SPH Formulation Research Articles

A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows

Abstract The present paper is concerned with the development of a new path-conservative meshless Lagrangian particle method (SPH), for the solution of non-conservative systems of hyperbolic partial differential equations, with applications to the shallow water equations, compressible and incompressible multi-phase flows. For shallow water flows, the proposed method is well-balanced. The starting point of our work is the SPH formulation of Vila and Ben Moussa. The method is based on Arbitrary-Lagrangian-Eulerian (ALE) numerical flux functions (Riemann solvers) and the scheme is rewritten in flux-difference form, thus ensuring at least zeroth order consistency for constant solutions at the discrete level. Furthermore, a smoothed velocity field is used, in order to obtain a final particle velocity that is consistent with the chosen ALE interface velocity. Starting from the formulation previously described, we then use a path-conservative discretization of the non-conservative terms, following the pioneering work of Castro and Pares. The scheme uses a generalized Roe-matrix that is computed as the path integral of the non-conservative terms along a prescribed integration path, which in this paper is chosen to be a simple straight-line segment. For the shallow water systems under consideration, the segment path leads to well-balanced schemes that exactly preserve the water at rest solution for arbitrary bottom topography. To the knowledge of the authors, this is the first well-balanced SPH scheme based on approximate Riemann solvers and the framework of path-conservative schemes for non-conservative hyperbolic systems. Three different approximate Riemann solvers have been investigated and compared in this paper: the Rusanov scheme, the Osher-type DOT scheme and the HLLEM method. A detailed comparison concerning the accuracy and computational efficiency of each Riemann solver is provided. Several test problems (both in 1D and 2D) are presented in the final part of the paper for different systems of non-conservative hyperbolic equations: the Baer Nunziato model of compressible multi-phase flows, the one- and two-layer shallow water equations and the Pitman & Le two-phase debris flow model. The numerical results for each system of equations have been compared with available reference solutions to verify the accuracy of the proposed scheme. For the shallow water type models under consideration the method proposed in this paper has furthermore been shown to be well-balanced up to machine precision.

Anisotropic diffusion in mesh-free numerical magnetohydrodynamics

We extend recently-developed mesh-free Lagrangian methods for numerical magnetohydrodynamics (MHD) to arbitrary anisotropic diffusion equations, including: passive scalar diffusion, Spitzer-Braginskii conduction and viscosity, cosmic ray diffusion/streaming, anisotropic radiation transport, non-ideal MHD (Ohmic resistivity, ambipolar diffusion, the Hall effect), and turbulent 'eddy diffusion.' We study these as implemented in the code GIZMO for both new meshless finite-volume Godunov schemes (MFM/MFV). We show the MFM/MFV methods are accurate and stable even with noisy fields and irregular particle arrangements, and recover the correct behavior even in arbitrarily anisotropic cases. They are competitive with state-of-the-art AMR/moving-mesh methods, and can correctly treat anisotropic diffusion-driven instabilities (e.g. the MTI and HBI, Hall MRI). We also develop a new scheme for stabilizing anisotropic tensor-valued fluxes with high-order gradient estimators and non-linear flux limiters, which is trivially generalized to AMR/moving-mesh codes. We also present applications of some of these improvements for SPH, in the form of a new integral-Godunov SPH formulation that adopts a moving-least squares gradient estimator and introduces a flux-limited Riemann problem between particles.

A truly incompressible smoothed particle hydrodynamics based on artificial compressibility method

In the present study, a truly incompressible smoothed particle hydrodynamics based on the artificial compressibility method for simulating steady and unsteady incompressible flows is proposed and assessed. The incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method proposed by Chorin in the Eulerian reference frame are written in a Lagrangian reference frame to provide an appropriate incompressible SPH algorithm. The proposed SPH formulation implemented here is based on an implicit dual-time stepping scheme to be capable of time-accurate analysis of unsteady flows. The advantage of the Artificial Compressibility-based Incompressible SPH (ACISPH) method proposed over the Weakly Compressible SPH (WCSPH) and the pseudo-compressibility SPH methods is that the ACISPH formulation is a truly incompressible SPH algorithm and it does not involve any approximate enforcement of the incompressibility condition that usually causes to use a large magnitude speed of sound implying a small time step in the computations. The approximate enforcement of the incompressibility condition used in the WCSPH method also causes spurious oscillations in the density and pressure fields which is not the case for the ACISPH formulation proposed. Unlike the projection SPH algorithm and the moving-particle semi-implicit (MPS) method formulation, the ACISPH formulation presented herein does not involve the iterative solution of a Poisson equation for the pressure field. The accuracy and robustness of the proposed incompressible SPH (ACISPH) are demonstrated by solving different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of particle resolution and the value of artificial compressibility parameter on the accuracy and convergence rate of the solution. To validate the results, different test cases are considered herein that are incompressible flows in a 2-D simple channel, a 2-D locally expanded channel, a 2-D cavity, the unsteady Couette flow with pressure gradient, the Taylor vortex problem and a dam break problem with dry bed. Results obtained for these cases are in good agreement with the available analytical and numerical results. The study shows the artificial compressibility-based ISPH (ACISPH) method proposed is accurate and robust for simulating steady and unsteady incompressible flow problems.

