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Particle-based meshless methods are commonly used in fluid dynamics and solid mechanics involving finite deformations, owing to their ability to break through the limitations imposed by mesh topology. Particle distribution usually plays a crucial role in determining the accuracy of simulation results for them. In this study, an error estimation was first conducted to ascertain the requirements for particle distribution necessary for accurate simulations. It was revealed that the sawtooth and chaotic particle distributions can significantly reduce numerical accuracy and even cause numerical instability in simulations. To address this issue, we propose a particle rearrangement approach including a pseudo hydrostatic pressure treatment to achieve body-fitted particles and a geometric smoothing method based on the metric of a point cloud unit to optimize chaotic particle distributions for improved regularity. The particles are iteratively moved according to their proximity to neighboring particles. Notably, the number of particles remains constant throughout the smoothing procedure, neither particles inserted or removed, nor particles overflow or volume expansion. This new approach facilitates the generation of body-fitted and relatively regular planar and surface particle distributions, meeting the requirements for arbitrarily complex shape particle distributions. The effectiveness of this method has been demonstrated through two-dimensional SPH simulations for problems of the flow around a cylinder, dam break, and Taylor–Green vortex.

In this paper we give a criterion to discriminate the entropy solution to quasi-linear equations of first order among weak solutions. This uniqueness statement is a generalization of Oleinik's criterion, which makes reference to the measure of the increasing character of weak solutions. The link between Oleinik's criterion and the entropy condition due to Kruzhkov is also clarified. An application of this analysis to the convergence of the particle method for conservation laws is also given by using the Filippov characteristics.

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite non-negative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.

The sound production by vortex rings is investigated by means of an axisymmetric vortex particle method. The predictions are first calibrated by analyzing the noise generated by steady vortex rings that are described by the analytical solutions of Fraenkel and Norbury. The noise produced by isolated vortex rings for both nominally steady and unsteady cores is then analyzed. For nominally steady cores, computed results indicate that the efficiency of sound radiation decreases as the slenderness parameter is reduced, and the acoustic signals reveal a dominant period that is approximately half the eddy turnover time. For unsteady cores, the amplitude of the radiated sound is substantially higher than that of similar steady rings. When the initial core vorticity distribution is nonuniform, complex internal motion may also occur within the core which is also reflected in the corresponding far-field acoustic signal. Finally, the effect of vortex stretching is analyzed based on computations of two coaxial corotating vortex rings.

This paper presents an overview on smoothed particle hydrodynamics (SPH), which is a meshfree, particle method of Lagrangian nature. In theory, the interpolation and approximations of the SPH method and the corresponding numerical errors are analyzed. The inherent particle inconsistency has been discussed in detail. It has been demonstrated that the particle inconsistency originates from the discrete particle approximation process and is the fundamental cause for poor approximation accuracy. Some particle consistency restoring approaches have been reviewed. In application, SPH modeling of general fluid dynamics and hyperdynamics with material strength have been reviewed with emphases on (1) microfluidics and microdrop dynamics, (2) coast hydrodynamics and offshore engineering, (3) environmental and geophysical flows, (4) high-explosive detonation and explosions, (5) underwater explosions, and (6) hydrodynamics with material strength including hypervelocity impact and penetration.

Driven by applications in the design of protective structure systems, the need to model high velocity impact is becoming of great importance. This paper presents a Smoothed Particle Hydrodynamics (SPH) procedure for 3D simulation of high velocity impacts where high rate hydrodynamics and material strength are of great concern. The formulations and implementations of the Johnson–Cook strength and damage model considering temperature effect, and Mie–Gruneison and Tilloton equations of state are discussed. The performance of the procedure is demonstrated through two example analyses, one modeling a cubic tungsten projectile penetrating a multi-layered target panel and the other involving a sphere perforating a thin plate. The results obtained, with comparisons made to both experimental results and other numerical solutions previously reported, show that our SPH-3D implementation is accurate and reliable for modeling the overall behavior of the high rate hydrodynamics with material strength.

This paper presents a new particle formulation for extreme material flow analyses in the bulk forming applications. The new formulation is first established by an introduction of a smoothed displacement field to the standard Galerkin formulation to eliminate zero-energy modes in conventional particle methods. The discretized system of linear equations is consistently derived and integrated using a direct nodal integration scheme. The linear formulation is next extended to the large deformation quasi-static analysis of inelastic materials. As quasi-static Lagrangian simulation proceeds in the severe deformation range, the analysis method is switched to explicit dynamics formulation and an adaptive Lagrangian kernel approach is preformed to reset the reference configuration and maintain the injective deformation mapping at the particles. Both nonconvex and convex meshfree approximations are investigated in this study. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.

In this paper, we developed a GPU parallelized Total Lagrangian Formation of Smoothed Particle Hydrodynamics (TLSPH) algorithm for 3D geometrical nonlinear structure analysis. The code was developed using NVDIA CUDA C++. Both the TLSPH and GPU parallelization algorithms are described in detail. Compared to the traditional FEM method for structure analysis, TLSPH method is much easier to be implemented and parallelized. In addition, as a meshless based method, there is no need to mesh the domain for TLSPH method. Also, the computational cost of TLSPH is much lower than the Weakly Compressible Smoothed Particle (WCSPH) method. By introducing GPU acceleration, we have significantly improved the code performance. Two benchmark test cases for 3D geometrical nonlinear structure analysis are carried out. The simulation results are compared with analysis results and the data obtained by Abaqus, which is a popularly-used software for structure analysis based on FEM method. In order to show the efficiency of GPU parallelization, a serial code based on the same TLSPH method is also developed as a reference. Results show GPU parallelization accelerates the code obviously. In summary, the GPU parallelized TLSPH method shows the potential to become an alternative way to deal with 3D geometrical nonlinear structure analysis.

