Smoothed Particle Hydrodynamics Algorithm Research Articles

A Runge–Kutta–Chebyshev SPH algorithm for elastodynamics

A stable and accurate Smoothed Particle Hydrodynamics (SPH) method is proposed for solving elastodynamics in solid mechanics. The SPH method is mesh-free, and it promises to overcome most of disadvantages of the traditional finite element techniques. The absence of a mesh makes the SPH method very attractive for those problems involving large deformations, moving boundaries and crack propagation. However, the conventional SPH method still has significant limitations that prevent its acceptance among researchers and engineers, namely the stability and computational costs. In approximating unsteady problems using the SPH method, attention should be given to the choice of time integration schemes as accuracy and efficiency of the SPH solution may be limited by the timesteps used in the simulation. This study presents an attempt to reconstruct an unconditionally stable SPH method for elastodynamics. To achieve this objective we implement an explicit Runge–Kutta Chebyshev scheme with extended stages in the SPH method. This time stepping scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. Numerical results are shown for several test problems in elastodynamics. For the considered elastic regimes, the obtained results demonstrate the ability of our new algorithm to better maintain the shape of the solution in the presence of shocks.

A new SPH-based approach to simulation of granular flows using viscous damping and stress regularisation

The smoothed particle hydrodynamics (SPH) method was recently extended to simulate granular materials by the authors and demonstrated to be a powerful continuum numerical method to deal with the post-flow behaviour of granular materials. However, most existing SPH simulations of granular flows suffer from significant stress oscillation during the post-failure process, despite the use of an artificial viscosity to damp out stress fluctuation. In this paper, a new SPH approach combining viscous damping with stress/strain regularisation is proposed for simulations of granular flows. It is shown that the proposed SPH algorithm can improve the overall accuracy of the SPH performance by accurately predicting the smooth stress distribution during the post-failure process. It can also effectively remove the stress oscillation issue in the standard SPH model without having to use the standard SPH artificial viscosity that requires unphysical parameters. The predictions by the proposed SPH approach show very good agreement with experimental and numerical results reported in the literature. This suggests that the proposed method could be considered as a promising continuum alternative for simulations of granular flows.

An improved weakly compressible SPH method for simulating free surface flows of viscous and viscoelastic fluids

In this paper, an improved weakly compressible smoothed particle hydrodynamics (SPH) method is proposed to simulate transient free surface flows of viscous and viscoelastic fluids. The improved SPH algorithm includes the implementation of (i) the mixed symmetric correction of kernel gradient to improve the accuracy and stability of traditional SPH method and (ii) the Rusanov flux in the continuity equation for improving the computation of pressure distributions in the dynamics of liquids. To assess the effectiveness of the improved SPH algorithm, a number of numerical examples including the stretching of an initially circular water drop, dam breaking flow against a vertical wall, the impact of viscous and viscoelastic fluid drop with a rigid wall, and the extrudate swell of viscoelastic fluid have been presented and compared with available numerical and experimental data in literature. The convergent behavior of the improved SPH algorithm has also been studied by using different number of particles. All numerical results demonstrate that the improved SPH algorithm proposed here is capable of modeling free surface flows of viscous and viscoelastic fluids accurately and stably, and even more important, also computing an accurate and little oscillatory pressure field.

A fluid simulation method based on two-dimensional smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) method is a kind of meshless method, which is used to solve the problem of fluid simulation without complex operations of the grids. To reduce the computational complexity, SPH method based on the two-dimensional shallow water equations is employed to establish a fluid model. In large scale scenes, taking into account the high computational complexity and the serious distortion problems, in this paper we introduce an improved two-dimensional SPH algorithm according to the shallow water equations. The proposed method with two-dimensional complexity is obtained by discretizing the two-dimensional shallow water equations with SPH, and the depth of water is introduced as the particle's property. The problem of increased amount of calculation cannot be well solved by using traditional neighboring particle search method based on tree structure. To improve the efficiency of search and simplify the search operation of neighborhood particles, in this paper we introduce a point-in-box search algorithm and put forward a neighboring particles searching method on the basis of dynamic grid. Besides, for large scale scenes, by considering that the virtual particle method provides slow computation speed with complex boundary condition, the type-one virtual particles are utilized to ensure that the borders can be calculated precisely by combining the punish force to prevent the phenomenon of particle boundary penetrating. Therefore, a method is further obtained to handle boundary condition efficiently by combining the virtual particles with punish force in this paper. In the process of rendering, the fluid surface is first determined by mapping and interpolating particles into regular grids without the complex reconstruction of surface in three-dimensional. Then, we utilize the bilinear interpolation method to deal with the problem of missing values, and the surface grids are further densified. With OpenSceneGraph three-dimensional render engine, OpenGL Shading Language is adopted to speed up the rendering speed, and in this way, the real-time fluid simulation of large scale scenes can be further achieved. With the basic KD tree searching method employed in the simulations, the comparative experiments are provided to verify effectiveness of the proposed searching method based on dynamic grid. Given the data set obtained from random points, experimental results demonstrate that the method in this paper can be used to solve the problem of neighboring particles searching in large scale scenes. To show the effectiveness of the proposed method on the basis of the virtual particles and the punish force, another experiment based on the collapsing of a water column is further provided. Besides, in this paper we conduct an experiment on a certain actual reservoir terrain to prove that the proposed method can be applied to fluid simulation of large scale scenes.

