Standard Smoothed Particle Hydrodynamics Formulation Research Articles

Modifications of SPH towards three-dimensional simulations of an icy moon with internal ocean

There are some traces of the existence of internal ocean in some icy moons, such as the vapor plumes of Europa and Enceladus. This implies a region of liquid water beneath the surface ice shell. Since liquid water would be essential for the origin of life, it is important to understand the development of these internal oceans, particularly their temperature distribution and evolution. The balance between tidal heating and radiative cooling is believed to sustain liquid water beneath an icy moon’s surface. We aim to simulate the tidal heating of an internal ocean in an icy moon using 3-dimensional numerical fluid calculations with the Smoothed Particle Hydrodynamics (SPH) method. We incorporated viscosity and thermal conduction terms into the governing equations of SPH. However, we encountered two issues while calculating rigid body rotation using SPH with a viscous term: (1) conventional viscosity formulations generated unphysical forces that hindered rotation, and (2) there was artificial internal energy partitioning within the layered structure, which was due to the standard SPH formulations. To address the first issue, we modified the viscosity formulation. For the second, we adopted Density Independent SPH (DISPH) developed in previous studies to improve behavior at discontinuous surfaces. Additionally, we implemented radiative cooling using an algorithm to define fluid surfaces via the particle method. We also introduced an equation of state accounting for phase transitions. With these modifications, we have refined the SPH method to encompass all necessary physical processes for simulating the evolution of icy moons with internal oceans.

Novel hydrodynamic schemes capturing shocks and contact discontinuities and comparison study with existing methods

We present a new hydrodynamic scheme named Godunov Density-Independent Smoothed Particle Hydrodynamics (GDISPH), that can accurately handle shock waves and contact discontinuities without any manually tuned parameters. This is in contrast to the standard formulation of smoothed particle hydrodynamics (SSPH), which requires the parameters for an artificial viscosity term to handle the shocks and struggles to accurately handle the contact discontinuities due to unphysical repulsive forces, resulting in surface tension that disrupts pressure equilibrium and suppresses fluid instabilities. While Godunov SPH (GSPH) can handle the shocks without the parameters by using solutions from a Riemann solver, it still cannot fully handle the contact discontinuities. Density-Independent Smoothed Particle Hydrodynamics (DISPH), one of several schemes proposed to handle contact discontinuities more effectively than SSPH, demonstrates superior performance in our tests involving strong shocks and contact discontinuities. However, DISPH still requires the artificial viscosity term. We integrate the Riemann solver into DISPH in several ways, yielding some patterns of GDISPH. The results of standard tests such as the one-dimensional Riemann problem, pressure equilibrium, Sedov–Taylor, and Kelvin–Helmholtz tests are favourable to GDISPH Case 1 and GDISPH Case 2, as well as DISPH. We conclude that GDISPH Case 1 has an advantage over GDISPH Case 2effectively handling shocks and contact discontinuities without the need for specific parameters or introducing any additional numerical diffusion.

An improved thermal model for SPH metal cutting simulations on GPU

This paper presents the first application of a higher-order Smoothed Particle Hydrodynamics (SPH) method to the thermal modeling of metal cutting problems. With this application, the heat transfer equation in the thermo-mechanical simulation of metal cutting is solved more accurately by addressing the consistency issue of standard SPH formulations. Furthermore, through a robust and effective surface-detection algorithm, this work enables the SPH cutting models to include heat loss thermal boundary conditions for the first time. Process forces, tool temperatures, and chip geometry are numerically investigated in machining a Ti6Al4V workpiece at two different cutting speeds. Several validation tests and sensitivity analyses are performed in high resolution, thanks to the runtime acceleration of SPH by parallel computing on Graphics Processing Units (GPUs). The results show that SPH simulations with the proposed thermal modeling approach achieve more realistic serrated chips in titanium cutting problems.

