Standard Smoothed Particle Hydrodynamics Research Articles

Geometrically nonlinear analysis of two-dimensional structures using an improved smoothed particle hydrodynamics method

Purpose – The purpose of this paper is to present an efficient smoothed particle hydrodynamics (SPH) method particularly adapted for the geometrically nonlinear analysis of structures. Design/methodology/approach – In order to resolve the inconsistency phenomenon which systematically occurs in the standard SPH method at the domain’s boundaries of the studied structure, the classical kernel function and its spatial derivatives were modified by the use of Taylor series expansion. The well-known tensile instabilities inherent to the Eulerian SPH formulation were attenuated by the use of the Total Lagrangian Formulation (TLF). Findings – In order to demonstrate the effectiveness of the present improved SPH method, several numerical applications involving geometrically nonlinear behaviors were carried out using the explicit dynamics scheme for the time integration of the PDEs. Comparisons of the obtained results using the present SPH model with analytical reference solutions and with those obtained using ABAQUS finite element (FE) commercial software, show its good accuracy and robustness. Practical implications – An additional application including a multilayered composite structure and involving buckling and delamination was investigated using the present improved SPH model and the results are compared to the FE results, they confirmed both the efficiency and the accuracy of the proposed method. Originality/value – An efficient 2D-continuum SPH model for the geometrically nonlinear analysis of thin and thick structures is proposed. Contrarily to the classical SPH approaches, here the constitutive material relations are used to link naturally the stresses and strains. The Total Lagrangian approach is investigated to alleviate the tensile instabilities problem, allowing at the same time to avoid the updating procedure of the neighboring particles search and therefore reducing CPU usage. The proposed approach is valid for isotropic and multilayered composites structures undergoing large transformations. CPU time savings and better results with the new 2D-continuum SPH formulation compared to the classical continuum SPH. The explicit dynamic scheme was used for time integration allowing a fast resolution algorithm even for highly nonlinear problems.

Fast SPH simulation for gaseous fluids

This paper presents a fast smoothed particle hydro-dynamics (SPH) simulation approach for gaseous fluids. Unlike previous SPH gas simulators, which solve the transparent air flow in a fixed simulation domain, the proposed approach directly solves the visible gas without involving the transparent air. By compensating the density and force calculation for the visible gas particles, we completely avoid the need of computational cost on ambient air particles in previous approaches. This allows the computational resources to be exclusively focused on the visible gas, leading to significant performance improvement of SPH gas simulation. The proposed approach is at least ten times faster than the standard SPH gas simulation strategy and is able to reduce the total particle number by 25–400 times in large open scenes. The proposed approach also enables fast SPH simulation of complex scenes involving liquid–gas transition, such as boiling and evaporation. A particle splitting and merging scheme is proposed to handle the degraded resolution in liquid–gas phase transition. Various examples are provided to demonstrate the effectiveness and efficiency of the proposed approach.

Smoothed particle hydrodynamics with smoothed pseudo-density

Abstract In this paper, we present a new formulation of smoothed particle hydrodynamics (SPH), which, unlike the standard SPH (SSPH), is well behaved at the contact discontinuity. The SSPH scheme cannot handle discontinuities in density (e.g., the contact discontinuity and the free surface), because it requires that the density of fluid is positive and continuous everywhere. Thus there is inconsistency in the formulation of the SSPH scheme at discontinuities of the fluid density. To solve this problem, we introduce a new quantity associated with particles and the “density” of that quantity. This “density” evolves through the usual continuity equation with an additional artificial diffusion term, in order to guarantee the continuity of the “density.” We use this “density,” or pseudo-density, instead of the mass density, to formulate our SPH scheme. We call our new method SPH with smoothed pseudo-density, and we show that it is physically consistent and can handle discontinuities quite well.

