Smoothed Particle Hydrodynamics Scheme Research Articles

Reproducing hydrodynamics and elastic objects: A hybrid mesh-free model framework for dynamic multi-phase flows

Reproducing Hydrodynamics and Elastic Objects (RHEO) is a new meshfree modeling framework for simulating complex multi-phase flows of fluids and solids. RHEO is implemented within the open-source Large-Scale Atomic/Molecular Massively Parallel Simulator particle dynamics code and couples a reproducing kernel smoothed particle hydrodynamics scheme for modeling fluid flow and heat transfer with a bonded particle model for modeling breakable elastic bodies. The resulting scheme provides a robust framework for simulating multi-phase material systems with complex and evolving boundaries and interfaces. RHEO collects many advanced mesh-free modeling features into a centralized, modular, and easily extensible package, including heat transport, reversible melting and solidification, particle distribution regularization, and user adjustable kernel accuracy. We report comprehensive tests of RHEO's accuracy and convergence for common benchmark flows of bulk fluids, boundary driven flows, and complex fluid/solid systems. A series of multi-phase systems are highlighted, including a bouncing water balloon, the fracture and flow of a brittle egg, melting of a free-standing solid beam, and conductive cooling and solidification of an extruded polymer during 3D printing.

Gaussian smoothed particle hydrodynamics: A high-order meshfree particle method

Smoothed particle hydrodynamics (SPH) has attracted significant attention in recent decades, and exhibits special advantages in modeling complex flows with multiphysics processes and complex phenomena. Its accuracy depends heavily on the distribution of particles, and will generally be lower if the particles are distributed non-uniformly. A high-order SPH scheme is proposed in the present work for simulating both compressible and incompressible flows. It uses a Gaussian quadrature rule to perform the particle approximation of SPH by introducing Gaussian nodes. Unfortunately, the Gaussian nodes hardly overlap with SPH particles due to the Lagrangian feature, and thus we use a high-order interpolation method to obtain the corresponding physical quantities at the Gaussian nodes. The accuracy and robustness of the proposed Gaussian SPH are demonstrated by several numerical tests, including the Sod problem, Poiseuille flow, Couette flow, cavity flow, Taylor–Green vortex and dam break flow, and a convergence analysis is also conducted to evaluate the effects of particle resolution and distribution for reconstructing a given function. The simulation results for each test case are in good agreements with the available analytical, experimental or numerical results, showing that the proposed Gaussian SPH method is accurate and reliable but expensive for simulating compressible and incompressible flow problems.

Particle-based modelling of laser powder bed fusion of metals with emphasis on the melting mode transition

The laser-beam powder bed fusion process for metals, commonly abbreviated as PBF-LB/M, is a widely used process for the additive manufacturing of parts. Numerical simulations are useful to identify optimal process parameters for different materials and to obtain detailed insights into process dynamics. The present work uses a single-phase incompressible Smoothed Particle Hydrodynamics (SPH) scheme to model PBF-LB/M which was found to reduce the required computational time and significantly stabilize the partially violent flow in the melt pool in comparison to a weakly compressible SPH approach. The laser-material interaction is realistically modelled by means of a ray tracing method. An approach to model the effective thermal coductivity of the powder bed is proposed. Excellent agreement between the simulation results and experimental X-ray analyses of the transition from conduction melting mode to keyhole mode including geometric properties of the vapor depression zone was found. These results prove the usability of SPH as a high precision simulation tool for PBF-LB/M.Graphic abstract

Extension of finite particle method simulating thermal-viscoelastic flow and fluid–rigid body interactional process in weakly compressible smoothed particle hydrodynamics scheme

In computational non-Newtonian fluid dynamics, heat transfer has obvious effects on motions of viscoelastic fluids, mechanical mechanism of elasticity, and flow regimes. This study suggests an extended numerical scheme of smoothed particle hydrodynamics and finite particle method within density smoothing (SPH_DSFPM), which involves the discretization of smoothed particle hydrodynamics (SPH) and finite particle method (FPM) within density smoothing (DS) in the weakly compressible flow scheme. A corrected particle shifting technique is incorporated to eliminate tensile instability and inhomogeneity near solid boundaries. A corrected dynamic solid coupled boundary is introduced to deal with casting molding within high-pressure operations, which has a good compatibility between virtual particle method and repulsive force model. Numerical results show that the present scheme has the nearly lower relative error (0.5%) than conventional SPH (2.6%) in the case of evolutionary thermal-viscoelastic Poiseuille flow and heat effects have active influences on velocity, pressure variations for viscoelastic fluid flow around periodic circular cylinders. Three different printing modes of moving printers significantly generate into differentiated forming regimes through high-pressure extrusion. Adaptive particle distributions possess robust flow evolutions, by which the shocked jets can be tracked well and the sinking velocities of wedge entering into solutions can be numerically probed well considering different cuspidal biting angles. In the case of macroscopic fluid–rigid body interactions, the statistical degree of deviation on probed forces with experiments is relatively 4.35% and that is 12.5% for SPH. The proposed numerical scheme has a good performance on improved accuracy, convergence, and stability for simulating transient thermal-viscoelastic flows.

