Numerical Study of Heat Diffusion Employing the Lagrangian Smoothed Particle Hydrodynamics Method: an Analysis of the Applicability of the Laplacian Operator and the Influence of the Boundaries on the Solution

Abstract

In this paper, the Smoothed Particle Hydrodynamics Method was applied for the study of heat diffusion in a homogeneous flat plate subjected to the Dirichlet boundary condition. The numerical simulations by the SPH method have been performed using different interpolation functions (kernels), discretizations of the domain by particles and a proposed Laplacian operator. On the other hand, the analytical solution was used to obtain a standard solution for the problem. In the inner regions of the domain, the convergence of the results occurred for the unique solution, when the steady state has been reached.  In SPH, the Laplacian operator was suitable to be employed in the study. Comparing SPH and analytical results, the lowest differences in temperatures have been found when the kernel degree was the highest and the number of particles used in the discretization of the domain was the highest too. In the regions near the boundaries was seen the phenomenon of particles inconsistency (due to the truncation of the smoothing function by the boundary) and the consequent emergence of the largest temperature differences near the bottom corners. To obtain better results it has been necessary to make boundaries corrections, in order to recover the consistency lost by the truncation of the compact support defined in the SPH method.
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