Smooth particle hydrodynamics

Welton, W. C. and S. B. Pope (1997). PDF model calculations of compressible turbulent flows using smoothed particle hydrodynamics. Journal of Computational Physics 134,150-168. 

Simulations of this scenario have been performed by Lee (Lee 2001 and references therein) using Newtonian gravity and polytropic equations of state of varying stiffness. Ruffert, Janka and Eberl performed similar simulations but with a detailed microphysics input (nuclear equation of state and neutrino leakage ). In our own simulations of NS-BH mergers we used a relativistic mean field equation of state together with three-dimensional smoothed particle hydrodynamics and a detailed, multiflavour neutrino treatment. 

J. S. Cintra and C. L. Tucker, Orthotropic closuere approximations for flow-induced fiber orientation, J. Rheol. 39, 1095-1122 (1995) C. V. Chaubal, A. Srinivasan, O. Egecioglu, and L. G. Leal, Smoothed particle hydrodynamics technique for the solution of kinetic theory problems. Part 1. Method, J. Non-Newtonian Fluid Mech. 70, 125-54 (1997). 

The smoothed-particle hydrodynamics (SPH) method is an empirical alternative to the two above discussed methods, since it is grid-free and the results are, therefore, independent of a prescribed coordinate system and numerical grid resolution. It is... 

Other methods include particle-based methods and also techniques that do not rely on the Navier-Stokes equation for flow solution. Particle-based methods are roughly divided into two main categories those that use particles in conjunction with a grid, namely, particle-in-cell (PIC) methods, and those that are massless, such as the smoothed particle hydrodynamics (SPH). The most striking feature of particle-based methods is their ease of implementation. They are essentially as easy to implement on 3D unstmcmred meshes as on 2D strucmred meshes. However, they are very demanding in terms of memory and processing power, which has caused their limited use in interfacial flows so far. 

The molecular dynamics, Monte Carlo methods, the Lagrangian probability density function (PDF) methods, and the Lattice Boltzmann (LBM) method are among these methods. Methods such as the smoothed particle hydrodynamics (SPH) and the vortex method initially developed as probabiUstic methods, but nowadays they are most frequently used as deterministic. 

Particle methods that require a predefinition of particles for their computations. Such methods require some kind of mesh to predefine the particles. Smoothed particle hydrodynamics method belongs to this category. 

In 1977, Lucy [1] and Gingold and Monaghan [2] independently introduced the so-called smoothed particle hydrodynamics method which is one of the oldest meshless methods. The method was originally developed for astrophysi-cal studies such as formation of asteroids and the evolution of galaxies and has now become a standard tool in this field. In recent years, SPH has been extensively used for various fluid flow problems including both compressible and incompressible flow regimes. The method has been recently used for simulation of the generalized non-Newtonian and viscoelastic fluid flow problems. The SPH method has been also... 

Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 30 543-574... 

In Figure 26.24 we display the snapshots from modeling of the R-T hydrodynamic instability using smoothed particle hydrodynamics in which 10 smoothed particles were modeled in about 3000 timesteps. 

Although originally designed for astrophysical problems [86], as shown in [120], SPH can also be used for modeling polymers in the macroscale. However, smoothed particle hydrodynamics does not include thermal fluctuations in the form of a random stress tensor and heat flux as in the Landau and Lifshitz theory of hydrodynamic fluctuations. Therefore, the validity of SPH to the study of complex fluids is problematic at scales where thermal fluctuations are important. 

FIGURE 26.24 The snapshots from a simulation using smoothed particle hydrodynamics of the Rayleigh-Taylor instahihty for gases. The size of the hox is approximately 1 cm. The dark grey system represents fluid which is 10 times heavier than the lighter one placed in the bottom (invisihle in this figure). Approximately 10 SPH particles have been modeled. 

The computational problems involving multi-million particle ensembles, found in modeling mesoscopic phenomena, were considered only recently as the typical problems. Rapid increase of computational power of modem processors and growing popularity of coarse-grained discrete particle methods, such as dissipative particle dynamics, fluid particle model, smoothed particle hydrodynamics and LEG, allow for the modehng of complex problems by using smaller shared-memory systems [101]. 

Within the context of cross-scaling systems, they are homogeneous with both microscopic molecular dynamics and macroscopic smoothed particle hydrodynamics techniques. From a numerical standpoint, we do not need to switch over from particles to a static grid with control parameters, such as the Knudsen number. 

Monaghan, J.J., Smoothed Particle Hydrodynamics, Annu. Rev. Astron. Astrophys., 30, 543,... 

Among several numerical methods, the finite element method (FEM) is a robust and thoroughly developed method. However, the accuracy and efficiency of the method rely on the quality of meshes or elements. Usually the FEM user has to spend considerable time in mesh creation or in fixing mesh problems. Therefore, some efforts have also been made to explore the applicability of meshless methods, such as the Smoothed Particle Hydrodynamics (SPH) method. 

Cleary P, Monaghan JJ (1999) Conduction modelling using smoothed particle hydrodynamics. J Comput Phys 148 227-264... 

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