Smoothed Particle Hydrodynamics for anisotropic dispersion in heterogeneous porous media

In the context of modeling solute transport through heterogeneous porous media, particle methods possess inherent advantages with respect to mesh-based (Eulerian) methods. In Smoothed Particle Hydrodynamics (SPH), particles represent fluid volumes exchanging concentrations with their neighbors to emulate hydrodynamic dispersion, and advection is simulated by the particles’ displacement. This crucially prevents problems that are otherwise typically associated with Eulerian advection schemes (especially at high grid-Péclet numbers), such as numerical diffusion. Despite the advantages of SPH, modeling dispersion with anisotropic coefficients remains a challenge for the approach, with studies reporting unphysical negative concentrations in conservative problems. This has likely hindered its practical use because dispersion is intrinsically anisotropic in porous media. This article provides a review and numerical evaluation of SPH for simulating dispersion, focusing on three formulations compatible with anisotropic dispersion coefficients. The analysis includes a scheme for which negative concentrations have been formerly reported, plus two more recently developed methods which are applied here for the first time to the problem of anisotropic dispersion in heterogeneous porous media. The SPH schemes are tested under different degrees of dispersion anisotropy for both homogeneous and heterogeneous velocity fields. The results indicate that the newer SPH schemes can produce accurate results without negative concentrations while considering anisotropic dispersion, providing a valid alternative to simulate solute transport through heterogeneous domains.

Three-dimensional analysis of vortex-lattice formation in rotating Bose–Einstein condensates using smoothed-particle hydrodynamics

Recently, we presented a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit time-integrating scheme; our SPH scheme could reproduce the vortex lattices and their formation processes in rotating quasi-two-dimensional (2D) BECs trapped in a 2D harmonic potential. In this study, we have successfully demonstrated a simulation of rotating 3D BECs trapped in a 3D harmonic potential forming ‘cigar-shaped’ condensates. We have found that our scheme can reproduce the following typical behaviors of rotating 3D BECs observed in the literature: (i) the characteristic shape of the lattice formed in the cross-section at the origin and its formation process, (ii) the stable existence of vortex lines along the vertical axis after reaching the steady state, (iii) a ‘cookie-cutter’ shape, with a similar lattice shape observed wherever we cut the condensate in a certain range in the vertical direction, (iv) the bending of vortex lines when approaching the inner edges of the condensate, and (v) the formation of vortex lattices by vortices entering from outside the condensate. Therefore, we further validated our scheme by simulating rotating 3D BECs.

Volume conservation issue within SPH models for long-time simulations of violent free-surface flows

Smoothed Particle Hydrodynamics (SPH) simulations of violent sloshing flows characterized by a strong fragmentation of free-surface can be affected by volume conservation errors. These errors can accumulate in time and preclude the possibility of using such models for long-time simulations. In the present work, different techniques to measure directly the particles’ volumes by their positions are displayed. Thanks to these algorithms, we discussed how volume errors develop when introducing stabilization terms in the standard SPH schemes. Further, we show that the control of these errors can be achieved through two different (non-alternative) stabilization terms: (i) enforcement of dynamic boundary conditions on thin fluid jets and drops; and (ii) a dynamical switch that can impose or not the time variation of the particle masses.To demonstrate the effectiveness of the numerical techniques, four different violent sloshing flows are investigated and compared with experimental data.

