J. J. Monaghan

Smoothed Particle Hydrodynamics

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.

An introduction to SPH

Abstract This paper gives the derivation of the equations for SPH (smoothed particle hydrodynamics) and describes their application to a wide variety of problems in compressible gas flow.

Shock simulation by the particle method SPH

Abstract The particle method SPH is applied to one-dimensional shock tube problems by incorporating an artificial viscosity into the equations of motion. When the artificial viscosity is either a bulk viscosity or the Von Neumann-Richtmyer viscosity, in a form analogous to that for finite differences, the results show either excessive oscillation or excessive smearing of the shock front. The reason for the excessive particle oscillation is that, in the standard form, the artificial viscosity cannot dampen irregular motion on the scale of the particle separation since that scale is usually less than the resolution of the interpolating kernel. We propose a new form of artificial viscosity which eliminates this problem. The resulting shock simulation has negligible oscillation and satisfactorily sharp discontinuities. Results with a gaussian interpolating kernel (with second-order errors) are shown to be greatly inferior to those with a super gaussian kernel (with fourth-order errors).

On the problem of penetration in particle methods

A method is described which prevents penetration when particle methods are used to simulate streams of fluid impinging on each other. The method does not produce dissipation but it does produce extra dispersion.

SPH elastic dynamics

The standard smoothed particle hydrodynamics (SPH) formulation of fluid dynamics can exhibit an instability called the tensile instability. This instability may occur with both positive and negative pressure. Usually the effects are small, but in the case of elastic or brittle solids the effects may be severe. Under tension, a brittle solid can fracture, but it is difficult to disentangle the physical fracture and fragmentation from the nonphysical clumping of SPH particles due to the tensile instability. Recently, one of us (JJM) has shown how this instability can be removed by an artificial stress which introduces negligible errors in long-wavelength modes. In this paper we show how the algorithm can be improved by basing the artificial stress on the signs of the principal stresses. We determine the parameters of the artificial stress from the dispersion relation for elastic waves in a uniform material. We apply the algorithm to oscillating beams, colliding rings and brittle solids. The results are in very good agreement with theory, and with other high-accuracy methods.

Why Particle Methods Work

The theme of this paper is that particle methods are closely related to both finite difference and spectral methods because the three methods can be considered special cases of interpolation by kernel estimation. The kernels for a number of special cases are given in detail, and the accuracy of the resulting interpolation is analyzed. A general procedure for deriving equations for numerical work from the equations of hydrodynamics is described. It is applied to the derivation of the SPH equations which conserve linear and angular momentum exactly.

Kernel estimates as a basis for general particle methods in hydrodynamics

The general theory of kernel estimation is discussed and applied to particle methods in hydrodynamics. We show that a simple form of estimation leads to a particle method which does not require a grid and satisfies the conservation laws very accurately. The merits of different kernels are examined, and numerical tests of their ability to reproduce known densities are described. Examples of the application of the new particle method to isothermal shocks, to the collapse of gas clouds, and to the tidal interaction of stars are described.

SPH simulation of multi-phase flow

Abstract This paper shows how to formulate the two phase flow of a dusty gas using SPH. The formulation is very general and can be easily extended to deal with gas, solid and liquid phases in each of which there may be several species.

An energy-conserving formalism for adaptive gravitational force softening in smoothed particle hydrodynamics and N-body codes

In this paper, we describe an adaptive softening length formalism for collisionless N-body and self-gravitating smoothed particle hydrodynamics (SPH) calculations which conserves momentum and energy exactly. This means that spatially variable softening lengths can be used in N-body calculations without secular increases in energy. The formalism requires the calculation of a small additional term to the gravitational force related to the gradient of the softening length. The extra term is similar in form to the usual SPH pressure force (although opposite in direction) and is therefore straightforward to implement in any SPH code at almost no extra cost. For N-body codes, some additional cost is involved as the formalism requires the computation of the density through a summation over neighbouring particles using the smoothing kernel. The results of numerical tests demonstrate that, for homogeneous mass distributions, the use of adaptive softening lengths gives a softening which is always close to the ‘optimal’ choice of fixed softening parameter, removing the need for fine-tuning. For a heterogeneous mass distribution (as may be found in any large-scale N-body simulation), we find that the errors on the least-dense component are lowered by an order of magnitude compared to the use of a fixed softening length tuned to the densest component. For SPH codes, our method presents a natural and an elegant choice of softening formalism which makes a small improvement to both the force resolution and the total energy conservation at almost zero additional cost.

SPH particle boundary forces for arbitrary boundaries

This paper is concerned with approximating arbitrarily shaped boundaries in SPH simulations. We model the boundaries by means of boundary particles which exert forces on a fluid. We show that, when these forces are chosen correctly, and the boundary particle spacing is a factor of 2 (or more) less than the fluid particle spacing, the total boundary force on a fluid SPH particle is perpendicular to boundaries with negligible error. Furthermore, the variation in the force as a fluid particle moves, while keeping a fixed distance from the boundary, is also negligible. The method works equally well for convex or concave boundaries. The new boundary forces simplify SPH algorithms and are superior to other methods for simulating complicated boundaries. We apply the new method to (a) the rise of a cylinder contained in a curved basin, (b) the spin down of a fluid in a cylinder, and (c) the oscillation of a cylinder inside a larger fixed cylinder. The results of the simulations are in good agreement with those obtained using other methods, but with the advantage that they are very simple to implement.