Lagrangian Smoothed Particle Hydrodynamics Research Articles

Improved Smoothed Particle Hydrodynamics for the Three-Dimensional Solid Impact Problems

Total Lagrangian-Smoothed Particle Hydrodynamics (TL-SPH) method is an important method to improve the tensile instability of traditional SPH, but it still faces some defects, such as insufficient accuracy at boundary region, difficulty in obtaining damage and applying interface conditions. Therefore, the high-order TL-SPH is applied for the elastic–plastic solid. A damage model is proposed by adaptively transforming the type of kernel function. Furthermore, a Riemann contact algorithm is introduced for different solids. The advantages of the methods in improving tensile instability and simulating damage are verified by the cases of the plates and the notched beams with an impact load.

Essentially non-hourglass SPH elastic dynamics

More than two decades ago, the numerical instabilities of particle clustering and non-physical fractures encountered in the simulations of typical elastic dynamics problems using updated Lagrangian smoothed particle hydrodynamics (ULSPH) was identified as tensile instability. Despite continuous efforts in the past, a satisfactory resolution for the simulations of these problems has remained elusive. In this paper, the concept of hourglass modes, other than tensile instability, is first explored for the discretization of shear force, arguing that the former may actually lead to the numerical instabilities in these simulations. Based on such concept, we present an essentially non-hourglass formulation by utilizing the Laplacian operator which is widely used in fluid simulations. Together with the dual-criteria time stepping, adopted into the simulation of solids for the first time to significantly enhance computational efficiency, a comprehensive set of challenging benchmark cases is used to showcase that our method achieves accurate and stable SPH elastic dynamics.

Smoothed Particle Hydrodynamics simulations of integral multi-mode and fractional viscoelastic models

To capture specific characteristics of non-Newtonian fluids, during the past years fractional constitutive models have become increasingly popular. These models are able to capture, in a simple and compact way, the complex behaviour of viscoelastic materials, such as the change in power-law relaxation pattern during the relaxation process of some materials. Using the Lagrangian Smoothed-Particle Hydrodynamics (SPH) method we can easily track particle history; this allows us to solve integral constitutive models in a novel way, without relying on complex tasks.Hence, we develop here a SPH integral viscoelastic method which is first validated for simple Maxwell or Oldroyd-B models under Small Amplitude Oscillatory Shear (SAOS) and start-up channel flows. By exploiting the structure of the integral method, a multi-mode Maxwell model is then implemented. Finally, the method is extended to include fractional constitutive models, validating the approach by comparing results with theory.

SPH numerical model of wave interaction with elastic thin structures and its application to elastic horizontal plate breakwater

This paper develops a Smoothed Particle Hydrodynamics (SPH) model to simulate wave interaction with elastic thin structures. This is achieved by coupling the δ-SPH and the Total Lagrangian SPH (TL-SPH) models for solving the fluid and structure phases, respectively. To this aim, a multiresolution algorithm within the ghost particles framework is introduced in order to treat the thin elastic components of the structure. The main idea of the proposed technology is to evaluate the hydrodynamic forces acting on the structure and the elastic dynamic at the finest solid particle resolution. Then, the information is transferred to the coarse solid particles, which act as a non-penetrating boundary condition for the fluid phase. The present enforcement is introduced with the aim of overcoming the disadvantages in terms of CPU time for heavy SPH simulations, in which the choice of the initial spatial resolution is driven by the presence of elastic thin structures immersed in the fluid environment. The present elastic thin-wall boundary treatment is verified through several numerical cases, in which it is proved that the proposed technology can block non-physical interaction of fluid particles located at opposite sides of the elastic thin-wall boundary, ensures the stability of the fluid pressure field, and reasonably predicts the motion of elastic structures when these are subjected to hydrodynamic loads. Successively, the proposed numerical model is applied to simulate wave interaction with an elastic horizontal plate breakwater. The hydrodynamic performances of the breakwater, with focus on the reflection and transmission coefficients, are analyzed. The results of elastic and rigid horizontal plate breakwaters are compared in order to better understand the effects of the structure deformation on the hydrodynamics. Finally, an analysis of the structural deformation of the breakwater is also reported.