A general smoothed particle hydrodynamics (SPH) formulation for coupled liquid flow and solid deformation in porous media

The method of smoothed particle hydrodynamics (SPH) has been recently developed to study the coupled flow-deformation problems in porous material and considerable success has been achieved comparing to traditional mesh-based method, especially for treating large deformation and post-failure. However, computational challenges remain for the hydro-mechanical boundary treatment as well as the accuracy and stability of the numerical scheme. It is shown that the use of conventional SPH operator for the solution of the coupled problem can lead to several issues including numerical instabilities, inaccuracies and unphysical particle clumping as well as particle disorders near the boundary. To address these issues, a general SPH scheme with enhanced accuracy for saturated/unsaturated porous material is proposed in this paper. An improved SPH formulation for seepage analysis is proposed to allow an accurate prediction of liquid flow in porous material. A new stabilisation technique that combines the use of a density diffusion term, a modified particle shifting algorithm, and a new viscous damping term is developed to further improve the accuracy, stability and robustness of the proposed method. The implementations of stress boundary conditions and hydraulic boundary conditions in SPH, such as confining stress, hydraulic head, infiltration/evaporation, and potential seepage face, using either wall boundary particles or free-surface domain particles, are discussed in detail. A range of benchmark examples is adopted to verify the validity of the present coupled framework. The proposed model is finally applied to simulate the failure process of embankment dam due to rapid drawdown. Results indicate that the methodology proposed herein can be a promising tool for the analysis of the coupled hydro-mechanical process in porous material involving large deformations.

Prediction modeling of coastal sediment transport using accelerated smooth particle hydrodynamics approach

A GPU-accelerated 3D smooth particle hydrodynamics (SPH) scheme is developed and applied to a coastal multi-phase liquid-sediment interaction and sediment transport. The SPH scheme's meshless design and the sediment's particle structure enable the modeling of the waves' interactions with the sediment particles beyond the limitation of the mesh-based methods. A Newtonian constitutive model is used to model the liquid phase, and the sediment transport is formulated based on the Herschel-Bulkley-Papanastasiou (HBP) model. The yield characteristics of the sediment phase are estimated using the Drucker-Prager yield criterion. Due to the parallelization of the solution on graphics processing units, the 3D SPH scheme's performance, which uses millions of particles, is improved. Good correlations were observed in the SPH predictions and experimental measurements, with a maximum difference of 4.85 %. The validated scheme is applied to formulate forecasting models for the coastline sediment transport. It is found that erosion and scouring are expected at the coastline region inclined to the direction of the sea waves, with a predicted mass erosion of about 60e3 kg in four years. The wave's velocity is also established to be directly proportional to the sediment transport. The proposed multi-phase SPH methodology is proven effective for sediment transport prediction.

A massive MPI parallel framework of smoothed particle hydrodynamics with optimized memory management for extreme mechanics problems

The dynamic failure process of structures under extreme loadings is very common in many fields of engineering and science. The smoothed particle hydrodynamics (SPH) method offers inherent benefits in dealing with complex interfaces and large material deformations in extreme mechanics problems. However, SPH simulations for 3D engineering applications are time-consuming. To address this issue, we introduce MPI (Message Passing Interface) in our SPH scheme to reduce computational time. Some optimizations are adopted to ensure the massive computation of the SPH method. In particular, an optimized memory management strategy is developed to control the memory footprint. With the present MPI-based massive parallelization of the SPH method, several validation examples are tested and analyzed. By comparing the present numerical results with the reference data, the dynamic failure process of complex structures subjected to extreme loadings like explosive and impact loadings can be well captured. A large number of particles, up to 2.04 billion, are adopted in the present simulations. The scaling tests show that the scalability of the massively parallel SPH program achieves a maximum parallel efficiency of 97% on 10020 CPU cores.

A regularized high-order diffusive smoothed particle hydrodynamics scheme without tensile instability

In the present work, we derive a novel high-order weakly compressible smoothed particle hydrodynamics scheme based on an accurate approximation of the pressure gradient and on the use of numerical Riemann fluxes. Specifically, a switch between non-conservative and conservative formulations of the pressure gradient is adopted close to the free surface, in order to fulfill the dynamic free-surface boundary condition and, at the same time, prevent the onset of the tensile instability in inner regions of the fluid domain. The numerical diffusion is obtained using Riemann solvers, with reconstruction/limitation of the left and right states derived from the Monotonic Upstream-centered Scheme for Conservation Laws technique. These allow for a high-order convergence rate of the diffusive terms that, for increasing spatial resolutions, results in a low numerical dissipation without tuning parameters. Regular particle distributions, which are crucial for the model accuracy, are obtained thanks to recent improvements in Particle Shifting Techniques. These are taken into account within the constitutive equations through a quasi-Lagrangian formalism. The energy balance of such a non-conservative formulation is derived, and an in-depth analysis of the term contributing to numerical dissipation is performed. The numerical investigation is carried out on several problems, illustrating the advantages of the present scheme with respect to conservative formulations. Since the proposed formulation does not intrinsically guarantee momenta conservation, the latter are monitored showing that the overall errors are generally small.

