Search algorithm,and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method

We first present a nonuniform box search algorithm with length of each side of the square box dependent on the local smoothing length, and show that it can save up to 70% CPU time as compared to the uniform box search algorithm. This is especially relevant for transient problems in which, if we enlarge the sides of boxes, we can apply the search algorithm fewer times during the solution process, and improve the computational efficiency of a numerical scheme. We illustrate the application of the search algorithm and the modified smoothed particle hydrodynamics (MSPH) method by studying the propagation of cracks in elastostatic and elastodynamic problems. The dynamic stress intensity factor computed with the MSPH method either from the stress field near the crack tip or from the J-integral agrees very well with that computed by using the finite element method. Three problems are analyzed. One of these involves a plate with a centrally located crack, and the other with three cracks on plates’s horizontal centroidal axis. In each case the plate edges parallel to the crack are loaded in a direction perpendicular to the crack surface. It is found that, at low strain rates, the presence of other cracks will delay the propagation of the central crack. However, at high strain rates, the speed of propagation of the central crack is unaffected by the presence of the other two cracks. In the third problem dealing with the simulation of crack propagation in a functionally graded plate, the crack speed is found to be close to the experimental one.     

Numerical simulation of reactive polymer flow during rotational molding using smoothed particle hydrodynamics method and experimental verification

A numerical study of flow behavior and the heat transfer process of reactive polymer in reactive rotational molding systems is carried out using the Smoothed Particle Hydrodynamics method (SPH). The flow of reactive polymer during rotational molding inside a horizontal rotating cylinder is modeled as the slightly compressible viscous fluid flow and with free surface. These simulations show the influence of process parameters on the flow of polymer. Especially, the influence of the change of viscosity on the flow, due to the chemical reactions, is simulated by using an adapted viscosity expression. For a constant rotational speed, several flow regimes are observed as follows: the fluid remains at the bottom of the mold with the formation of a thin layer on the surface if the viscosity of reactive polymer is very low (less than 850 Pa.s), the presence of cascades is shown (after the point of maximum height) if the viscosity is higher (850 to 980 Pa.s), and the fluid moves at the same speed as the mold if the viscosity is sufficiently high (approximately 1230 Pa.s). The heat transfer between the mold and reactive polymer is also simulated. The results of SPH simulations are then compared to the experimental results conducted on polyurethane (TPU) system based on 1,6-hexamethylene diisocyanate (HDI, Sigma Aldrich, France), polyethylene glycol (PEG 1000) as macrodiol, 1,3-propanediol (PDO), and dibutyl tin dilaurate (DBTDL).     

Numerical modeling of Kelvin–Helmholtz instability using smoothed particle hydrodynamics

This paper presents a Smoothed Particle Hydrodynamics (SPH) solution for the Kelvin–Helmholtz Instability (KHI) problem of an incompressible two‐phase immiscible fluid in a stratified inviscid shear flow with interfacial tension. The time‐dependent evolution of the two‐fluid interface over a wide range of Richardson number (Ri) and for three different density ratios is numerically investigated. The simulation results are compared with analytical solutions in the linear regime. Having captured the physics behind KHI, the effects of gravity and surface tension on a two‐dimensional shear layer are examined independently and together. It is shown that the growth rate of the KHI is mainly controlled by the value of the Ri number, not by the nature of the stabilizing forces. It was observed that the SPH method requires a Richardson number lower than unity (i.e. Ri?0.8) for the onset of KHI, and that the artificial viscosity plays a significant role in obtaining physically correct simulation results that are in agreement with analytical solutions. The numerical algorithm presented in this work can easily handle two‐phase fluid flow with various density ratios. Copyright © 2011 John Wiley & Sons, Ltd.     

Calculation of oblique impact and fracture of tungsten cubes using smoothed particle hydrodynamics

The oblique impact and fracture of tungsten cubes is simulated using the Smoothed Particle Hydrodynamics code MAGI. A series of experiments with well defined and repeatable results provide the basis for validation of an evolving scalar damage model within SPH.     

