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Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating fluid flows. It was developed by Gingold and Monaghan (1977) and Lucy (1977) initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a mesh-free Lagrangian method (where the coordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as the density. == Method == The smoothed-particle hydrodynamics (SPH) method works by dividing the fluid into a set of discrete elements, referred to as particles. These particles have a spatial distance (known as the "smoothing length", typically represented in equations by ), over which their properties are "smoothed" by a ''kernel function''. This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within the range of the kernel. For example, using Monaghan's popular cubic spline kernel the temperature at position depends on the temperatures of all the particles within a radial distance of . The contributions of each particle to a property are weighted according to their distance from the particle of interest, and their density. Mathematically, this is governed by the kernel function (symbol ). Kernel functions commonly used include the Gaussian function and the cubic spline. The latter function is exactly zero for particles further away than two smoothing lengths (unlike the Gaussian, where there is a small contribution at any finite distance away). This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles. The equation for any quantity at any point is given by the equation : where is the mass of particle , is the value of the quantity for particle , is the density associated with particle , denotes position and is the kernel function mentioned above. For example, the density of particle () can be expressed as: : where the summation over includes all particles in the simulation. Similarly, the spatial derivative of a quantity can be obtained easily by virtue of the linearity of the derivative (del, ). : Although the size of the smoothing length can be fixed in both space and time, this does not take advantage of the full power of SPH. By assigning each particle its own smoothing length and allowing it to vary with time, the resolution of a simulation can be made to automatically adapt itself depending on local conditions. For example, in a very dense region where many particles are close together the smoothing length can be made relatively short, yielding high spatial resolution. Conversely, in low-density regions where individual particles are far apart and the resolution is low, the smoothing length can be increased, optimising the computation for the regions of interest. Combined with an equation of state and an integrator, SPH can simulate hydrodynamic flows efficiently. However, the traditional artificial viscosity formulation used in SPH tends to smear out shocks and contact discontinuities to a much greater extent than state-of-the-art grid-based schemes. The Lagrangian-based adaptivity of SPH is analogous to the adaptivity present in grid-based adaptive mesh refinement codes. In some ways it is actually simpler because SPH particles lack any explicit topology relating them, unlike the elements in FEM. Adaptivity in SPH can be introduced in two ways; either by changing the particle smoothing lengths or by splitting SPH particles into 'daughter' particles with smaller smoothing lengths. The first method is common in astrophysical simulations where the particles naturally evolve into states with large density differences.〔http://arxiv.org/abs/astro-ph/9512078〕 However, in hydrodynamics simulations where the density is often (approximately) constant this is not a suitable method for adaptivity. For this reason particle splitting can be employed, with various conditions for splitting ranging from distance to a free surface 〔http://dl.acm.org/citation.cfm?id=1568695〕 through to material shear.〔http://www.ase.uc.edu/~liugr/Storage/Journal%20Papers/2006/JA_2006_09.pdf〕 Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics. The particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance tree gravity code,〔 ("The Parallel k-D Tree Gravity Code" ); ("PKDGRAV (Parallel K-D tree GRAVity code" ) use a kd-tree gravity simulation. 〕 particle mesh, or particle-particle particle-mesh. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Smoothed-particle hydrodynamics」の詳細全文を読む スポンサード リンク
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