Open Access Open Access  Restricted Access Subscription or Fee Access

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

(*) Corresponding author

Authors' affiliations



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.
Copyright © 2014 Praise Worthy Prize - All rights reserved.


SPH Method; Heat Diffusion; Laplacian Operator; Boundaries; Particles Inconsistency

Full Text:



Abas, Z.A., Salleh, S., Rahman, A.F.N.A., Basiron, H., Hasan Basari, A.S., Hassim, N., Improvement of element creation procedure for generating initial triangular unstructured mesh for radiative heat transfer modelling, (2013) International Review on Modelling and Simulations (IREMOS), 6 (5), pp. 1649-1656.

El Khaoudi, F., Gueraoui, K., Driouich, M., Sammouda, M., Numerical and theoretical modeling of natural convection of nanofluids in a vertical rectangular cavity, (2014) International Review on Modelling and Simulations (IREMOS), 7 (2), pp. 350-355.

Ben Mabrouk, S., Ben Ahmed, H., Numerical simulation of compressible thermo-buoyant flow in a partially opened enclosure with localized heater from below, (2013) International Review on Modelling and Simulations (IREMOS), 6 (1), pp. 214-223.

Cleary, P.W. & Monaghan J.J., Conduction Modelling Using Smoothed Particle Hydrodynamics. Journal of Computational Physics, 148: 227-264, 1999.

Jeong, J.H., Jhon M.S., Halow J.S. & van Osdol J., Smoothed Particle Hydrodynamics: Applications to Heat Conduction. Computer Physics Communications, 153: 71-84, 2003.

Schwaiger, H.F., An implicit corrected SPH formulation for thermal duffusion with linear free surface boundary conditions. International Journal for Numerical Methods in Engineering, 75: 647-671, 2008.

Rook, R., Yildiz M. & Dost S., Modeling Transient Heat Transfer Using SPH and Implicit Time Integration. Numerical Heat Transfer, Part B: Fundamentals, 51:1, 1-23, 2007.

Xu, J., Heat Transfer with Explicit SPH Method in LS-DYNA. Proceedings of the 12th International LS-DYNA Users Conference. Computer Technologies. Detroit, USA, 2012.

Chen, J.K., Beraun, J. E. & Carney, T.C., A Corrective Smoothed Patricle Method for Boundary Value Problems in Heat Conduction. International Journal for Numerical Methods in Engineering, 46: 231-252, 1999.;2-k

Liu, M.B. & Liu, G.R., Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments. Archives of Computational Methods in Engineering, 17: 25-76, 2010.

Lucy, L.B., Numerical approach to testing the fission hypothesis. Astronomical Journal, 82: 1013-1024, 1977.

Liu, M.B., Liu, G.R. & Lam K.Y., Constructing smoothing functions in smoothed particle hydrodynamics with applications. Journal of Computational and Applied Mathematics, 155: 263-284, 2003.

Morris, J.P., Fox, P.J. & Zhu, Y., Modeling Low Reynolds Number Incompressible Flows Using SPH. Journal of Computational Physics, 136: 214-226, 1997.

Pletcher, R.H., Tanehill, J.C. & Anderson D.A., Computational Fluid Mechanics and Heat Transfer (CRC Press, 2013).


  • There are currently no refbacks.

Please send any question about this web site to
Copyright © 2005-2024 Praise Worthy Prize