Numerical and statistical method for solving the heat exchange problems in the heat-protective constructions of the honeycomb type

A method for determining the thermal state of a heat insulation panel of the honeycomb type is proposed in the paper. The use of heat-shielding materials of this type is a promising direction in the design of high-speed aircraft. The considered heat transfer is described by the boundary value problem for the heat equation. It is assumed that the ther- mal diffusivity coefficients of the materials of the panel are the given constants, and the heat exchange process occurs only due to thermal conductivity. The proposed method is based on the probability representation of the solution of the boundary value problem, which is an expectation of the functional of the random process of the diffusion type, and the numerical simulating this random process. In earlier papers the authors offered to perform the computations by the Euler method. As the heat exchange occurs in inhomogeneous medium, we use smoothing discontinuous coefficients of the heat equation, based on the integral averaging, in numerical simulation of trajectories of the random process. The simulated random process coincides with Wiener process in subareas in which the medium is homogeneous. This fact means that a significant reduction in computational costs can be achieved by using the method of the random walks on spheres in these subareas. In the paper we propose to use the random walks on spheres in the cells of the honeycomb panel and the Euler method in moving on the frame and its vicinity. The calculations performed according to the data of the physical experiment have shown the high efficiency of the proposed combined method in comparison with calcu- lations performed by only the Euler method.