Traditional Smoothed Particle Hydrodynamics Research Articles

Investigations on the fracture mechanisms of Z-shaped fissured rock-like specimens

Complex shapes of cracks exist in natural rock masses, however, present research has simplified it into straight line-shaped fissures. In view of this, three-dimensional (3D) printing is employed to make Z-shaped fissured specimens with different main fissure angle α and secondary fissure angle β. The compress fracture experiments are conducted, and full-field strain distributions during crack propagations are obtained using Digital Image Correlation (DIC) technology. Traditional Smoothed Particle Hydrodynamics (SPH) method is improved to examine damage process as well as fracture mechanisms in Z-shaped fissured specimens. It can be concluded that: Wing Crack (WC), Main Crack (MC), Outer Crack (OC) and Inner Crack (IC) are observed in the experiment. The secondary fissure angle β guides the initiation of MC, while the main fissure angle α guides the initiation of WC. Stress–strain curve of 3D printing samples with Z-shaped fissures shows similar trends to that of real rock, experiencing four typical parts. Numerical results show high similarities with experimental results. We have also discussed the crack initiation mechanisms in various main fissure angle α and secondary fissure angle β in the end, and concluded the stress distribution characters. The findings of this research will offer some references for correct understanding of the crack evolution processes and fracture mechanics of Z-shaped fissured rock masses.

Smoothed-Interface SPH Model for Multiphase Fluid-Structure Interaction

Numerical simulation of multiphase flows with large density ratios presents considerable challenges. Traditional Smoothed Particle Hydrodynamics (SPH) methods, while efficient in tracking multi-phase interfaces, often suffer from non-physical gaps near interfaces, compromising accuracy. In this paper, we develop an advanced multiphase model with enhanced interface continuity and stability, offering a reliable approach to both multiphase flows and fluid-structure interaction. The model employs equivalent continuity equations across different phases and refines the treatment of pressure and its gradients at interfaces through a smoothing technique, which effectively eliminates the non-physical gaps prevalent in the conventional SPH multiphase models. Moreover, an efficient and accurate method for calculating interface normal vectors and identifying interfaces is proposed. Based on the method, we introduce a series of techniques, including an improved interface force model, a particle displacement strategy, and a new particle penetration detection scheme, all of which significantly improve the interface continuity and the overall simulation accuracy of multiphase flows. Finally, the proposed multiphase SPH model is validated through several numerical examples, including two-dimensional static water, standing waves, dam break, oscillating multi-fluid, and wedge entry, as well as three-dimensional sphere entry. Comparative results with the traditional model indicate that the multiphase SPH model in this work ensures interface continuity and stability in long-period simulations, even under intense flow conditions. These results suggest that the smoothed-interface SPH multiphase model can eliminate non-physical gaps at interfaces, greatly enhancing interface continuity and stability, and highlighting its potential for accurate multiphase flow simulations.

An improved algorithm for Finite Particle Method considering Lagrange-type remainder

The Finite Particle Method (FPM) is a significant improvement to the traditional Smoothed Particle Hydrodynamics method (SPH), which can greatly improve the computational accuracy of the entire computational domain. However, unstable calculation results and long computational time are still major obstacles to the development of FPM. Based on matrix decomposition, the fundamental equations of the traditional Finite Particle Method are rewritten and a Generalized Finite Particle Method (GFPM) is derived by introducing Lagrange-type remainder. By deriving and rewriting the fundamental equations, the GFPM method can be theoretically proven to be always stable. Numerical examples show that the GFPM method can utilize a smaller computational scale to achieve the same computational accuracy as the FPM method, with a corresponding reduction in computational time. Finally, the GFPM method is applied to a one-dimensional stress wave propagation problem and a one-dimensional heat conduction problem, and the computational results are compared with those of the SPH method and the FPM method, which verify that the GFPM has higher computational accuracy and stability.

A modified weakly compressible smoothed particle hydrodynamics mixture model for accurate simulation of wave and porous structure interaction

In this study, a modified weakly compressible smoothed particle hydrodynamics (WCSPH) mixture model was developed to more accurately simulate the interaction between waves and porous structures. In this model, we enhanced the governing equations of the traditional WCSPH mixture model by introducing Darcy velocity, apparent density, and an adjustable smoothing length. This refinement ensures that the modified model effectively maintains the conservation of fluid volume in seepage simulations. Additionally, this paper proposes a permeable interface treatment technique that replaces traditional smoothed particle hydrodynamics interpolation with finite element shape function interpolation, significantly enhancing computational efficiency. At the same time, we also introduced and revised a particle shifting technique, which further increases the computational precision of the model. The modified WCSPH mixture model was then applied to simulate several physical experiments, including the dam-break wave propagation in a permeable dam, the attenuation of solitary waves on a permeable riverbed, the propagation of the solitary wave on a submerged porous structure, and the breaking process of waves passing through permeable breakwaters. Through comparison with the experimental data and other numerical results, the current model was comprehensively verified from various aspects, such as fluid volume conservation, wave evolution in and around the porous structure, and pressure distribution characteristics. The results confirm the excellent performance of the current model in simulating the interaction between waves and porous structures.

