Smoothed Particle Hydrodynamics Scheme Research Articles

Numerical Simulation of Squeezed Flow of a Viscoplastic Material by a Three-step Smoothed Particle Hydrodynamics Method

In the current work, the mesh free Smoothed Particle Hydrodynamics (SPH) method, was employed to numerically investigate the transient flow of a viscoplastic material. Using this method, large deformation of the sample and its free surface boundary were captured without the cumbersome process of the grid generation. This three-step SPH scheme employs an explicit predictor-corrector technique and the incompressibility characteristic of the material was guaranteed by solving a pressure Poisson equation. The Papanastasiou constitutive model was also utilized in the simulations to study the compression of the sample under both constant load and constant velocity conditions. The no-slip boundary condition was satisfied by projecting the velocity of the viscoplastic material on the wall particles. In order to validate the fidelity of this numerical method during the compression of the samples, the resultant load at constant velocity as well as the height change of the sample for a constant load were computed and compared with other published results. The results indicated that this method could be employed as a reliable technique to simulate such highly deformable viscoplastic deformation of the materials.

A theoretical mesh-free scheme to model viscous drop interactions: a particle-based method

Here a Lagrangian mesh-free formalism is presented to simulate the coalescence process between three unequal-sized liquid drops in the three-dimensional space. The surface tension forces acting on the surface of the drops cause the formation of a circular flat section when the droplets collide. The effect of polydispersity on the collision dynamics is simulated using a set of droplets with radius around 30 μm. It is important to see that the inhomogeneous distribution of the droplets size results in very important changes on the drops dynamics. The smoothed particle hydrodynamics scheme proposed here can be used to model situations where a continuum phase is included in the problem. The velocity vector fields are computed for each situation, and it can be seen that in the zone of contact between the droplets, there is an increment of the velocity value. This is due to the pressure distribution inside the drops.

NUMERICAL MODELING OF TSUNAMI-INDUCED HYDRODYNAMIC FORCES ON ONSHORE STRUCTURES USING SPH

In this paper, the simulation of the violent impact of tsunami-like bores with a square column is performed using a single-phase, weakly compressible three-dimensional Smoothed Particle Hydrodynamics (SPH) model. In order to avoid large fluctuations in the pressure field and to obtain accurate simulations of the hydrodynamic forces, a Riemann solver-based formulation of the SPH method is utilized. Large-scale physical experiments conducted by the authors are reproduced using the numerical model. Time-histories of the water surface elevation as well as time-histories of the pressure distribution and net total force acting on the column are successfully compared. As observed in previous breaking wave impact studies, results show that the magnitude and duration of the impulsive force at initial bore impact depend on the degree of entrapped air in the bore-front. Although ensuring a stable pressure field, the Riemann solver-based SPH scheme is believed to induce excessive numerical diffusion, as sudden and large water surface deformations, such as splashing at initial bore impact, are marginally reproduced. To investigate this particular issue, the small-scale physical experiment of Kleefsman et al. (2005) is also considered and modeled.

Calibrating an updated smoothed particle hydrodynamics scheme within gcd+

We adapt a modern scheme of smoothed particle hydrodynamics (SPH) to our tree N-body/SPH galactic chemodynamics code GCD+. The applied scheme includes imple- mentations of the artificial viscosity switch and artificial thermal conductivity pro- posed by Morris & Monaghan (1997), Rosswog & Price (2007) and Price (2008), to model discontinuities and Kelvin-Helmholtz instabilities more accurately. We first present hydrodynamics test simulations and contrast the results to runs undertaken without artificial viscosity switch or thermal conduction. In addition, we also explore the different levels of smoothing by adopting larger or smaller smoothing lengths, i.e. a larger or smaller number of neighbour particles, Nnb. We demonstrate that the new version of GCD+ is capable of modelling Kelvin-Helmholtz instabilities to a simi- lar level as the mesh code, Athena. From the Gresho vortex, point-like explosion and self-similar collapse tests, we conclude that setting the smoothing length to keep the number of neighbour particles as high as Nnb~58 is preferable to adopting smaller smoothing lengths. We present our optimised parameter sets from the hydrodynamics tests.

