# An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates
## 5 Results and Discussion  
### Validation Model  
For the verification of the accuracy of the proposed methods, the temperature distribution and the effective conductivity in a benchmark problem were calculated with the MLPG methods and compared with FD results. A solid sphere was placed at the center of a cubic domain with relative size corresponding to a volume fraction \(f_{p}=0.0042\). Dirichlet boundary conditions were applied for the temperature at all sides of the cube imposing heat flux in the vertical direction. The temperature distribution was obtained by the analytical solution of the problem of a single spherical inclusion in a matrix, Equation (9). The problem was then solved numerically with the FD and the MLPG methods. The MPLG method was used with the application either of the MLS approach, or of the DC PSE operators. Figure 5 shows the estimated error in the temperature profiles obtained by each method as a function of the total number of nodes. The error is defined as the weighted Euclidean norm of each approach and the analytical solution (\(L_{2}=\sqrt{\frac{\sum_{i=1}^{n}(T_{i}-T_{m})^{2}}{n}}\), \(i=\) FD, MLPG). The comparison was made over a wide range of thermal conductivity ratios of the particle (\(k_{p}\)) and the matrix (\(k_{f}\)), namely, between 0.01 and 100.  
The solution was attempted using normal grids, with relatively low uniform resolution (\(11\times 11\times 11\)), up to a relatively high one (\(81\times 81\times 81\)), with convergence at \(10^{-6}\) precision globally. For cases with local refinement, grids of 11, 21, 41, and 81 nodes per dimension are used as initial density, with double resolution near the interfaces All the methods converge practically onto the analytical solution, as the maximum errors are of the order of \(10^{-4}\). For all \(\mathsf{k}_{r}\) cases, the MLPG method with local refinement converged at the same level of accuracy using one order of magnitude less nodes compared to FD (\(3\times 10^{3},2\times 10^{4}\) respectively, Figure 5d). This reduced substantially the computational cost. Moreover, the FD method showed higher errors in most cases.  
The volume fraction of the particle was very small (\(f_{p}=0.0042\)), so the effective conductivity could be accurately calculated from Equation (10) at the dilute limit. The superiority of the MLPG methods  
Figure 5: Weighted Euclidean norm of the deviation of the numerical approaches from the analytical solution as a function of the total number of nodes for conductivity ratio (\(\mathsf{k}_{r}\)) (**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, FD: finite differences.  
in view of their accuracy becomes more apparent in Figure 6, where the error on the effective thermal conductivity is presented. Local refinement clearly offers better description of the effective conductance.