Improvements of the Brownian walkers method towards the modeling of conduction-radiation coupling

This work directly follows the one presented in a previous article of V. Gonneau, D. Rochais and F. Enguehard (2022) [1]. This contribution describes new results and improvements to existing methods for the modeling of transient thermal conduction within a heterogeneous medium by the movement of Brownian walkers. The material structure is voxelized, and each walker transports an elementary enthalpy during its displacement within the structure. This enthalpy transport associated with the displacement of the walkers represents the conductive flow and makes it possible to simulate transient conduction with a quantitative stochastic approach.

This article presents a new method for accounting for Dirichlet type boundary conditions that is quite efficient and that allows to relax quite substantially a constraint of maximum value of the time step of the Brownian walker simulations that had previously been pointed out in [1]. Other boundary condition treatments are also upgraded, in terms of speed (for Neumann type conditions) or generalization (for Robin type conditions). Then, a first non-linear conduction-radiation coupling model in an optically thick and gray semi-transparent medium is solved by Brownian walkers through two approaches: the first one based on a global conductivity function of temperature and allowing to confirm the validity of a stochastic transmission criterion at the interface of two constituents within a heterogeneous medium, and the second one based on the radiative volume power field. This second approach is preceded by the description and the validation of a procedure for the management of a volume power field using Brownian walkers. This procedure introduces the notion of negative Brownian walkers, which prove necessary for representing negative local values of the volume power field as is frequently the case with the radiative volume power.