Smooth particle hydrodynamics simulation of damage induced by a spherical indenter scratching a viscoplastic material

We present an implementation of a (mesh-free) smooth particle hydrodynamics (SPH) algorithm, intended for the application to solid bodies, and use it to simulate scratch-induced surface damage on an elasto-viscoplastic material. If conventional SPH is used to simulate solids, an unphysically high artificial viscosity is required to damp strong oscillatory modes and thus stabilize the system. To overcome these intrinsic difficulties associated with modeling solid bodies, the recently implemented so-called total-Lagrangian SPH utilizes an hourglass control scheme similar to what is known from finite element algorithms. Elasto-viscoplastic material properties are modeled by using the Mie–Grüneisen equation of state and the Johnson–Cook model. The material parameters are selected to reproduce the strain-stress behavior of annealed oxygen-free high conductivity copper. The spherical indenter is modeled as a rigid sphere. The topographies of the calculated scratch-induced surface damage are in excellent agreement with experimental scratch tests carried out at comparable normal loads. We also calculate the real contact area between the indenter and the surface, allowing a sound estimation of the scratch hardness.

Time Domain Simulation of Sound Waves Using Smoothed Particle Hydrodynamics Algorithm with Artificial Viscosity

Smoothed particle hydrodynamics (SPH), as a Lagrangian, meshfree method, is supposed to be useful in solving acoustic problems, such as combustion noise, bubble acoustics, etc., and has been gradually used in sound wave computation. However, unphysical oscillations in the sound wave simulation cannot be ignored. In this paper, an artificial viscosity term is added into the standard SPH algorithm used for solving linearized acoustic wave equations. SPH algorithms with or without artificial viscosity are both built to compute sound propagation and interference in the time domain. Then, the effects of the smoothing kernel function, particle spacing and Courant number on the SPH algorithms of sound waves are discussed. After comparing SPH simulation results with theoretical solutions, it is shown that the result of the SPH algorithm with the artificial viscosity term added attains good agreement with the theoretical solution by effectively reducing unphysical oscillations. In addition, suitable computational parameters of SPH algorithms are proposed through analyzing the sound pressure errors for simulating sound waves.

Numerical simulation of the complete rock blasting response by SPH–DAM–FEM approach

Rock mass blasting is a complex process which involves the coupling of both discontinuous and continuous media. This paper aims to provide a numerical simulator capable of reproducing the complete blasting response in rock mass by incorporating large deformations, damage distribution and blasting vibration with a single numerical model. A Smooth Particle Hydrodynamics (SPH) algorithm and a modified blasting damage model were introduced into the explicit finite element code LS-DYNA with a subroutine interface in order to develop the coupled SPH–DAM–FEM numerical simulator. The coupled SPH technology with the modified Damage model is used to simulate the near zone in the blasting process and the FEM to capture the far field response of the rock. The complete dynamic response of a presplit blast and a bench blast was investigated with the developed coupled model in two case studies regarding a blasting excavation of high rock slopes in China. The discontinuous characteristic of the blast process, large deformations and damage were analyzed carefully. The reproduction of the damage contour and vibration was compared with site measurements. From the analysis conducted, it is shown that the results produced by the developed coupled model match quite well with field observations. This demonstrates the capability of the proposed coupled model. To complete the analysis, a sensitivity analysis of the main SPH parameters was studied parametrically to obtain a better understanding and expose more capabilities and applications of the proposed coupled approach.

