Smoothed Particle Hydrodynamics SPH Method

SPH was introduced thirty years ago to simulate astrophysical fluid d)mamics (Lucy, 1977). It is based on the idea that a continuous field. Air), can be represented by a superposition of smooth beU-shaped functions, lT( r-r,j) (usually referred to as the smoothing function or weighting function) centered on a set of the points, r,-. The approximation A(r) is expressed by its neighbored particles in a domain Sir (Meakin and Tartakovsky, 2009)  

The SPH method has been also used to simulate pore-scale dissolution of the trapped non-aqueous phase liquids, pore-scale miscible non-reactive flows (Tartakovsky and Meakin, 2005 Zhu et al, 2002), and single- and multi-component reactive transport and precipitation (Tartakovsky et al, 2007). Recently, Jiang et al. (2008) developed a meso-scale SPH model for 

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. 

In SPH, the fluid is discretized into a finite number of moving points, or particles , where any physical quantity/(x) associated with the particle at the position X is interpolated using function values at neighboring particles wifliin a small local support domain of the position x, i.e.. 

To simulate confined fluid flow such as mold filling, one usually models the solid boundaries by solid particles . These particles are fixed or moving with the 

The explicit integration methods, such as leapfrog, prediction-correction or Runge-Kutta methods, are usually used to integrate SPH equations for fluid flows. The explicit time integration is conditionally stable. The time step should satisfy the convective stabihty constraint, i.e., the so-caUed Courant-Friedrichs-Lewy (CFL) condition, 

If the diffusion term is integrated explicitly, the time step should satisfy the diffusive stability constraint. 

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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-based simulation techniques include atomistic MD and coarse-grained molecular dynamics (CG-MD). Accelerated dynamics methods, such as hyperdynamics and replica exchange molecular dynamics (REMD), are very promising for circumventing the timescale problem characteristic of atomistic simulations. Structure and dynamics at the mesoscale level can be described within the framework of coarse-grained particle-based models using such methods as stochastic dynamics (SD), dissipative particle dynamics (DPD), smoothed-particle hydrodynamics (SPH), lattice molecular dynamics (LMD), lattice Boltzmann method (IBM), multiparticle collision dynamics (MPCD), and event-driven molecular dynamics (EDMD), also referred to as collision-driven molecular dynamics or discrete molecular dynamics (DMD). 

Recently, a new type of phase separation called viscoelastic phase separation was observed in polymer solutions or dynamically asymmetric fluid mixtures [1-3]. It is an interesting feature of this phenomenon that network-like domains of more viscous phase emerge in a transient regime. It has little been understood what ingredient of physics is crucial to this phenomenon. Various numerical approaches have been made for the phase separation phenomena in binary fluid systems in the last decade [4-6]. Most of these studies have been concerned with classical fluids and have not involved viscoelasticity. A new numerical model was recently proposed by the author [7] based upon the two-fluid model [8,9] using the method of smoothed-particle hydrodynamics (SPH) [10,11]. In this model the Lagrangian picture for fluid is adopted and the viscoelastic effect can easily be incorporated. In this paper we carry out a computer simulation for the viscoelastic phase separation in polymer solutions with this model. 

A technique, called smoothed particle hydrodynamics (SPH) has been introduced to the Al-body problem. This involves the combination of the two methods of conventional few-body integrators for each body with its immediate nearest neighbors and the overall computation of the distant gravitational potential via FFTs. SPH and FFT methods have also been merged with hierarchical tree searching techniques to improve their speed. Computation of many-body effects, while not important for satellite dynamics, which depend primarily on the central field approximation, may be important in the study of asteroid and comet orbital evolution and also applies to the... 

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... 

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