A comparison of numerical stability for ESPH and TLSPH for dynamic brittle fracture

Dynamic brittle fracture is a numerically challenging problem that involves crack nucleation, formation, propagation, and material fragmentation. In this work, we use two forms of Smoothed Particle Hydrodynamics (SPH), namely Eulerian SPH (ESPH) and Total Lagrangian SPH (TLSPH) augmented with the pseudo-spring or virtual-link analogy for seamless modelling of crack formation, subsequent propagation, and material fragmentation. Being particle-based in nature, SPH is naturally capable of capturing finite deformation in materials, and the pseudo-spring or virtual-link analogies provide modelling of multiple discrete cracks without any additional condition, such as visibility criteria. We simulate the crack branching and propagation in a brittle polymeric material subjected to biaxial tensile loading with a pre-existing central notch. The numerical results using ESPH and TLSPH agree with the previously published experimental and numerical results. We have also simulated the dynamic fragmentation of a cylinder and compared the results. This work shows the capability of both ESPH and TLSPH frameworks to model dynamic brittle fracture especially crack branching and curving. It is also observed that the ESPH and TLSPH frameworks present similar results for minor material deformation problems. However, the ESPH framework shows better stability and capability for finite material deformation problems.

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.

A novel Eulerian SPH shallow water model for 2D overland flow simulations

A 2D Eulerian meshless shallow water model based on smoothed particle hydrodynamics (SPH) is proposed to study overland flow. Rain on a dry bed to generate a runoff flow is one common hydraulic phenomenon. However, it is difficult for existing Lagrangian SPH shallow water models to present the flow because no computational particle exists on a dry bed. To overcome this difficulty, the Eulerian form of the 2D shallow water equations is used to describe flow behavior in the proposed model. An HLLC approximate Riemann solver is applied to calculate fluxes between two neighboring particles to smear out the instabilities caused by the discretization of the nonlinear convective term and the discontinuity of wet-dry beds. Additionally, we use a surface reconstruction method (SRM) to ensure that the proposed model has well-balanced and positivity-preserving properties. Bed friction and infiltration effects are also analytically considered. Eight case studies involving various overland flows with wet-dry transitions, rainfall, infiltration, bed friction and complex topography are adopted to test the proposed model capacity. The first-order convergence rate is obtained in the convergence analysis. Against the exact and measured results, in the analytical case studies, the relative root-mean-square errors of depth and velocity are approximately 0.02% to 0.3% and 1.3% to 1.7%, respectively. In the experimental case studies, the relative root-mean-square errors of depth and discharge (velocity) are approximately 3.8% and 1.2% to 6.1%, respectively. The good agreement demonstrates between the results shows the reliability of the proposed 2D Eulerian SPH shallow water model for real-world overland flows.

Simulation of droplet bouncing on flexible substrate in 2D and 3D with WC-TL SPH method

Droplet impacts on super-hydrophobic surfaces play an important role in many fields. Previous studies have focused on rigid substrates and ignored the effect of substrate deformation. Simulating droplet impacts on a flexible substrate is a challenge to numerical methods due to the coupling of free surface, elasticity, large deformation and surface tension. In this study, a fully meshfree model of fluid–solid interaction is developed based on the smoothed particle hydrodynamics (SPH) method. The liquid droplet is modeled by the weakly compressible (WC) SPH method, and the flexible substrate is modeled by the total Lagrangian (TL) SPH method. Liquid surface tension is modeled by the continuum surface force (CSF) method through reconstructing the free surface of the droplet. The surface geometry reconstruction process is given for both 2D and 3D cases, enabling the model to simulate the bouncing of droplets in both 2D and 3D cases. Furthermore, in order to reduce the computational cost of 3D simulation, the TL-SPH model combined with shell theory established the TL-SPH-shell model, thus achieving the simulation of 3D droplet impact on thin plates. For verification and testing, the established WC-TL SPH model is used to simulate the process of droplet impact on various substrates, including fixed-fixed and curved beams, micro-pillar substrates, and cantilever beams. The results show that the spreading, retraction and bouncing processes of the droplets interact with the deformation and recovery of the flexible substrate, accompanied by large deformation and dynamic characteristics, which can be effectively simulated by the model.