Ghost SPH for animating water

We propose a new ghost fluid approach for free surface and solid boundary conditions in Smoothed Particle Hydrodynamics (SPH) liquid simulations. Prior methods either suffer from a spurious numerical surface tension artifact or drift away from the mass conservation constraint, and do not capture realistic cohesion of liquid to solids. Our Ghost SPH scheme resolves this with a new particle sampling algorithm to create a narrow layer of ghost particles in the surrounding air and solid, with careful extrapolation and treatment of fluid variables to reflect the boundary conditions. We also provide a new, simpler form of artificial viscosity based on XSPH. Examples demonstrate how the new approach captures real liquid behaviour previously unattainable by SPH with very little extra cost.

Comparative study on the accuracy and stability of SPH schemes in simulating energetic free-surface flows

Free-surface flows are of significant interest in Computational Fluid Dynamics (CFD). However, modelling them especially when the free-surface breaks or impacts on solid walls can be challenging for many CFD techniques. Smoothed Particle Hydrodynamics (SPH) has been reported as a robust and stable method when applied to these problems. In modelling incompressible flows using the SPH method an equation of state with a large sound speed is typically used. This weakly compressible approach (WCSPH) results in a stiff set of equations with a noisy pressure field and stability issues at high Reynolds number. As a remedy, the incompressible SPH (ISPH) technique was introduced, which uses a pressure projection technique to model incompressibility. Although the pressure field calculated by ISPH is smooth, the stability of the technique is still an open discussion. An alternative approach is to use an acoustic Riemann solver and replace the particle velocities and pressures by pressures and velocities determined from a Riemann solver. This technique is equivalent to the Godunov method in Eulerian techniques and so will be called the Godunov SPH method (GSPH). However, since the acoustic Riemann solver is a first order approximation of the Riemann solution, it is highly dissipative and cannot be employed in energetic free-surface flows without modification. In this paper, the GSPH method is modified by using the HLLC (Harten Lax and van Leer-Contact) Riemann solver. The accuracy of the modified GSPH technique is further improved by utilising the MUSCL (Monotone Upstream-centred Schemes for Conservation Laws) scheme with Slope–Limiter. This modified GSPH method along with the WCSPH and ISPH techniques are used to study non-linear sloshing flow. The accuracy, stability and efficiency of the techniques are assessed and the results are compared with experimental data.

Particle packing algorithm for SPH schemes

Using some intrinsic features of the Smoothed Particle Hydrodynamics (SPH) schemes, an innovative algorithm for the initialization of the particle distribution has been defined. The proposed particle packing algorithm allows a drastic reduction of the numerical noise due to particle resettlement during the early stages of the flow evolution. Moreover, thanks to its structure, it can be easily derived starting from whatever SPH scheme and applies under the hypotheses that the fluid is weakly-compressible or incompressible as well. A broad range of numerical test cases proved this tool to be fast, robust and reliable also for complex geometrical configurations.

극대변형 해석을 위한 SPH 수치기법 개발 및 ExLO 코드 연계

This paper describes the development of SPH(Smooth Particle Hydrodynamics) scheme and integration into the multi-material shock physics code(ExLO) for the purpose of the application to the extreme large deformation problems. SPH numerical scheme has been extended into the fluid dynamics and the high-speed impact events, such as space structure protection against space debris and meteorite catering. Like other hydrocodes, SPH scheme also solves the conservation equations with the constitutive equation including equation of state. The benchmark problem, Taylor-Impact test, was simulated and the predictions show good agreements with both the published numerical data and experimental data. Currently, the contact treatment between materials is under development.

Error estimation in smoothed particle hydrodynamics and a new scheme for second derivatives

Several schemes for discretization of first and second derivatives are available in Smoothed Particle Hydrodynamics (SPH). Here, four schemes for approximation of the first derivative and three schemes for the second derivative are examined using a theoretical analysis based on Taylor series expansion both for regular and irregular particle distributions. Estimation of terms in the truncation errors shows that only the renormalized (the first-order consistent) scheme has acceptable convergence properties to approximate the first derivative. None of the second derivative schemes has the first-order consistency. Therefore, they converge only when the particle spacing decreases much faster than the smoothing length of the kernel function. In addition, using a modified renormalization tensor, a new SPH scheme is presented for approximating second derivatives that has the property of first-order consistency. To assess the computational performance of the proposed scheme, it is compared with the best available schemes when applied to a 2D heat equation. The numerical results show at least one order of magnitude improvement in accuracy when the new scheme is used. In addition, the new scheme has higher-order convergence rate on regular particle arrangements even for the case of only four particles in the neighborhood of each particle.