An arbitrarily Lagrangian–Eulerian SPH scheme with implicit iterative particle shifting procedure

This work presents an Arbitrarily Lagrangian–Eulerian Smoothed Particle Hydrodynamics (ALE-SPH) scheme for Navier–Stokes equations where the particle distribution is optimized by an Implicit Iterative Particle Shifting (IIPS) correction capable of improving the accuracy of the SPH spatial operators. Here two different possible approaches, based respectively on Moving Least Square (MLS) and on convective fluxes have been adopted to embed the IIPS formulation in an ALE-SPH scheme. The theoretical second-order convergence rate has been achieved for the Taylor–Green flow at Reynolds number Re=100. Results obtained with the proposed numerical scheme for the Moving box and the Impinging jet test cases confirm that the accuracy is significantly improved, with respect to existing quasi-Lagrangian ALE-SPH schemes.

Direct numerical simulation of three-dimensional isotropic turbulence with smoothed particle hydrodynamics

This study presents an investigation of the capability of smoothed particle hydrodynamics (SPH) to simulate three-dimensional isotropic turbulence. The effect of the error introduced by the particle disorder is assessed by comparing the standard Lagrangian SPH with an Eulerian adaptation. For the free decay of isotropic turbulence in a triple periodic box, the Eulerian SPH shows very good agreement with the reference solution, while the particle disorder in Lagrangian simulations yields an incorrect prediction of turbulent energy spectra. For the first time, a SPH investigation on linearly forced isotropic turbulence is also conducted with a focus on how the numerical dissipation affects the obtained solution. It is found that by using a Godunov-type SPH scheme for the continuity equation and by employing Roe's approximate solver for the Riemann problem at the interface of each neighboring particle, a stable solution is obtained, which is also in agreement with the results predicted by the theory of homogeneous isotropic turbulence. The efficacy of the particle shifting technique applied to turbulent SPH flows is studied in the end. Numerical findings indicate that corrective terms derived from the arbitrary Lagrangian–Eulerian theory are essential for a proper estimation of turbulence characteristics.

An improved SPH scheme for the 3D FEI (Fluid-Elastomer Interaction) problem of aircraft tire spray

The problem of aircraft tire spray produced by taking off or landing on wet runways is widely concerned. When simulating this kind of FEI(Fluid-Elastomer Interaction) problem with traditional SPH((Smoothed Particle Hydrodynamics) method, there are some drawbacks in both fluid and elastomer calculation. For the elastomer simulation, the TL-SPH(Total Lagrangian-SPH) is used to deal with the deformation of tire under complicated motion and load. For the fluid simulation, a new SPH scheme of Riemann solver combined with transport velocity is proposed to solve the problem of particle clumping when the free surface is suffering from violent splash. Then the tire spray model by unified SPH is established, in which both the tire deformation and the spray pattern are validated by experiments. Finally, based on the large-scale tire spray simulation, the phenomenon of tire hydroplaning is studied.

SPH modelling of dissipative sloshing flows under violent vertical harmonic excitation

In the present work the damping effect of sloshing flows in tanks subjected to vertical harmonic oscillations is modelled through the Smoothed Particle Hydrodynamics (SPH) numerical method. The prediction of energy dissipation in these problems is of interest, among the others, in the aeronautic field to appropriately address sloshing-induced loads on aircraft wings. To this purpose, an enhanced SPH scheme is applied and extensively validated through a comparison with a recent experimental campaign in which a partially filled tank is subjected to harmonic vertical accelerations ranging from 0.25 g up to 6 g. Conversely to previous works addressing the same phenomenon, in the present work long-time simulations are performed spanning over several periods of oscillations. This approach allows comparing the predicted energy dissipation in terms of averages computed over several tank oscillations, thus providing a robust numerical outcome. The numerical scheme is tested over a large matrix of different frequencies and accelerations covering a wide range of flow regimes and spanning from mildly-deformed free surface to violent shaken flow. Both 2D and 3D numerical frameworks are considered and compared to the experimental reference. It is shown that the 3D solver is able, in most of the cases, to recover the experimental rate of dissipated energy with errors comparable to the intrinsic uncertainties of the problem whereas the 2D solution significantly under-predicts the damping in high-energy sloshing regimes.

