A kernel gradient free (KGF) SPH method

The finite particle method (FPM) is a modified SPH method with high order accuracy while retaining the advantages of SPH in modeling problems with free surfaces, moving interfaces, and large deformations. In both SPH and FPM, kernel gradient is necessary in kernel and particle approximation of a field function and its derivatives. In this paper, a new FPM is presented, which only involves kernel function itself in kernel and particle approximation. The kernel gradient is not necessary in the whole computation, and this approach is thus referred to as a kernel gradient free (KGF) SPH method. This is helpful when a kernel function is not differentiable or the resultant kernel gradient is not sufficiently smooth, and thus it is more general in selecting a kernel function. Moreover, different from the original FPM with an asymmetric corrective matrix, in the new FPM, the resultant corrective matrix is symmetric, and this is advantageous in particle approximations. A series of numerical examples have been conducted to show the efficiencies of KGF‐SPH including one‐dimensional mathematical tests of polynomial functions with equal or variable smoothing length and two‐dimensional incompressible fluid flow of shear cavity. It is found that KGF‐SPH is comparable with FPM in accuracy and is flexible as SPH. Copyright © 2015 John Wiley & Sons, Ltd.     

An improved KGF‐SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows

The kernel gradient free (KGF) smoothed particle hydrodynamics (SPH) method is a modified finite particle method (FPM) which has higher order accuracy than the conventional SPH method. In KGF‐SPH, no kernel gradient is required in the whole computation, and this leads to good flexibility in the selection of smoothing functions and it is also associated with a symmetric corrective matrix. When modeling viscous incompressible flows with SPH, FPM or KGF‐SPH, it is usual to approximate the Laplacian term with nested approximation on velocity, and this may introduce numerical errors from the nested approximation, and also cause difficulties in dealing with boundary conditions. In this paper, an improved KGF‐SPH method is presented for modeling viscous, incompressible fluid flows with a novel discrete scheme of Laplacian operator. The improved KGF‐SPH method avoids nested approximation of first order derivatives, and keeps the good feature of ‘kernel gradient free’. The two‐dimensional incompressible fluid flow of shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, the original KGF‐SPH and improved KGF‐SPH. The numerical results show that the improved KGF‐SPH with the novel discrete scheme of Laplacian operator are more accurate than SPH, and more stable than FPM and the original KGF‐SPH. Copyright © 2015 John Wiley & Sons, Ltd.     

An assessment of particle methods for approximating anisotropic dispersion

We derive a smoothed particle hydrodynamics (SPH) approximation for anisotropic dispersion that only depends upon the first derivative of the kernel function and study its numerical properties. In addition, we compare the performance of the newly derived SPH approximation versus an implementation of the particle strength exchange (PSE) method and a standard finite volume method for simulating multiple scenarios defined by different combinations of physical and numerical parameters. We show that, for regularly spaced particles, given an adequate selection of numerical parameters such as kernel function and smoothing length, the new SPH approximation is comparable with the PSE method in terms of convergence and accuracy and similar to the finite volume method. On other hand, the performance of both particle methods (SPH and PSE) decreases as the degree of disorder of the particle increases. However, we demonstrate that in these situations the accuracy and convergence properties of both particle methods can be improved by an adequate choice of some numerical parameters such as kernel core size and kernel function. Copyright © 2012 John Wiley & Sons, Ltd.     

固体介质中SPH方法的拉伸不稳定性问题研究进展

光滑粒子流体动力学法(smoothed particle hydrodynamics, SPH)是一种基于核估计的无网格Lagrange数值方法.它用粒子方程离散流体动力学的连续方程, 既可以处理有限元难于处理的大变形和严重扭曲问题, 又可以处理有限差分法不易处理的自由边界和材料界面的问题, 在固体力学中的冲击、爆炸和裂纹模拟中具有广阔的发展前景.但是, 该算法的拉伸不稳定性(tensile instability)问题是它在固体力学领域中应用的最大障碍.对SPH稳定性分析表明, 算法不稳定性的条件仅与应力状态和核函数的2阶导数有关.目前, 应力点法(stress points)、Lagrange核函数法、人工应力法(artificialstress)、修正光滑粒子法(corrective smoothed particle method, CSPM)和守恒光滑法(conservativesmoothing)以及其他一些方法成功地改善了SPH的拉伸不稳定性, 但是每一种方法都不能彻底解决SPH的拉伸不稳定性问题.本文介绍了SPH法的方程和Von Neumann稳定性分析的思想, 以及国内外在这几个方面的研究成果及其最新进展, 同时指出目前研究中存在的问题和研究的方向.      

