Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> B. SMOOTHING FUNCTIONS <s> A finite-size particle scheme for the numerical solution of two- and three-dimensional gas dynamical problems of astronomical interest is described and tested. The scheme is then applied to the fission problem for optically thick protostars. Results are given, showing the evolution of one such protostar from an initial state as a single, rotating star to a final state as a triple system whose components contain 60% of the original mass. The decisiveness of this numerical test of the fission hypothesis and its relevance to observed binaries are briefly discussed. <s> BIB001 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> B. SMOOTHING FUNCTIONS <s> Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks. <s> BIB002
In SPH, the smoothing function is used by a kernel for particle approximation. The smoothing function is extremely important in SPH because it determines the pattern to interpolate and define the threshold of the support area of a particle. Like the mathematical approach in hydrodynamics, the SPH algorithm involves the concept of ''smooth kernel,'' which can be understood in this manner, and the properties of the particles are ''diffused'' to the surroundings and gradually become small as the distance increases. The function of attenuation with distance is called the ''smooth kernel'' function, and the maximum influence radius is the ''smooth kernel radius.'' When SPH method is used for fluid simulation, the stability, accuracy and speed of calculation depend highly on the selected smooth kernel function. Different smoothing functions have been used in SPH, and their major requirements or properties have been analyzed as follows BIB002 : (a) Regularization conditions: where κ is the constant of the smooth function at a point and κh determines the effective range of the smooth function (i.e., the support domain) (c) Nonnegativity: In general meshless methods, the weight function can be negative. In the study of hydrodynamic problems, the traditional smoothing function usually uses non-negative function. Negative smoothing functions may lead to negative physical parameters, such as negative energy, density and mass, which violate physical properties in the calculation process.. (d) Attenuation: The smoothing function value of particles should monotonously decrease with the increase of distance from particles. On the contrary, the interaction force decreases with the increase of distance between two interacting particles. (e) Dirac Functionality: (f) Even Functionality: This property indicates that the same distance from a given particle to different particles at the different positions has the same effect on a given particle. Lucy in the original SPH paper BIB001 used a bell-shaped function. Gingold and Monaghan  selected Gaussian kernel to simulate nonspherical stars. The most frequently used smoothing function is the cubic B-spline function  . Table 1 lists the existing appropriate smooth functions.