Numerical modelling of the effect of using multi-explosives on the explosive forming of steel cones

Modelling and analysis of underwater explosive forming process by using FEM and SPH formulation is presented in this work. The explosive forming of a steel cone is studied. The model setup includes a low carbon steel plate, plate holder, forming die as well as water and C4 explosive.The effect of multiple explosives on rate of targets deformation has been studied. Four different multi-explosives models have been developed and compared to the single explosive model. The formability of the steel plate based on forming limit failure criteria has been investigated. Aspects such as shape of plates deformation and thickness of the plate during the forming process have been examined.The model results indicate that a multi-explosives model does not always guarantee a faster rate of target deformation without central explosive. On the other hand the model results indicate that the multi-explosives setup is capable of preventing crack failure of the steel plate during the forming process which would occur if a single explosive model was used.

The effects of AGN feedback and SPH formulation on black hole growth in galaxies

The effects of AGN feedback and SPH formulation on black hole growth in galaxies

A novel SPH method for sedimentation in a turbulent fluid

A novel method for simulating sedimentation is described and applied to the sedimentation of dust in a turbulent fluid. We assume the dust grains are sufficiently numerous that they may be treated as a fluid and modelled by SPH particles. A different set of SPH particles describes the fluid. The equations of motion are therefore similar to those of Monaghan and Kocharyan [14] with the exception that the sedimentation of dust onto a solid surface is treated as if the surface mimics a sink for the dust fluid. The continuity equation for the dust then contains a sink term that can be modelled in the SPH formulation by allowing the mass of each SPH dust particle to decrease when it is sufficiently close to the boundary. We apply this method both to sedimentation in a nearly static fluid, and to sedimentation in a turbulent fluid. In the latter case we produce the turbulence by both a mechanical stirrer and by a stochastic algorithm. Our results agree very closely with the experiments of Martin and Nokes.

Boosting the accuracy of SPH techniques: Newtonian and special-relativistic tests

We study the impact of different discretization choices on the accuracy of SPH and we explore them in a large number of Newtonian and special-relativistic benchmark tests. As a first improvement, we explore a gradient prescription that requires the (analytical) inversion of a small matrix. For a regular particle distribution this improves gradient accuracies by approximately ten orders of magnitude and the SPH formulations with this gradient outperform the standard approach in all benchmark tests. Second, we demonstrate that a simple change of the kernel function can substantially increase the accuracy of an SPH scheme. While the "standard" cubic spline kernel generally performs poorly, the best overall performance is found for a high-order Wendland kernel which allows for only very little velocity noise and enforces a very regular particle distribution, even in highly dynamical tests. Third, we explore new SPH volume elements that enhance the treatment of fluid instabilities and, last, but not least, we design new dissipation triggers. They switch on near shocks and in regions where the flow --without dissipation-- starts to become noisy. The resulting new SPH formulation yields excellent results even in challenging tests where standard techniques fail completely.

Numerical Simulation of Crater Creating Process in Dynamic Replacement Method by Smooth Particle Hydrodynamics

Abstract A theoretical base of SPH method, including the governing equations, discussion of importance of the smoothing function length, contact formulation, boundary treatment and finally utilization in hydrocode simulations are presented. An application of SPH to a real case of large penetrations (crater creating) into the soil caused by falling mass in Dynamic Replacement Method is discussed. An influence of particles spacing on method accuracy is presented. An example calculated by LS-DYNA software is discussed. Chronological development of Smooth Particle Hydrodynamics is presented. Theoretical basics of SPH method stability and consistency in SPH formulation, artificial viscosity and boundary treatment are discussed. Time integration techniques with stability conditions, SPH+FEM coupling, constitutive equation and equation of state (EOS) are presented as well.

