# An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates
## 5 Results and Discussion
### Single Arrangement and Fully Dispersed Particles  
The convergence of the numerical methods in a system of fully dispersed particles and in a single arrangement of spheres is examined in this section. The results of the meshless technique were validated with the output of the well-known, FE method-based, commercial software Comsol Multiphysics(r). In the case of fully dispersed particles, the particle number was set to one hundred (100) and their volume fraction \(f_{p}\) to 0.1. The effective thermal conductivity of the system is presented in Figure 7 for varying discretization. The thermal conductivity ratio of the particles and the matrix was equal to \(\mathrm{k_{r}}=0.01\), Figure 7a, and \(\mathrm{k_{r}}=100\), Figure 7b. Figure 7 presents the convergence of the MLPG and the  
Figure 6: Percentage error on the effective thermal conductivity as a function of the total number of nodes for 4 conductivity ratios (\(\mathrm{k_{r}}\)) (**a**) 1/100 (**b**) 1/10, (**c**) 10, (**d**) 100. Red lines: MLPG with MLS. Black lines: MLPG with DC PSE. Continuous lines: normal grids. Dashed lines: grids locally refined. Blue lines: FD. Abbreviations; MLPG: meshless local Petrov-Galerkin, MLS: moving least squares, DC PSE: discretization-corrected particle strength exchange, ref: local refinement, FD: finite differences. FE methods, as well as the predictions of [PERSON]'s DEM relationship, Equation (13). Regarding the discretization, the simulations start with a coarse grid of 31 nodes per dimension, i.e., ca. \(3\times 10^{4}\) total nodes, and reach a dense grid of 151 nodes per dimension (\(3.4\times 10^{6}\) nodes in total). The methods with local refinement initially have a grid with 10 nodes per dimension, with a fourfold discretization near the interfaces, reaching 80 nodes per dimension and \(5\times 10^{6}\) total nodes. The results using a typical FE approach were in agreement with the predictions of the analytical expression and the meshless approach. Free tetrahedral elements were created and five boundary layers along the interfaces were used. The element size varied from 7% to 0.02% of the domain size for an extremely coarse and an extremely fine mesh, respectively. FE solutions converge up to \(10^{-6}\) precision globally. Initially, one can observe that the converged results of the simulations were very close to the predictions of the effective medium theory (EMT). Also, local refinement improved the solution in all cases. It is obvious that the use of MLPG with DC PSE and local refinement offered more stable description for the effective conductivity among the MLPG methods examined here. The DC PSE with local refinement appears to have had practically the same convergence behavior as the FE method, having a minimum discretization of 50 nodes per dimension and total amount of nodes \(5\times 10^{5}\), and provided a converged solution with an error less than 1%. The thermal conductivity results for a single arrangement of spheres, Figure 8, are shown in Figure 9 for varying volume fraction of the inclusions. The MLPG method refers to the MLPG using the DC PSE operators and fourfold refinement near the interfaces. A grid with maximum resolution of 200 nodes per dimension proved sufficient for all cases. The problem was solved for a wide range of conductivity ratios. The simulations were in good agreement with the DEM approximation for low conductivity ratios, Figure 9a-c. In contrast, when the inclusions have much higher conductivity than the matrix (\(\mathrm{k_{r}}=100\)), Figure 9d, a steep increase of the effective conductivity in the region where \(f_{p}=0.524\) was observed. At this volume fraction, a highly conductive path was created. This effect  
Figure 7: Effective thermal conductivity as a function of the total number of nodes for two different conductivity ratios (\(\mathrm{k_{r}}\)) (**a**) 1/100 (**b**) 100. Red lines: MLPG with MLS. Black lines: MLPG with DC PSE. Continuous lines: normal grids. Dashed lines: grids locally refined. Blue lines: FE.