A physically consistent AI-based SPH emulator for computational fluid dynamics

Abstract The integration of artificial intelligence (AI) into computational fluid dynamics (CFD) has significantly expanded the scope of fluid modeling, allowing enhanced analysis capabilities and improved simulation performance. While Eulerian methods already benefit extensively from AI, notably in reliable weather prediction, the application of AI to Lagrangian methods remains less consolidated. Smoothed particle hydrodynamics (SPH) is a Lagrangian mesh-less numerical method for CFD with well-established advantages for the simulation of highly dynamic free-surface flows. Here, we explore an application of AI to SPH simulations, utilizing an artificial neural network (ANN) to estimate hydrodynamic forces between particle pairs, learning from SPH-simulated results. A model of this nature, which emulates the mathematical representation of physics, is termed an emulator. We examine the physical significance of the emulator, presenting its applications in benchmark tests, assessing its faithfulness to traditional SPH simulations, and highlighting its ability to generalize and simulate test cases with varying levels of complexity beyond its training data.

Research on the Water Entry of the Fuselage Cylindrical Structure Based on the Improved SPH Model

During aircraft landing on water, the intense impact load may lead to significant local deformation of the fuselage skin. Ensuring the aircraft’s integrity and reliability is of paramount importance. This paper investigates the fuselage skin’s dynamic response during water entry. In the simulation of complex water entry problems, the smoothed particle hydrodynamics (SPH) method can fully leverage the advantages of the particle method. However, the traditional SPH method still suffers from the drawbacks of tensile instability, significantly affecting the computational accuracy. Therefore, this paper first introduces the improved SPH model addressing fluid and solid tensile instability issues. Furthermore, the Riemann-based contact algorithm at the fluid–solid interface is also demonstrated. Based on the above improved SPH model, the simulation of water entry of the elastic cylinder is performed to validate the efficacy of the improved SPH model. Then, the dynamic response characteristics of elastic fuselage skin and the skin–stringer–floor–column structure when it enters the water are analyzed, including the deformation features and slamming force. Lastly, based on the presented damage model, a study is conducted on the water entry of the metallic elastic–plastic skin–stringer–floor–column structure, analyzing the locations of failure and providing guidance for the structural safety design of engineering.

Derivation of an improved smoothed particle hydrodynamics model for establishing a three-dimensional numerical wave tank overcoming excessive numerical dissipation

This paper presents an improved smoothed particle hydrodynamics (SPH) model through a rigorous mathematical derivation based on the principle of virtual work, aiming at establishing a three-dimensional numerical wave tank overcoming excessive numerical dissipation that has been usually encountered in traditional SPH models in practical applications. In order to demonstrate the accuracy and convergence of the new scheme, the viscous damping of a standing wave is first investigated as a quantitative validation, with particular attention on emphasizing (1) its physical rationality with respect to energy conservation and (2) its ability to alleviate wave over-attenuation even using fewer neighbors compared with the traditional δ-SPH model. Subsequently, several fully three-dimensional engineering problems, with respect to water wave propagation and the interaction with structures, are investigated to demonstrate the effectiveness of the new scheme in alleviating wave over-attenuation. It is demonstrated that the present model can be performed with relatively few neighbors (i.e., higher computational efficiency) to obtain accurate and convergent numerical results for those SPH simulations involving long-term and long-distance water wave propagation.

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

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

Using an improved SPH algorithm to simulate thermo-hydro-mechanical-damage coupling problems in rock masses

The Thermo-Hydro-Mechanical-Damage (THMD) process inside rock structures is a complex multi-field coupling problem, which is also regarded as the hotspot worldwide. In view of this situation, a fracture mark ξ is introduced to reflect the “active” or “failure” mode of Smoothed Particle Hydrodynamics (SPH) particles, which can simulate rock progressive cracking characteristics. The solid-water interaction modes and the damaged particles - water particles transformation algorithms are embedded into traditional SPH program to simulate the water-stress coupling problems. The heat conduction equations are programmed into SPH, which can reach the goal of modeling the temperature-stress coupling problems. Results show that: By coupling the parameter transformations between water and solid particles, the thermo-hydro-mechanical-damage simulation is then realized. Three numerical examples are carried out to validate the improved THMD method. Finally, an engineering numerical model of a typical tunnel is established, and the THMD coupling processes under complex conditions are simulated, which shows that the improved SPH method can simulate rock THMD multi-field coupling problems. The research results will help understanding the rock THMD coupling mechanisms, meanwhile, it can also promote the applications of SPH method into THMD modeling.