Resolving mixing in smoothed particle hydrodynamics

Standard formulations of smoothed particle hydrodynamics (SPH) are unable to resolve mixing at fluid boundaries. We use an error and stability analysis of the generalised SPH equations of motion to prove that this is due to two distinct problems. The first is a leading order error in the momentum equation. This should decrease with increasing neighbour number, but does not because numerical instabilities cause the kernel to be irregularly sampled. We identify two important instabilities: the clumping instability and the banding instability, and we show that both are cured by a suitable choice of kernel. The second problem is the local mixing instability (LMI). This occurs as particles attempt to mix on the kernel scale, but are unable to due to entropy conservation. The result is a pressure discontinuity at boundaries that pushes fluids of different entropy apart. We cure the LMI by using a weighted density estimate that ensures that pressures are single valued throughout the flow. This also gives a better volume estimate for the particles, reducing errors in the continuity and momentum equations. We demonstrate mixing in our new Optimised Smoothed Particle Hydrodynamics (OSPH) scheme using a Kelvin Helmholtz instability (KHI) test with density contrast 1:2, and the 'blob test' - a 1:10 density ratio gas sphere in a wind tunnel - finding excellent agreement between OSPH and Eulerian codes.

SPH simulation of free surface flow over a sharp-crested weir

In this paper the numerical simulation of a free surface flow over a sharp-crested weir is presented. Since in this case the usual shallow water assumptions are not satisfied, we propose to solve the problem using the full weakly compressible Navier–Stokes equations with the Tait equation of state for water. The numerical method used consists of the new meshless Smooth Particle Hydrodynamics (SPH) formulation proposed by Ferrari et al. (2009) [8], that accurately tracks the free surface profile and provides monotone pressure fields. Thus, the unsteady evolution of the complex moving material interface (free surface) can been properly solved. The simulations involving about half a million of fluid particles have been run in parallel on two of the most powerful High Performance Computing (HPC) facilities in Europe. The validation of the results has been carried out analysing the pressure field and comparing the free surface profiles obtained with the SPH scheme with experimental measurements available in literature [18]. A very good quantitative agreement has been obtained.

A comparison of hydrodynamic techniques for modelling collisions between main-sequence stars

An Eulerian total variation diminishing (TVD) code and a Lagrangian smoothed particle hydrodynamics (SPH) code are used to simulate the off-axis collision of equal-mass main-sequence stars in order to address the question of whether stellar mergers can produce a remnant star where the interior has been replenished with hydrogen due to significant mixing. Each parent main-sequence star is chosen to be found near the turn-off, with hydrogen depleted in the core, and is modelled with a M = 0.8 M ⊙ realistic stellar model and as a n = 3 polytrope. An ideal fluid description with adiabatic index y = 5/3 is used for all hydrodynamic calculations. We found good agreement between the simulations for the polytropic case, with the remnant showing strong, non-local mixing throughout. In the interior quarter of the mass, ∼35 per cent is mixed in from larger radii and on average the remnant is ∼50 per cent fully mixed. For the realistic model, we found less mixing, particularly in the interior and in the SPH simulation. In the inner quarter, ∼20 per cent of the contained mass in the TVD case, but only ∼3 per cent in the SPH one is mixed in from outside. The simulations give consistent results for the overall profile of the merger remnant and the amount of mass-loss, but the differences in mixing suggests that the intrinsic difference between grid and particle based schemes remains a possible artefact. We conclude that both the TVD and SPH schemes can be used equally well for problems that are best suited to their strengths and that care should be taken in interpreting results about fluid mixing.

