Advertisement

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

Abstract

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.

Keywords

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Crespo, A.J.C., Gómez-Gesteira, M., Dalrymple, R.A.: Boundary conditions generated by dynamic particles in SPH methods. CMC-Comput. Mater. Contin. 5, 173–184 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Gomez-Gesteira, M., Rogers, B.D., Crespo, A.J.C., Dalrymple, R.A., Narayanaswamy, M.: User Guide for SPHysics Code. https://wiki.manchester.ac.uk/sphysics/index.php?title=Special:UserLogin&returnto=SPHYSICS+2D+Download+v2.2 (2010). Accessed 27 April 2018
  3. 3.
    Coveney, P.V., Boon, J.P., Succi, S.: Bridging the gaps at the physics–chemistry–biology interface. Philos. Trans. R. Soc. A 3, 1–2 (2016).  https://doi.org/10.1098/rsta.2016.0335 Google Scholar
  4. 4.
    Delgado-Buscalioni, R., Coveney, P.V., Riley, G.D., Ford, R.W.: Hybrid molecular-continuum fluid models: implementation within a general coupling framework. Philos. Trans. R. Soc. A 363(1833), 1975–85 (2005).  https://doi.org/10.1098/rsta.2005.1623 ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Borgh, M.K., Lockerby, D.A., Reese, J.M.: Fluid simulations with atomistic resolution: a hybrid multiscale method with field-wise coupling. J. Comput. Phys. 255, 49–165 (2013).  https://doi.org/10.1016/j.jcp.2013.08.022 MathSciNetGoogle Scholar
  6. 6.
    Mukhopadhyay, S., Abraham, J.: A particle-based multiscale model for submicron fluid flows. Phys. Fluids 21, 027102 (2009).  https://doi.org/10.1063/1.3073041 ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Ren, W., Weinan, E.: Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys. 204, 1–26 (2005).  https://doi.org/10.1016/j.jcp.2004.10.001 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Stalter, S., Yelash, L., Emamy, N., Statt, A., Hanke, M., Lukáčová-Medvid’ová, M., Virnau, P.: Molecular dynamics simulations in hybrid particle-continuum schemes: pitfalls and caveats. Comput. Phys. Commun. 224, 198–208 (2018).  https://doi.org/10.1016/j.cpc.2017.10.016 ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Yamaguchi, T., Ishikawa, T., Imai, Y., Matsuki, N., Xenos, M., Deng, Y., Bluestein, D.: Particle-based methods for multiscale modeling of blood flow in the circulation and in devices: challenges and future directions. Ann. Biomed. Eng. 38(3), 1225–35 (2010).  https://doi.org/10.1007/s10439-010-9904-x CrossRefGoogle Scholar
  10. 10.
    Sih, G.C. (ed.): Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano: Application to Biology, Physics, Material Science, Mechanics, Structural and Processing Engineering. Springer, Dordrecht (2007).  https://doi.org/10.1007/978-1-4020-5062-6 Google Scholar
  11. 11.
    Petsev, N.D., Leal, L.G., Shell, M.S.: Multiscale simulation of ideal mixtures using smoothed dissipative particle dynamics. J. Chem. Phys. 144, 084115 (2016).  https://doi.org/10.1063/1.4942499 ADSCrossRefGoogle Scholar
  12. 12.
    Moeendarbary, E., NG, T.Y., Zangeneh, M.: Dissipative particle dynamics: introduction, methodology and complex fluid applications—a review. Int. J. Appl. Mech. 1(4), 737–763 (2009).  https://doi.org/10.1142/S1758825109000381 CrossRefGoogle Scholar
  13. 13.
    Liu, M.B., Liu, G.R., Zhou, L.W., Chang, J.Z.: Dissipative particle dynamics (DPD): an overview and recent developments. Arch. Comput. Methods Eng. 22(4), 529–556 (2015).  https://doi.org/10.1007/s11831-014-9124-x MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fraga Filho, C.A.D.: An algorithmic implementation of physical reflective boundary conditions in particle methods: collision detection and response. Phys. Fluids (2017).  https://doi.org/10.1063/1.4997054 Google Scholar
  15. 15.
    Eslamian, A., Khayat, M.: Numerical studies to propose a ghost particle removed SPH (GR-SPH) method. Appl. Math. Model. 42, 71–99 (2017).  https://doi.org/10.1016/j.apm.2016.09.026 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Liu, G.R., Liu, M.B.: Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Valizadeh, A., Shafieefar, M., Monaghan, J.J., Salehi Neyshabouri, S.A.: Modeling two-phase flows using SPH method. J. Appl. Sci. 8(21), 3816–3826 (2008)Google Scholar
  18. 18.
    Fraga Filho, C.A.D., Chacaltana, J.T.A., Pinto, W.J.N.: Meshless Lagrangian SPH method applied to isothermal lid-driven cavity flow at low-Re numbers. Comput. Part. Mech. (2018).  https://doi.org/10.1007/s40571-018-0183-x Google Scholar
  19. 19.
    Fraga Filho, C.A.D., Pezzin D.F., Chacaltana J.T.A.: A numerical study of heat diffusion using the Lagrangian particle SPH method and the Eulerian finite-volume method: analysis of convergence, consistency and computational cost. In: Sund’en B., Brebbia C.A. (eds.) Proceedings of the Thirteenth International Conference on Simulation and Experiments in Heat and Mass Transfer (Heat Transfer XIII), WIT Press, Southampton, UK, pp. 15–26 (2014)Google Scholar
  20. 20.