An entropy-stable updated reference Lagrangian smoothed particle hydrodynamics algorithm for thermo-elasticity and thermo-visco-plasticity

This paper introduces a novel upwind Updated Reference Lagrangian Smoothed Particle Hydrodynamics (SPH) algorithm for the numerical simulation of large strain thermo-elasticity and thermo-visco-plasticity. The deformation process is described via a system of first-order hyperbolic conservation laws expressed in referential description, chosen to be an intermediate configuration of the deformation. The linear momentum, the three incremental geometric strains measures (between referential and spatial domains), and the entropy density of the system are treated as conservation variables of this mixed coupled approach, thus extending the previous work of the authors in the context of isothermal elasticity and elasto-plasticity. To guarantee stability from the SPH discretisation standpoint, appropriate entropy-stable upwinding stabilisation is suitably designed and presented. This is demonstrated via the use of the Ballistic free energy of the coupled system (also known as Lyapunov function), to ensure the satisfaction of numerical entropy production. An extensive set of numerical examples is examined in order to assess the applicability and performance of the algorithm. It is shown that the overall algorithm eliminates the appearance of spurious modes (such as hour-glassing and non-physical pressure fluctuations) in the solution, typical limitations observed in the classical Updated Lagrangian SPH framework.

Investigation of Nanoscale Scratching on Copper with Conical Tools Using Particle-Based Simulation

In this study, a modeling approach based on smooth particle hydrodynamics (SPH) was implemented to simulate the nanoscale scratching process using conical tools with different negative rake angles. The implemented model enables the study of the topography of groove profiles, scratching forces, and the residual plastic strain beneath the groove. An elastoplastic material model was employed for the workpiece, and the tool–workpiece interaction was defined by a contact model adopted from the Hertz theory. An in-house Lagrangian SPH code was implemented to perform nano-scratching simulations. The SPH simulation results were compared with nanoscale scratching experimental data available in the literature. The simulation results revealed that the normal force was more dominant compared to the cutting force, in agreement with experimental results reported for a conical tip tool with a 60° negative rake angle. In addition, the simulated groove profile was in good agreement with the groove profile produced in the aforementioned experiment. The numerical simulations also showed that the normal and cutting forces increased with the increase in the scratching depth and rake angle. Although the cutting and ploughing mechanisms were noticed in nano-scratching, the ploughing mechanism was more dominant for increased negative rake angles. It was also observed that residual plastic strain exists below the groove surface, and that the plastically deformed layer thickness beneath a scratched groove is larger for more negative values of the tool rake angle and higher scratching depths.

An essentially non-hourglass formulation for total Lagrangian smoothed particle hydrodynamics

The total Lagrangian smoothed particle hydrodynamics (TL-SPH) for elastic solid dynamics suffers from hourglass modes which can grow and lead to the failure of simulation for problems with large deformation. To address this long-standing issue, we present an essentially non-hourglass formulation based on volumetric-deviatoric stress decomposition. Inspired by the fact that the artifact of nonphysical zigzag particle distribution induced by the hourglass modes is mainly characterized by shear deformation and the standard SPH discretization for the viscous term in the Navier–Stokes (NS) equation, the present formulation computes the action of shear stress directly through the Laplacian of displacement other than from the divergence of shear stress. A comprehensive set of challenging benchmark cases are simulated to demonstrate that, while improving accuracy and computational efficiency, the present formulation is able to essentially suppress the hourglass modes and achieves very good numerical stability with a single general effective parameter. In addition, the deformation of a practically relevant stent structure is simulated to demonstrate the potential of the present method in the field of biomechanics.

On methodology and application of smoothed particle hydrodynamics in fluid, solid and biomechanics.

