In this review the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed. Emphasis is placed on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.

https://iopscience.iop.org/article/10.1088/0034-4885/68/8/R01/pdf

Smoothed particle hydrodynamics

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks.

Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments

Many-particle charged-particle plasma simulations using spatial meshes for the electromagnetic field solutions, particle-in-cell (PIC) merged with Monte Carlo collision (MCC) calculations, are coming into wide use for application to partially ionized gases. The author emphasizes the development of PIC computer experiments since the 1950s starting with one-dimensional (1-D) charged-sheet models, the addition of the mesh, and fast direct Poisson equation solvers for 2-D and 3-D. Details are provided for adding the collisions between the charged particles and neutral atoms. The result is many-particle simulations with many of the features met in low-temperature collision plasmas; for example, with applications to plasma-assisted materials processing, but also related to warmer plasmas at the edges of magnetized fusion plasmas. >

Particle-in-Cell Charged Particle Simulations, plus Monte Carlo Collisions with Neutral Atoms, PIC-MCC

https://users.monash.edu.au/~dprice/ndspmhd/price-spmhd.pdf

Smoothed particle hydrodynamics and magnetohydrodynamics

Smoothed particle hydrodynamics (SPH) has proven to be a useful numerical tool in studying a number of different astrophysical problems. Still, used on other problems, such as the modeling of low-β MHD systems, the method has so far not performed as well as one might have hoped. The present work has been motivated by the desire to accurately model strong hydrodynamic and magnetohydrodynamic shocks, and a key issue has therefore been to achieve a near-optimal representation of the simulated system at all times. Using SPH, this means combining the Lagrangian nature of the method with a smoothing-length profile that varies in both space and time. In this paper, a scheme containing two novel features is proposed. First, the scheme assumes a piecewise constant smoothing-length profile. To avoid substantial errors near the steps in the profile, alternative forms of the SPH equations of motion are used. Second, a predictive attitude toward optimizing the particle distribution is introduced by activating a mass, momentum, and energy conservation regularization process at intervals. The scheme described has been implemented in a new code called regularized smoothed particle hydrodynamics (RSPH), and test results for a number of standard hydrodynamic and magnetohydrodynamic tests in one and two dimensions using this code are presented.

https://iopscience.iop.org/article/10.1086/323228/pdf

Regularized Smoothed Particle Hydrodynamics: A New Approach to Simulating Magnetohydrodynamic Shocks

Low‐frequency electrostatic turbulence generated by the ion–ion beam instability was investigated experimentally in a double‐plasma device. Real time signals were recorded and examined by a conditional statistical analysis. Conditionally averaged potential distributions reveal the formation and propagation of structures with a relatively long lifetime. Various methods for making a conditional analysis are discussed and compared. The results are discussed with reference to ion phase space vortices and clump formation in collisionless plasmas.

Conditional Eddies in Plasma Turbulence

In the BLAAMANN device a weakly ionized hydrogen plasma is produced by electrons accelerated from a hot, negatively biased tungsten filament and confined in a toroidal magnetic field of strength up to 0.4 T. The plasma is turbulent, with relative fluctuation levels in ne, phi and Te of 10% or more. The time-averaged state exhibits nested toroidal surfaces of constant potential and pressure, which requires an anomalous cross-field current to remove the space-charge injected by the cathode and the charge accumulated due to the Del B- and curvature drifts. Typical plasma parameters are ne approximately 1016 m-3, Te approximately 1-20 eV, Ti approximately 1 eV. The cross-field diffusion coefficient is typically Dperpendicular to approximately 30 m2 s$ -1 approximately 104*Dperpendicular to classical approximately 101*Dperpendicular to Bohm. Evidence is presented in support of the hypothesis that the plasma goes turbulent because it needs to develop an anomalous current channel, and this turbulence in turn determines the plasma transport and the time-averaged state.

http://iopscience.iop.org/article/10.1088/0741-3335/36/7/002/pdf

Confinement and turbulent transport in a plasma torus with no rotational transform

The dynamics of jetstreams have been studied by following the evolution of a simulation particle population. The self-gravitational field of the stream has been neglected, the individual small bodies interacting through hard, partially inelastic collisions only, no fragmentation or accreaction taking place. The chosen collision model — the beta-model — is given by Equation (4). The collisions will quickly establish a Rayleigh distribution of both eccentricities and inclinations, the average amplitude of the oscillations associated with eccentricity being up by approximately a factor √2. The velocity distribution inside the stream is highly non-Maxwellian. If the collisions are sufficiently inelastic, the stream is focused, the individual orbits becoming more and more circular (for azimuthally symmetric streams) and ecliptic. In the opposite case the collisions destroy the stream configuration rapidly. The beta collisional model is unable to produce any radial focusing.

Numerical Simulation of Jetstreams. II: The Two Dimensional Case

Field and particle data recorded on the geostationary satellite GEOS 2 are used to investigate the electric and magnetic signatures of a substorm characterized by a dispersionless injection of energetic electrons and ions. Three types of field variations are observed: (1) Long-period oscillations with period of ∼ 300 s, interpreted as oscillations of entire field lines. These oscillations develop as second harmonic standing waves and correspond to coupled shear Alfven-slow magnetosonic modes. They grow after the most active period of the breakup. (2) Short-period transient oscillations with periods of ∼ 45–65 s, interpreted as wave modes trapped in a current layer which develops prior to the substorm breakup and is disrupted at breakup. These oscillations also correspond to a coupled shear Alfven-slow magnetosonic mode (coupled via magnetic field curvature effects in a high-β plasma). The short-period transient oscillations are only observed during the most active period of the breakup. (3) A nonoscillatory sharp increase observed on both the parallel magnetic component and the energetic ion flux, averaged over one satellite rotation, interpreted as evidence for the fast magnetosonic mode which in view of the simultaneous large impulsive increase in the azimuthal electric field, appears to propagate radially outwards, transporting the substorm breakup downtail.

Jan Trulsen

Papers

Regularized Smoothed Particle Hydrodynamics: A New Approach to Simulating Magnetohydrodynamic Shocks

Conditional Eddies in Plasma Turbulence

Confinement and turbulent transport in a plasma torus with no rotational transform

Numerical Simulation of Jetstreams. II: The Two Dimensional Case

Characterization of low frequency oscillations at substorm breakup