Smooth Particle Hydrodynamics with nonlinear Moving-Least-Squares WENO reconstruction to model anisotropic dispersion in porous media

Most numerical schemes applied to solve the advection–diffusion equation are affected by numerical diffusion. Moreover, unphysical results, such as oscillations and negative concentrations, may emerge when an anisotropic dispersion tensor is used, which induces even more severe errors in the solution of multispecies reactive transport. To cope with this long standing problem we propose a modified version of the standard Smoothed Particle Hydrodynamics (SPH) method based on a Moving-Least-Squares-Weighted-Essentially-Non-Oscillatory (MLS-WENO) reconstruction of concentrations. This scheme formulation (called MWSPH) approximates the diffusive fluxes with a Rusanov-type Riemann solver based on high order WENO scheme. We compare the standard SPH with the MWSPH for different a few test cases, considering both homogeneous and heterogeneous flow fields and different anisotropic ratios of the dispersion tensor. We show that, MWSPH is stable and accurate and that it reduces the occurrence of negative concentrations compared to standard SPH. When negative concentrations are observed, their absolute values are several orders of magnitude smaller compared to standard SPH. In addition, MWSPH limits spurious oscillations in the numerical solution more effectively than classical SPH. Convergence analysis shows that MWSPH is computationally more demanding than SPH, but with the payoff a more accurate solution, which in addition is less sensitive to particles position. The latter property simplifies the time consuming and often user dependent procedure to define the initial dislocation of the particles.

Energy balance in the δ-SPH scheme

Abstract An in-depth analysis of the energy balance in the δ -SPH model has been carried on. In comparison to the standard SPH scheme, the mechanical energy equation of the δ -SPH variant is characterized by a further term that is generally dissipative and is related to the diffusive operator inside the continuity equation. The behaviour and the structure of such a term have been studied in detail and a number of specifically conceived test cases have been considered, highlighting that the dissipative term is generally small and it mainly acts when spurious high-frequency acoustic components are excited. In spite of such a dissipation mechanism, the δ -SPH appears more accurate than the standard SPH scheme even in simulating inviscid fluids.

The Taylor-SPH Meshfree Method: Basis and Validation

This paper presents the basis and validation of the Taylor-SPH meshless method formulated in terms of stresses and velocities which can be applied to Solid Dynamic problems. The proposed method consists of applying first the time discretization by means of a Taylor series expansion in two steps and a corrected SPH method for the space discretization. In order to avoid numerical instabilities, two different sets of particles are used in the time discretization. To validate the Taylor-SPH method, it has been applied to solve the propagation of shock waves in elastic materials and the results have been compared with those obtained with a corrected SPH discretization combined with a 4th order Runge-Kutta time integration. The Taylor-SPH method is shown to be stable, robust and efficient and it provides more accurate results than those obtained with the standard SPH along with the Runge-Kutta time integration scheme. Numerical dispersion and diffusion are eliminated and only a reduced number of particles is required to obtain accurate results.

SPH Simulation of Acoustic Waves: Effects of Frequency, Sound Pressure, and Particle Spacing

Acoustic problems consisting of multiphase systems or with deformable boundaries are difficult to describe using mesh-based methods, while the meshfree, Lagrangian smoothed particle hydrodynamics (SPH) method can handle such complicated problems. In this paper, after solving linearized acoustic equations with the standard SPH theory, the feasibility of the SPH method in simulating sound propagation in the time domain is validated. The effects of sound frequency, maximum sound pressure amplitude, and particle spacing on numerical error and time cost are then subsequently discussed based on the sound propagation simulation. The discussion based on a limited range of frequency and sound pressure demonstrates that the rising of sound frequency increases simulation error, and the increase is nonlinear, whereas the rising sound pressure has limited effects on the error. In addition, decreasing the particle spacing reduces the numerical error, while simultaneously increasing the CPU time. The trend of both changes is close to linear on a logarithmic scale.