SPH方法在模拟线弹性波传播中的运用

通过对固体中波动问题的模拟建立了一种光滑粒子法的新形式,一种运用SPH的核函数的类似有限体积法的计算方法。通过对统计体积的修正以及对边界粒子的核函数修正,较好地解决了SPH方法中长期以来制约其被广泛应用的主要问题之一边界条件的表述。在此基础上成功地在光滑粒子法中实现了透射边界条件的模拟。同时利用反卷积修正使得较大粒子间距下的计算结果的精度大大提高。这种方法不但保持了SPH的简单性,而且很容易实现应力边界条件。     

A new version of smoothed point method to simulate linear elastic wave propagation

By using the kernel function of the smoothed particle hydrodynamics (SPH) and modification of statistical volumes of the boundary points and their kernel functions, a new version of smoothed point method is established for simulating elastic waves in solid. With the simplicity of SPH kept, the method is easy to handle stress boundary conditions, especially for the transmitting boundary condition. A result improving by de-convolution is also proposed to achieve high accuracy under a relatively large smooth length. A numerical example is given and compared favorably with the analytical solution.     

求解Helmholtz方程基于核重构思想的最小二乘配点法

基于核重构思想构造近似函数，将配点法和最小二乘原理相结合对微分方程进行离散， 建立了Helmholtz方程的最小二乘配点格式，并分别研究了Helmholtz方程的波传播问题和 边界层问题. 通过数值算例可以发现，给出的数值计算结果非常接近于精确解，计算精度明显高于SPH 法的数值结果，且随着节点数目的增加，其精确度越来越高，具有良好的收敛性.     

完全拉格朗日SPH在冲击问题中的改进和应用

光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)在模拟固体大变形、破碎和裂纹扩展等问题中有天然的优势, 但SPH固有的拉伸不稳定缺陷是SPH在计算固体力学领域进一步应用的一大障碍. 完全拉格朗日SPH (total Lagrangian-SPH, TL-SPH)方法是一种有效的改善拉伸不稳定的措施, 但其仍面临边界区域精度低、界面条件难以施加、损伤裂纹难以模拟等缺陷. 因此, 首先将可达到二阶精度的高阶SPH方法与TL-SPH耦合, 为了节省高阶方法的计算量, 进一步简化粒子选取模式, 提出TL-SFPM (TL-simplified finite particle method)方法; 其次, 将可提高界面精度的DFPM (discontinuous finite particle method)方法与TL-SPH结合, 并提出一种基于黎曼解的界面接触算法, 通过在不同材料粒子间建立黎曼模型求解不同材料间的相互作用, 分别应用于流体?固体接触和固体?固体接触中; 再者, 为了捕捉固体受外载荷后的损伤程度及破坏模式, 提出一种完全拉格朗日框架下的粒子损伤破坏模型; 最后, 通过流?固冲击的带弹性挡板溃坝算例和固?固冲击的子弹撞击靶板算例验证提出的TL-SFPM方法、界面接触算法和损伤破坏模型的合理性和精确性, 进一步扩展TL-SPH方法在计算固体冲击问题中的应用.      

Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.     

Hermite梯度重构核近似配点法及其在功能梯度材料板的静力分析中的应用

由于直接配点法在求解边值问题时边界上的求解精度较低,本文提出了Hermite梯度重构核近似配点法（HGCM）来改进边界求解精度。重构核近似是无网格法中一种常用的近似函数,但是其在求解高阶导数时格式复杂且非常耗时。HGCM采用梯度重构核近似构建形函数的任意高阶导数,提高了计算效率;通过Hermite配点法构建离散方程,提高了边界求解精度。这种方法在求解对应变系数四阶偏微分方程的功能梯度材料板的静力问题时精度高,计算效率高,并可进一步推广应用于高阶偏微分方程描述的边值问题。     

An SPH model for free surface flows with moving rigid objects

This paper presents a computational model for free surface flows interacting with moving rigid bodies. The model is based on the SPH method, which is a popular meshfree, Lagrangian particle method and can naturally treat large flow deformation and moving features without any interface/surface capture or tracking algorithm. Fluid particles are used to model the free surface flows which are governed by Navier–Stokes equations, and solid particles are used to model the dynamic movement (translation and rotation) of moving rigid objects. The interaction of the neighboring fluid and solid particles renders the fluid–solid interaction and the non‐slip solid boundary conditions. The SPH method is improved with corrections on the SPH kernel and kernel gradients, enhancement of solid boundary condition, and implementation of Reynolds‐averaged Navier–Stokes turbulence model. Three numerical examples including the water exit of a cylinder, the sinking of a submerged cylinder and the complicated motion of an elliptical cylinder near free surface are provided. The obtained numerical results show good agreement with results from other sources and clearly demonstrate the effectiveness of the presented meshfree particle model in modeling free surface flows with moving objects. Copyright © 2013 John Wiley & Sons, Ltd.     