Implementation of three-dimensional physical reflective boundary conditions in mesh-free particle methods for continuum fluid dynamics: Validation tests and case studies

Mesh-free particle methods applied to continuum fluid dynamics still use fictitious, ghost or virtual particles, and/or molecular dynamics concepts in the treatment of the boundaries. The aim of this paper is to present the implementation of the physical reflective boundary conditions (RBC), based on Newton’s restitution law and the foundations of analytical geometry, in the treatment of contours in a three-dimensional (3D) domain. In a previous paper [C. A. D. Fraga Filho, “An algorithmic implementation of physical reflective boundary conditions in particle methods: Collision detection and response,” Phys. Fluids 29, 113602 (2017)], RBC validation tests and simulation results were presented for a two-dimensional (2D) domain. The current work presents the validation of the collision detection and response algorithm employed for the RBC and its application in two cases (hydrostatics and hydrodynamics) in a 3D domain. Following an analysis of the simulation results, the applicability of the RBC to 3D continuum fluid dynamic problems is verified. In the hydrostatics case, a still liquid (Newtonian, incompressible, uniform, and isothermal) inside an immobile reservoir is studied. The fluid flow is modeled using an improved Smoothed Particle Hydrodynamics (SPH) formulation utilizing a modified concept of pressure. The simulation results are excellent, showing complete agreement with the analytical solution and the nonmotion of the particles throughout the simulation time. In the hydrodynamics case, 3D dam break flow modeling is carried out using the standard SPH formulation. Results provided by the RBC and standard SPH modeling are compared with the literature data demonstrating good agreement with the experimental findings.

LOQUAT: an open-source GPU-accelerated SPH solver for geotechnical modeling

Smoothed particle hydrodynamics (SPH) is a meshless method gaining popularity recently in geotechnical modeling. It is suitable to solve problems involving large deformation, free-surface, cracking and fragmentation. To promote the research and application of SPH in geotechnical engineering, we present LOQUAT, an open-source three-dimensional GPU accelerated SPH solver. LOQUAT employs the standard SPH formulations for solids with two geomechnical constitutive models which are the Drucker–Prager model and a hypoplastic model. Three stabilization techniques, namely, artificial viscosity, artificial pressure and stress regularization are included. A generalized boundary particle method is presented to model static and moving boundaries with arbitrary geometry. LOQUAT employs GPU acceleration technique to greatly increase the computational efficiency. Numerical examples show that the solver is convergent, stable and highly efficient. With a mainstream GPU, it can simulate large scale problems with tens of millions of particles, and easily performs more than one thousand times faster than serial CPU code.

Smoothed particle hydrodynamics numerical model for modeling an oscillating water chamber

This paper presents the application and the validation of the SPH numerical model SPHyCE, based on a standard Smoothed Particle Hydrodynamics formulation, for modeling an onshore oscillating water chamber, comparing results of amplification factor and phase lag with those obtained by a mesh-based RANS model. Firstly, an analysis of the results for different resolutions (i.e. particle dimension) showed the independence of results with refinement. Secondly, the amplification factor of mean free surface elevation inside the oscillating water chamber and the phase lag between time series of free surface elevation outside and inside the water chamber were analysed. A good agreement was achieved between the mesh free SPHyCE model and the mesh-based RANS model. SPHyCE model was finally applied to analyse the hydrodynamic fluid flow inside and outside the water chamber and to quantify the forces on the vertical wall and the amplification factor for different incident wave heights for a rough sea state with complex wave breaking, impact loads and overtopping, contributing to the future applications of SPH models for development and studies of OWC devices.

A COMPARISON OF SPH ARTIFICIAL VISCOSITIES AND THEIR IMPACT ON THE KEPLERIAN DISK

ABSTRACT Hydrodynamical simulations of rotating disks play important roles in the field of astrophysical and planetary science. Smoothed particle hydrodynamics (SPH) has been widely used for such simulations. However, it has been known that when using SPH, a cold and thin Kepler disk breaks up due to the unwanted angular momentum transfer. Two possible reasons have been suggested for this breaking up of the disk; the artificial viscosity (AV) and the numerical error in the evaluation of pressure gradient in SPH. Which one is dominant is still unclear. In this paper, we investigate the reason for this rapid breaking up of the disk. We implemented most of the popular formulations of AV and switches, and measured the angular momentum transfer due to both AV and the error of SPH’s estimate of the pressure gradient. We found that the angular momentum transfer due to AV at the inner edge triggers the breaking up of the disk. We also found that the classical von Neumann–Richtmyer–Landshoff type AV with a high-order estimate for can maintain the disk for ∼100 orbits even when used with the standard formulation of SPH.