On the accuracy of SPH formulations with boundary integral terms

Handling boundaries in smoothed particle hydrodynamics (SPH) is believed to be one of the most critical problems in studies and applications of SPH. In the last two decades, excellent progress has been made in exploring the evaluation of the boundary integral terms directly to improve SPH approximation on the boundary. However, because of the fundamental difficulties, formulations in this regard are difficult to be made consistent when using SPH particle approximations, which significantly affect the solution accuracy. In this work, an intensive investigation is made in carefully analyzing the order of accuracy of the SPH approximation when boundary integral terms are included. Both kernel consistent and particle consistent boundary integral formulations of SPH are examined in great detail. Notably, a novel particle consistent boundary integral formulation for first-order derivatives is derived. An approximation of this new formulation, which is weakly consistent, is also discussed. The theoretical analysis and numerical experiments show that the SPH with particle consistent boundary integral formulation outperforms significantly in terms of accuracy compared to the correction factors used in traditional SPH implementations.

Smoothed particle hydrodynamics method for free surface flow based on MPI parallel computing

In the field of computational fluid dynamics (CFD), smoothed particle hydrodynamics (SPH) is very suitable for simulating problems with large deformation, free surface flow and other types of flow scenarios. However, traditional smoothed particle hydrodynamics methods suffer from the problem of high computation complexity, which constrains their application in scenarios with accuracy requirements. DualSPHysics is an excellent smoothed particle hydrodynamics software proposed in academia. Based on this tool, this paper presents a largescale parallel smoothed particle hydrodynamics framework: parallelDualSPHysics, which can solve the simulation of large-scale free surface flow. First, an efficient domain decomposition algorithm is proposed. And the data structure of DualSPHysics in a parallel framework is reshaped. Secondly, we proposed a strategy of overlapping computation and communication to the parallel particle interaction and particle update module, which greatly improves the parallel efficiency of the smoothed particle hydrodynamics method. Finally, we also added the pre-processing and post-processing modules to enable parallelDualSPHysics to run in modern high performance computers. In addition, a thorough evaluation shows that the 3 to 120 million particles tested can still maintain more than 90% computing efficiency, which demonstrates that the parallel strategy can achieve superior parallel efficiency.

An improved high order smoothed particle hydrodynamics method for numerical simulations of selective laser melting process

Metal selective laser melting (SLM) is an important Additive Manufacturing (AM) technology, which directly melts and solidifies metal parts by heating metal or alloy powder with laser. High-fidelity numerical simulations could provide tools for insight investigations of the SLM processes. In this study, an improved high order smoothed particle hydrodynamics (SPH) method was developed to simulate the melting and solidification process of metal selective laser melting. In the improved high order SPH model, the improved kernel gradient correction (KGC) is developed to overcome the non-conservation problem of traditional KGC. And an improved surface tension model is developed to improve the stability and accuracy of modeling the surface tension. In order to establish the numerical model for simulating the SLM forming process, the Gaussian heat source model and melting latent heat model are introduced. Moreover, particle shifting technology integrated with X-SPH technology are applied to improve the uniform of particle distributions stability of calculation. A number of test examples are investigated with the improved high order SPH, and compared with results from other approaches including traditional SPH. From the obtained numerical results, we can conclude the improved high order SPH is effective in modeling the melting and solidification process of SLM.

Application of the Godunov-Type Corrective Smoothed Particle Method to Impulsive Load Studies

The smoothed particle hydrodynamics (SPH) method is based on the kernel particle approximation, which is sensitive to the uniformity of the SPH particle distribution in the computational domain; that is, all SPH particles must be distributed evenly in the computational domain. These factors significantly influence the practical application of the SPH method. Meanwhile, calculating the sum near the boundaries of the computational domain may cause boundary defect problems since there are insufficient particles in the support domain, thus often resulting in relatively high errors in numerical simulation results near boundaries. To address these problems, the kernel particle approximation discrete process was corrected based on the traditional SPH method, and the corrected SPH method, the Godunov-type corrective smoothed particle method (CSPM), was formulated by introducing Riemann decomposition. In this study, the traditional SPH method and Godunov-type CSPM method were applied in a comparative study of discontinuous function problems, 1D shock tubes and 1D detonation waves. According to the analysis results, the Godunov-type CSPM method can not only effectively improve the calculation accuracy and compatibility of the traditional SPH method in discontinuous shock wave problems but also increase the accuracy of the traditional SPH method in capturing strong discontinuities.