    Lee, E.-S., Moulinec, C., Xu, R., Violeau, D., Laurence, D., Stansby, P.: Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. J. Comput. Phys. 227(18), 8417–8436 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gomez-Gesteira, M., Rogers, B.D., Crespo, A.J.C., Dalrymple, R.A., Narayanaswamy, M., Dominguez, J.M.: SPHysics—development of a free surface fluid solver—part 1: theory and formulations. Comput. Geosci. 48, 289–299 (2012).  https://doi.org/10.1016/j.cageo.2012.02.029 ADSCrossRefGoogle Scholar
  22. 22.
    Dirac, P.M.A.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, New York (1958)zbMATHGoogle Scholar
  23. 23.
    Vorobyev, A.: A Smoothed Particle Hydrodynamics Method for the Simulation of Centralized Sloshing Experiments. KIT Scientific Publishing, Germany (2013)Google Scholar
  24. 24.
    Korzilius, S.P., Kruisbrink, A.C.H., Schilders, W.H.A.: Momentum conserving methods that reduce particle clustering in SPH. CASA-report 14-15, Eindhoven University of Technology, The Netherlands (2014)Google Scholar
  25. 25.
    Goffin, L., Erpicum, S., Dewals, B.J., Pirotton, M., Archambeau, P.: Validation of a SPH model for free surface flows. In: Proceedings of the 3rd SimHYDRO Conference, pp. 11–13, Nice, France (2014)Google Scholar
  26. 26.
    Fourtakas, G., Vacondio, R., Rogers, B.D.: On the approximate zeroth and first-order consistency in the presence of 2-D irregular boundaries in SPH obtained by the virtual boundary particle methods. Int. J. Numer. Methods Fluids 78, 475–501 (2015).  https://doi.org/10.1002/fld.4026 ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Fraga Filho, C.A.D., Chacaltana, J.T.A.: Study of fluid flows using smoothed particle hydrodynamics: the modified pressure concept applied to quiescent fluid and dam breaking. In: Proceedings of the XXXVI Iberian Latin-American Congress on Computational Methods in Engineering, Rio de Janeiro, Brazil (2015). https://ssl4799.websiteseguro.com/swge5/PROCEEDINGS/PDF/CILAMCE2015-0071.pdf. Accessed 26 Dec 2017
  28. 28.
    Kleefsman, K.M.T., Fekken, G., Veldman, A.E.P., Iwanowski, B., Buchner, B.: Volume-of-fluid based simulation method for wave impact problems. J. Comput. Phys. 206, 363–393 (2005).  https://doi.org/10.1016/j.jcp.2004.12.007 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Newman, J.N.: Marine Hydrodynamics. MIT Press, Boston (1977)CrossRefGoogle Scholar
  30. 30.
    Fraga Filho, C.A.D.: Study of gravity-inertial phase of spreading of oil on a calm sea employing the Lagrangian particle method smoothed particle hydrodynamics, Ph.D. Thesis, Federal University of Espírito Santo, Brazil (2014). http://cfd.mace.manchester.ac.uk/sph/SPH_PhDs/2014/Carlos_Alberto_DUTRA_FRAGA_FILHO_PhD_Thesis_2014.pdf. Accessed 27 April 2018
  31. 31.
    Cruchaga, M.A., Celentano, D.J., Tezduyar, T.E.: Collapse of a liquid column: numerical simulation and experimental validation. Comput. Mech. 39, 453–476 (2007).  https://doi.org/10.1007/s00466-006-0043-z CrossRefzbMATHGoogle Scholar
  32. 32.
    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).  https://doi.org/10.1002/(SICI)1097-0207(19990920)46:2%3c231::AID-NME672%3e3.0.CO;2-K CrossRefzbMATHGoogle Scholar
  33. 33.
    Liu, M.B., Liu, G.R.: Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch. Comput. Methods Eng. 17, 25–76 (2010).  https://doi.org/10.1007/s11831-010-9040-7 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Vaughan, G.L., Healy, T.R., Bryan, K.R., Sneyd, A.D., Gorman, R.M.: Completeness, conservation and error in SPH for fluids. Int. J. Numer. Methods Fluids 56, 37–62 (2008).  https://doi.org/10.1002/fld.1530 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Quinlan, N.J., Basa, M., Lastiwka, M.: Truncation error in mesh-free particle methods. Int. J. Num. Methods Eng. 66, 2064–2085 (2006).  https://doi.org/10.1002/nme.1617 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kulasegaram, S., Bonet, J., Lewis, R.W., Profit, M.: A variational formulation based contact algorithm for rigid boundaries in two-dimensional SPH applications. Comput. Mech. 33, 316–325 (2004).  https://doi.org/10.1007/s00466-003-0534-0 CrossRefzbMATHGoogle Scholar
  37. 37.
    Ferrand, M., Laurence, D.R., Rogers, B.D., Violeau, D., Kassiotis, C.: Unified semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method. Int. J. Numer. Meth. Fluids 71(4), 446–472 (2012).  https://doi.org/10.1002/fld.3666 MathSciNetCrossRefGoogle Scholar
  38. 38.
    Leroy, A., Violeau, D., Ferrand, M., Kassiotis, C.: Unified semi-analytical wall boundary conditions applied to 2-D incompressible SPH. J. Comput. Phys. 261, 106–129 (2014).  https://doi.org/10.1016/j.jcp.2013.12.035 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Mayrhofer, A., Ferrand, M., Kassiotis, C., Violeau, D., Morel, F.: Unified semianalytical wall boundary conditions in SPH: analytical extension to 3-D. Numer. Algor. 68, 15–34 (2015).  https://doi.org/10.1007/s11075-014-9835-y CrossRefzbMATHGoogle Scholar

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

Personalised recommendations