Abstract
Particle methods are a robust and versatile computational tool for the simulation of continuous and discrete physical systems ranging from Fluid Mechanics to Biology and Social Sciences. In advection dominated problems particle methods can be considered as the method of choice due to their inherent robustness, stability and Lagrangian adaptivity. At the same time however, smooth particle methods encounter major difficulties in simulating the equations they set out to discretize when their computational elements fail to overlap, a condition necessary for their convergence [2]. A number of ad-hoc parameters and artificial dissipation techniques are often introduced in techniques such as Smoothed Particle Hydrodynamics (SPH) [15, 19] in order to remedy these difficulties.
In the present paper we demonstrate that the convergence of smooth particle methods can be ensured by a periodic remeshing of the particles using high-order interpolation kernels. This procedure retains the Lagrangian character and stability of particle methods and enables the control of their accuracy [5, 9, 16, 17] while introducing numerical dissipation at levels well below those introduced by temporal discretizations.
In addition, remeshing enables two major improvements over grid-free particle methods: First by exploiting the regularity of the remeshed particles, it reduces by at least an order of magnitude their computational cost [6, 10] and facilitates their massively parallel implementation. Second, remeshing enables the development of consistent multiresolution techniques such as wavelet-particle methods [4]. This approach has been implemented efficiently in massively parallel computer architectures allowing for unprecedented vortex dynamics simulations using billions of particles.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. T. Beale, On the accuracy of vortex methods at large times, in Proc. Workshop on Comput. Fluid Dyn. and React. Gas Flows, IMA, Univ. of Minnesota, 1986, B. E. et al., (ed)., Springer-Verlag, New York, 1988, p. 19.
J. T. Beale and A. Majda, Vortex methods I: convergence in 3 dimensions, Mathematics of Computation, 39 (1982) pp. 1–27.
M. Bergdorf, G.-H. Cottet, and P. Koumoutsakos, Multilevel adaptive particle methods for convection-diffusion equations, Multiscale Model. Simul., 4 (2005), pp. 328–357.
M. Bergdorf and P. Koumoutsakos, A Lagrangian Particle-Wavelet method, Multiscale Modeling & Simulation: A SIAM Interdisciplinary Journal, 5 (2006), pp. 980–995.
A. Chaniotis, D. Poulikakos, and P. Koumoutsakos, Remeshed smoothed particle hydrodynamics for the simulation of viscous and heat conducting flows J. Comput. Phys., 182 (2002), pp. 67–90.
P. Chatelain, G.-H. Cottet, and P. Koumoutsakos, PMH: Particle Mesh Hydrodynamics, International Journal of Modern Physics C, 18 (2007), pp. 610–618.
P. Chatelain, A. Curioni, M. Bergdorf, D. Rossinelli, W. Andreoni, and P. Koumoutsakos, Billion vortex particle direct numerical simulations of aircraft wakes, Computer Methods in Applied Mechanics and Engineering, 197 (2008), pp. 1296–1304.
R. Cocle, L. Dufresne, and G. Winckelmans, Investigation of multiscale subgrid models for les of instabilities and turbulence in wake vortex systems, Lecture Notes in Computational Science and Engineering, 56 (2007), pp. 141–159.
G.-H. Cottet and P. Koumoutsakos, Vortex Methods, Theory and Practice, Cambridge University Press, 2000.
G.-H. Cottet and L. Weynans, Particle methods revisited: a class of high-order finite-difference schemes, C. R. Acad. Sci. Paris, Sér. I, 343 (2006), pp. 51–56.
P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations, part 1: The case of an isotropic viscosity, Mathematics of Computation, 53 (1989), pp. 485–507.
D. A. Durston, S. M. Walker, D. M. Driver, S. C. Smith, and Ö. Savas, Wake vortex alleviation flow field studies, J. Aircraft, 42 (2005), pp. 894–907.
J. D. Eldredge, T. Colonius, and A. Leonard A vortex particle method for two dimensional compressible flow, J. Comp. Phys., 179 (2002), pp. 371–399.
F. Gibou, R. Fedkiw, R. Caflisch, and S. Osher A level set approach for the numerical simulation of dendritic growth, SIAM J. Sci. Comput., 19 (2003), pp. 183–199.
R. Gingold and J. Monaghan, Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. Roy. Astron. Soc., 181 (1977) p. 375.
P. Koumoutsakos, Inviscid axisymmetrization of an elliptical vortex, J. Comput. Phys., 138 (1997), pp. 821–857.
-P. Koumoutsakos, Multiscale flow simulations using particles, Annu. Rev. Fluid Mech., 37 (2005), pp. 457–487.
J. Liandrat and P. Tchamitchian, Resolution of the 1D regularized burgers equation using a spatial wavelet approximation, ICASE Report 90-83 NASA Langley Research Center, 1990.
J. J. Monaghan, Smoothed particle hydrodynamics, Reports on Progress in Physics, 68 (2005), pp. 1703–1759.
J. M. Ortega and Ö. Savas, Rapidly growing instability mode in trailing multiple-vortex wakes, AIAA Journal, 39 (2001), pp. 750–754.
I. F. Sbalzarini, J. H. Walther, M. Bergdorf, S. E. Hieber, E. M. Kotsalis, and P. Koumoutsakos, PPM a highly efficient parallel particle mesh library for the simulation of continuum systems, J. Comput. Phys., 215 (2006), pp. 566–588.
R. Stuff, The near-far relationship of vortices shed from transport aircraft, in AIAA Applied Aerodynamics Conference, 19th, Anaheim, CA, AIAA, ed., AIAA, 2001, pp. AIAA-2001–AIAA-2429.
E. Stumpf, Study of four-vortex aircraft wakes and layout of corresponding aircraft configurations, J. Aircraft, 42 (2005), pp. 722–730.
G. Winckelmans, Vortex methods, in Encyclopedia of Computational Mechanics, E. Stein, R. De Borst, and T. J. Hughes, eds., vol. 3, John Wiley and Sons, 2004.
G. Winckelmans, R. Cocle, L. Dufresne, and R. Capart, Vortex methods and their application to trailing wake vortex simulations, C. R. Phys., 6 (2005), pp. 467–486.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chatelain, P., Bergdorf, M., Koumoutsakos, P. (2008). Large Scale, Multiresolution Flow Simulations Using Remeshed Particle Methods. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations IV. Lecture Notes in Computational Science and Engineering, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79994-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-79994-8_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79993-1
Online ISBN: 978-3-540-79994-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)