From continuum mechanics to SPH particle systems and back: systematic derivation and convergence

Joep H.M. Evers; Iason A. Zisis; Bas J. van der Linden; Manh Hong Duong

doi:10.1002/zamm.201600077

From continuum mechanics to SPH particle systems and back: systematic derivation and convergence

Joep H.M. Evers, Iason A. Zisis, Bas J. van der Linden, Manh Hong Duong

Center for Analysis, Scientific Computing and Applications

Research output: Contribution to journal › Article › Academic › peer-review

12 Citations (Scopus)

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Abstract

In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method [27], and with the equation treated by Di Lisio et al. in [9], respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.

Original language	English
Pages (from-to)	106-133
Number of pages	28
Journal	Zeitschrift für Angewandte Mathematik und Mechanik
Volume	98
Issue number	1
DOIs	https://doi.org/10.1002/zamm.201600077
Publication status	Published - 1 Jan 2018

Funding

Acknowledgements We thank Adrian Muntean, Mark Peletier, and Fons van de Ven (TU Eindhoven, The Netherlands) for fruitful discussions and useful comments. J.H.M. Evers acknowledges the support of the Netherlands Organisation for Scientific Research (NWO), Graduate Programme 2010, and of an AARMS Postdoctoral Fellowship. For I.A. Zisis, this research was carried out under project number M11.4.10412 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl). For B.J. van der Linden it is a research activity of the Laboratory of Industrial Mathematics in Eindhoven LIME bv (www.limebv.nl). M. H. Duong was supported by ERC Starting Grant 335120.

Keywords

convergence rate
measure-valued equations
principle of least action
Smoothed Particle Hydrodynamics
Wasserstein distance

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10.1002/zamm.201600077

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abstract = "In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method [27], and with the equation treated by Di Lisio et al. in [9], respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.",

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AU - Evers, Joep H.M.

AU - Zisis, Iason A.

AU - van der Linden, Bas J.

AU - Duong, Manh Hong

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From continuum mechanics to SPH particle systems and back: systematic derivation and convergence

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