp. \(\Gamma _{q}\)) the temperature (resp. the flux) is imposed. The paper addresses a numerical method for the resolution of this problem. It combines a geometrical definition of the model which is adapted for the radiation component and a finite element method which is adapted for the convection part. The authors first rewrite the boundary condition for the heat flux in a discretized way. This discretized equation involves view factors \(F_{i-j}\) which are the fraction of the uniform diffusion radiation which leaves the surface element \(A_{i}\) and which directly reaches the surface element \(A_{j}\). This view factor \(F_{i-j}\) is given as \(F_{i-j}=\frac{1}{A_{i}}\int_{A_{i}}\int_{A_{j}}\frac{\cos (\theta _{i})\cos (\theta _{j})}{\tau s_{i-j}^{2}}V_{i-j}dA_{i}dA_{j}\) where \(\theta _{i}\) is the angle between the normal \(\overrightarrow{n_{i}}\) and the line which connects \(dA_{i}\) and \(dA_{j}\), \(s_{i-j}\) is the distance between \( dA_{i}\) and \(dA_{j}\) and \(V_{i-j}\) is the visibility function equal to 1 if \( dA_{i}\) and \(dA_{j}\) can see each other and 0 otherwise. In order to compute these view factors, the authors first introduce a random method consisting of a stochastic ray tracing. From \(A_{i}\) a large number of rays are generated, some of them reaching \(A_{j}\). The value of the view factor is close to some quotient of such rays. The authors also introduce a parametric method which uses the mesh of the body. In the last part of their paper, the authors present some results of the numerical computations based on these methods. They especially study the cases of concentric specular spheres and of a simplified model of a satellite consisting of a tank, of an electronic box, of a cylinder, of a bar and of two panels. In each case, they give the value of the view factors and the CPU times which are required for these computations.