Smoothed particle hydrodynamics (SPH), as one of the earliest meshfree methods, has broad prospects in modeling a wide range of problems in engineering and science, including extremely large deformation problems such as explosion and high velocity impact. This paper aims to provide a comprehensive overview on the recent advances of SPH method in the fields of fluid, solid, and biomechanics. First, the theory of SPH is described, and improved algorithms of SPH with high accuracy are summarized, such as the finite particle method (FPM). Techniques used in SPH method for simulating fluid, solid and biomechanics problems are discussed. The δ-SPH method and Godunov SPH (GSPH) based on the Riemann model are described for handling instability issues in fluid dynamics. Next, the interface contact algorithm for fluid-structure interaction is also discussed. The common algorithms for improving the tensile instability and the framework of total Lagrangian SPH are examined for challenging tasks in solid mechanics. In terms of biomechanics, the governing equations and the coupling forces based on SPH method are exemplified. Then, various typical engineering applications and recent advances are elaborated. The application of fluid mainly depicts the interaction between fluid and rigid body as well as elastomer, while some complicated fluid-structure interaction ocean engineering problems are also presented. In the aspect of solid dynamics, galaxy, geotechnical mechanics, explosion and impact, and additive manufacturing are summarized. Furthermore, the recent advancements of SPH method in biomechanics, such as hemodynamically and gut health, are discussed in general. In addition, to overcome the limitations of computational efficiency and computational scale, the multiscale adaptive resolution, the parallel algorithm and the automated mesh generation are addressed. The development of SPH software in China and abroad is also summarized. Finally, the challenging task of SPH method in the future is summarized. In future research work, the establishment of multi-scale coupled SPH model and deep learning technology in solid and biodynamics will be the focus of expanding the engineering applications of SPH methods.

Modelling the Lyman-α forest with Eulerian and SPH hydrodynamical methods

ABSTRACT We compare two state-of-the-art numerical codes to study the overall accuracy in modelling the intergalactic medium and reproducing Lyman-α forest observables for DESI and high-resolution data sets. The codes employ different approaches to solving both gravity and modelling the gas hydrodynamics. The first code, Nyx, solves the Poisson equation using the Particle-Mesh (PM) method and the Euler equations using a finite-volume method. The second code, CRK-HACC , uses a Tree-PM method to solve for gravity, and an improved Lagrangian smoothed particle hydrodynamics (SPH) technique, where fluid elements are modelled with particles, to treat the intergalactic gas. We compare the convergence behaviour of the codes in flux statistics as well as the degree to which the codes agree in the converged limit. We find good agreement overall with differences being less than observational uncertainties, and a particularly notable ≲1 per cent agreement in the 1D flux power spectrum. This agreement was achieved by applying a tessellation methodology for reconstructing the density in CRK-HACC instead of using an SPH kernel as is standard practice. We show that use of the SPH kernel can lead to significant and unnecessary biases in flux statistics; this is especially prominent at high redshifts, z ∼ 5, as the Lyman-α forest mostly comes from lower-density regions that are intrinsically poorly sampled by SPH particles.

An improved recoil pressure boundary condition for the simulation of deep penetration laser beam welding using the SPH method

Deep penetration laser beam welding is simulated using the mesh-free Lagrangian Smoothed Particle Hydrodynamics (SPH) method. The physical phenomena modeled are the elastic material behavior of the solid phase, the fluid dynamics of the liquid phase including temperature-dependent surface tension, heat transfer by heat conduction, and the phase transitions melting, solidification, and evaporation. The energy input of the laser beam is determined by a ray-tracing scheme that calculates the absorbed irradiance distribution in the capillary. Based on the locally absorbed irradiance, an improved recoil pressure model is introduced and compared with a temperature-dependent model frequently used in the literature. To transfer the momentum of the recoil pressure to the surrounding melt, two boundary conditions based on either dynamically detected boundary particles or surface forces are presented and validated using simple examples. The results show that the recoil pressure is a key factor for the formation of a deep capillary and the acceleration of the melt around the capillary. Comparison of capillary geometry and melt pool size from simulation and experiment shows good agreement for the irradiance-dependent recoil pressure model.

