Smoothed-particle hydrodynamics

Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method (where the co-ordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as density.

Method

Advantages By construction, SPH is a meshfree method, which makes it ideally suited to simulate problems dominated by complex boundary dynamics, like free surface flows, or large boundary displacement. The lack of a mesh significantly simplifies the model implementation and its parallelization, even for many-core architectures. SPH can be easily extended to a wide variety of fields, and hybridized with some other models, as discussed in Modelling Physics. As discussed in section on weakly compressible SPH, the method has great conservation features. The computational cost of SPH simulations per number of particles is significantly less than the cost of grid-based simulations per number of cells when the metric of interest is related to fluid density (e.g., the probability density function of density fluctuations). This is the case because in SPH the resolution is put where the matter is.

Limitations Setting boundary conditions in SPH such as inlets and outlets and walls is more difficult than with grid-based methods. In fact, it has been stated that "the treatment of boundary conditions is certainly one of the most difficult technical points of the SPH method". This challenge is partly because in SPH the particles near the boundary change with time. Nonetheless, wall boundary conditions for SPH are available. The computational cost of SPH simulations per number of particles is significantly larger than the cost of grid-based simulations per number of cells when the metric of interest is not (directly) related to density (e.g., the kinetic-energy spectrum). Therefore, overlooking issues of parallel speedup, the simulation of constant-density flows (e.g., external aerodynamics) is more efficient with grid-based methods than with SPH.

Examples

Fluid dynamics

Smoothed-particle hydrodynamics is being increasingly used to model fluid motion as well. This is due to several benefits over traditional grid-based techniques. First, SPH guarantees conservation of mass without extra computation since the particles themselves represent mass. Second, SPH computes pressure from weighted contributions of neighboring particles rather than by solving linear systems of equations. Finally, unlike grid-based techniques, which must track fluid boundaries, SPH creates a free surface for two-phase interacting fluids directly since the particles represent the denser fluid (usually water) and empty space represents the lighter fluid (usually air). For these reasons, it is possible to simulate fluid motion using SPH in real time. However, both grid-based and SPH techniques still require the generation of renderable free surface geometry using a polygonization technique such as metaballs and marching cubes, point splatting, or 'carpet' visualization. For gas dynamics it is more appropriate to use the kernel function itself to produce a rendering of gas column density (e.g., as done in the SPLASH visualisation package). One drawback over grid-based techniques is the need for large numbers of particles to produce simulations of equivalent resolution. In the typical implementation of both uniform grids and SPH particle techniques, many voxels or particles will be used to fill water volumes that are never rendered. However, accuracy can be significantly higher with sophisticated grid-based techniques, especially those coupled with particle methods (such as particle level sets), since it is easier to enforce the incompressibility condition in these systems. SPH for fluid simulation is being used increasingly in real-time animation and games where accuracy is not as critical as interactivity. Recent work in SPH for fluid simulation has increased performance, accuracy, and areas of application:

B. Solenthaler, 2009, develops Predictive-Corrective SPH (PCISPH) to allow for better incompressibility constraints M. Ihmsen et al., 2010, introduce boundary handling and adaptive time-stepping for PCISPH for accurate rigid body interactions K. Bodin et al., 2011, replace the standard equation of state pressure with a density constraint and apply a variational time integrator R. Hoetzlein, 2012, develops efficient GPU-based SPH for large scenes with Fluids v.3 N. Akinci et al., 2012, introduce a versatile boundary handling and two-way SPH-rigid coupling technique that is completely based on hydrodynamic forces; the approach is applicable to different types of SPH solvers M. Macklin et al., 2013 simulates incompressible flows inside the Position Based Dynamics framework, for bigger timesteps N. Akinci et al., 2013, introduce a versatile surface tension and two-way fluid-solid adhesion technique that allows simulating a variety of interesting physical effects that are observed in reality J. Kyle and E. Terrell, 2013, apply SPH to Full-Film Lubrication A. Mahdavi and N. Talebbeydokhti, 2015, propose a hybrid algorithm for implementation of solid boundary condition and simulate flow over a sharp crested weir S. Tavakkol et al., 2016, develop curvSPH, which makes the horizontal and vertical size of particles independent and generates uniform mass distribution along curved boundaries W. Kostorz and A. Esmail-Yakas, 2020, propose a general, efficient and simple method for evaluating normalization factors near piecewise-planar boundaries Colagrossi et al., 2019, study flow around a cylinder close to a free-surface and compare with other techniques

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