Artificial viscosity in smoothed particle hydrodynamics simulation of sound interference

The artificial viscosity has been widely used in reducing unphysical oscillations in the Smoothed Particle Hydrodynamics (SPH) simulations. However, the effects of artificial viscosity on the SPH simulation of sound interference have not been discussed in existing literatures. This paper analyzes the effects and gives some suggestions on the choice of computational parameters of the artificial viscosity in the sound interference simulation. First, a standard SPH code for simulating sound interference in the time domain is built by solving the linearized acoustic wave equations. Second, the Monaghan type artificial viscosity is used to optimize the SPH simulation. Then the SPH codes with and without the artificial viscosity are both used to simulate the sound interference and the numerical solutions are compared with the theoretical results. Finally, different values of computational parameters of the artificial viscosity are used in the simulation in order to determine the appropriate values. It turns out that the numerical solutions of SPH simulation of sound interference agree well with the theoretical results. The artificial viscosity can improve the accuracy of the sound interference simulation. The appropriate values of computational parameters of the artificial viscosity are recommended in this paper.

Equalizing resolution in smoothed-particle hydrodynamics calculations using self-adaptive sinc kernels

The smoothed-particle hydrodynamics (SPH) technique is a numerical method for solving gas-dynamical problems. It has been applied to simulate the evolution of a wide variety of astrophysical systems. The method has a second-order accuracy, with a resolution that is usually much higher in the compressed regions than in the diluted zones of the fluid. In this work, we propose and check a scheme to balance and equalize the resolution of SPH between high- and low-density regions. This method relies on the versatility of a family of interpolators called Sinc kernels, which allows increasing the interpolation quality by varying only a single parameter (the exponent of the Sinc function). The scheme is checked and validated through a number of numerical tests, from standard one-dimensional Riemann problems in shock tubes, to multidimensional simulations of explosions, hydrodynamic instabilities and the collapse of a sun-like polytrope. The analysis of the hydrodynamical simulations suggests that the scheme devised to equalizing accuracy improves the treatment of the post-shock regions and, in general, of the rarefacted zones of fluids while causing no harm to the growth of hydrodynamic instabilities. The method is robust and easy to implement with a low computational overload. It conserves mass, energy, and momentum and reduces to the standard SPH scheme in regions of the fluid that have smooth density gradients.

Numerical simulation of plunging wave breaker impact by a modified Turbulent WCSPH method

In this paper, the mesh-less weakly compressible smoothed particle hydrodynamics (WCSPH) method was used to solve the continuity and momentum equations with laminar viscosity and SPS turbulent model. To correct the pressure field and improve the accuracy of the free surface profile, modification of kernel and gradient of kernel was implemented to WCSPH model. The modified method, namely mSPH-T-K, was also equipped with periodic smoothing of the density using the modified kernel. To validate the modified model, the pressure field and wave front position of the 2D dam break flow were compared with those of experimental data, standard SPH method and mSPH-T method, which is the turbulence SPH method without modification of kernel and its gradients. Furthermore, the dam break on a wet bed was performed by the model. The results showed the well agreement of the modified model results of free surface with this benchmark. Finally, transformation of plunging breaker wave on a ramp was modeled with both mSPH-T and modified model. The results showed that the modified model has a good agreement with experimental data.

Numerical simulation of 2D sloshing waves using SPH with diffusive terms

A numerical model based on the Smoothed Particle Hydrodynamics (SPH) method is developed which can appropriately simulate two-dimensional (2D) sloshing waves starting from smooth harmonic to violent breaking waves in a partially filled rectangular water container. This is achieved by employing a single set of numerical tools for solving the fluid flow problem under given boundary conditions. Recently, δ-SPH model has been proven to be superior to standard SPH for solving fluid flow problems covering a broad range of Reynolds (Re) numbers including violent fluid motion involving free surface fragmentation. Present work extends this further by applying δ-SPH to non-violent sloshing waves with specific focus in capturing higher order harmonics as generally predicted by an analytical or potential flow based model. An in depth analysis has been carried out to identify major factors affecting the present δ-SPH model from predicting higher order harmonics. The role of XSPH factor as used in SPH in smoothening of velocity has further been revisited to investigate its’ effect on pressure, particularly in violent sloshing flows. Inter-comparison has been performed with few other SPH versions (namely, fully incompressible SPH (ISPH) and f-SPH, apart from standard weakly compressible SPH (WCSPH)) for selected cases. The model prediction for different physical quantities of sloshing has been found to be in good agreement with results from other experimental, numerical and analytical based studies. Based on the present study, suitable scales for the tuning factors have been proposed for the development of robust SPH models which is expected to simulate sloshing waves defined by relatively broader d/L (ratio of initial water depth to tank length) ranges.

