Smoothed-particle hydrodynamics

Second, SPH computes pressure from weighted contributions of neighboring particles rather than by solving linear systems of equations.

Recent work in SPH for fluid simulation has increased performance, accuracy, and areas of application: Smoothed-particle hydrodynamics's adaptive resolution, numerical conservation of physically conserved quantities, and ability to simulate phenomena covering many orders of magnitude make it ideal for computations in theoretical astrophysics.

Incorporating other astrophysical processes which may be important, such as radiative transfer and magnetic fields is an active area of research in the astronomical community, and has had some limited success.

The main advantage of SPH in this application is the possibility of dealing with larger local distortion than grid-based methods.

This feature has been exploited in many applications in Solid Mechanics: metal forming, impact, crack growth, fracture, fragmentation, etc.

In particular, mesh alignment is related to problems involving cracks and it is avoided in SPH due to the isotropic support of the kernel functions.

[30] Over the past years, different corrections have been introduced to improve the accuracy of the SPH solution, leading to the RKPM by Liu et al.[31] Randles and Libersky[32] and Johnson and Beissel[33] tried to solve the consistency problem in their study of impact phenomena.

Dyka et al.[34][35] and Randles and Libersky[36] introduced the stress-point integration into SPH and Ted Belytschko et al.[37] showed that the stress-point technique removes the instability due to spurious singular modes, while tensile instabilities can be avoided by using a Lagrangian kernel.

Recent improvements in understanding the convergence and stability of SPH have allowed for more widespread applications in Solid Mechanics.

as: These particles interact through a kernel function with characteristic radius known as the "smoothing length", typically represented in equations by

By assigning each particle its own smoothing length and allowing it to vary with time, the resolution of a simulation can be made to automatically adapt itself depending on local conditions.

For example, in a very dense region where many particles are close together, the smoothing length can be made relatively short, yielding high spatial resolution.

The above SPH governing equations can be derived from a least action principle, starting from the Lagrangian of a particle system: where

The most straightforward boundary model is neglecting the integral, such that just the bulk interactions are taken into account, This is a popular approach when free-surface is considered in monophase simulations.

Such technique is based on populating the compact support across the boundary with so-called ghost particles, conveniently imposing their field values.

[50] Along this line, the integral neglect methodology can be considered as a particular case of fluid extensions, where the field, A, vanish outside the computational domain.

The main benefit of this methodology is the simplicity, provided that the boundary contribution is computed as part of the bulk interactions.

[51][50][52] On the other hand, deploying ghost particles in the truncated domain is not a trivial task, such that modelling complex boundary shapes becomes cumbersome.

To overcome undesired errors at the free surface through kernel truncation, the density formulation can again be integrated in time.

Hence, pressure information travels fast compared to the actual bulk flow, which leads to very small Mach numbers

[57] This phenomenon is caused by the nonlinear interaction of acoustic waves and by fact that the scheme is explicit in time and centered in space .

In long time simulations, the use of the filtering procedure may lead to the disruption of the hydrostatic pressure component and to an inconsistency between the global volume of fluid and the density field.

[60] In Antuono et al. [61] a correction to the diffusive term of Molteni[56] was proposed to remove some inconsistencies close to the free-surface.

In this case the adopted diffusive term is equivalent to a high-order differential operator on the density field.

[62] The scheme is called δ-SPH and preserves all the conservation properties of the SPH without diffusion (e.g., linear and angular momenta, total energy, see [63] ) along with a smooth and regular representation of the density and pressure fields.

In the third group there are those SPH schemes which employ numerical fluxes obtained through Riemann solvers to model the particle interactions.

By comparing both it can be seen that the intermediate velocity and pressure from the inter-particle averages amount to implicit dissipation, i.e. density regularization and numerical viscosity, respectively.

In general, the description of hydrodynamic flows require a convenient treatment of diffusive processes to model the viscosity in the Navier–Stokes equations.

Introduced by Monaghan and Gingold [68] the artificial viscosity was used to deal with high Mach number fluid flows.

are given by and The anti-symmetric property of the derivative of the kernel function will ensure the momentum conservation for each pair of interacting particles