The giant impact simulations with density independent smoothed particle hydrodynamics

At present, the giant impact (GI) is the most widely accepted model for the origin of the Moon. Most of the numerical simulations of GI have been carried out with the smoothed particle hydrodynamics (SPH) method. Recently, however, it has been pointed out that standard formulation of SPH (SSPH) has difficulties in the treatment of a contact discontinuity such as a core–mantle boundary and a free surface such as a planetary surface. This difficulty comes from the assumption of differentiability of density in SSPH. We have developed an alternative formulation of SPH, density independent SPH (DISPH), which is based on differentiability of pressure instead of density to solve the problem of a contact discontinuity. In this paper, we report the results of the GI simulations with DISPH and compare them with those obtained with SSPH. We found that the disk properties, such as mass and angular momentum produced by DISPH is different from that of SSPH. In general, the disks formed by DISPH are more compact: while formation of a smaller mass moon for low-oblique impacts is expected with DISPH, inhibition of ejection would promote formation of a larger mass moon for high-oblique impacts. Since only the improvement of core–mantle boundary significantly affects the properties of circumplanetary disks generated by GI and DISPH has not been significantly improved from SSPH for a free surface, we should be very careful when some conclusions are drawn from the numerical simulations for GI. And it is necessary to develop the numerical hydrodynamical scheme for GI that can properly treat the free surface as well as the contact discontinuity.

A smoothed particle hydrodynamics (SPH) model for simulating surface erosion by impacts of foreign particles

This paper presents a smoothed particle hydrodynamics modeling procedure to simulate surface erosion by impacts. The present SPH is improved based on the standard SPH formulation, by developing SPH formulations for the constitutive relations to describe plastic behavior and ductile fracture process. The improved SPH method is first applied to simulate impacts of particles on ductile targets. Two modified schemes in terms of density correction and kernel gradient correction are adopted to improve the accuracy of the SPH approximation. The present SPH model is applied to simulate the erosion process by angular particles. The results are compared with available experimental data, and good agreements have been achieved in terms of the crater profile and particle kinematic results.

A Particle Model for Prediction of Cement Infiltration of Cancellous Bone in Osteoporotic Bone Augmentation

Femoroplasty is a potential preventive treatment for osteoporotic hip fractures. It involves augmenting mechanical properties of the femur by injecting Polymethylmethacrylate (PMMA) bone cement. To reduce the risks involved and maximize the outcome, however, the procedure needs to be carefully planned and executed. An important part of the planning system is predicting infiltration of cement into the porous medium of cancellous bone. We used the method of Smoothed Particle Hydrodynamics (SPH) to model the flow of PMMA inside porous media. We modified the standard formulation of SPH to incorporate the extreme viscosities associated with bone cement. Darcy creeping flow of fluids through isotropic porous media was simulated and the results were compared with those reported in the literature. Further validation involved injecting PMMA cement inside porous foam blocks — osteoporotic cancellous bone surrogates — and simulating the injections using our proposed SPH model. Millimeter accuracy was obtained in comparing the simulated and actual cement shapes. Also, strong correlations were found between the simulated and the experimental data of spreading distance (R2 = 0.86) and normalized pressure (R2 = 0.90). Results suggest that the proposed model is suitable for use in an osteoporotic femoral augmentation planning framework.