IISPH‐FLIP for incompressible fluids

AbstractWe propose to use Implicit Incompressible Smoothed Particle Hydrodynamics (IISPH) for pressure projection and boundary handling in Fluid‐Implicit‐Particle (FLIP) solvers for the simulation of incompressible fluids. This novel combination addresses two issues of existing SPH and FLIP solvers, namely mass preservation in FLIP and efficiency and memory consumption in SPH. First, the SPH component enables the simulation of incompressible fluids with perfect mass preservation. Second, the FLIP component efficiently enriches the SPH component with detail that is comparable to a standard SPH simulation with the same number of particles, while improving the performance by a factor of 7 and significantly reducing the memory consumption. We demonstrate that the proposed IISPH‐FLIP solver can simulate incompressible fluids with a quantifiable, imperceptible density deviation below 0.1%. We show large‐scale scenarios with up to 160 million particles that have been processed on a single desktop PC using only 15GB of memory. One‐ and two‐way coupled solids are illustrated.

A compatibly differenced total energy conserving form of SPH

SUMMARYWe describe a modified form of smoothed particle hydrodynamics (SPH) in which the specific thermal energy equation is based on a compatibly differenced formalism, guaranteeing exact conservation of the total energy. We compare the errors and convergence rates of the standard and compatible SPH formalisms on a variety of shock test problems with analytic answers. We find that the new compatible formalism reliably achieves the expected first‐order convergence for these analytic shock tests and, in all cases, improves the accuracy of the numerical solution over the standard formalism. We also examine the performance of our new formalism on a more complicated applied problem: the diversion of an asteroid by a kinetic impactor. We find the compatible discretization demonstrates measurable improvement in the convergence of properties such as the deflection velocity in this kind of applied problem as well. Copyright © 2014 John Wiley & Sons, Ltd.

A new class of Moving-Least-Squares WENO–SPH schemes

We present a new class of meshless Lagrangian particle methods based on the SPH formulation of Vila and Ben Moussa, combined with a new weighted essentially non-oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The key idea is to produce for each particle first a set of high order accurate Moving-Least-Squares (MLS) reconstructions on a set of different reconstruction stencils. Then, these reconstructions are combined with each other using a non-linear WENO technique in order to capture at the same time discontinuities and to maintain accuracy and low numerical dissipation in smooth regions. The numerical fluxes between interacting particles are subsequently evaluated using this MLS–WENO reconstruction at the midpoint between two particles, in combination with a Riemann solver that provides the necessary stabilization of the scheme based on the underlying physics of the governing equations. We propose the use of two different Riemann solvers: the Rusanov flux and an Osher-type flux. The use of monotone fluxes together with a WENO reconstruction ensures accuracy, stability, robustness and an essentially non-oscillatory solution without the artificial viscosity term usually employed in conventional SPH schemes.To our knowledge, this is the first time that the WENO method, which has originally been developed for mesh-based schemes in the Eulerian framework on fixed grids, is extended to meshfree Lagrangian particle methods like SPH in multiple space dimensions.We test the new algorithm on two dimensional blast wave problems and on the classical one-dimensional Sod shock tube problem for the Euler equations of compressible gas dynamics. We obtain a good agreement with the exact or numerical reference solution in all cases and an improved accuracy and robustness compared to existing standard SPH schemes.

Development of the meshless finite volume particle method with exact and efficient calculation of interparticle area

The Finite Volume Particle Method (FVPM) is a meshless method based on a definition of interparticle area which is closely analogous to cell face area in the classical finite volume method. In previous work, the interparticle area has been computed by numerical integration, which is a source of error and is extremely expensive. We show that if the particle weight or kernel function is defined as a discontinuous top-hat function, the particle interaction vectors may be evaluated exactly and efficiently. The new formulation reduces overall computational time by a factor between 6.4 and 8.2. In numerical experiments on a viscous flow with an analytical solution, the method converges under all conditions. Significantly, in contrast with standard FVPM and SPH, error depends on particle size but not on particle overlap (as long as the computational domain is completely covered by particles). The new method is shown to be superior to standard FVPM for shock tube flow and inviscid steady transonic flow. In benchmarking on a viscous multiphase flow application, FVPM with exact interparticle area is shown to be competitive with a mesh-based volume-of-fluid solver in terms of computational time required to resolve the structure of an interface.

