SPH Formulation Research Articles

A Riemann-based SPH formulation for modelling elastoplastic soil behaviour using a Drucker–Prager model

The present paper aims at proposing and investigating a Riemann-based SPH formulation to simulate the elastoplastic behaviour of soils undergoing large deformations, using a Drucker–Prager model. Basing on the pioneer work from Parshikov and Medin (Parshikov and Medin, 2002), a Riemann solver is used to maintain regular fields while being free of tuning parameters. By contrast to the work in Parshikov and Medin (2002) where piecewise constant reconstructions were employed, piecewise linear reconstructions are preferred in this work to reduce the numerical diffusion. A Particle Shifting Technique (PST) is used to maintain regular particle distributions and consequently accurate SPH interpolations. To the best of the author’s knowledge, the use of a Riemann solver specific to solid mechanics with a pressure-dependent elastoplastic Drucker–Prager yield surface to model the behaviour of the material represents a novelty with respect to the existing literature. A Boundary Integral Method (BIM) initially derived for fluid dynamics (Ferrand et al., 2013; Chiron et al., 2019) is adapted to solid mechanics in order to handle complex geometries. It allows to deal with wall treatment without using fictitious particles, and shows satisfactory results even in sharp angle regions. The ability of the proposed Riemann-based formulation to simulate accurately elastoplastic problems and its robustness are examined through several test cases in plane strain conditions. Attention is paid to the capacity of the formulation to mitigate the occurrence of Tensile Instability (TI) with respect to other schemes, for which additional treatment is required to treat this issue, such as the additional artificial stress method (Gray et al., 2001).

A reliable SPH(2) formulation for Darcy–Forchheimer–Brinkman equation using a density-based particle shifting in the ALE description

A reliable SPH(2) formulation for Darcy–Forchheimer–Brinkman equation using a density-based particle shifting in the ALE description

A three-dimensional smoothed particle hydrodynamics analysis of multiple retrogressive landslides in sensitive soil

Numerical modelling and analysis of three-dimensional multiple retrogressive landslides of Saint-Jude landslide in sensitive clay is conducted. The investigations of the three-dimensional effects in determining the extent of landslides are concerned. To accurately simulate the extremely large deformation caused by progressive soil failure, a special form of SPH formulation known as finite particle method (FPM) is adopted. A hypoelastic–plastic model with strain softening is implemented to capture the key feature of the mechanical behaviour of the sensitive clay in undrained condition. The multiple progressive failure process, the final run-out stretch, and the retrogression distance of the Saint-Jude landslide are compared quantitatively with the data from site investigation. Furthermore, three-dimensional effects and influences of various factors, including slope morphology and soil properties, are discussed for better understanding of the development of retrogressive landslides in sensitive soil.

A SPH Formulation for Coupled Thermal-Hydraulic-Mechanical Processes of Saturated Porous Media

A SPH Formulation for Coupled Thermal-Hydraulic-Mechanical Processes of Saturated Porous Media

Delamination link-ups in composite laminates due to multiple hail impacts

Damages to aircraft fuselages by ice impactors are categorized as barely visible impact damage (BVID). As the application of composite laminates in manufacturing aircraft fuselage is increasing rapidly, studying hail impact on composite laminates seems crucial. Investigation of variation of delamination area and link-up area for different impact scenarios between ice and carbon fibre prepreg composite plates is the main goal of this study. Link-up refers to joining of delaminated areas created by separate impacts at relatively close impact locations. To this aim, multiple impacts at 1 and up to 6 impact locations and with different impactor energies were simulated using SPH formulation, and delamination areas were quantified. The results showed that as the spacing L between two impact locations increases from 0 to 4R (where R denotes the impactor radius), the total delamination area first shows an increase up to a peak point (usually at R<L<2R) after which it shows a large drop, and after L≈2R, the total delamination area level remains almost constant. The large drop is related to disappearance of the link-up phenomenon beyond a certain spacing. Moreover, it was observed that increasing the number of impact locations increases the delamination area almost linearly. As for the impactor energy influence, the results showed that as the impact energy increases, the threshold spacing for link-up increases. Moreover, for the case of impact at two locations and constant total energy level, two impactors with identical energy level lead to higher delamination area as compared to impactors with non-identical energy levels.