Investigation of S-SPH for Hypervelocity Impact Calculations

In this paper, we investigate the Symmetric-SPH (S-SPH) method. S-SPH restores n-order consistency to the SPH formulation while remaining fully conservative by using a Taylor expansion of field variables to fit the kernel function. It has potential for HVI problems because it enables the ability to perform accurate stress and state calculations. We implement S-SPH in the Velodyne hydro-structural solver and evaluate its performance over a series of numerical examples including flyer impact, fragment penetration, and Taylor Rod impact. Idealized contact algorithms are employed to eliminate uncertainty in the flyer and Taylor impact problems while an advanced Lagrange multiplier algorithm is used for the fragment penetration test. We use the CTH hydrocode to provide a baseline response for each of the examples due to its ability to effectively handle penetration, hydrodynamic deformation, and shock propagation. Direct numerical comparisons are used to eliminate uncertainty from material characterizations, equation of state (EOS) models, and mesh resolution. We identify strengths and shortcomings of the S-SPH method and evaluate its utility for classes of HVI problems. We also compare the performance against SPH to compare relative accuracy, computational cost, and stability.

Multiple-Fluid SPH Simulation Using a Mixture Model

This article presents a versatile and robust SPH simulation approach for multiple-fluid flows. The spatial distribution of different phases or components is modeled using the volume fraction representation, the dynamics of multiple-fluid flows is captured by using an improved mixture model, and a stable and accurate SPH formulation is rigorously derived to resolve the complex transport and transformation processes encountered in multiple-fluid flows. The new approach can capture a wide range of real-world multiple-fluid phenomena, including mixing/unmixing of miscible and immiscible fluids, diffusion effect and chemical reaction, etc. Moreover, the new multiple-fluid SPH scheme can be readily integrated into existing state-of-the-art SPH simulators, and the multiple-fluid simulation is easy to set up. Various examples are presented to demonstrate the effectiveness of our approach.

Modeling of Water Flows around a Circular Cylinder with the SPH Method

Abstract The paper describes the SPH modeling of a plane problem of fluid flow around a rigid circular cylinder. In the model considered, the cylinder is placed in a rectangular fluid domain at a certain distance from a horizontal plane boundary, and it is subjected to fluid flow forces. The fluid motion is induced by a piston type generator. The generator - fluid system starts to move from rest at a certain moment of time. The work aims at a discrete description of the fluid flow around the cylinder and, at the same time, calculation of the pressure distribution along the circumference of the cylinder and the resultant of the pressure on the cylinder. In order to solve the initial value problem considered, a new SPH formulation of boundary conditions on the cylinder surface is proposed which match the physical condition for the fluid velocity at this boundary. For a viscous fluid, an approximate description of the stress tensor is formulated which allows to reduce the differentiation of field functions to the first order in calculating the shear forces in the SPH approach.

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.

Viscous bubbly flows simulation with an interface SPH model

The multi-fluid SPH formulation by Grenier et al. (2009) is extended to study practical problems where bubbly flows play an important role for production processes of the offshore industry. Since these flows are dominated by viscous and surface tension effects, the proposed formulation includes specific models of these physical effects and validations on specific benchmark test cases are carefully performed. The numerical outputs are validated against analytical and numerical reference solutions, and accuracy and convergence of the proposed numerical model are assessed. This model is then used to simulate viscous incompressible bubbly flows of increasing complexity. These flows include the evolution of isolated bubbles, the merging of two bubbles, and the separation process in a bubbly flow. In each case, results are compared to reference solutions and the influence of the Bond number on these interfacial flow evolutions is investigated in detail.

Considering Artificial Viscosity in a SPH Model for Simulation of Transom Waves

In the current article, a SPH model based on artificial viscosity is utilized to simulate transom stern flow. In this context, Navier-Stokes equations are discretized in SPH formulation. Furthermore, predictor-corrector algorithm is implemented to solve the discretized equation. Due to density fluctuations, a density filter approach is also included. Finally, a rectangular body is considered as transom and is towed by a constant velocity. Moreover, simulations are performed at two different draft based Froude numbers at which complex physical phenomena have been observed. The obtained results are compared against both experimental and numerical solutions existing in literature. Some capabilities of SPH solution are also illustrated.