Numerical simulation of heat transfer in gas-particle two-phase flow with smoothed discrete particle hydrodynamics

Heat transfer between particles and that between gas phase and particle phase in gas-particle two-phase flow cannot be ignored. Smoothed discrete particle hydrodynamics, as a new method for solving the gas-particle two-phase flow, has been used in simulating the aerolian sand transport successfully. Based on the smoothed discrete particle hydrodynamics method, a heat conduction model is presented in this paper and is used to simulate the heat transfer processes and the particle evaporation in gas-particle two-phase flow. Firstly, the equations to be solved are presented in which the energy equations are introduced for each phase and the second derivative item in conduction is treated by combining a standard smoothed particle hydrodynamics first derivative with a finite difference approximation of a first derivative. The heat conduction between particle and gas is computed from temperature difference and heat transfer coefficient. The disc-type particle cluster problem and bubble fluidized bed are simulated and the results are in close agreement with the two fluid model simulation results. The vaporization law for discrete phase droplet is used to deal with the particle evaporation and then a jet evaporation is simulated. Numerical results all show a good agreement with the discrete particle model results. It is indicated that the new method is of good accuracy and practical applicability.

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.

A critical investigation of smoothed particle hydrodynamics applied to problems with free‐surfaces

SUMMARYIn this paper, an in‐depth study of SPH method, in its original weakly compressible version, is achieved on dedicated 2D and 3D free‐surface flow test cases. These rather critical prototype problems shall constitute suitable test cases to get through when building a free‐surface SPH model. The present work aims at investigating various numerical aspects of this method, often little mentioned in literature. In particular, a great care is paid to the dynamic part of the solution, which is critical to the local hydrodynamic load prediction. The role of numerical errors in the development of acoustic frequencies in the pressure signals is discussed, as well as the influence of the choice of the sound velocity. On the shown test problems, it is also evidenced that some numerical tools are crucial to ensure the robustness and accuracy of the standard SPH method. The convergence of our model is heuristically proved on these nonlinear prototype tests, showing at the same time the very satisfactory level of accuracy reached. Through these tests, some other numerical specificities of the SPH method are discussed, such as the self‐redistribution of the particles occurring during the Lagrangian evolution. A higher order model is also proposed, and its advantages and drawbacks are discussed. Copyright © 2013 John Wiley & Sons, Ltd.

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.

A DENSITY-INDEPENDENT FORMULATION OF SMOOTHED PARTICLE HYDRODYNAMICS

The standard formulation of the smoothed particle hydrodynamics (SPH) assumes that the local density distribution is differentiable. This assumption is used to derive the spatial derivatives of other quantities. However, this assumption breaks down at the contact discontinuity. At the contact discontinuity, the density of the low-density side is overestimated while that of the high-density side is underestimated. As a result, the pressure of the low (high) density side is over (under) estimated. Thus, unphysical repulsive force appears at the contact discontinuity, resulting in the effective surface tension. This tension suppresses fluid instabilities. In this paper, we present a new formulation of SPH, which does not require the differentiability of density. Instead of the mass density, we adopt the internal energy density (pressure), and its arbitrary function, which are smoothed quantities at the contact discontinuity, as the volume element used for the kernel integration. We call this new formulation density independent SPH (DISPH). It handles the contact discontinuity without numerical problems. The results of standard tests such as the shock tube, Kelvin-Helmholtz and Rayleigh-Taylor instabilities, point like explosion, and blob tests are all very favorable to DISPH. We conclude that DISPH solved most of known difficulties of the standard SPH, without introducing additional numerical diffusion or breaking the exact force symmetry or energy conservation. Our new SPH includes the formulation proposed by Ritchie & Thomas (2001) as a special case. Our formulation can be extended to handle a non-ideal gas easily.