Dynamic response of bird strike on aluminium foam-based sandwich panels

This study attempts to establish an experimental–numerical framework to simulate the dynamic response of aluminium foam-based sandwich panels subjected to bird strike. The numerical model is developed with the non-linear dynamic finite element code PAM-CRASH, where the smooth particle hydrodynamics (SPH) algorithm is used to model the bird, an elastic–plastic material model with isotropic damage is used to describe aluminium skin and the Deshpande–Fleck foam model is used to describe the foam core. Mechanical tests of the skin and foam materials as well as bird-strike tests of a double sandwich panel are conducted and the experimental results are used to calibrate the model parameters. The bird-strike simulation results show reasonably good agreement with test data, indicating the simulation method is capable of predicting the dynamic response of aluminium foam-based sandwich panels in the event of bird strike. Finally a series of parametric studies are conducted to examine the effects of foam thickness and arrangement on the deformation behaviour and energy-absorbing ability of the sandwich panels to illustrate the potential application of the numerical model in the design of aluminium foam-based sandwich structures.

Formation of Prestellar Cores via Non-Isothermal Gas Fragmentation

AbstractSheet-like clouds are common in turbulent gas and perhaps form via collisions between turbulent gas flows. Having examined the evolution of an isothermal shocked slab in an earlier contribution, in this work we follow the evolution of a sheet-like cloud confined by (thermal) pressure and gas in it is allowed to cool. The extant purpose of this endeavour is to study the early phases of core-formation. The observed evolution of this cloud supports the conjecture that molecular clouds themselves are three-phase media (comprising viz. a stable cold and warm medium, and a third thermally unstable medium), though it appears, clouds may evolve in this manner irrespective of whether they are gravitationally bound. We report, this sheet fragments initially due to the growth of the thermal instability (TI) and some fragments are elongated, filament-like. Subsequently, relatively large fragments become gravitationally unstable and sub-fragment into smaller cores. The formation of cores appears to be a three stage process: first, growth of the TI leads to rapid fragmentation of the slab; second, relatively small fragments acquire mass via gas-accretion and/or merger and third, sufficiently massive fragments become susceptible to the gravitational instability and sub-fragment to form smaller cores. We investigate typical properties of clumps (and smaller cores) resulting from this fragmentation process. Findings of this work support the suggestion that the weak velocity field usually observed in dense clumps and smaller cores is likely seeded by the growth of dynamic instabilities. Simulations were performed using the smooth particle hydrodynamics algorithm.

Density-Independent Smoothed Particle Hydrodynamics for a Non-Ideal Equation of State

Abstract The smoothed particle hydrodynamics (SPH) method is a useful numerical tool to study a variety of astrophysical and planetlogical problems. However, it turned out that the standard SPH algorithm has problems in dealing with hydrodynamical instabilities. This problem is due to the assumption that the local density distribution is differentiable. In order to solve this problem, a new SPH formulation, which does not require the differentiability of the density, was proposed. This new SPH method improved the treatment of hydrodynamical instabilities. This method, however, is applicable only to the equation of state (EOS) of an ideal gas. In this paper, we describe how to extend the new SPH method to non-ideal EOS. We present the results of various standard numerical tests for non-ideal EOS. Our new method works well for non-ideal EOS. We conclude that our new SPH can handle hydrodynamical instabilities for an arbitrary EOS, and that it is an attractive alternative to the standard SPH.

Bulk viscosity effects in event-by-event relativistic hydrodynamics

Bulk viscosity effects on the collective flow harmonics in heavy-ion collisions are investigated, on an event-by-event basis, using a newly developed 2+1 Lagrangian hydrodynamic code named v-usphydro, which implements the smoothed particle hydrodynamics algorithm for viscous hydrodynamics. A new formula for the bulk viscous corrections present in the distribution function at freeze-out is derived, starting from the Boltzmann equation for multi-hadron species. Bulk viscosity is shown to enhance the collective flow Fourier coefficients from ${v}_{2}({p}_{T})$ to ${v}_{5}({p}_{T})$ when ${p}_{T}\ensuremath{\sim}1$--2.5 GeV, even when the bulk viscosity to entropy density ratio, $\ensuremath{\zeta}/s$, is significantly smaller than $1/(4\ensuremath{\pi})$.