An artificial damping method for total Lagrangian SPH method with application in biomechanics

Introducing artificial damping into the momentum equation to enhance the numerical stability of the smoothed particle hydrodynamics (SPH) method for large strain dynamics has been well established, however, implementing appropriate damping term in the constitutive model as an alternative stabilization strategy in the context of the SPH method is still not investigated. In this paper, we present a simple stabilization procedure by introducing an artificial damping term into the second Piola–Kirchhoff stress to enhance the numerical stability of the SPH method in total Lagrangian formulation. The key idea is to reformulate the constitutive equation by adding a Kelvin–Voigt (KV) type damper with a scaling factor imitating a von Neumann–Richtmyer type artificial viscosity to alleviate the spurious oscillation in the vicinity of sharp spatial gradients. The proposed method is shown to effectively eliminate the appearance of spurious non-physical instabilities and easy to be implemented into the original total Lagrangian SPH formulation. After validating the numerical stability and accuracy of the present method through a set of benchmark tests with very challenging cases, we demonstrate its applications and potentials in the field of biomechanics by simulating the deformation of complex stent structures and the electromechanical excitation–contraction of the realistic left ventricle.

GPU-based SPH-DEM Method to Examine the Three-Phase Hydrodynamic Interactions between Multiphase Flow and Solid Particles

Particulate flows in a gas-liquid multiphase mixture (i.e., three-phase flows) are frequently encountered in various fields as well as the chemical engineering, resource engineering, ocean engineering, and nuclear engineering domains. Thus, the numerical modeling for such three-phase flows is essential to evaluate and enhance the fluidic design of the existing industrial devices or increase the engineering safety/efficiency. In terms of the solid particle phase, the discrete element method (DEM) has demonstrated outstanding capability in numerically modeling solid–solid interactions. For multiphase CFD model, several numerical techniques, ranging from the Eulerian–Eulerian two-fluid model to a fully resolved DNS solver, have been coupled with the DEM to model the three-phase flow. Most of the existing research on three-phase particulate flows was based on coupled grid-based CFD and DEM techniques. Considering that many three-phase particulate flows in industries are driven by a vapor/gas phase that forms a sharp interface with the liquid, coupling DEM with particle-based Lagrangian CFD methods, which can handle multi-phase flows without any interface capturing or associated correlations, can also be a good option or alternative for modeling such three-phase flows. In this respect, a three-phase flow solver was established in this study by coupling the DEM with a Lagrangian smoothed particle hydrodynamics (SPH) model while adopting the unresolved phase coupling between the two techniques. In order to overcome the low computational efficiency of these two numerical methods, GPU-based acceleration is adopted for speedup of SPH-DEM coupled solver due to its high compatibility for Lagrangian methods offered by thousands of GPU processors. The results of a scalability analysis indicated that the accelerated version exhibited an enhanced performance especially when many Lagrangian nodes are used in simulation. Several benchmark simulations were conducted to demonstrate the capability and applicability of the SPH-DEM solver in various multi-phase applications associated with particulate flows. The results of numerical simulations validated the feasibility of the momentum exchange model, and the potential and applicability of the SPH-DEM coupled solver for three-phase flow was demonstrated through a three-phase SPH-DEM simulation on the gas-driven leveling behavior of a particulate bed.

A New Updated Reference Lagrangian Smooth Particle Hydrodynamics algorithm for isothermal elasticity and elasto-plasticity