Dynamic Cores in Hydrostatic Disguise

We discuss the column density profiles of in three-dimensional smoothed particle hydrodynamics (SPH) numerical simulations of turbulent molecular clouds. The SPH scheme allows us to perform a high spatial resolution analysis of the density maxima (cores) at scales between ~0.003 and 0.3 pc. We analyze simulations in three different physical conditions: large-scale driving (LSD), small-scale driving (SSD), and random Gaussian initial conditions without driving (GC), each one at two different time steps: just before self-gravity is turned on (t0) and when gravity has been operating such that 5% of the total mass in the box has been accreted into cores (t1). For this data set, we perform Bonnor-Ebert fits to the column density profiles of cores found by a clump-finding algorithm. We find that, for the particular fitting procedure we use, 65% of the cores can be matched to Bonnor-Ebert (BE) profiles, and of these, 47% correspond to stable equilibrium configurations with ξmax < 6.5, even though the cores analyzed in the simulations are not in equilibrium but instead are dynamically evolving. The temperatures obtained with the fitting procedure vary between 5 and 60 K (in spite of the simulations being isothermal, with T = 11.3 K), with the peak of the distribution being at T = 11 K and most clumps having fitted temperatures between 5 and 30 K. Central densities obtained with the BE fit tend to be smaller than the actual central densities of the cores. We also find that for the LSD and GC cases, there are more BE-like cores at t0 than at t1 with ξmax ≤ 20, while in the case of SSD, there are more such cores at t1 than at t0. We interpret this as a consequence of the stronger turbulence present in the cores of run SSD, which prevents good BE fits in the absence of gravity, and delays collapse in its presence. Finally, in some cases we find substantial superposition effects when we analyze the projection of the density structures, even though the scales over which we project are small (0.18 pc). As a consequence, different projections of the same core may give very different values of the BE fits. Finally, we briefly discuss recent results claiming that Bok globule B68 is in hydrostatic equilibrium, stressing that they imply that this core is unstable by a wide margin. We conclude that fitting BE profiles to observed cores is not an unambiguous test of hydrostatic equilibrium and that fit-estimated parameters such as mass, central density, density contrast, temperature, or radial profile of the BE sphere may differ significantly from the actual values in the cores.

The origin of the rebrightening in soft X-ray transient outbursts

We present a model of an outburst of the soft X-ray transient A0620‐003. A two-dimensional time-dependent smoothed particle hydrodynamics scheme is used to simulate the evolution of the accretion disc through a complete outburst. The scheme includes the full tidal potential of the binary and a simple treatment of the thermal-viscous disc instability. In the case where the mass accretion rate on to the primary determines the fraction of the disc that can be kept in a hot, high-viscosity state by the resulting X-ray emission, we find that the shape of the X-ray light curve is ultimately determined by the relative sizes of the irradiated and unirradiated parts of the disc and the growth time-scale of the tidal instability. The model accounts for the rebrightening that has been observed in the light curves of A0620‐003 and several other transients. The primary maximum and subsequent decline are caused by the accretion of gas within the irradiated portion of the disc, while the secondary maximum is caused by the accretion of gas in the outer part of the disc, which is initially shadowed from the central X-rays, but subject to tidal forces. We propose that tidal effects at the disc edge can be sufficient to drive accretion on a time-scale shorter than that expected for a standard alpha-viscosity disc. The final decay is subsequently controlled by the gradual retreat of the irradiated portion of the disc. If the entire disc is kept in the high-viscosity state by the irradiation, no rebrightening is possible.

Analysis of ceramic/metal armour systems

A combined numerical and experimental study for the analysis of ceramic/metal composite armour system against 40.7 g steel projectiles has been performed. The ballistic performance of the add-on lightweight armours was examined by varying the thickness of tiles, while maintaining equal areal density of the system. A numerical study using smoothed particle hydrodynamics scheme is promising since the major distinguishing features of composite armour systems such as, projectile erosion, crack propagation, ceramic conoid formation and failure of backing plate, are successfully captured. Simulation results for ballistic limits appear to match fairly well with the test values and reveal an optimum value of the front plate to back plate thickness ratio.

