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

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Evers, Joep H. M., Zisis, Iason A., Linden, Bas J. van der and Duong, Manh Hong (2018) From continuum mechanics to SPH particle systems and back : systematic derivation and convergence. ZAMM - Journal of Applied Mathematics and Mechanics, 98 (1). pp. 106-133.

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Official URL: https://doi.org/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 (e.g. Monaghan), and with the equation treated by Di Lisio et al., 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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Faculty of Science, Engineering and Medicine > Science > Physics
Journal or Publication Title: ZAMM - Journal of Applied Mathematics and Mechanics
Publisher: Wiley
ISSN: 1521-4001
Official Date: January 2018
Dates:
DateEvent
January 2018Published
14 August 2017Available
26 June 2017Accepted
Volume: 98
Number: 1
Page Range: pp. 106-133
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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