On the adoption of the Monte Carlo method to solve one-dimensional steady state thermal diffusion problems for non-uniform solids

Abstract

The present paper is focussed on the investigation of the potential adoption of the Monte Carlo method to solve one-dimensional, steady state, thermal diffusion problems for continuous solids characterised by an isotropic, space-dependent conductivity tensor and subjected to non-uniform heat power deposition.To this purpose the steady state form of Fourier's heat diffusion equation relevant to a continuous, heterogeneous and isotropic solid, undergoing a space-dependent heat power density has been solved in a closed analytical form for the general case of Cauchy's boundary conditions. The thermal field obtained has been, then, put in a peculiar functional form, indicating that it might be obtained performing statistical averages by means of a well-posed distribution function, adopting a numerical approach based on the Monte Carlo method.Some test cases have been considered and the very good agreement between their analytical solutions and the results obtained by means of the proposed numerical procedure are presented and critically discussed.