An SPH stress correction algorithm based on the quartic piecewise smooth kernel function

<p indent="0mm">Smoothed particle hydrodynamics (SPH) is a popular adaptive Lagrangian mesh-free particle method. For the tensile instability problem in traditional SPH, according to the sufficient condition of tensile instability, a new quartic piecewise smooth kernel function with non-negative second derivative is proposed, and an SPH stress correction algorithm is obtained in this study. The numerical results show that the modified SPH algorithm can effectively describe the formation processes of 2D and 3D vdW liquid drops. Then, based on the modified SPH algorithm, the effects of different physical parameters on the particle distribution of 2D circular droplets are further discussed. The results show that, with the appropriate temperature or initial particle spacing, the boundary particles of circular droplets are neither clustered nor scattered, and as the fluid temperature or initial particle spacing increases, the boundary particles will be scattered. Conversely, the particle distribution of circular droplets becomes more uniform with the increase in the smooth length proportion coefficient.

A coupled thermo-mechanical bond-based smoothed particle dynamics model for simulating thermal cracking in rocks

A smoothed particle dynamics (SPD) method was proposed for crack propagation analysis. The computational efficiency of this method was significantly better than the traditional smoothed particle hydrodynamics (SPH) method. In the SPD method, the motion equations were approximated by a Lagrange kernel determined by the coordinates of the material in the initial configuration. A coupled thermo-mechanical bond-based smoothed particle dynamics (TM-BB-SPD) model was proposed for simulating the thermal cracking process of rocks. In the SPD program, the thermal cracking behaviors of rock, such as crack initiation and propagation, were dynamically tracked by capturing the damage state of virtual bonds. The robustness and accuracy of the proposed SPD method and TM-BB-SPD model were verified using two benchmark examples. The numerical results were in good agreement with the experimental results and analytical solutions. In addition, the effects of the thermal expansion coefficient and homogeneity index on the mode of thermal cracking in the rock disc were also investigated. The numerical results were also in good agreement with previous experimental observations. Moreover, the numerical results showed that with an increase in the homogeneity index, the mode of crack propagation in rock gradually changed from non-smooth to smooth types of fracture.

Magnetorotational instability with smoothed particle hydrodynamics

The magnetorotational instability (MRI) is an important process in driving turbulence in sufficiently ionized accretion disks. It has been extensively studied using simulations with Eulerian grid codes, but remains fairly unexplored for meshless codes. Here, we present a thorough numerical study on the MRI using the smoothed particle magnetohydrodynamics method with the geometric density average force expression. We performed 37 shearing box simulations with different initial setups and a wide range of resolution and dissipation parameters. We show, for the first time, that MRI with sustained turbulence can be simulated successfully with smoothed-particle hydrodynamics (SPH), with results consistent with prior work with grid-based codes, including saturation properties such as magnetic and kinetic energies and their respective stresses. In particular, for the stratified boxes, our simulations reproduce the characteristic “butterfly” diagram of the MRI dynamo with saturated turbulence for at least 100 orbits. On the contrary, traditional SPH simulations suffer from runaway growth and develop unphysically large azimuthal fields, similar to the results from a recent study with meshless methods. We investigated the dependency of MRI turbulence on the numerical Prandtl number (Pm) in SPH, focusing on the unstratified, zero net-flux case. We found that turbulence can only be sustained with a Prandtl number larger than ∼2.5, similar to the critical values for the physical Prandtl number found in grid-code simulations. However, unlike grid-based codes, the numerical Prandtl number in SPH increases with resolution, and for a fixed Prandtl number, the resulting magnetic energy and stresses are independent of resolution. Mean-field analyses were performed on all simulations, and the resulting transport coefficients indicate no α-effect in the unstratified cases, but an active αω dynamo and a diamagnetic pumping effect in the stratified medium, which are generally in agreement with previous studies. There is no clear indication of a shear-current dynamo in our simulation, which is likely to be responsible for a weaker mean-field growth in the tall, unstratified, zero net-flux simulation.

