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

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.

The Mathematics of Smoothed Particle Hydrodynamics (SPH) Consistency

“…Therefore, the joint limit N → ∞, h → 0, and n → ∞, with n/N → 0, is a necessary requirement to achieve consistency and self-consistency [13,28]. This complies with the suggestion advanced by Evers et al [50] that h → 0 as N → ∞ if the sum interpolant is taken over all particles of the system, as in Eq. 56.…”

“…Convergence of measure-valued solutions was first studied by Di Lisio et al [33], who proved the convergence of the SPH method with the use of measures in combination with the Wasserstein distance. The measure-valued formulation of Evers et al [50] not only generalizes Di Lisio et al [33] work in that the SPH-particle approach is just a special case because it can be applied to a class of approximating measures that is much broader than just sums of Dirac delta distributions. An important concluding remark from this study is that a favorable approach is to have h depending on the total number of particles N in such a way that h − O(N −1/n ) (where n denotes dimension) as N → ∞, and hence the joint limit N → ∞ and h → 0 must hold simultaneously.…”

“…Litvinov et al [11] found that standard SPH will achieve better error properties and improve on the partition of unity by relaxing a particle distribution under a constant pressure field and invariant particle volume. The use of the principle of least action to derive the equations of motion with a regularized density field in the context of measures has been illustrated by Evers et al [50]. Depending on the order in which regularization and the principle of least action are applied two different equations can be derived, whose discrete counterparts coincide with the traditional scheme used in SPH.…”

Review of smoothed particle hydrodynamics: towards converged Lagrangian flow modelling

“…High-order and highly accurate smoothed particle hydrodynamics While theoretical general proofs of convergence are being established (e.g. [25,26]), the practical convergence properties of particle methods and SPH started to receive attention in the mid-1990s with seminal works by Belytschko et al [27], Liu et al [28,29], Dilts [30,31], Bonet & Lok [32], Welton [33], Vila [34] and others looking at the role of the truncation error and the use of correction techniques to ensure polynomial consistency up to nth order and convergence. While these approaches certainly provided an option to correct for the SPH error and reproduce functions consistently, in practice they have been rarely used because SPH was viewed wholly as a Lagrangian method, with construction and solution of the correction matrices a costly operation at each time step.…”

Grand challenges for Smoothed Particle Hydrodynamics numerical schemes

“…Cartesian) arrays of particles, SPH can be shown to converge in numerical experiments with rates of convergence matching theoretical error measures extremely well. Evers et al [32] derived the rate of convergence of SPH numerical scheme using the least action principle. Franz & Wendland have recently provided a mathematical proof of convergence of SPH for a specific barotropic fluid and under certain properties of the underlying kernel [39].…”