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From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence
Author(s) -
Evers Joep H.M.,
Zisis Iason A.,
Linden Bas J.,
Duong Manh Hong
Publication year - 2018
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201600077
Subject(s) - mathematics , regularization (linguistics) , convergence (economics) , action (physics) , smoothed particle hydrodynamics , classical mechanics , equations of motion , conservative force , mathematical analysis , continuum mechanics , context (archaeology) , particle system , physics , mechanics , computer science , paleontology , quantum mechanics , artificial intelligence , economics , biology , economic growth , operating system
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.