Numerical Algorithms

, Volume 68, Issue 1, pp 15–34 | Cite as

Unified semi-analytical wall boundary conditions in SPH: analytical extension to 3-D

  • Arno Mayrhofer
  • Martin Ferrand
  • Christophe Kassiotis
  • Damien Violeau
  • François-Xavier Morel
Original Paper


Solid wall boundary conditions have been an area of active research within the context of Smoothed Particle Hydrodynamics (SPH) for quite a while. Ferrand et al. (Int. J. Numer. Methods Fluids 71(4), 446–472, 2012) presented a novel approach using a renormalization factor in the SPH approximation. The computation of this factor depends on an integral along the boundary of the domain and in their original paper Ferrand et al. gave an analytical formulation for the 2-D case using the Wendland kernel. In this paper the formulation will be extended to 3-D, again providing analytical formulae. Due to the boundary being two dimensional a domain decomposition algorithm needs to be employed in order to obtain special integration domains. For these the analytical formulae will be presented when using the Wendland kernel. The algorithm presented within this paper is applied to several academic test-cases for which either analytical results or simulations with other methods are available. It will be shown that the present formulation produces accurate results and provides a significant improvement compared to approximative methods.


Fluid mechanics Smoothed particle hydrodynamics Unified semi-analytical Wall boundary conditions 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Arno Mayrhofer
    • 1
  • Martin Ferrand
    • 2
  • Christophe Kassiotis
    • 3
    • 4
  • Damien Violeau
    • 2
    • 4
  • François-Xavier Morel
    • 5
  1. 1.School of MACEUniversity of ManchesterManchesterUK
  2. 2.EDF R&DParisFrance
  3. 3.Ecole des Ponts ParisTechChamps-sur-MarneFrance
  4. 4.Saint-Venant Laboratory for HydraulicsUniversité Paris-EstParisFrance
  5. 5.Sequans CommunicationsParisFrance

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