Application of a modified Monte Carlo method for the simulation of heat conduction in a rectangular slab

Monte Carlo method has been used to study heat conduction problems. It is grid-free inimplementation, unlike the conventional Finite Element and Finite Difference Method. However forMonte Carlo method, solutions of desired field of interest can only be obtained one after the other,unlike the others that can be obtained simultaneously. Therefore a modified method has beendeveloped which benefits from the simplicity of the Monte Carlo method and also provides full fielddescription of temperature at one computer run.The Modified Monte Carlo first obtains sample values of temperature in the domain of interestusing the conventional Monte Carlo technique. Thereafter a histogram table is constructed which isthen used to predict values for the other parts of the field unknown. The number of clusters obtainedfrom the prediction is noted. The prediction that has minimal clusters is adjudged the best. Thetechnique was tested on grid configurations such as 3 x 3 and 4 x 4 grids. Only isothermal steady-state2-D cases were considered. When compared with the results of ordinary Monte Carlo method, the modified techniqueincurred a maximum error of 4% for 4 x 4 grids. Larger grids were obtained by seamless stitching ofsmaller grids and as such no growth in errors was noticed. For the 3 x 3 grids, the sample size wasabout 55% while only 25% of the domain was sampled using 4 x 4 grids. Using the modified technique, the errors incurred even with the 4 x 4 grids was only 4%. Thetechnique can therefore be used for simulation of steady-state heat conduction.