On the Galerkin formulation of the smoothed particle hydrodynamics method

In this paper, we propose a Galerkin‐based smoothed particle hydrodynamics (SPH) formulation with moving least‐squares meshless approximation, applied to free surface flows. The Galerkin scheme provides a clear framework to analyse several procedures widely used in the classical SPH literature, suggesting that some of them should be reformulated in order to develop consistent algorithms. The performance of the methodology proposed is tested through various dynamic simulations, demonstrating the attractive ability of particle methods to handle severe distortions and complex phenomena. Copyright © 2004 John Wiley & Sons, Ltd.     

An improvement for tensile instability in smoothed particle hydrodynamics

A corrective Smoothed-Particle Method (CSPM) is proposed to address the tensile instability and, boundary deficiency problems that have hampered full exploitation of standard smoothed particle hydrodynamics (SPH). The results from applying this algorithm to the 1-D bar and 2-D plane stress problems are promising. In addition to the advantage of being a gridless Lagrangian approach, improving the above two major obstacles in standard SPH makes it attractive for applications in computational mechanics.     

Development of smoothed particle hydrodynamics method and its application in the hydrodynamics of condensed matter

An improved smoothed particle hydrodynamics (SPH) method is described; in this method, the solution to the Riemann problem in strength media is described. Generalization of this approach to solving heat conduction problems is performed. The improved SPH method is used to solve a wide range of problems. Problems of heat conduction and volume energy release accompanied by spallation effects, simulation of high speed perforation, and propagation of failure waves in brittle materials are considered. Shock wave compression of porous materials and diffraction of detonation waves in heterogeneous explosives are simulated on the mesostructure scale.     

Simulation of particle-laden flow in a Humphrey spiral concentrator using dust-liquid smoothed particle hydrodynamics

This paper demonstrates that our extended smoothed particle hydrodynamics (SPH) model can successfully simulate multiphase flow in a Humphrey spiral concentrator (HSC) with two phases: powder and water. The powder phase in the model was assumed to be a continuum, as the spacing between particles in this state is much smaller than the typical length scale of flow. Further investigation was conducted on the influences of various design factors of the HSC, including the descent angle and curvature profile of the trough, during the separation of a binary mineral particle mixture.The model was validated by comparing the simulated results with the experimental results of Loveday and Cilliers (1994) as well as those of a novel lab-scale-experiment using a miniature of the HSC. The proposed SPH model accurately simulated dusty liquid flow in the HSC in both cases with an acceptable degree of accuracy relative to the experimental results. These studies are expected to be useful in future optimizations of HSC design and operating conditions.     

Discrete-element modelling and smoothed particle hydrodynamics: potential in the environmental sciences

Particle-based simulation methods, such as the discrete-element method and smoothed particle hydrodynamics, have specific advantages in modelling complex three-dimensional (3D) environmental fluid and particulate flows. The theory of both these methods and their relative advantages compared with traditional methods will be discussed. Examples of 3D flows on realistic topography illustrate the environmental application of these methods. These include the flooding of a river valley as a result of a dam collapse, coastal inundation by a tsunami, volcanic lava flow and landslides. Issues related to validation and quality data availability are also discussed.     

Modified smoothed particle hydrodynamics method and its application to transient problems

A modification to the smoothed particle hydrodynamics method is proposed that improves the accuracy of the approximation especially at points near the boundary of the domain. The modified method is used to study one-dimensional wave propagation and two-dimensional transient heat conduction problems.This work was supported by the ONR grant N00014-98-1-0300 and the ARO grant DAAD19-01-1-0657 to Virginia Polytechnic Institute and State University, and the AFOSR MURI grant to Georgia Tech that awarded a subcontract to Virginia Polytechnic Institute and State University. Opinions expressed in the paper are those of authors and not of the funding agencies.     

