Boundary Conditions Generated by Dynamic Particles in SPH Methods

Smoothed Particle Hydrodynamics is a purely Lagrangian method that can be applied to a wide variety of fields. The foundation and properties of the so called dynamic boundary particles (DBPs) are described in this paper. These boundary particles share the same equations of continuity and state as the moving particles placed inside the domain, although their positions and velocities remain unaltered in time or are externally prescribed. Theoretical and numerical calculations were carried out to study the collision between a moving particle and a boundary particle. The boundaries were observed to behave in an elastic manner in absence of viscosity. They allow the fluid particles to approach till a critical distance depending on the energy of the incident particle. In addition, a dam break confined in a box was used to check the validity of the approach. The good agreement between experiments and numerical results shows the reliability of DBPs.     

SPH方法在剪切式碰撞能量吸收器中的应用

剪切式碰撞能量吸收器是提高汽车被动安全性的一种实用新型设计。建立了剪切式碰撞能量吸收器数值分析模型,采用SPH方法数值模拟了碰撞吸能器的碰撞过程,研究了其碰撞吸能特性,并通过不同碰撞速度的台车碰撞试验进行了试验验证研究。数值模拟结果同试验结果相符,表明SPH方法在碰撞吸能器性能研究中是行之有效的数值计算方法。同时分析了在碰撞吸能器的复杂碰撞情况下,SPH方法相比有限元法的优势。     

SPH with the multiple boundary tangent method

In this article, we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics (SPH) method. Benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries. As well, current SPH boundary approaches do not properly treat curved boundaries in complicated flow domains. These drawbacks have been remedied in a new boundary treatment method presented in this article, called the multiple boundary tangent (MBT) approach. In this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions. The two benchmark problems chosen are the lid‐driven cavity problem, and flow over a cylinder. The SPH solutions using the MBT approach and the results from literature are in very good agreement. These solutions involved solid boundaries, but the approach presented herein should be extendable to time‐evolving, free‐surface boundaries. Copyright © 2008 John Wiley & Sons, Ltd.     

充液容器跌落过程的SPH模拟方法

采用SPH和FEM耦合的方法,考虑流固耦合效应的影响,对典型矩形薄壁充液容器的跌落过程进行数值仿真。通过对容器的变形、自由液面的运动、液体对容器的动压变化、容器的应力状态和所受动态激励的情况进行分析,说明SPH方法在充液容器的冲击问题研究中是行之有效的数值计算方法。同时数值算例的结果对充液容器的设计和试验具有一定的参考价值。     

A robust weakly compressible SPH method and its comparison with an incompressible SPH

This paper presents a comparative study for the weakly compressible (WCSPH) and incompressible (ISPH) smoothed particle hydrodynamics methods by providing numerical solutions for fluid flows over an airfoil and a square obstacle. Improved WCSPH and ISPH techniques are used to solve these two bluff body flow problems. It is shown that both approaches can handle complex geometries using the multiple boundary tangents (MBT) method, and eliminate particle clustering‐induced instabilities with the implementation of a particle fracture repair procedure as well as the corrected SPH discretization scheme. WCSPH and ISPH simulation results are compared and validated with those of a finite element method (FEM). The quantitative comparisons of WCSPH, ISPH and FEM results in terms of Strouhal number for the square obstacle test case, and the pressure envelope, surface traction forces, and velocity gradients on the airfoil boundaries as well as the lift and drag values for the airfoil geometry indicate that the WCSPH method with the suggested implementation produces numerical results as accurate and reliable as those of the ISPH and FEM methods. Copyright © 2011 John Wiley & Sons, Ltd.     