Monolithic Friction and Contact Handling for Rigid Bodies and Fluids Using SPH

AbstractWe propose a novel monolithic pure SPH formulation to simulate fluids strongly coupled with rigid bodies. This includes fluid incompressibility, fluid–rigid interface handling and rigid–rigid contact handling with a viable implicit particle‐based dry friction formulation. The resulting global system is solved using a new accelerated solver implementation that outperforms existing fluid and coupled rigid–fluid simulation approaches. We compare results of our simulation method to analytical solutions, show performance evaluations of our solver and present a variety of new and challenging simulation scenarios.

FSISPH: An SPH formulation for impacts between dissimilar materials

We present an SPH formulation with several new features designed to better model the fully-compressible interaction of dissimilar materials. We developed the new method to simulate the atmospheric entry and break-up of small celestial bodies in planetary atmospheres. The formulation uses a unity-based, density-energy discretization of the hydrodynamic conservation laws with linear-corrected kernel gradients. To account for variations in compressibility, we use an HLLC approximate Riemann solver to adjust the velocity gradient at material interfaces. To handle large transverse velocity discontinuities, we introduce a simple slip interface model that limits the artificial viscosity at material interfaces. Diffusion is optionally applied through the velocity gradient and this allows the density and specific thermal energy to evolve in a manner more consistent with the first law of thermodynamics in comparison to other more direct diffusion schemes. We also introduce a material-local second-order artificial conduction scheme used to smooth the specific thermal energy field. Material damage fits neatly under this framework by treating the damage front as a material interface. The method has been implemented as a solver, FSISPH, within the code, Spheral++, and is publicly available on github. We test our new solver on a number of classic shock, mixing, and multi-material problem. The components we outline can significantly improve accuracy of SPH for problems with sharp contact discontinuities.

Fsisph: An Sph Formulation for Impacts between Dissimilar Materials

Fsisph: An Sph Formulation for Impacts between Dissimilar Materials

Simulation of projectile impact on steel plate-lined, reinforced concrete panels using the smooth particle hydrodynamics formulation

Data from the Tsubota et al. (1993) experiments provided the basis for a numerical study that investigated the impact response of steel-plate lined, reinforced concrete panels using the SPH formulation in LS-DYNA. The simulated tests involved 50 mm (1.97 in), 70 mm (2.76 in), and 90 mm (3.54 in) thick reinforced concrete (RC) panels with steel liners and one 50-mm thick benchmark RC panel. Three of the five panels had a steel liner attached to the back face and one had a steel liner on both faces. The panels were normally impacted by a 39.6 mm (1.56 in) diameter projectile at a velocity of 170 m/s (6693 in/s). Reasonable predictions of observed damage, including perforation, liner fracture or bulging, and concrete scabbing were achieved using the MAT072R3 concrete material model. The effectiveness of adding steel liners to a concrete panel to prevent perforation and scabbing resulting from projectile impact was investigated using the numerical model and MAT072R3. Installing a steel liner on the back face of a panel, with a reinforcement ratio equal to that of the internal reinforcement, is an effective method to mitigate scabbing but has little effect on perforation resistance.

An alternative SPH formulation: ADER-WENO-SPH

We present a new class of fully-discrete one-step SPH schemes based on a mesh-free ADER (Arbitrary DERivatives in space and time) reconstruction on moving particles in multiple space dimensions. In particular, the new SPH scheme computes mesh-free and local high order accurate polynomials in space and time to evaluate numerical fluxes at the midpoint between two interaction particles with a proper Riemann solver within the general SPH framework of Vila (1999) for nonlinear systems of hyperbolic conservation laws. The new scheme has been carefully tested against reference solutions for both the compressible Euler and the magneto-hydrodynamics (MHD) equations. The capability of the proposed scheme to accurately capture shocks and rarefaction waves for 1D and 2D problems with minimal amount of diffusion has been demonstrated. Via numerical evidence it has been shown that the new fully-discrete one-step ADER-WENO-SPH method is computationally more efficient than WENO-SPH schemes based on classical Runge–Kutta time-stepping. This is mainly due to the fact that with ADER timestepping the expensive stencil and neighbor search needs to be done only once per time step, while with Runge–Kutta time integrators the neighbor and stencil search is needed in each Runge–Kutta stage again.

