Accurate numerical simulation of radiative heat transfer with application to crystal growth

We present an accurate and cost-effective numerical method to investigate thermomagnetic problems arising in crystal growth applications. The governing equations are the quasi-static, time-harmonic, axisymmetric Maxwell equations coupled with an energy conservation equation. Radiant energy transfer is modeled by an integral equation yielding a strongly nonlinear and non-local problem. Conformal finite elements are used to discretize the partial differential equations and a discontinuous Galerkin method to discretize the integral equation. A key aspect of the present methodology is to introduce an appropriate renormalization of the view factor matrix so that singularities near re-entrant corners are resolved and optimal convergence rates are recovered. In addition, this renormalization guarantees, under some assumptions, that the discrete problem is well-posed. Computational aspects related to the evaluation of view factors in axisymmetric enclosures are also addressed. We consider a ray-search method involving an initial bracketing of the view angle interval followed by local azimuthal refinement near shadowing obstacles. The impact of renormalization on solution accuracy is assessed on reactors with convex and non-convex enclosures. Numerical results are also compared with previous work. Finally, we consider an industrial prototype reactor involving several non-convex radiating surfaces.