应用SPH方法模拟近水面爆炸

应用无网格光滑粒子法对近水面爆炸问题进行了数值模拟,模拟再现爆轰产物、水以及空气多种介质间的相互作用过程。相对于传统的基于网格的数值方法,SPH方法的无网格、Lagrangian性质能够自然、简单地处理爆炸产生的大变形、多介质作用等问题。通过自编程序,选用有效合理的SPH理论方法再现了近水面爆炸的复杂物理现象,定性分析了近水面爆炸的密度和压强的分布情况。通过数值模拟,近水面爆炸的复杂物理现象得到了合理再现,同时验证说明了光滑粒子法能有效地对可压缩多介质流进行计算模拟。     

一种SPH流体仿真边界校正方法

使用光滑粒子流体动力学方法进行流体仿真,并提出一种边界校正方法。使用快速泊松盘采样算法对容器边界进行采样,生成边界粒子,对边界粒子质量进行差值估算,计算边界粒子对流体粒子的作用力,以此来仿真流体与边界的相互作用。该方法可避免穿刺、滞留等现象的发生。通过实验验证了该算法的正确性。     

Normalized SPH with stress points

Smoothed particle hydrodynamics is extended to a normalized, staggered particle formulation with boundary conditions. A companion set of interpolation points is introduced that carry the stress, velocity gradient, and other derived field variables. The method is stable, linearly consistent, and has an explicit treatment of boundary conditions. Also, a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement. Test problems show that these improvements lead to increased accuracy and stability. Published in 2000 by John Wiley & Sons, Ltd.     

SPH simulations of flow around a periodic array of cylinders confined in a channel

In this paper, we present a numerical study on the performance of SPH in the case of a very viscous flow of a Newtonian liquid around a linear array of cylinders confined in a channel. This specific flow problem, being characterized by a complex mixing of both shear and extensional behaviour, allows to quantify systematically the accuracy of the standard SPH in, up to now, rarely considered complex geometries. Global accuracy tests based on the estimation of the dimensionless drag force acting on the cylinder as well as the inspection of the local velocity profiles are considered and compared with reference solutions. In agreement with previous findings, the impact of two numerical parameters, namely the smoothing length h and particle spacing Δx, is discussed and found to be crucial for the overall order of convergence of the method. In particular, accurate results can be obtained which are in very good agreement with standard mesh‐based methods provided that the number of neighbours is chosen properly. The present results, being based on a detailed convergence analysis in a complex flow problem, justify the applicability of the SPH method to more complex wall‐bounded flows upon critical choice of the model parameters and at the same time can serve as a useful benchmark test for further modelling improvement in the field. Copyright © 2010 John Wiley & Sons, Ltd.     

弹性力学静力问题的SPH方法

光滑质点流体动力学方法(Smoothed Particle Hydrodynamics,SPH)是纯Lagrangian方法,可用于模拟流体或固体的静动力学问题。不需网格系统即可进行空间导数计算,可避免Lagrangian网格在处理结构变形计算时的缠结和扭曲问题。但经典SPH方法计算二阶以上导数时易引起计算失败。该文提出一种改进的SPH方法,既可避免二阶导数的计算失败,又可提高二阶导数的精度。据此计算了均布荷载作用下两端固结梁的变形问题。经与ANSYS计算结果比较,该方法的计算足够精确。虽以弹性力学小变形问题为例,但结论可推广到大变形情形。     

Stabilized updated Lagrangian corrected SPH for explicit dynamic problems

Smooth particle hydrodynamics with a total Lagrangian formulation are, in general, more robust than finite elements for large distortion problems. Nevertheless, updating the reference configuration may still be necessary in some problems involving extremely large distortions. However, as discussed here, a standard updated formulation suffers the presence of zero‐energy modes that are activated and may completely spoil the solution. It is important to note that, unlike an Eulerian formulation, the updated Lagrangian does not present tension instability but only zero‐energy modes. Here a stabilization technique is incorporated to the updated formulation to obtain an improved method without any mechanisms and which is capable to solve problems with extremely large distortions. Copyright © 2006 John Wiley & Sons, Ltd.     