Magnetic induction and electric potential smoothed particle magnetohydrodynamics for incompressible flows

SummaryIn order to solve incompressible, nonideal magnetohydrodynamic (MHD) free‐surface flows, two weakly compressible smoothed particle hydrodynamics models, with and without the consideration of magnetic induction, are developed. The SPH formulation for magnetic induction magnetohydrodynamics (SPMHD), which is popular in astrophysical studies, is applied for the first time to incompressible free‐surface MHD flows, such as liquid metal flows, with the consideration of nonideal MHD effects and boundaries with arbitrary electric conductivity. An SPMHD implementation using the inductionless approximation is also proposed for both electrically conductive and insulating boundaries, in which a Poisson equation is solved to compute the Lorentz force instead of evolving the magnetic induction equation. Both proposed methods are validated against MHD benchmarks, including free‐surface MHD cases. The proposed inductionless SPMHD implementation has the advantages of stability and relaxed time‐step restrictions, but is only accurate at a low range of Hartmann numbers. For high Hartmann number problems, magnetic induction SPMHD model is more accurate. The computational efficiency and conservation error of the two models are compared and discussed.

An axisymmetric multiphase SPH model for the simulation of rising bubble

3-D SPH simulations are often time-consuming due to the massive particle numbers. For the problem which has axisymmetric characteristics, the solving of the entire three-dimensional domain can be simplified which saves computational cost dramatically. In this paper, a novel derivation for the SPH formulation in cylindrical coordinate is presented, which is straightforward and physically meaningful. A multiphase numerical model is developed based on the derived formulation including the discretization formulas of surface tension force, viscous force, interface sharpness force and the treatment of high density ratio near the multiphase interface, as well as an improved anti-penetration treatment close to the axis. The effectiveness and robustness of the proposed multiphase SPH model are demonstrated via the classical surface-tension test and bubble rising problems at different conditions. The results are compared with analytical solutions, the experimental data and other numerical results. The promising results of the novel axisymmetric SPH model demonstrate a great potential for an efficient SPH modeling of complex axisymmetric multiphase flows. Dramatic efficiency boost is achieved by comparing the axisymmetric simulation with 3-D simulation in terms of CPU time.

Accelerating Surface Tension Calculation in SPH via Particle Classification and Monte Carlo Integration

Surface tension has a strong influence on the shape of fluid interfaces. We propose a method to calculate the corresponding forces efficiently. In contrast to several previous approaches, we discriminate to this end between surface and non-surface SPH particles. Our method effectively smooths the fluid interface, minimizing its curvature. We make use of an approach inspired by Monte Carlo integration to estimate local normals as well as curvatures, based on which the force can be calculated. We compare different sampling schemes for the Monte Carlo approach, for which a Halton sequence performed best. Our overall technique is applicable, but not limited to 2D and 3D simulations, and can be coupled with any common SPH formulation. It outperforms prior approaches with regard to total computation time per time step in dynamic scenes. Additionally, it is adjustable for higher quality in small scale scenes with dominant surface tension effects.

An axisymmetric solid SPH solver with consistent treatment of particles close to the symmetry axis

A meshless smoothed particle hydrodynamics solid solver is developed in order to study fluid–structure interactions and to predict cavitation erosion. The solid solver is developed in-house in an axisymmetric configuration. The existing SPH methods dedicated to solid materials do not allow a consistent treatment of particles close to the symmetry axis, so a density correction scheme is proposed here to derive new density and momentum equations for solid mechanics in axisymmetric SPH formulation. The new SPH equations are coded in the solver, and the SPH solid solver is then validated against FEM results, which shows excellent agreement.