A SPH-based particle method for simulating 3D transient free surface flows of branched polymer melts

In this article we present a particle method based on smoothed particle hydrodynamics (SPH) for simulating three-dimensional (3D) transient free surface flows of branched polymer melts which are governed by the single equation version of the eXtended Pom-Pom (XPP) model. In order to remedy the so-called tensile instability which results in particle clustering and unphysical fracture in the state of fluid stretching, both artificial stress and artificial viscosity are incorporated into the momentum equation. For facilitating the implementation of the wall boundary condition in 3D space, an enhanced treatment of solid boundaries is proposed in this paper. The proposed SPH algorithm is validated by the excellent agreement between the numerical and the analytical solutions of Hagen-Poiseuille flow of an Oldroyd-B fluid. To demonstrate the ability of the numerical method in simulating free surface flows of branched polymer melts, the fall and impact on a rigid plate of a XPP fluid droplet is investigated for a wide range of the relevant parameters. Specifically, the convergence of the numerical results is discussed by three particle sizes of different levels of refinement. To provide some additional applications of 3D free surface flows of branched polymer melts, the challenging jet buckling and rod-climbing effect of viscoelastic fluids are further simulated and analyzed. All numerical results establish the capability of the SPH-based particle method to simulate 3D transient free surface flows of branched polymer melts in an entirely mesh-free framework.

RESEARCH DEVELOPMENTS OF SMOOTHED PARTICLE HYDRODYNAMICS METHOD AND ITS COUPLING WITH FINITE ELEMENT METHOD

The merits and shortcomings are existed simultaneously in finite element method (FEM) and smoothed particle hydrodynamics (SPH) method, in which these methods are established based on Lagrange system. In order to makes use of high computational efficiency of FEM and advantages of SPH method, such as naturally simulation of large deformation of material, the coupling algorithm of FEM with SPH method is presented to calculate region of small deformation by FEM and simulate region of large deformation by SPH method. In this paper, the research and application of FEM, SPH and FEM-SPH coupling algorithm are reviewed. And the problems existing in these methods are also discussed. Finally, the computational accuracy and efficiency of FEM, SPH and FEM-SPH coupling algorithm are investigated by a numerical example, which is a reference for research fellows.

Simulation of 2-D Dam Break Using Improved Incompressible Smoothed Particle Hydrodynamics Based on Projection Method

This paper deals with numerical modeling of water flow which is generated by the break of a dam. The problem is solved by applying a new Incompressible Smoothed Particle Hydrodynamics (ISPH) algorithm based on projection method. The proposed ISPH model has two steps. In the first step, the incompressibility of fluid is maintained regarding to the changes of intermediate and initial particles densities at the first half-time step (stability step). In the second step, by computing the divergence of the intermediate secondary velocity at the second half-time step (accuracy step), the incompressibility is satisfied completely. In fact, by using this method both stability and accuracy are increased. The simulation illustrates the formation and subsequent propagation of a wave over the horizontal plane. It is shown that the model predictions compare well with experimental data.

Balancing the source terms in a SPH model for solving the shallow water equations

A shallow flow generally features complex hydrodynamics induced by complicated domain topography and geometry. A numerical scheme with well-balanced flux and source term gradients is therefore essential before a shallow flow model can be applied to simulate real-world problems. The issue of source term balancing has been exhaustively investigated in grid-based numerical approaches, e.g. discontinuous Galerkin finite element methods and finite volume Godunov-type methods. In recent years, a relatively new computational method, smooth particle hydrodynamics (SPH), has started to gain popularity in solving the shallow water equations (SWEs). However, the well-balanced problem has not been fully investigated and resolved in the context of SPH. This work aims to discuss the well-balanced problem caused by a standard SPH discretization to the SWEs with slope source terms and derive a corrected SPH algorithm that is able to preserve the solution of lake at rest. In order to enhance the shock capturing capability of the resulting SPH model, the Monotone Upwind-centered Scheme for Conservation Laws (MUSCL) is also explored and applied to enable Riemann solver based artificial viscosity. The new SPH model is validated against several idealized benchmark tests and a real-world dam-break case and promising results are obtained.

Examining a modified algorithm of smoothed particle hydrodynamics for a high velocity perforation of an aluminum beam

The wide spreading of utilizing of smoothed particle hydrodynamics (SPH) for numerical studies of the complex and high rate deformations of continuums, led the current study to gain a more reliable simulation by employing a modified compressible smoothed particle hydrodynamics (MCSPH) algorithm which could be a more accurate and stable technique in high tension regions, in despite of incompressible standard SPH. The main feature of the modified compressible SPH algorithm relies on a three steps solution procedure to calculate the pressure gradient, the deviatoric stress tensor, and the body forces separately. This algorithm is free of any artificial viscosity in its formulations, as well as welcoming to compressible effects which permits the pressure shock waves in high rate plastic deformation. To examine the accuracy of the algorithm, a benchmark problem of colliding rubber cylinders was simulated first and then a high velocity perforation process of an aluminum beam struck by a rigid projectile was simulated in various projectile speeds, and the failure response of the beam in each case was accompanied by crack propagation process. The prominent capability of the utilized MCSPH can be more illustrated when it was used in simulation of thickness crack propagation a tiny crack paths and defragmentation which can be encountered as a not easy numerical case study. The adequate assurance has been more fortified when the results were compared to those reported from a Finite Element method study.

