Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> II. SMOOTHED PARTICLE HYDRODYNAMICS (SPH) <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> II. SMOOTHED PARTICLE HYDRODYNAMICS (SPH) <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
A. BASIC THEORY OF SPH SPH was originally developed by Lucy BIB001 , Gingold and Monaghan  to simulate astrophysics, but it also has wide applicability in other dynamic simulations. SPH is an interpolation method based on particle system, which enables the physical field defined at discrete points to be evaluated at any position in space. The method uses a radially symmetric smooth kernel function, also known as a kernel function, which distributes the field in the local neighborhood of each particle, which is called the support region. The physical field A at position r can be calculated by interpolating the weights of all particles in the support domain: Among them, j denotes any particle in the support domain at the current position r, m j denotes the mass of particle j, r j denotes the position of particle j, ρ j denotes the density of particle j, A j denotes a physical field at the position r j , and W (r-r j , h) is a smooth kernel function, where h is the radius of the support domain, also known as the smooth core radius. If W is even, standardized and second-order, then there are: In most dynamic equations, the derivative of physical field to position will appear, which needs to be solved. When using SPH method, the derivatives of the physical field loaded by particles only affect the smooth kernel function because the physical field loaded by particles only changes with time from the Lagrangian perspective. The gradient of physical field A can be expressed as: The Laplace operator of physical field A can be expressed as: Approximate schematic of SPH particles in a two-dimensional space. W is a smooth function and supports domain Kh, and p is the surface of the calculation area. BIB002 . In SPH simulation, every fixed time step t, the position and velocity of each particle will be calculated once for image rendering. After obtaining the acceleration of fluid particles at the current time, the appropriate numerical method can be selected to estimate the velocity and position of particles at the next time, so that the whole simulation process can be continued. SPH method can deal with the flow system with complex wall, free surface, multi-phase moving interface, and large deformation and breakage of solid relatively simply. As a particle method, mass is strictly conserved. In addition, its concept is concise, it is easy to add various special physical or chemical effects, and it can be directly extended from two-dimensional system to three-dimensional system, which is easy to program. Because of the short-range interaction between particles, it has good parallelism and can make full use of cluster system for large-scale computation. Figure 1 shows a schematic representation of the SPH particle approximation in a two-dimensional space. Techniques for deriving SPH formulations for the Navier-Stokes (N-S) equations are reviewed in a previous article BIB002 .