SPH原理、发展现状及热传导问题模型

对光滑粒子流体动力学方法（smoothed particle hydrodynamics,SPH）的基本技术原理及发展现状进行了综述,利用SPH方法对两个典型二维非线性动力学算例进行了数值模拟。同时,利用国外学者提出的方法初步探索了有关热传导问题的SPH模型。     

The use of smoothed particle hydrodynamics for simulating crystal growth from solution

Most numerical simulations for crystal growth processes have been performed using the finite volume and finite element techniques. Both of these methods require adaptive meshes to track the growth and dissolution interfaces. In addition, in the growth of ternary systems the solid and liquid phases must be solved simultaneously, which requires iterations at the interfacial boundaries between solid and liquid regions. Such iterations slow down simulations tremendously, and also lead to numerical instabilities. In order to address these issues, an alternative mesh-free, Lagrangian method, known as smoothed particle hydrodynamics (SPH) has been investigated. For the simulation, the liquid phase diffusion (LPD) growth of Si_xGe_1−x has been selected, since both experimental and finite volume simulation results were available for comparison. This comparison between finite volume and SPH simulations shows that although SPH has the potential to accurately model the crystal growth process, the number of SPH particles required for accurate predictions is upwards of 60,000 particles for the Reynolds number of the LPD system. This high number of particles translates to a computational time of approximately six times longer than the equivalent finite volume simulations. However, future improvements made to the relatively young SPH method may overcome such computational difficulties.     

Fatigue crack growth simulation in heterogeneous material using s-version FEM

A fully automatic fatigue crack growth simulation system is developed using the s-version Finite Element Method (s-FEM). This system is extended to fractures in heterogeneous materials. In a heterogeneous material, the crack tip stress field has a mixed-mode condition, and the crack growth path is affected by inhomogeneous materials and mixed-mode conditions. Stress intensity factors (SIFs) in the mixed-mode condition are evaluated using the virtual crack closure method (VCCM). The criteria for the crack growth amount and crack growth path are based on these SIFs, and the growing crack configurations are obtained. At first, the basic problem is solved, and the results are compared with some results available in the literature. It is shown that this system gives an adequate accurate estimation of the SIFs. Then, two-dimensional fatigue crack growth problems are simulated using this system. The first example is a plate with an interface between hard and soft materials. The cracks tend to grow in soft materials through the interface. A second example is a plate with distributed hard inclusions. The crack takes a zig-zag path by propagating around the hard inclusions. In each case, the crack growth path changes in a complicated manner. Changes of the SIFs values are also shown and discussed.     

Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems

We discuss the symmetric smoothed particle hydrodynamics (SSPH) method for generating basis functions 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 particle hydrodynamics (MSPH), the reproducing kernel particle method (RKPM), and the moving least squares (MLS) methods. For finding kernel estimates of derivatives of a function, the SSPH method does not use derivatives of the kernel function while other methods do, instead the SSPH method uses basis functions different from those employed to approximate 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 derivatives of the function. Results computed for a sine function defined on a one-dimensional domain reveal that the L ², the H ¹ and the H ² error norms of the kernel estimates of a function 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 sample 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 magnitudes 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 kernel function in the SSPH method. We use the revised super Gauss kernel function to find the displacement field in a linear elastic rectangular plate with a circular hole at its centroid and subjected to tensile loads on two opposite edges. Results given by the SSPH and the MSPH methods agree very well with the analytic solution of the problem. However, results computed with the SSPH method have smaller error norms than those obtained from the MSPH method indicating that the former will give a better solution than the latter. The SSPH method is also applied to study wave propagation in a linear elastic bar.     