A Dual-Support Smoothed Particle Hydrodynamics for Weakly Compressible Fluid Inspired By the Dual-Horizon Peridynamics

A dual-support smoothed particle hydrodynamics (DS-SPH) that allows variable smoothing lengths while satisfying the conservations of linear momentum, angular momentum and energy is developed. The present DS-SPH is inspired by the dual-support, a concept introduced from dual-horizon peridynamics from the authors and applied here to SPH so that the unbalanced interactions between the particles with different smoothing lengths can be correctly considered and computed. Conventionally, the SPH formulation employs either the influence domain or the support domain. The concept of dual-support identifies that the influence domain and the support domain involves the duality and should be simultaneously in the SPH formulation when variable smoothing lengths are used. The DS-SPH formulation can be implemented into conventional SPH codes with minimal changes and also without compromising the computational efficiency. A number of numerical examples involving weakly compressible. fluid are presented to demonstrate the capability of the method

Bending analysis of planar structures made of functionally graded material by a meshless method

ABSTRACTA meshless Smoothed Hydrodynamic Particle (SPH) model is established to simulate the bending deformation of 2D structures made of Functionally Graded Materials (FGMs). The material property is assumed to be varied continuously through the thickness direction according to the power-law distribution. Problems in the classical SPH formulation, like inconsistency and tensile instability are solved by introducing several techniques. Several benchmarks are selected to estimate the performance of the proposed SPH model.

A SPH elastic-viscoplastic model for granular flows and bed-load transport

An elastic-viscoplastic model (Ulrich, 2013) is combined to a multi-phase SPH formulation (Hu and Adams, 2006; Ghaitanellis et al., 2015) to model granular flows and non-cohesive sediment transport. The soil is treated as a continuum exhibiting a viscoplastic behaviour. Thus, below a critical shear stress (i.e. the yield stress), the soil is assumed to behave as an isotropic linear-elastic solid. When the yield stress is exceeded, the soil flows and behaves as a shear-thinning fluid. A liquid-solid transition threshold based on the granular material properties is proposed, so as to make the model free of numerical parameter. The yield stress is obtained from Drucker–Prager criterion that requires an accurate computation of the effective stress in the soil. A novel method is proposed to compute the effective stress in SPH, solving a Laplace equation. The model is applied to a two-dimensional soil collapse (Bui et al., 2008) and a dam break over mobile beds (Spinewine and Zech, 2007). Results are compared with experimental data and a good agreement is obtained.

An Eulerian–Lagrangian incompressible SPH formulation (ELI-SPH) connected with a sharp interface

An Eulerian–Lagrangian incompressible SPH (ELI-SPH) formulation is proposed that improves accuracy over a fully Lagrangian formulation for many problems. This develops the original formulation of Lind and Stansby (2016) by providing a sharp interface rather than a transition zone. This is generally convenient and avoids any need for Arbitrary Lagrangian Eulerian (ALE) particle mass correction. It also enables a simple, accurate solid boundary condition in a Lagrangian formulation by having the interface close to the solid boundary with an Eulerian fluid domain typically three particles thick with mirror particles. Particle regularisation is necessary in a Lagrangian domain and we apply a general form based on Fick’s shifting which is modified at the interface by ignoring Eulerian particles and using mirror particles to give zero concentration gradient and hence zero shifting across the interface, avoiding spurious migration. Continuity is enforced at the interface as part of the combined Eulerian–Lagrangian domain. The formulation is validated against the analytical solution for Taylor–Green vortices, vortex spin down in a box, and propagating waves. The use of mixed kernel order in the Eulerian domain is also demonstrated.

Axis-symmetrical Riemann problem solved with standard SPH method. Development of a polar formulation with artificial viscosity

This paper presents the development of a cylindrical SPH formulation based on previous study of the literature (Petschek et al) with an explicit formulation for the artificial viscosity. The entire development is explained to propose a formulation adapted to solve Euler equations in the case of a Riemann problem with axis-symmetric conditions. Thus, the artificial viscosity is constructed to find smooth solutions of well-known Riemann problems such as Sod, Noh and Sedov problems. Numerical results are compared to exact solutions and observations are made on numerical parameters influence. This study contributes to validate the axis-symmetrical formulation for pure hydrodynamics tests.

Improving Accuracy of SPH Method Using Voronoi Diagram

In this paper, the mesh-less SPH formulation is modified based on Voronoi diagram to approximate region influence of computational nodal points to achieve higher accuracy. The accuracy of the proposed method is examined for a 2-D elliptical partial differential equation with known analytical solution using both regular and irregular node distributions. In addition, a comparison between the accuracy is accomplished for conventional SPH and the proposed method. The numerical results indicate the accuracy of the proposed method over SPH method especially for irregular node distributions.