An approach for the coupled simulation of machining processes using multibody system and smoothed particle hydrodynamics algorithms

The prediction accuracy of a simulation method is limited by its theoretical background. This fact can lead to disadvantages regarding the simulation quality when investigating systems of high complexity, e.g. containing components showing a fairly different behavior. To overcome this limitation, co-simulation approaches are used more and more, combining the advantages of different simulation disciplines. That is why we propose a new strategy for the dynamic simulation of cutting processes. The method couples Lagrangian particle methods, such as the smoothed particle hydrodynamics (SPH) method, and multibody system (MBS) tools using co-simulations. We demonstrate the capability of the new approach by providing simulation results of an orthogonal cutting process and comparing them with experimental data.

Influence of Smoothing Length and Virtual Particles on SPH Accuracy

 Abstract—Using a well known 2-D transient heat conduction benchmark, a parametric study is conducted to identify the best values for the smoothing length (h) as function of particles spacing in order to minimize the error in the temperature distribution particularly at the center of the domain when applying the Smooth Particle Hydrodynamics (SPH) algorithm. Also the effect of the virtual particles on the numerical solution has been investigated. The results showed that the predefined smoothing length can have significant influence on the accuracy of the prediction and using the optimum value leads to minimum error in temperature prediction values. The results obtained with and without virtual particles has been presented and compared with the analytical solution.

SPH simulations of three-dimensional non-Newtonian free surface flows

In this paper, the smoothed particle hydrodynamics (SPH) method is extended to deal with three-dimensional (3D) non-Newtonian flows with complex free surfaces, in which the viscosity is modeled using the Cross model. In order to alleviate the so-called tensile instability which leads to particle clustering and unphysical fracture in fluid stretching, an artificial stress term is particularly incorporated into the momentum equation. For convenience in implementation of wall boundary condition in 3D, an enhanced treatment of solid boundaries is proposed to improve the computational performance. Parallelization is also developed to ensure affordable computational time of simulations involving millions of particles. The proposed SPH algorithm is validated by solving the Hagen–Poiseuille flow and comparing the SPH results with the available analytical solutions. To demonstrate the ability of the numerical method in simulating 3D non-Newtonian flows with free surfaces, three challenging engineering applications, including the impacting droplet, molding injection of a thin plate mold and a Z-shaped mold, and jet buckling, are investigated. It is found that the shear-thinning behavior can be well displayed in all cases, and the proposed SPH algorithm is stable and fairly accurate and agrees well with the available data.

On the evolution of irradiated turbulent clouds: a comparative study between modes of triggered star formation

Here we examine the evolution of irradiated clouds using the Smoothed Particle Hydrodynamics ({\small SPH}) algorithm coupled with a ray-tracing scheme that calculates the position of the ionisation-front at each timestep. We present results from simulations performed for three choices of {\small IR}-flux spanning the range of fluxes emitted by a typical {\small B}-type star to a cluster of {\small OB}-type stars. The extent of photo-ablation, of course, depends on the strength of the incident flux and a strong flux of {\small IR} severely ablates a {\small MC}. Consequently, the first star-formation sites appear in the dense shocked layer along the edges of the irradiated cloud. Radiation-induced turbulence readily generates dense filamentary structure within the photo-ablated cloud although several new star-forming sites also appear in some of the densest regions at the junctions of these filaments. Prevalent physical conditions within a {\small MC} play a crucial role in determining the mode, i.e., filamentary as compared to isolated pockets, of star-formation, the timescale on which stars form and the distribution of stellar masses. The probability density functions ({\small PDF}s) derived for irradiated clouds in this study are intriguing due to their resemblance with those presented in a recent census of irradiated {\small MC}s. Furthermore, irrespective of the nature of turbulence, the protostellar mass-functions({\small MF}s) derived in this study follow a power-law distribution. When turbulence within the cloud is driven by a relatively strong flux of {\small IR} such as that emitted by a massive {\small O}-type star or a cluster of such stars, the {\small MF} approaches the canonical form due to Salpeter, and even turns-over for protostellar masses smaller than $\sim$0.2 M$_{\odot}$.