This paper presents a new Updated Reference Lagrangian Smooth Particle Hydrodynamics (SPH) algorithm for the analysis of large deformation isothermal elasticity and elasto-plasticity. Taking as point of departure a Total Lagrangian setting and considering as referential configuration an intermediate configuration of the deformation process, the equation of conservation of linear momentum and three geometric conservation laws (for the deformation gradient, its cofactor and its determinant) are re-written leading to a very generic system of first order conservation laws. The key feature of the formulation is a suitable multiplicative decomposition of the conservation variables, leading to a very simple final set of equations with striking similarities to the conventional Total Lagrangian system albeit re-written in terms of alternative Referential Updated conservation variables which are evolved in time. Taking advantage of this new Updated Reference Lagrangian formalism, a second order entropy-stable SPH upwinding stabilisation method will be introduced. With respect to previous publications by the group, a new three-stage Runge–Kutta time integration method is implemented in order to increase the CFL stability restriction. Finally, and to demonstrate the robustness and applicability of the methodology, a wide spectrum of challenging problems will be presented and compared, including some benchmark three-dimensional large deformation elasto-plasticity problems. To facilitate its ease of implementation, the paper explores the use of a series of novel expressions for the evaluation of kernels and the gradients of kernels to provide the SPH user the freedom to choose amongst various options, all leading to equally convincingly robust results.

On the equivalence of Eulerian Smoothed Particle Hydrodynamics, Total Lagrangian Smoothed Particle Hydrodynamics and molecular dynamics simulations for solids

Significant advances in nanoscale research have enabled the continuous miniaturization of devices. With the reduction in the size of the devices, it is important to identify if the continuum scale methods remain applicable to such small-scale systems. Motivated by this, the present work tries to understand the equivalence, or its lack thereof, of the continuum scale Eulerian Smoothed Particle Hydrodynamics (ESPH) and Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) with the atomistic scale molecular dynamics (MD) simulations. The equivalence is studied using four simple problems — (i) uniaxial tensile testing of a beam, (ii) stress profile in a pre-notched plate under small extension, (iii) head-on collision of two elastic rubber-like rings, and (iv) large deformation of a cantilever beam subjected to an impact at the free end. Using MD simulation data as the pseudo-experimental data, we show that both ESPH and TLSPH provide results that are qualitatively and quantitatively in agreement with the MD simulations if the properties at the continuum scale are obtained directly from the MD simulations, and the same initial conditions are chosen. The comparisons are based on the stress–strain behavior, the distribution of normal and shear stresses, the temporal evolution of the variables such as kinetic energy, etc.

Large deformation analysis of geomaterials using stabilized total Lagrangian smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) based on Eulerian kernels is widely-used in large deformation analysis of geomaterials. Despite being popular, it has tensile instability and rank-deficiency; thus, it needs several numerical treatments to be stable. In this work, a stabilized total Lagrangian SPH method is presented, which is inherently free of tensile instability. A stiffness-based hourglass control algorithm is employed to cure the instability caused by rank-deficiency. Periodic update of reference configuration is employed to allow the modeling of large deformation. Numerical examples are presented to show the performance of the method. The comparison between the presented method and the conventional smoothed particle hydrodynamics are discussed. The influences of hourglass control and configuration update are also investigated. It is found that the presented method is robust and can model large deformation and plastic flows in geomaterials. Particularly, the stabilized method delivers better stress results.

SPH–FV coupling algorithm for solving multi-scale three-dimensional free-surface flows

In two previous papers we presented an algorithm for coupling the Finite Volume (FV) method for the solution of 2D Navier–Stokes equations discretized on block structured Eulerian grids with the weakly-compressible Lagrangian Smoothed Particle Hydrodynamics (SPH) method. This coupling procedure exploits the SPH method to discretize flow regions close to free-surfaces and the Finite Volume approach to resolve both the bulk flow and the wall regions, where grid stretching can be favourably used. The information exchange between the two numerical schemes is established through overlapping zones. In the present paper this coupling paradigm is extended to a 3D framework. To this purpose, the extension of the algorithms for particle creation/deletion on the interfaces are described and issues related to free-surface intersection with each sub-domain boundary are addressed. Moreover, a new coupling procedure that simplifies the algorithm is proposed and tested. Effectiveness and accuracy achieved by the coupled solver are tested on challenging problems involving large free surface deformations and vorticity generation like the 3D flow past a cylinder below a free surface, the breaking wave generated by a ship bow in forward motion and the impact of a flat plate on the water surface.