Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> E. OTHER IMPROVEMENTS <s> A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element-free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C∞, the solution and its derivatives are also C∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two-dimensional examples of advection-diffusion equations and compressible Euler equations are also presented together with 2D multiple-scale decompositions. <s> BIB001 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> E. OTHER IMPROVEMENTS <s> The Smooth-Particle-Hydrodynamics (SPH) method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of continuum mechanics as in the finite-element method. This derivation is modified to replace the SPH interpolant with the Moving-Least-Squares (MLS) interpolant of Lancaster and Saulkaskas, and define a new particle volume which ensures thermodynamic compatibility. A variable-rank modification of the MLS interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required. A surprise benefit of the Galerkin SPH derivation is a theoretical justification of a common ad hoc technique for variable-h SPH. The new MLSPH method is conservative if an anti-symmetric quadrature rule for the stiffness matrix elements can be supplied. In this paper, a simple one-point collocation rule is used to retain similarity with SPH, leading to a non-conservative method. Several examples document how MLSPH renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the SPH method: spurious boundary effects, erroneous rates of strain and rotation and tension instability. Two of these examples are non-linear Lagrangian patch tests with analytic solutions with which MLSPH agrees almost exactly. The examples also show that MLSPH is not absolutely stable if the problems are run to very long times. A linear stability analysis explains both why it is more stable than SPH and not yet absolutely stable and an argument is made that for realistic dynamic problems MLSPH is stable enough. The notion of coherent particles, for which the numerical stability is identical to the physical stability, is introduced. The new method is easily retrofitted into a generic SPH code and some observations on performance are made. Copyright © 1999 John Wiley & Sons, Ltd. <s> BIB002 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> E. OTHER IMPROVEMENTS <s> Combining the kernel estimate with the Taylor series expansion is proposed to develop a Corrective Smoothed Particle Method (CSPM). This algorithm resolves the general problem of particle deficiency at boundaries, which is a shortcoming in Standard Smoothed Particle Hydrodynamics (SSPH). In addition, the method’s ability to model derivatives of any order could make it applicable for any time-dependent boundary value problems. An example of the applications studied in this paper is unsteady heat conduction, which is governed by second-order derivatives. Numerical results demonstrate that besides the capability of directly imposing boundary conditions, the present method enhances the solution accuracy not only near or on the boundary but also inside the domain. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States. <s> BIB003 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> E. OTHER IMPROVEMENTS <s> This paper describes the applications of a finite particle method (FPM) to modeling incompressible flow problems. FPM is a meshfree particle method in which the approximation of a field variable and its derivatives can be simultaneously obtained through solving a pointwise matrix equation. A set of basis functions is employed to obtain the coefficient matrix through a sequence of transformations. The finite particle method can be used to discretize the Navier–Stokes equation that governs fluid flows. The incompressible flows are modeled as slightly compressible via specially selected equations of state. Four numerical examples including the classic Poiseuille flow, Couette flow, shear driven cavity and a dam collapsing problem are presented with comparisons to other sources. The numerical examples demonstrate that FPM is a very attractive alternative for simulating incompressible flows, especially those with free surfaces, moving interfaces or deformable boundaries. <s> BIB004 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> E. OTHER IMPROVEMENTS <s> We discuss the symmetric smoothed particle hydrodynamics (SSPH) method for generating basis func- tions for a meshless method. It admits a larger class of kernel functions than some other methods, including the smoothed particle hydrodynamics (SPH), the modified smoothed parti- cle hydrodynamics (MSPH), the reproducing kernel particle method (RKPM), and the moving least squares (MLS) meth- ods. For finding kernel estimates of derivatives of a function, the SSPH method does not use derivatives of the kernel func- tion while other methods do, instead the SSPH method uses basis functions different from those employed to approxi- mate the function. It is shown that the SSPH method and the RKPM give the same value of the kernel estimate of a function but give different values of kernel estimates of deriv- atives of the function. Results computed for a sine function defined on a one-dimensional domain reveal that the L 2 ,t he H 1 and the H 2 error norms of the kernel estimates of a func- tion computed with the SSPH method are less than those found with the MSPH method. Whereas the L 2 and the H 2 norms of the error in the estimates computed with the SSPH method are less than those with the RKPM, the H 1 norm of the error in the RKPM estimate is slightly less than that found with the SSPH method. The error norms for a sam- ple problem computed with six kernel functions show that their rates of convergence with an increase in the number of uniformly distributed particles are the same and their magni- tudes are determined by two coefficients related to the decay rate of the kernel function. The revised super Gauss function has the smallest error norm and is recommended as a ker- nel function in the SSPH method. We use the revised super <s> BIB005
Given that the conventional SPH is insufficient, several new high-precision methods were proposed as follows. Reproducing kernel particle method BIB001 : a correction function is introduced and applied for regeneration conditions, which satisfy the boundary compatibility conditions and improve the solution accuracy for large deformation analysis. Moving least squares particle dynamics BIB002 : the method has a tight support domain of the weight function, which is obtained by the moving least squares interpolation of the shape functions and the shape functions for solving fluid mechanics equations. Corrected smooth BIB003 and finite particle methods BIB004 : by a Taylor series expansion, traditional discrete SPH way is used to describe the correction. Symmetric smooth particle dynamics BIB005 : the SSPH method was proven highly suitable for elastic solid simulation by constructing a new basis function. However, the construction of its base function is overly complex, and this complexity should be difficult to popularize to the free surface of the simulation.