Simulation of high velocity impact in fluid-filled containers using finite elements with adaptive coupling to smoothed particle hydrodynamics

This paper presents a numerical study on the simulation of impacts of projectiles on fluid-filled containers. The type of impact investigated leads to hydrodynamic ram (HRAM) and complete failure of the container shell. Two different numerical approaches are compared which are both implemented in a research hydrocode: a pure Lagrangian discretization with Finite Elements (FE) and element erosion, and a coupled adaptive FE/SPH discretization. The numerical results are compared with two reference experiments. The principal phenomenology including the container deformation could be modeled well with both methods. The coupled FE/SPH approach was superior in the reproduction of the projectile’s observed residual velocity, it is, however, computationally more expensive.     

Dynamic simulation of damage‐fracture transition in smoothed particles hydrodynamics shells

This paper presents a meshless method for the modeling of shell‐type structures in fast dynamics. The model is based on the Mindlin–Reissner theory and takes into account material and geometric nonlinearities. The phenomena that occur prior to rupture are dealt with using damage laws, while the rupture itself is represented through the introduction of sharp discontinuities. The method does not represent cracks explicitly, which makes the treatment of multicracking easier. The time discretization is carried out in the framework of explicit dynamics, and the spatial discretization is handled through the smoothed particles hydrodynamics method and the use of moving least square functions. The capabilities of the method are demonstrated using cracking, puncturing and fragmentation examples. Copyright © 2012 John Wiley & Sons, Ltd.     

Transient Fokker–Planck–Kolmogorov equation solved with smoothed particle hydrodynamics method

Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function ψ (pdf). The time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient Fokker–Planck–Kolmogorov (FPK) equation. The FPK equation is a conservation equation of a hypothetical or abstract fluid, which models the transport of probability. This paper presents a generalized formalism for the resolution of the transient FPK equation by using the well‐known mesh‐free Lagrangian method, smoothed particle hydrodynamics). Numerical implementation shows notable advantages of this method in an unbounded state space: (1) the conservation of total probability in the state space is explicitly written; (2) no artifact is required to manage far‐field boundary conditions; (3) the positivity of the pdf is ensured; and (4) the extension to higher dimensions is straightforward. Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions, even slightly dispersed distributions. The FPK equation is solved without any a priori knowledge of the stationary distribution, just a precise representation of the initial distribution is required.Copyright © 2013 John Wiley & Sons, Ltd.     

Smoothed particle hydrodynamics for numerical simulation of underwater explosion

 Underwater explosion arising from high explosive detonation consists of a complicated sequence of energetic processes. It is generally very difficult to simulate underwater explosion phenomena by using traditional grid-based numerical methods due to the inherent features such as large deformations, large inhomogeneities, moving interface and so on. In this paper, a meshless, Lagrangian particle method, smoothed particle hydrodynamics (SPH) is applied to simulate underwater explosion problems. As a free Lagrangian method, SPH can track the moving interface between the gas produced by the explosion and the surrounding water effectively. The meshless nature of SPH overcomes the difficulty resulted from large deformations. Some modifications are made in the SPH code to suit the needs of underwater explosion simulation in evolving the smoothing length, treating solid boundary and material interface. The work is mainly focused on the detonation of the high explosive, the interaction of the explosive gas with the surrounding water, and the propagation of the underwater shock. Comparisons of the numerical results for three examples with those from other sources are quite good. Major features of underwater explosion such as the magnitude and location of the underwater explosion shock can be well captured. Received: 2 April 2002 / Accepted: 20 September 2002     

基于二阶Godunov格式的SPH方法模拟弹塑性波的传播

针对一阶Godunov格式的SPH方法的计算精度和激波分辨率不高的问题,提出了二阶Godunov格式的SPH方法。新方法在求解相互作用的粒子间黎曼问题时,认为粒子内物理量呈线性分布,用线性插值后求得的值作为黎曼问题的初始值,然后把黎曼解和Taylor展开引入到SPH方法中。应用新方法对一维弹塑性应力波的传播进行了数值模拟,并与一阶Godunov格式的SPH方法进行比较.计算结果显示新方法有效地提高了计算精度和激波分辨率,同时验证了它的稳定性。