Continuum Mechanics and Thermodynamics

, Volume 31, Issue 2, pp 475–489 | Cite as

On the boundary conditions in Lagrangian particle methods and the physical foundations of continuum mechanics

  • Carlos Alberto Dutra Fraga FilhoEmail author
Original Article


This paper aims to discuss the boundary conditions techniques employed in the solution of physical and engineering problems using the Lagrangian particle approach in continuum mechanics. Simulations using different boundary treatment techniques have been performed, and in addition, a review of the literature on the techniques commonly employed in SPH simulations was also performed. The fictitious (virtual/ghost), the dynamic particles and the artificial repulsive forces are widely employed in problem-solving, mixing concepts of molecular and continuum scales, being inadequate for the use in continuum scale (considering the disrespect of the classical physics laws). On the other hand, the reflective boundary conditions, based on fundamentals of Physics (Newton’s restitution law) and analytic geometry, are in accordance with the continuum hypothesis and their use is recommended in continuum mechanics problems, out of the molecular scale. In many boundary conditions employed in particle methods, models of repulsive molecular forces in conjunction with virtual particles are being used without theoretical foundation in continuum mechanics and classical Physics. Purely computational solutions employing fictitious particles and artificial molecular forces should be avoided, and methods that respect the continuum theory, such as the reflective boundary conditions, should be employed on the macroscopic scale. Both hydrostatic and hydrodynamics analysis have been performed. An excellent agreement with the analytical solution or experimental results have been achieved. From the results found, the applicability of the reflexive boundary technique in the continuum domain, discretized by particles, was verified.


Continuum mechanics Particle method SPH Virtual particles Reflective boundary conditions Dynamic boundary conditions 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Development, Implementation and Application of Computational Tools for Problem Solving in Engineering Research GroupFederal Institute of Education, Science and Technology of Espírito Santo, Mechanical Engineering CoordinationJucutuquara, VitóriaBrazil

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