Convergence of AMR and SPH simulations – I. Hydrodynamical resolution and convergence tests

We compare the results for a set of hydrodynamical tests performed with the adaptive mesh refinement finite volume code, mg, and the smoothed particle hydrodynamics (SPH) code, seren. The test suite includes shock tube tests, with and without cooling, the non-linear thin-shell instability and the Kelvin–Helmholtz instability. The main conclusions are the following. (i) The two methods converge in the limit of high resolution and accuracy in most cases. All tests show good agreement when numerical effects (e.g. discontinuities in SPH) are properly treated. (ii) Both methods can capture adiabatic shocks and well-resolved cooling shocks perfectly well with standard prescriptions. However, they both have problems when dealing with under-resolved cooling shocks, or strictly isothermal shocks, at high Mach numbers. The finite volume code only works well at first order and even then requires some additional artificial viscosity. SPH requires either a larger value of the artificial viscosity parameter, αAV, or a modified form of the standard artificial viscosity term using the harmonic mean of the density, rather than the arithmetic mean. (iii) Some SPH simulations require larger kernels to increase neighbour number and reduce particle noise in order to achieve agreement with finite volume simulations (e.g. the Kelvin–Helmholtz instability). However, this is partly due to the need to reduce noise that can corrupt the growth of small-scale perturbations (e.g. the Kelvin–Helmholtz instability). In contrast, instabilities seeded from large-scale perturbations (e.g. the non-linear thin shell instability) do not require more neighbours and hence work well with standard SPH formulations and converge with the finite volume simulations. (iv) For purely hydrodynamical problems, SPH simulations take an order of magnitude longer to run than finite volume simulations when running at equivalent resolutions, i.e. when they both resolve the underlying physics to the same degree. This requires about two to three times as many particles as the number of cells.

Moving mesh cosmology: tracing cosmological gas accretion

We investigate the nature of gas accretion onto haloes and galaxies at z=2 using cosmological hydrodynamic simulations run with the moving mesh code AREPO. Implementing a Monte Carlo tracer particle scheme to determine the origin and thermodynamic history of accreting gas, we make quantitative comparisons to an otherwise identical simulation run with the smoothed particle hydrodynamics (SPH) code GADGET-3. Contrasting these two numerical approaches, we find significant physical differences in the thermodynamic history of accreted gas in haloes above 10^10.5 solar masses. In agreement with previous work, GADGET simulations show a cold fraction near unity for galaxies forming in massive haloes, implying that only a small percentage of accreted gas heats to an appreciable fraction of the virial temperature during accretion. The same galaxies in AREPO show a much lower cold fraction, <20% in haloes above 10^11 solar masses. This results from a hot gas accretion rate which, at this same halo mass, is an order of magnitude larger than with GADGET, while the cold accretion rate is also lower. These discrepancies increase for more massive systems, and we explain both as due to numerical inaccuracies in the standard formulation of SPH. We also observe that the relatively sharp transition from cold to hot mode dominated accretion, at a halo mass of ~10^11, is a consequence of comparing past gas temperatures to a constant threshold value independent of virial temperature. Examining the spatial distribution of accreting gas, we find that gas filaments in GADGET tend to remain collimated and flow coherently to small radii, or artificially fragment and form a large number of purely numerical "blobs". Similar gas streams in AREPO show increased heating and disruption at 0.25-0.5 virial radii and contribute to the hot gas accretion rate in a manner distinct from classical cooling flows.

Subsonic turbulence in smoothed particle hydrodynamics and moving-mesh simulations

Highly supersonic, compressible turbulence is thought to be of tantamount importance for star formation processes in the interstellar medium. Likewise, cosmic structure formation is expected to give rise to subsonic turbulence in the intergalactic medium, which may substantially modify the thermodynamic structure of gas in virialized dark matter halos and affect small-scale mixing processes in the gas. Numerical simulations have played a key role in characterizing the properties of astrophysical turbulence, but thus far systematic code comparisons have been restricted to the supersonic regime, leaving it unclear whether subsonic turbulence is faithfully represented by the numerical techniques commonly employed in astrophysics. Here we focus on comparing the accuracy of smoothed particle hydrodynamics (SPH) and our new moving-mesh technique AREPO in simulations of driven subsonic turbulence. To make contact with previous results, we also analyze simulations of transsonic and highly supersonic turbulence. We find that the widely employed standard formulation of SPH yields problematic results in the subsonic regime. Instead of building up a Kolmogorov-like turbulent cascade, large-scale eddies are quickly damped close to the driving scale and decay into small-scale velocity noise. Reduced viscosity settings improve the situation, but the shape of the dissipation range differs compared with expectations for a Kolmogorov cascade. In contrast, our moving-mesh technique does yield power-law scaling laws for the power spectra of velocity, vorticity and density, consistent with expectations for fully developed isotropic turbulence. We show that large errors in SPH's gradient estimate and the associated subsonic velocity noise are ultimately responsible for producing inaccurate results in the subsonic regime. In contrast, SPH's performance is much better for supersonic turbulence. [Abridged]