A saturated discrete particle model and characteristic-based SPH method in granular materials

Based on the discrete particle model for solid-phase deformation of granular materials consisting of dry particulate assemblages, a discrete particle–continuum model for modelling the coupled hydro-mechanical behaviour in saturated granular materials is developed. The motion of the interstitial fluid is described by two parallel continuum schemes governed by the averaged incompressible N–S equations and Darcy's law, respectively, where the latter one can be regarded as a degraded case of the former. Owing to the merits in both Lagrangian and mesh-free characters, the characteristic-based smoothed particle hydrodynamics (SPH) method is proposed in this paper for modelling pore fluid flows relative to the deformed solid phase that is modelled as packed assemblages of interacting discrete particles. It is assumed that the formulation is Lagrangian with the co-ordinate system transferring with the movement of the solid particles. The assumed continuous fluid field is discretized into a finite set of Lagrangian (material) points with their number equal to that of solid particles situated in the computational domain. An explicit meshless scheme for granular materials with interstitial water is formulated. Numerical results illustrate the capability and performance of the present model in modelling the fluid–solid interaction and deformation in granular materials saturated with water. Copyright © 2007 John Wiley & Sons, Ltd.     

SPH truncation error in estimating a 3D derivative

Following the procedure proposed in Quinlan et al. (Int. J. Numer. Meth. Engng. 2006; 66 :2064–2085) for a 1D generic derivative, a 3D formulation of the Smoothed Particle Hydrodynamics (SPH) truncation error (ε_T) has been derived and validated. We have then underlined the differences between traditional SPH simulations, which are not consistent, and estimations using renormalization, a first‐order consistency technique. The consistency order is here defined as the highest degree of a generic polynomial function, which can be exactly reproduced by an SPH approximation. Under the homogeneous conditions assumed in our analyses renormalization generally reduces the relative truncation error by 1 or 2 orders of magnitude, both at inner points and boundary locations. Due to renormalization the error tends to a lowest constant value as the kernel support size (h) goes to zero, while in general with no consistency the error behaves like 1/h. In contrast to formulations without any consistency estimations, using renormalization there is a weak dependence of the error on the absolute value of the displacement of the particles from their volume barycentre (δ). In addition, for simulations with renormalization, the best choice for the kernel function seems to be the closest to Dirac's delta, while for the ones with no consistency, the preferences are altered. Furthermore, we observe that renormalization reduces the number of neighbors that are necessary to obtain a discretization error that is negligible with respect to the integral error. Copyright © 2011 John Wiley & Sons, Ltd.     

Truncation error in mesh‐free particle methods

A truncation error analysis has been developed for the approximation of spatial derivatives in smoothed particle hydrodynamics (SPH) and related first‐order consistent methods such as the first‐order form of the reproducing kernel particle method. Error is shown to depend on both the smoothing length h and the ratio of particle spacing to smoothing length, Δx/h. For uniformly spaced particles in one dimension, analysis shows that as h is reduced while maintaining constant Δx/h, error decays as h² until a limiting discretization error is reached, which is independent of h. If Δx/h is reduced while maintaining constant h (i.e. if the number of neighbours per particle is increased), error decreases at a rate which depends on the kernel function's smoothness. When particles are distributed non‐uniformly, error can grow as h is reduced with constant Δx/h. First‐order consistent methods are shown to remove this divergent behaviour. Numerical experiments confirm the theoretical analysis for one dimension, and indicate that the main results are also true in three dimensions. This investigation highlights the complexity of error behaviour in SPH, and shows that the roles of both h and Δx/h must be considered when choosing particle distributions and smoothing lengths. Copyright © 2005 John Wiley & Sons, Ltd.     

基于罚函数SPH新方法的水模拟充型过程的数值分析

同传统的网格法相比,光滑粒子流体动力学(SPH)方法非常适合于求解大变形和自由表面流动问题.阐述了SPH理论及其应用,并用罚函数处理流体与壁面的相互作用,以解决传统SPH本质边界条件不易施加的问题.对水模拟的允型过程实验进行数值分析,并和文献实验结果以及传统SPH进行对比,最终表明仿真结果与实验非常吻合,比传统SPH方...     