Simulating multi-phase sloshing flows with the SPH method

In this paper, the intense sloshing flow is simulated by Smoothed Particle Hydrodynamics (SPH) method successfully. First, the traditional SPH method is used for the simulation of sloshing process; and the obtained results for free surface shape and pressure evolution are in good agreement with the experimental data. Then, the δ-SPH method and particle shifting technique (PST) are introduced in the traditional SPH model to improve its stability and accuracy in simulating sloshing flow. Air phase does have an inescapable influence on water phase in intense sloshing process. Thus, an improved multiphase SPH model with pressure correction algorithm is utilized for interfacial sloshing simulation. The stability, accuracy and conservation of pressure correction algorithm are validated by the cases of hydrostatic test and dam breaking test. Besides, the comparison results show that unphysical pressure transmission at the interface between two fluids with large density ratio is effectively suppressed. Finally, the effect of air is considered in the multiphase flow simulation of sloshing process with presented multiphase δ-SPH and pressure correction utilized. The phenomenon that air phase is involved into inner water flow and then moves, rolls and rises in it is replicated accurately. It is found that both the shape of free surface and the air cavities during the slamming phase are well captured, which proves that the integrated multiphase SPH is effectively utilized for the simulation of such multiphase problem as sloshing.

A consistency-driven particle-advection formulation for weakly-compressible smoothed particle hydrodynamics

The traditional smoothed particle hydrodynamics is not 0th-order consistent and suffers from tensile instability which induces void regions when negative pressure presents in the flow. In order to improve the consistency and stability of the method, we propose a consistency-driven particle-advection formulation, which regularizes the particle configuration by local particle consistency instead of using background pressure. With the target of the normalization condition, the modification of particles position can be evaluated by gradient descent method according to the error between unity and the integral of kernel. In addition, the error term is modified to be negative to further improve the effectiveness and avoid the attraction between each pair particles. Note that in present formulation no physical-related parameter is introduced. A number of challenging test cases including lid-driven cavity, taylor-green vortex and fluid–structure interactions are investigated to validate the accuracy and robustness of the present method.

Numerical Analysis of the Hydraulic Fracturing of Pressure Tunnel Lining Based on the 2P-IKSPH Method

In order to investigate the fracture mechanisms of the pressure tunnel lining under water-stress coupling, based on the traditional smoothed particle hydrodynamics (SPH) method, the solid-water particle interaction method, and the particle damage conversion algorithm are proposed to realize the hydraulic fracturing process, which is called the 2P-IKSPH method. The “particle domain searching method,” the “birth-and-death particle method,” and the “group discrimination searching method” have also been proposed to realize the simulations of complex processes of excavation, lining, and operation of the hydraulic tunnel. Taking the Guzeng hydraulic tunnel as an engineering example, the hydraulic fracturing of tunnel lining under different conditions is numerically simulated. Results show the following: (1) the 2P-IKSPH method can dynamically reflect the stress wave propagation processes of surrounding rock and the damage process of tunnel lining. (2) The lining damage mainly occurs on the vault and the arch foot. (3) The critical internal water pressure increases with the increase of the tunnel buried depth and the thickness of lining, but increases first and then decreases with the increase of the surrounding rock mass grade. The research results can provide some references for the optimization designs of tunnel lining and reinforcement of similar projects. Meanwhile, developing 3D parallelization program based on 2P-IKSPH will be the future research directions.

Hydrodynamics with complex boundary motions by non-inertial SPH method and its application in attitude-liquid-control coupled dynamics of a liquid-filled quadrotor UAV

Smoothed Particle Hydrodynamics (SPH) is a fully Lagrangian method that does not require the use of any mesh. Due to its advantages, SPH method is usually adopted to simulate flows with free surfaces. Based on traditional SPH, non-inertial SPH as a new method is proposed recently to deal with cases when the liquid is subjected to inertial force for the motion of reference frame. This study finds that the transport velocity of liquid particles has additional effects on the solution of momentum equation in Navier-Stokes equations. Comparisons and validations are conducted to quantificationally investigate this influence in this paper. Furthermore, the modified non-inertial SPH method is applied to simulate the attitude-liquid-control coupled dynamics of a liquid-filled quadrotor unmanned aerial vehicle (UAV). Parallelization technology and particle searching method based on KD-tree algorithm are used to speed up the simulation. Coupled model of the liquid-filled quadrotor is built firstly, and the liquid sloshing effect on the position & attitude of UAV and the control output of power system are then simulated and discussed. Results prove that the liquid sloshing will deteriorate the trajectory tracking performance, especially the tracking error and rotor output.