Abstract
Smoothed Particle Hydrodynamics is a meshless particle method able to evaluate unknown field functions and relative differential operators. This evaluation is done by performing an integral representation based on a suitable smoothing kernel function which, in the discrete formulation, involves a set of particles scattered in the problem domain. Two fundamental aspects strongly characterizing the development of the method are the smoothing kernel function and the particle distribution. Their choice could lead to the so-called particle inconsistency problem causing a loose of accuracy in the approximation; several corrective strategies can be adopted to overcome this problem. This paper focuses on the numerical behaviors of SPH with respect to the consistency restoring problem and to the particle distribution choice, providing useful hints on how these two aspects affect the goodness of the approximation and moreover how they mutually influence themselves. A series of numerical studies are performed approximating 1D, 2D and 3D functions validating this idea.
Similar content being viewed by others
References
Ala G., Francomano E., Tortorici A., Toscano E., Viola F.: A smoothed particle interpolation scheme for transient electromagnetic simulation. IEEE Trans. Magn. 42, 647–650 (2006)
Ala G., Francomano E., Tortorici A., Toscano E., Viola F.: Corrective meshless particle formulations for time domain Maxwell’s equations. J. Comput. Appl. Math. 210, 34–46 (2007)
Belytschko T., Krongauz Y., Dolbow J., Gerlach C.: On the completeness of meshfree methods. Int. J. Numer. Methods Eng. 43, 785–819 (1998)
Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P.: Meshless methods: an overview and recent developments. Comput. Meth. Appl. Mech. Eng. 139, 3–47 (1996)
Bonet J., Lok T.S.L.: Variational and momentum preservation aspects of smooth particle hydrodynamics formulations. Comput. Meth. Appl. Mech. Eng. 180, 97–115 (1999)
Bonet J., Kulasegaram S.: A simplified approach to enhance the performance of smooth particle hydrodynamics methods. Appl. Math. Comput. 126, 133–155 (2002)
Chen J.K., Beraun J.E., Carney T.C.: A corrective smoothed particle method for boundary value problems in heat conduction. Int. J. Numer. Methods Eng. 46, 231–252 (1999)
G. Di Blasi, E. Francomano, A. Tortorici, E. Toscano, On the Consistency Restoring in SPH. in Proceedings of the CMMSE (International Conference on Computational and Mathematical Methods in Science and Engineering) (2009)
Fulk D.A., Quinn D.W.: An analysis of 1-D smoothed particle hydrodynamics kernels. J. Comput. Phys. 126, 165–180 (1996)
Gingold R.A., Monaghan J.J.: Smoothed particle hydrodynamics: theory and application to nonspherical stars. Mon. Not. Roy. Astron. Soc. 181, 375–389 (1985)
Hernquist L., Katz N.: TreeSPH: a unification of SPH with the hierarchical tree method. Astrophys. J. Suppl. Ser. 70, 419–446 (1989)
Laguna P.: Smoothed particle interpolation. Astrophys. J. 439, 814–821 (1994)
Lastiwka M., Quinlan N., Basa M.: Adaptive particle distribution for smoothed particle hydrodynamics. Int. J. Numer. Methods Fluids 47, 1403–1409 (2005)
Liu G.R., Liu M.B.: Smoothed Particle Hydrodynamics—a Mesh-Free Particle Method. World Scientific Publishing, Singapore (2003)
Liu G.R.: Mesh Free Methods—Moving beyond the Finite Element Method. CRC Press, Boca Raton (2003)
Liu M.B., Liu G.R.: Restoring particle consistency in smoothed particle hydrodynamics. Appl. Numer. Math. 56, 19–36 (2006)
Liu M.B., Liu G.R., Lam K.Y.: Constructing smoothing functions in smoothed particle hydrodynamics with applications. J. Comput. Appl. Math. 155, 263–284 (2003)
Monaghan J.J.: An introduction to SPH. Comput. Phys. Commun. 48, 89–96 (1988)
Monaghan J.J.: Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30, 543–574 (1992)
Monaghan J.J., Lattanzio J.C.: A refined particle method for astrophysical problems. Astron. Astrophys. 149, 135–143 (1985)
J.P. Morris, Analysis of Smoothed Particle Hydrodynamics with Applications. Ph.D. Thesis, Monash University, (1996)
Shapiro P.R., Martel H., Villumsen J.V., Owen J.M.: Adaptive smoothed particle hydrodynamics, with application to cosmology: methodology. Astrophys. J. Suppl. Ser. 103, 269–330 (1996)
D. Shepard. A Two Dimensional Function for Irregularly Spaced Data. in Proceedings of the ACM National Conference, (1968)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Di Blasi, G., Francomano, E., Tortorici, A. et al. Exploiting numerical behaviors in SPH. J Math Chem 48, 128–136 (2010). https://doi.org/10.1007/s10910-009-9642-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-009-9642-1