Total Lagrangian SPH modelling of necking and fracture in electromagnetically driven rings

This paper describes research on the prediction of necking and failure in metals at very high strain rates. The model developed in this paper uses a total Lagrangian SPH formulation with a normalised kernel. The detailed data from electromagnetically driven ring experiments by Zhang and Ravi-Chandar (Int J Fract 142:183–217, 2006) is used to evaluate the accuracy of the model predictions. In order to correctly model fracture in the total Lagrangian SPH formulation a visibility criterion based on a truncated cone has been implemented to remove particles obscured by a failed particle. A Johnson-Cook plasticity model is used in combination with a Lemaitredamage model to describe the plastic deformation and fracture of the rings. The effect of Joule heating due to the current induced in the ring is taken into account in the constitutive model. The acceleration due to the ring currents was implemented in the SPH code as a body force. The results demonstrate that this type of model is capable of predicting the number of fragments as well as the time of fracture. In agreement with experimental data, the model also predicts arrested necks and bending in the fragments.

A Boundary Integral SPH Formulation: Consistency and Applications to ISPH and WCSPH

One of the historical problems appearing in SPH formulations is the inconsistencies coming from the inappropriate implementation of boundary conditions. In this work, this problem has been investigated; instead of using typical methodologies such as extended domains with ghost or dummy particles where severe inconsistencies are found, we included the boundary terms that naturally appear in the formulation. First, we proved that in the 1D smoothed continuum formulation, the inclusion of boundary integrals allows for a consistent O(h) formulation close to the boundaries. Second, we showed that the corresponding discrete version converges to a certain solution when the discretization SPH parameters tend to zero. Typical tests with the first and second derivative operators confirm that this boundary condition implementation works consistently. The 2D Poisson problem, typically used in ISPH, was also studied, obtaining consistent results. For the sake of completeness, two practical applications, namely, the duct flow and a sloshing tank, were studied with the results showing a rather good agreement with former experiments and previous results.

Simulation of free falling rigid body into water by a stabilized incompressible SPH method

A stabilized incompressible smoothed particles hydrodynamics (ISPH) method is utilized to simulate free falling rigid body into water domain. Both of rigid body and fluid domain are modeled by SPH formulation. The proposed source term in the pressure Poisson equation contains two terms; divergence of velocity and density invariance. The density invariance term is multiplied by a relaxed parameter for stabilization. In addition, large eddy simulation with Smagorinsky model has been introduced to include the eddy viscosity effect. The improved method is applied to simulate both of free falling vessels with different materials and water entry-exit of horizontal circular cylinder. The applicability and efficiency of improved method is tested by the comparisons with reference experimental results.

Modelling shear flows with smoothed particle hydrodynamics and grid-based methods

Given the importance of shear flows for astrophysical gas dynamics, we study the evolution of the Kelvin-Helmholtz instability (KHI) analytically and numerically. We derive the dispersion relation for the two-dimensional KHI including viscous dissipation. The resulting expression for the growth rate is then used to estimate the intrinsic viscosity of four numerical schemes depending on code-specific as well as on physical parameters. Our set of numerical schemes includes the Tree-SPH code VINE, an alternative SPH formulation developed by Price (2008), and the finite-volume grid codes FLASH and PLUTO. In the first part, we explicitly demonstrate the effect of dissipation-inhibiting mechanisms such as the Balsara viscosity on the evolution of the KHI. With VINE, increasing density contrasts lead to a continuously increasing suppression of the KHI (with complete suppression from a contrast of 6:1 or higher). The alternative SPH formulation including an artificial thermal conductivity reproduces the analytically expected growth rates up to a density contrast of 10:1. The second part addresses the shear flow evolution with FLASH and PLUTO. Both codes result in a consistent non-viscous evolution (in the equal as well as in the different density case) in agreement with the analytical prediction. The viscous evolution studied with FLASH shows minor deviations from the analytical prediction.

An Hamiltonian interface SPH formulation for multi-fluid and free surface flows

In the present work a new SPH model for simulating interface and free surface flows is presented. This formulation is an extension of the one discussed in Colagrossi and Landrini (2003) and is related to the one proposed by Hu and Adams (2006) to study multi-fluid flows. The new SPH scheme allows an accurate treatment of the discontinuity of quantities at the interface (such as the density), and permits to model flows where both interfaces and a free surface are present. The governing equations are derived following a Lagrangian variational principle leading to an Hamiltonian system of particles. The proposed formulation is validated on test cases for which reference solutions are available in the literature.

Full SPH fluid–shell interaction for leakage simulation in explicit dynamics

AbstractThis paper is devoted to the modeling of fluid leakage through a shell in case of impact loading. The modeling of the fluid and the shell is based on SPH formulation. The proposed model is devoted to the prediction of failure of a shell filled with fluid. This paper is devoted to the numerical modeling of fluid–structure interaction. The pinballs technique is used for contact analysis. Numerical predictions are compared with analytical as well as with an original experiment. Copyright © 2009 John Wiley & Sons, Ltd.