修正的SPH方法模拟强制旋转动边界问题研究

采用移动最小二乘(movingleastsquares,MLS)修正的光滑粒子流体动力学(smoothedparticlehydrodynamics,SPH)算法模拟了一个强制旋转动边界问题模型。提出了一种施加强制旋转粒子动边界方案。阐述了修正方法的原理并给出了具体的修正操作方法。同时,还构建了静边界模型,并分别使用修正的SPH以及商业软件FLUENT计算。对计算结果的比较分析表明：该修正SPH方法,能够消除压力振荡;该方法及动边界处理方案可以有效计算该强制动边界问题,为进一步计算更加复杂模型奠定了理论基础。同时,该方法对于不同领域中的动边界问题也具有一定参考价值。     

Development of computationally efficient augmented Lagrangian SPH for incompressible flows and its quantitative comparison with WCSPH simulating flow past a circular cylinder

In Lagrangian particle-based methods such as smoothed particle hydrodynamics (SPH), computing totally divergence-free velocity field in a flow domain with the smallest error possible is the most critical issue, which might be achieved through solving pressure Poisson equation implicitly with higher particle resolutions. However, implicit solutions are computationally expensive and may be particularly challenging in the solution of multiphase flows with highly nonlinear deformations as well as fluid-structure interaction problems. Augmented Lagrangian SPH (ALSPH) method is a new alternative algorithm as a prevalent pressure solver where the divergence-free velocity field is achieved by iterative calculation of velocity and pressure fields. This study investigates the performance of the ALSPH technique by solving a challenging flow problem such as two-dimensional flow around a cylinder within the Reynolds number range of 50 to 500 in terms of improved robustness, accuracy, and computational efficiency. The same flow conditions are also simulated using the conventional weakly compressible SPH (WCSPH) method. The results of ALSPH and WCSPH solutions are not only compared in terms of numerical validation/ verification studies, but also rigorous investigations are performed for all related physical flow characteristics, namely, hydrodynamic coefficients, frequency domain analyses, and velocity divergence fields.     

基于细分粒子的刚体破裂动画

为了模拟固体在外力作用下产生的破裂现象,提出一种采用细分粒子的刚体破裂的模拟算法.该算法首先将固体的四面体网格绑定到一系列离散的粒子上;再利用光滑粒子流体动力学（SPH）对线弹性力学方程进行离散求解;并采用粒子细分的算法来进行开裂面的生成和延展.最后实现了多个固体现象的模拟,如砖块碎落、砖墙受力倒塌等.文中算法可适用于刚体脆性破碎的动画应用.     

光学玻璃材料动态冲击试验与鸟撞研究

针对多光谱硫化锌(zinc sulfide,ZnS)光学玻璃材料用于飞行器时的鸟撞问题进行了研究。对多光谱硫化锌玻璃进行了中高应变率下的压缩试验获得其材料属性。鸟体采用光滑粒子流体动力学(smoothed particle hydrodynamics,SPH)方法建模,引入Gruneisen状态方程定义鸟体本构模型。对建立的鸟体模型进行撞击铝板的仿真并进行试验验证,验证结果表明建立的鸟体模型具有较高的精度,可以用于其它碰撞情景下的仿真。建立鸟撞玻璃平板的模型对多光谱硫化锌玻璃的耐鸟撞性能进行预测分析,仿真结果表明ZnS玻璃撞击部位和边缘约束的地方容易达到极限应变并发生失效;另外,随着鸟体撞击角度的减小玻璃表面的接触力峰值也逐渐减小。该研究所得结果可以为鸟体SPH方法的数值模拟以及多光谱硫化锌玻璃在飞行器上的应用提供参考。