Simulation of container filling process with two inlets by improved smoothed particle hydrodynamics (SPH) method

In this article, an improved smoothed particle hydrodynamics (SPH) method is proposed to simulate the filling process with two inlets. Improvements are achieved by deriving a corrected kernel gradient of SPH and a density re-initialisation. In addition, a new treatment of solid wall boundaries is presented. Thus, the improved SPH method has higher accuracy and better stability, and conserves both linear and angular momentums. The validity of the new boundary treatment is shown by simulating the spin-down problem. The bench tests are also presented to demonstrate the performance of the improved SPH method. Then the filling process with a single inlet is simulated to show the ability to capture complex-free surface of the proposed method. Finally, the filling process with two inlets is numerically investigated. The numerical results show that the filling patterns are affected significantly by Reynolds number, aspect ratio of the container and the location of the inlets.     

应力波数值计算中的SPH方法

对一维波动方程的SPH(smoothed particle hydrodynamics)格式和有限差分格式进行比较,并采用SPH法模拟了一维应力/应变波, 获得1个可衡量SPH法模拟应力波准确性的重要指标。结果表明,SPH法模拟应力波传播中采用的光滑长度必须不小于粒子间距;采用B-样条核函数和高斯型核函数能够获得良好的应力波图像,而二次型核函数不能,因此二次型核函数不适用于冲击动力学的数值计算。     

非等温黏弹性复杂流动的改进SPH方法模拟

非等温黏弹性流体广泛存在于自然界和工业生产中,准确预测黏弹性流体的非等温流动机理和复杂流变特性有着重要的应用价值.文章提出一种改进的光滑粒子流体动力学(smoothed particle hydrodynamics,SPH)方法对非等温黏弹性复杂流动进行了数值模拟,其中流体的黏弹特性通过eXtended Pom-Pom本构模型来表征.为了提高模拟结果的精度,采用了一种核函数梯度的修正算法;为了灵活地施加边界条件,发展了边界粒子和虚拟粒子相联合的边界处理方法;为了消除流动过程中的拉伸不稳定性,施加了粒子迁移技术.运用改进SPH方法数值模拟了液滴撞击固壁和F型腔注塑成型问题,通过与Basilisk软件得到的结果进行比较验证了改进SPH方法求解非等温黏弹性流体的有效性.通过利用不同粒子初始间距进行计算,评价了改进SPH方法的数值收敛性.研究了非等温流动相较于等温流动的不同流动特征,深入分析了不同热流变参数对流动过程的影响.数值结果表明,文章提出的改进SPH方法可稳定、准确地描述非等温黏弹性复杂流动的传热机理、复杂流变特性和自由面变化特性.     

Application of particle splitting method for both hydrostatic and hydrodynamic cases in SPH

Smoothed particle hydrodynamics(SPH) method with numerical diffusive terms shows satisfactory stability and accuracy in some violent fluid–solid interaction problems. However, in most simulations, uniform particle distributions are used and the multi-resolution, which can obviously improve the local accuracy and the overall computational efficiency, has seldom been applied. In this paper, a dynamic particle splitting method is applied and it allows for the simulation of both hydrostatic and hydrodynamic problems. The splitting algorithm is that, when a coarse(mother) particle enters the splitting region, it will be split into four daughter particles, which inherit the physical parameters of the mother particle. In the particle splitting process,conservations of mass, momentum and energy are ensured. Based on the error analysis, the splitting technique is designed to allow the optimal accuracy at the interface between the coarse and refined particles and this is particularly important in the simulation of hydrostatic cases. Finally, the scheme is validated by five basic cases, which demonstrate that the present SPH model with a particle splitting technique is of high accuracy and efficiency and is capable for the simulation of a wide range of hydrodynamic problems.Smoothed particle hydrodynamics(SPH)method with numerical diffusive terms shows satisfactory stability and accuracy in some violent fluid–solid interaction problems.However,in most simulations,uniform particle distributions are used and the multi-resolution,which can obviously improve the local accuracy and the overall computational efficiency,has seldom been applied.In this paper,a dynamic particle splitting method is applied and it allows for the simulation of both hydrostatic and hydrodynamic problems.The splitting algorithm is that,when a coarse(mother)particle enters the splitting region,it will be split into four daughter particles,which inherit the physical parameters of the mother particle.In the particle splitting process,conservations of mass,momentum and energy are ensured.Based on the error analysis,the splitting technique is designed to allow the optimal accuracy at the interface between the coarse and refined particles and this is particularly important in the simulation of hydrostatic cases.Finally,the scheme is validated by five basic cases,which demonstrate that the present SPH model with a particle splitting technique is of high accuracy and efficiency and is capable for the simulation of a wide range of hydrodynamic problems.     

