Distribution of temperature in a single lens due to absorption of light and heat conduction: an adaptive solver

摘要

We develop an algorithm for the solution of the stationary heat-equation in a single lens due to absorption of light, heat-conduction and transfer of the heat to the environment while we assume rotational symmetry for the whole situation. The proceeding is based on an easy to implement finite difference scheme, which is best suited for rectangular areas. Therefore, we have to transform the heat equation and the boundary conditions from the original domain, i.e. the surface of section of the lens by the aid of tensor methods to a rectangle. So the algorithm generates a grid, which adopts automatically to the actual shape of the lens. In this sense, we characterize the method as adaptive. In the examples, we investigate the effect of a high-transmission glass on the distribution of temperature and further demonstrate the adjustment to a realistic lens shape with a strong deviation from a spherical surface in form of a kink near the edge. We compare the results with a simple model for the distribution of temperature and show the strong dependency of the results on the transmission of the materials.