# An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates
## 6 Conclusions  
A novel numerical method that combines the discretization-corrected particle strength exchange (DC PSE) operators with the meshless local Petrov-Galerkin (MLPG) technique was developed in order to simulate heat transport phenomena in particulate systems. Local refinement of the grid was shown to improve the accuracy of the calculations at reference geometries. An alternative, more efficient approach of interpolation was introduced, increasing the stability of the method and reducing the computational cost for grid construction. An automated method for grid refinement was introduced in multiple discretization levels inside the folds of the structures bypassing the necessity for user intervention. Moreover, the use of cubic domains to calculate local integrals, aiming at full geometry coverage with minimal overlapping, produced adequately stable and accurate results. In addition, two different versions of the MLPG and the finite differences (FD) methods were compared in terms of the accuracy of the solutions. One version of the method used the moving least squares (MLS) approach for the trial function, while the other one used the DC PSE method. The DC PSE operators, which were used for the first time in this work as trial functions for the MLPG method, enhanced the solution stability and accuracy. It appears that the MLPG methods outweighed the FD one in the accuracy of the calculation for both the temperature distribution and the effective thermal  
Figure 12: Effective thermal conductivity as a function of the fractal dimension for four different conductivity ratio (\(\mathrm{k}_{x}\)) values, (**a**) 1/100, (**b**) 1/10, (**c**) 10, (**d**) 100. Blue dots: BA. Red dots: DLA. Yellow dots: BA 2p. Purple dots: DLA 2p. Black lines: Maxwell model. Abbreviations; BA: ballistic aggregation, DLA: diffusion limited aggregation. conductivity. Additionally, the use of DC PSE operators reduced the computational cost required, as solutions converged using a smaller number of nodes. The MLPG performance was tested against the performance of the finite element (FE) method-based software for the calculation of the effective conductivity in systems of fully dispersed spheres. The MLPG method with DC PSE operators and fourfold discretization near the interfaces had a convergence similar to that of the FE method. Experimental data from the literature were used for the sake of comparison with simulations. Sample structures were reconstructed with the Monte Carlo technique, and the heat transport problem was solved therein. The results of the MLPG method were in excellent agreement with experimental data. An important side outcome of this work is the observation that the differential effective medium (DEM) approximation predicted well the conductivity of unconsolidated granular media. Finally, the convergence of the method for dense granular structures was studied, where the extension of particle overlapping significantly influenced the conduction, and therefore, the thermal conductivity. Aggregate structures with different morphological properties were reconstructed, and the effective conductivity was calculated with elevated accuracy, a result of great value in practical applications. For particles with conductivity lower than that of the bulk medium, the effective conductivity was not affected by aggregation, and the results were in full agreement with the Maxwell approximation. Conversely, for highly conducting particles, aggregation enhanced conduction. The thermal conductivity was a function of the fractal dimension of the aggregates. It was found that, as the fractal dimension decreased, the thermal conductivity increased monotonically. Conceptualization, N.P.K., E.D.S. and V.N.B. ; methodology, N.P.K., N.B. and E.D.S. ; software, N.P.K. and N.B. ; supervision, V.N.B. ; validation, N.P.K. and N.B. ; visualization, N.P.K. ; writing--original draft, N.P.K. ; writing--review & editing, N.B., E.D.S. and V.N.B. All authors have read and agreed to the published version of the manuscript.