Numerical model of the Gross–Pitaevskii equation for rotating Bose–Einstein condensates using smoothed-particle hydrodynamics

This study proposed a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit real-time integration scheme. Specifically, the Gross–Pitaevskii equation was described as a complex representation to obtain a pair of time-dependent equations, which were then solved simultaneously following discretization based on SPH particle approximation. We adopt the fourth-order Runge–Kutta method for time evolution. We performed simulations of a rotating Bose gas trapped in a harmonic potential, showing results that qualitatively agreed with previously reported experiments and simulations. The geometric patterns of formed lattices were successfully reproduced for several cases, for example, the hexagonal lattice observed in the experiments of rotating BECs. Consequently, it was confirmed that the simulation began with the periodic oscillation of the condensate, which attenuated and maintained a stable rotation with slanted elliptical shapes; however, the surface was excited to be unstable and generated ripples, which grew into vortices and then penetrated inside the condensate, forming a lattice. We confirmed that each branch point of the phase of wavefunctions corresponds to each vortex. These results demonstrate our approach at a certain degree of accuracy. In conclusion, we successfully developed a new SPH scheme for the simulations of vortex lattice formation in rotating BECs.

Energy balance in quasi-Lagrangian Riemann-based SPH schemes

The Smoothed Particle Hydrodynamics (SPH) method suffers from the presence of irregular particle distributions inherent in its Lagrangian nature. A way to circumvent this problem is to consider a quasi-Lagrangian SPH scheme and use a Particle Shifting Technique (PST). In this framework, we include an approximate Riemann solver to represent the particle interaction and obtain two different quasi-Lagrangian Riemann-based SPH schemes: one is a mass-constant SPH model whereas the other is derived from an ALE formalism. These schemes are examined and validated by focusing on their energy balance. In particular, the energy contribution provided by the terms associated to Riemann solver and PST is studied from both a theoretical and numerical point of view. The consistency and the diffusive properties of these energy terms are investigated on test cases involving confined and free-surface flows. Albeit the investigation is performed for a specific Riemann solver and PST, the proposed methodology can be easily extended to other formulations.

Investigations on a high order SPH scheme using WENO reconstruction

Theoretically, a 2nd order convergence can be reached with the Smoothed Particle Hydrodynamics (SPH) method. However, depending on the spatial disorder of particles, the order of convergence observed in practice can be lower than one. In this paper, a methodology is proposed for the reconstruction of high order numerical fluxes in Riemann-SPH formulations for weakly-compressible flows, so as to increase the global order of convergence of the scheme. This methodology is based on the use of one-dimensional Weighted Essentially Non-Oscillatory (WENO) reconstructions applied at each pair of interacting particles, and each 1D WENO stencil is completed using Moving-Least-Squares (MLS) reconstructions. It is shown that a 6th order convergence can be reached with the proposed SPH-WENO scheme. The gain in accuracy and convergence properties of this scheme is shown and discussed through its application to various one-dimensional and two-dimensional test cases. In particular, the influence of the number of neighbor particles and of the spatial particle disorder is studied. Finally, it is shown that the proposed high-order SPH scheme provides a better accuracy to CPU time ratio than usual Riemann-SPH schemes. The treatment of boundary conditions on rigid walls with the proposed scheme is also discussed.

Globally time-reversible fluid simulations with smoothed particle hydrodynamics

This paper describes an energy-preserving and globally time-reversible code for weakly compressible smoothed particle hydrodynamics (SPH). We do not add any additional dynamics to the Monaghan's original SPH scheme at the level of ordinary differential equation, but we show how to discretize the equations by using a corrected expression for density and by invoking a symplectic integrator. Moreover, to achieve the global-in-time reversibility, we have to correct the initial state, implement a conservative fluid-wall interaction, and use the fixed-point arithmetic. Although the numerical scheme is reversible globally in time (solvable backwards in time while recovering the initial conditions), we observe thermalization of the particle velocities and growth of the Boltzmann entropy. In other words, when we do not see all the possible details, as in the Boltzmann entropy, which depends only on the one-particle distribution function, we observe the emergence of the second law of thermodynamics (irreversible behavior) from purely reversible dynamics.

Techniques for second-order convergent weakly compressible smoothed particle hydrodynamics schemes without boundaries

Despite the many advances in the use of weakly compressible smoothed particle hydrodynamics (SPH) for the simulation of incompressible fluid flow, it is still challenging to obtain second-order convergence even for simple periodic domains. In this paper, we perform a systematic numerical study of convergence and accuracy of kernel-based approximation, discretization operators, and weakly compressible SPH (WCSPH) schemes. We explore the origins of the errors and issues preventing second-order convergence despite having a periodic domain. Based on the study, we propose several new variations of the basic WCSPH scheme that are all second-order accurate. Additionally, we investigate the linear and angular momentum conservation property of the WCSPH schemes. Our results show that one may construct accurate WCSPH schemes that demonstrate second-order convergence through a judicious choice of kernel, smoothing length, and discretization operators in the discretization of the governing equations.