Substantial stream-disc overflow found in three-dimensional SPH simulations of cataclysmic variables

We study numerically the interaction of the infalling gas stream and the rim of the accretion disc in cataclysmic variables. The simulations were performed with a smoothed particle hydrodynamics scheme with high spatial resolution. Parameters of the systems AM CVn, OY Car, DQ Her, U Gem and IP Peg were used for the simulations. The simulations cover a wide range of orbital periods, mass ratios and mass transfer rates, as well as different thermal states of the accretion disc. The main result of this study is that the accretion stream is not stopped at the impact region (the bright spot at the outer rim of the disc). In fact, after undergoing the shock interaction, most of the matter is deflected vertically and flows in a more or less diffuse stream to inner parts of the disc, hitting the disc surface close to the circularization radius at orbital phase 0.5. This is a common feature in all systems for all simulated parameters. This stream overflow can cause the X-ray absorption dips observed in cataclysmic variables (CVs) and low-mass X-ray binaries (LMXBs) around orbital phase 0.7, if the inclination is at least 65°. Under certain circumstances, namely a sudden increase of the mass transfer rate from the secondary or a rather small disc, parts of the overflowing stream bounce off the disc surface after hitting it at orbital phase ≈0.5. Another absorption region can be expected around orbital phase 0.2. In our simulations most of the infalling matter reaches the inner disc very quickly. This must alter the evolution of the quiescent disc and the outburst behaviour considerably compared with purely viscous transport of the material through the disc from the outer rim, and therefore should be taken into account in dwarf nova outburst cycle calculations. To our knowledge, the consequences of such a massive stream overflow for the dwarf nova outburst cycle have not been considered yet.

Smoothed Particle Hydrodynamics in cosmology: a comparative study of implementations

We analyse the performance of 12 different implementations of Smoothed Particle Hydrodynamics (SPH) using seven tests designed to isolate key hydrodynamic elements of cosmological simulations which are known to cause the SPH algorithm problems. In order, we consider a shock tube, spherical adiabatic collapse, cooling flow model, drag, a cosmological simulation, rotating cloud-collapse and angular momentum transport. In the implementations special attention is given to the way in which force symmetry is enforced in the equations of motion. We study in detail how the hydrodynamics are affected by different implementations of the artificial viscosity including those with a shear-correction modification. We present an improved first-order smoothing-length update algorithm that is designed to remove instabilities that are present in simple forward prediction algorithms. Gravity is calculated using the adaptive particle–particle, particle–mesh algorithm. For all tests we find that the artificial viscosity is the single most important factor distinguishing the results from the various implementations. The shock tube and adiabatic collapse problems show that the artificial viscosity used in the hydra code prior to version 4.0 performs relatively poorly for simulations involving strong shocks when compared to a more standard artificial viscosity. The shear-correction term is shown to reduce the shock-capturing ability of the algorithm and to lead to a spurious increase in angular momentum in the rotating cloud-collapse problem. For the disc stability test, the shear-corrected and previous hydra artificial viscosities are shown to reduce outward angular momentum transport. The cosmological simulations produce comparatively similar results, with the fraction of gas in the hot and cold phases varying by less than 10 per cent amongst the versions. Similarly, the drag test shows little systematic variation amongst versions. The cooling flow tests show that implementations using the force symmetrization of Thomas & Couchman are more prone to accelerate the overcooling instability of SPH, although the problem is generic to SPH. The second most important factor in code performance is the way force symmetry is achieved in the equation of motion. Most results favour a kernel symmetrization approach. The exact method by which SPH pressure forces are included in the equation of motion appears to have comparatively little effect on the results. Combining the equation of motion presented by Thomas & Couchman with a modification of the Monaghan & Gingold artificial viscosity leads to an SPH scheme that is both fast and reliable.