An improved SPH method for saturated soils and its application to investigate the mechanisms of embankment failure: Case of hydrostatic pore‐water pressure

SUMMARYThe method of smoothed particle hydrodynamics (SPH) has recently been applied to computational geomechanics and has been shown to be a powerful alternative to the standard numerical method, that is, the finite element method, for handling large deformation and post‐failure of geomaterials. However, very few studies apply the SPH method to model saturated or submerged soil problems. Our recent studies of this matter revealed that significant errors may be made if the gradient of the pore‐water pressure is handled using the standard SPH formulation. To overcome this problem and to enhance the SPH applications to computational geomechanics, this article proposes a general SPH formulation, which can be applied straightforwardly to dry and saturated soils. For simplicity, the current work assumes hydrostatic pore‐water pressure. It is shown that the proposed formulation can remove the numerical error mentioned earlier. Moreover, this formulation automatically satisfies the dynamic boundary conditions at a submerged ground surface, thereby saving computational cost. Discussions on the applications of the standard and new SPH formulations are also given through some numerical tests. Furthermore, techniques to obtain the correct SPH solution are also proposed and discussed throughout. As an application of the proposed method, the effect of the dilatancy angle on the failure mechanism of a two‐sided embankment subjected to a high groundwater table is presented and compared with that of other solutions. Finally, the proposed formulation can be considered a basic formulation for further developments of SPH for saturated soils. Copyright © 2011 John Wiley &amp; Sons, Ltd.

SPH Simulation of Sediment Flushing Induced by a Rapid Water Flow

This paper shows an advanced application of the smoothed particle hydrodynamics (SPH) method to the numerical modeling of noncohesive sediment flushing aiming at the setup of a reliable engineering tool for the prediction of the coupled water-sediment dynamics at the bottom of an artificial reservoir. Both liquid and granular materials are modeled as weakly compressible viscous fluids, whose motion results from the numerical solution of the continuity and momentum equations discretized according to standard SPH formulation. The effect of two alternative erosion criteria on the description of the failure mechanism of bottom sediments is analyzed. These criteria are based, respectively, on Mohr-Coulomb yielding criterion and Shields theory. A sensitivity analysis is performed in order to assess, for both criteria, the influence of the model parameters on the simulation of the erosion process; the method is eventually validated by comparing numerical results with the experimental data obtained in a two-dimen...

Particle-based model to simulate the micromechanics of biological cells

This paper is concerned with addressing how biological cells react to mechanical impulse. We propose a particle based model to numerically study the mechanical response of these cells with subcellular detail. The model focuses on a plant cell in which two important features are present: (1) the cell's interior liquidlike phase inducing hydrodynamic phenomena, and (2) the cell wall, a viscoelastic solid membrane that encloses the protoplast. In this particle modeling framework, the cell fluid is modeled by a standard smoothed particle hydrodynamics (SPH) technique. For the viscoelastic solid phase (cell wall), a discrete element method (DEM) is proposed. The cell wall hydraulic conductivity (permeability) is built in through a constitutive relation in the SPH formulation. Simulations show that the SPH-DEM model is in reasonable agreement with compression experiments on an in vitro cell and with analytical models for the basic dynamical modes of a spherical liquid filled shell. We have performed simulations to explore more complex situations such as relaxation and impact, thereby considering two cell types: a stiff plant type and a soft animal-like type. Their particular behavior (force transmission) as a function of protoplasm and cell wall viscosity is discussed. We also show that the mechanics during and after cell failure can be modeled adequately. This methodology has large flexibility and opens possibilities to quantify problems dealing with the response of biological cells to mechanical impulses, e.g., impact, and the prediction of damage on a (sub)cellular scale.