Adaptive smoothed particle hydrodynamics for high strain hydrodynamics with material strength

This paper presents the implementation of an adaptive smoothed particle hydrodynamics (ASPH) method for high strain Lagrangian hydrodynamics with material strength. In ASPH, the isotropic kernel in the standard SPH is replaced with an anisotropic kernel whose axes evolve automatically to follow the mean particle spacing as it varies in time, space, and direction around each particle. Except for the features inherited from the standard SPH, ASPH can capture dimension-dependent features such as anisotropic deformations with a more generalized elliptical or ellipsoidal influence domain. Two numerical examples, the impact of a plate against a rigid surface and the penetration of a cylinder through a plate, are investigated using both SPH and ASPH. The comparative studies show that ASPH has better accuracy than the standard SPH when being used for high strain hydrodynamic problems with inherent anisotropic deformations. PACS 46.15.-x, 83.10.Rs, 83.50.-v     

An improved SPH model for multiphase flows with large density ratios

This paper presents a new smoothed particle hydrodynamics (SPH) model for simulating multiphase fluid flows with large density ratios. The new SPH model consists of an improved discretization scheme, an enhanced multiphase interface treatment algorithm, and a coupled dynamic boundary treatment technique. The presented SPH discretization scheme is developed from Taylor series analysis with kernel normalization and kernel gradient correction and is then used to discretize the Navier‐Stokes equation to obtain improved SPH equations of motion for multiphase fluid flows. The multiphase interface treatment algorithm involves treating neighboring particles from different phases as virtual particles with specially updated density to maintain pressure consistency and a repulsive interface force between neighboring interface particles into the pressure gradient to keep sharp interface. The coupled dynamic boundary treatment technique includes a soft repulsive force between approaching fluid and solid particles while the information of virtual particles are approximated using the improved SPH discretization scheme. The presented SPH model is applied to 3 typical multiphase flow problems including dam breaking, Rayleigh‐Taylor instability, and air bubble rising in water. It is demonstrated that inherent multiphase flow physics can be well captured while the dynamic evolution of the complex multiphase interfaces is sharp with consistent pressure across the interfaces.     

多相流动的光滑粒子流体动力学方法研究综述

光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)具有粒子方法的无网格和全拉格朗日特征, 适用于具有界面大变形、不连续性和多物理场的多相流的高精度模拟. SPH方法模拟多相流已有大量报道, 具体的实现方式也大不相同. 本文首先阐述了采用SPH方法模拟流体的基本控制方程, 以及求解过程中需要考虑的流体压力求解、表面张力、固体边界等问题. 整理和总结了基于SPH方法进行多相流模拟的主要实现方式: (1)双流体模型的拉格朗日求解器: 两相离散为两组独立SPH粒子, 并用显式相间作用耦合两相; (2)多相SPH方法: SPH方法对多相流模拟的自然延伸, 相间作用由SPH参数隐式描述; (3) SPH与其他离散方法的耦合: 差异较大的两相各自采用不同离散方法, 发挥不同拉格朗日方法的优点; (4) SPH和基于网格方法的耦合: 网格方法处理简单的单相流动主体, 获得精度和效率间的平衡. 另外, 还在模拟参数物理化等方面论述了与SPH方法模拟多相流相关的一些改进和修正方法, 并在最后讨论和建议了提高多相流SPH模拟效率和精度的措施.      

基于核重构思想的最小二乘配点型无网格方法

介绍重构核点法的基本原理和近似函数的构造方法，并基于核重构思想，应用配点法和最小二乘原理，离散微分方程，建立求解的代数方程，提出了一种基于核重构思想的最小二乘配点型无网格方法．与一般配点法相比，该方法的系数矩阵是有对称正定的，计算精度高，稳定性好．该方法的实施不需要背景网格，不需要进行高斯积分，与Galerkin法相比，具有计算量小、边界条件处理简单的特点，是一种真正的无网格法．对该方法构造过程中的近似函数及其导数的计算、修正函数的计算及方法的实现等问题进行了探讨．文中结合若干典型算例，检验了该方法的有效性．     

An enhanced treatment of boundary conditions in implicit smoothed particle hydrodynamics简

In this work, an enhanced treatment of the solid boundaries is proposed for smoothed particle hydrodynamics with implicit time integration scheme （Implicit SPH）. Three types of virtual particles, i.e., boundary particles, image particles and mirror particles, are used to impose boundary conditions. Boundary particles are fixed on the solid boundary, and each boundary particle is associated with two fixed image particles inside the fluid domain and two fixed mirror particles outside the fluid domain. The image particles take the flow properties through fluid particles with moving least squares （MLS） interpolation and the properties of mirror particles can be obtained by the corresponding image particles. A repulsive force is also applied for boundary particles to prevent fluid particles from unphysical penetra- tion through solid boundaries. The new boundary treatment method has been validated with five numerical examples. All the numerical results show that Implicit SPH with this new boundary-treatment method can obtain accurate results for non-Newtonian fluids as well as Newtonian fluids, and this method is suitable for complex solid boundaries and can be easily extended to 3D problems.