Smoothed Particle Hydrodynamics in cosmology: a comparative study of implementations

We analyse the performance of 12 different implementations of Smoothed Particle Hydrodynamics (SPH) using seven tests designed to isolate key hydrodynamic elements of cosmological simulations which are known to cause the SPH algorithm problems. In order, we consider a shock tube, spherical adiabatic collapse, cooling flow model, drag, a cosmological simulation, rotating cloud-collapse and angular momentum transport. In the implementations special attention is given to the way in which force symmetry is enforced in the equations of motion. We study in detail how the hydrodynamics are affected by different implementations of the artificial viscosity including those with a shear-correction modification. We present an improved first-order smoothing-length update algorithm that is designed to remove instabilities that are present in simple forward prediction algorithms. Gravity is calculated using the adaptive particle–particle, particle–mesh algorithm. For all tests we find that the artificial viscosity is the single most important factor distinguishing the results from the various implementations. The shock tube and adiabatic collapse problems show that the artificial viscosity used in the hydra code prior to version 4.0 performs relatively poorly for simulations involving strong shocks when compared to a more standard artificial viscosity. The shear-correction term is shown to reduce the shock-capturing ability of the algorithm and to lead to a spurious increase in angular momentum in the rotating cloud-collapse problem. For the disc stability test, the shear-corrected and previous hydra artificial viscosities are shown to reduce outward angular momentum transport. The cosmological simulations produce comparatively similar results, with the fraction of gas in the hot and cold phases varying by less than 10 per cent amongst the versions. Similarly, the drag test shows little systematic variation amongst versions. The cooling flow tests show that implementations using the force symmetrization of Thomas & Couchman are more prone to accelerate the overcooling instability of SPH, although the problem is generic to SPH. The second most important factor in code performance is the way force symmetry is achieved in the equation of motion. Most results favour a kernel symmetrization approach. The exact method by which SPH pressure forces are included in the equation of motion appears to have comparatively little effect on the results. Combining the equation of motion presented by Thomas & Couchman with a modification of the Monaghan & Gingold artificial viscosity leads to an SPH scheme that is both fast and reliable.

Particle Methods in Astrophysical Fluid Dynamics

Particle methods play an important role in the study of a wide variety of astrophysical fluid dynamics problems. The different methods currently in use are all variants of the socalled Smoothed Particle Hydrodynamics (SPH) scheme introduced by Lucy and Gingold & Monaghan more than twenty years ago. This paper presents a complete introduction to SPH in its modern form, and discusses some of the main numerical properties of the scheme. In particular, the convergence properties of SPH are studied, as a function of the number of particles N and the number of interacting neighbors NN , using a simple analysis based on sound waves. It is shown that consistency of SPH (i.e., convergence towards a physical solution) requires both N → ∞ and NN → ∞, with the smoothing length h → 0, i.e., NN/N → 0.

Tests of Spurious Transport in Smoothed Particle Hydrodynamics

We have performed a series of systematic tests to evaluate quantitatively the effects of spurious transport in three-dimensional smoothed particle hydrodynamics (SPH) calculations. Our tests investigate (i) particle diffusion, (ii) shock heating, (iii) numerical viscosity, and (iv) angular momentum transport. The effects of various program parameters on spurious mixing and on viscosity are investigated. The results are useful for quantifying the accuracy of the SPH scheme, especially for problems where shear flows or shocks are present, as well as for problems where true hydrodynamic mixing is relevant. In particular, the particle diffusion coefficients we measure can be used to help estimate the spurious fluid mixing in SPH applications. We examine the different forms of artificial viscosity (AV) which have been proposed by Monaghan, by Hernquist and Katz, and by Balsara. Our tests suggest a single set of values for the AV parameters which are appropriate in a large number of situations: α≈0.5, β≈1 for the classical AV of Monaghan, α≈β≈0.5 for the Hernquist and Katz AV, and α≈β≈γ/2 for the Balsara AV (where γ is the adiabatic index). We also discuss how these choices should be modified depending on the goals of the particular application. For instance, if spurious particle mixing is not a concern and only weak shocks (Mach number M⋦2) are expected during a calculation, then a smaller value of α is appropriate. Somewhat larger values for α and β may be preferable if an accurate treatment of high Mach number shocks (M⋧10) is required. We find that both the Hernquist and Katz and Balsara forms introduce only small amounts of numerical viscosity. Furthermore, both Monaghan's and Balsara's AV do well at treating shocks and at limiting the amount of spurious mixing. For these reasons, we endorse the Balsara AV for use in a broad range of applications.