Modelling discontinuities and Kelvin–Helmholtz instabilities in SPH

In this paper we discuss the treatment of discontinuities in smoothed particle hydrodynamics (SPH) simulations. In particular we discuss the difference between integral and differential representations of the fluid equations in an SPH context and how this relates to the formulation of dissipative terms for the capture of shocks and other discontinuities. This has important implications for many problems, in particular related to recently highlighted problems in treating Kelvin–Helmholtz instabilities across entropy gradients in SPH. The specific problems pointed out by Agertz et al. [O. Agertz, B. Moore, J. Stadel, D. Potter, F. Miniati, J. Read, L. Mayer, A. Gawryszczak, A. Kravtsov, Å. Nordlund, F. Pearce, V. Quilis, D. Rudd, V. Springel, J. Stone, E. Tasker, R. Teyssier, J. Wadsley, R. Walder, Fundamental differences between SPH and grid methods, MNRAS 380 (2007) 963–978] are shown to be related in particular to the (lack of) treatment of contact discontinuities in standard SPH formulations which can be cured by the simple application of an artificial thermal conductivity term. We propose a new formulation of artificial thermal conductivity in SPH which minimises dissipation away from discontinuities and can therefore be applied quite generally in SPH calculations.

A shock-capturing SPH scheme based on adaptive kernel estimation

Here we report a method that converts standard smoothed particle hydrodynamics (SPH) into a working shock-capturing scheme without relying on solutions to the Riemann problem. Unlike existing adaptive SPH simulations, the present scheme is based on an adaptive kernel estimation of the density, which combines intrinsic features of both the kernel and nearest neighbor approaches in a way that the amount of smoothing required in low-density regions is effectively controlled. Symmetrized SPH representations of the gas dynamic equations along with the usual kernel summation for the density are used to guarantee variational consistency. Implementation of the adaptive kernel estimation involves a very simple procedure and allows for a unique scheme that handles strong shocks and rarefactions the same way. Since it represents a general improvement of the integral interpolation on scattered data, it is also applicable to other fluid-dynamic models. When the method is applied to supersonic compressible flows with sharp discontinuities, as in the classical one-dimensional shock-tube problem and its variants, the accuracy of the results is comparable, and in most cases superior, to that obtained from high quality Godunov-type methods and SPH formulations based on Riemann solutions. The extension of the method to two- and three-space dimensions is straightforward. In particular, for the two-dimensional cylindrical Noh’s shock implosion and Sedov point explosion problems the present scheme produces much better results than those obtained with conventional SPH codes.

Conduction Modelling Using Smoothed Particle Hydrodynamics

Heat transfer is very important in many industrial and geophysical problems. Because these problems often have complicated fluid dynamics, there are advantages in solving them using Lagrangian methods like smoothed particle hydrodynamics (SPH). Since SPH particles become disordered, the second derivative terms may be estimated poorly, especially when materials with different properties are adjacent. In this paper we show how a simple alteration to the standard SPH formulation ensures continuity of heat flux across discontinuities in material properties. A set of rules is formulated for the construction of isothermal boundaries leading to accurate conduction solutions. A method for accurate prediction of heat fluxes through isothermal boundaries is also given. The accuracy of the SPH conduction solutions is demonstrated through a sequence of test problems of increasing complexity.

STRESS POINTS FOR TENSION INSTABILITY IN SPH

In this work, the stress-point approach, which was developed to address tension instability and improve accuracy in Smoothed Particle Hydrodynamics (SPH) methods, is further extended and applied for one-dimensional (1-D) problems. Details of the implementation of the stress-point method are also given. A stability analysis reveals a reduction in the critical time step by a factor of 1/√2 when the stress points are located at the extremes of the SPH particle. An elementary damage law is also introduced into the 1-D formulation. Application to a 1-D impact problem indicates far less oscillation in the pressure at the interface for coarse meshes than with the standard SPH formulation. Damage predictions and backface velocity histories for a bar appear to be quite reasonable as well. In general, applications to elastic and inelastic 1-D problems are very encouraging. The stress-point approach produces stable and accurate results. © 1997 by John Wiley & Sons, Ltd.