In computational fluid dynamics there have been many attempts to combine the advantages of having a fixed mesh, on which to carry out spatial calculations, with using particles moving according to the velocity field. These ideas in fact go back to particle-in-cell methods, proposed about 60 years ago. Of course, some procedure is needed to transfer field information between particles and mesh. There are many possible choices for this “assignment”, or “projection”. Several requirements may guide this choice. Two well-known ones are conservativity and stability, which apply to volume integrals of the fields. An additional one is here considered: preservation of information. This means that assignment from the particles onto the mesh and back should yield the same field values when the particles and the mesh coincide in position. The resulting method is termed “mass” assignment, due to its strong similarities with the finite element method. Several procedures are tested, including the well-known FLIP, on three scenarios: simple 1D convection, 2D convection of Zalesak’s disk, and a CFD simulation of the Taylor–Green periodic vortex sheet. Mass assignment is seen to be clearly superior to other methods.

A particle method, or a gridless Lagrangian method, shows the high performance in describing the complicated behavior of water surface with the fragmentation and coalescence of water. In this paper, a wave overtopping process on a vertical seawall is numerically simulated on the basis of the Navier–Stokes equation with surface-tension term, which is discretized by the MPS (moving particle semi-implicit) method belonging to the category of the particle method. An improvement of the listing process of neighboring particle is introduced to reduce the computational load. Wave overtopping process in the experiments are well reproduced by the MPS method. The predictions of the MPS method of the overtopping volume agree well with the experimental results.

A Lagrangian numerical simulation of breaking waves is performed by the moving particle semi-implicit (MPS) method, in which the Navier-Stokes equation is discritized based on the interaction of particles. The Eulerian numerical solvers of the Navier-Stokes equation with the volume of fluid (VOF) method have difficulties in the calculation of the free surface due to the existence of the numerical diffusion derived from the advection terms. To attenuate the numerical diffusion, the procedures of the calculation of the cells involving the free surface should be very complicated one in Eulerian models. While, the MPS method is free from the numerical diffusion, hence it can calculate the free surface clearly, even under the existence of the fragmentation and the coalescence of fluid. In this study, the breaking waves are simulated on the several bottom configurations, namely a uniform slope, a permeable uniform slope and a vertical wall with small step on its foot. The time series of the water surface profiles and the velocity fields are displayed to show the performance of the MPS method in the wave breaking simulations.

The multi-scale structures of complex flows in chemical engineering have been great challenges to the design and scaling of such systems, and multi-scale modeling is the natural way in response. Particle methods (PMs) are ideal constituents and powerful tools of multi-scale models, owing to their physical fidelity and computational simplicity. Especially, pseudo-particle modeling (PPM, Ge & Li, 1996; Ge & Li, 2003) is most suitable for molecular scale flow prediction and exploration of the origin of multi-scale structures; macro-scale PPM (MaPPM, Ge & Li, 2001) and similar models are advantageous for meso-scale simulations of flows with complex and dynamic discontinuity, while the lattice Boltzmann model is more competent for homogeneous media in complex geometries; and meso-scale methods such as dissipative particle dynamics are unique tools for complex fluids of uncertain properties or flows with strong thermal fluctuations. All these methods are favorable for seamless interconnection of models for different scales.

However, as PMs are not originally designed as either tools for complexity or constituents of multi-scale models, further improvements are expected. PPM is proposed for microscopic simulation of particle-fluid systems as a combination of molecular dynamics (MD) and direct simulation Monte-Carlo (DSMC). The collision dynamics in PPM is identical to that of hard-sphere MD, so that mass, momentum and energy are conserved to machine accuracy. However, the collision detection procedure, which is most time-consuming and difficult to be parallelized for hard-sphere MD, has been greatly simplified to a procedure identical to that of soft-sphere MD. Actually, the physical model behind such a treatment is essentially different from MD and is more similar to DSMC, but an intrinsic difference is that in DSMC the collisions follow designed statistical rules that are reflection of the real physical processes only in very limited cases such as dilute gas.

PPM is ideal for exploring the mechanism of complex flows ab initio. In final analysis, the complexity of flow behavior is shaped by two components on the micro-scale: the relative displacements and interactions of the numerous molecules. Adding to the generality of the characteristics of complex system as described by Li and Kwauk (2003), we notice that complex structures or behaviors are most probably observed when these two components are competitive and hence they must compromise, as in the case of emulsions and the so-called soft-matter that includes most bio-systems. When either displacement or interaction is dominant, as in the case of dilute gas or solid crystals, respectively, complexity is much less spectacular. Most PMs consist explicitly of these two components, which is operator splitting in a numerical sense, but it is physically more meaningful and concise in PPM.

The properties of the pseudo-particle fluid are in good conformance to typical simple gas (Ge et al., 2003; Ge et al., 2005). The ability of PPM to describe the dynamic transport process on the micro-scale in heterogeneous particle-fluid systems has been demonstrated in recent simulations (Ge et al., 2005). Especially, the method has been employed to study the temporal evolution of the stability criterion in the energy minimization multi-scale model (Li & Kwauk, 1994), which confirms its monotonously asymptotic decreasing as the model has assumed (Zhang et al., 2005). Massive parallel processing is also practiced for simulating particle-fluid systems in PPM, indicating an optimistic prospect to elevate the computational limitations on their wider applications, and exploring deeper underlying mechanism in complex particle-fluid systems.

The 2011 off the Pacific coast of Tohoku Earthquake was one of the most powerful earthquakes on record in Japan and the huge tsunami caused by the earthquake inflicted extensive damage to the coastal areas of the Tohoku region. To form safe coastal areas, countermeasures against disaster should be developed considering not only tangible infrastructures including breakwater and bridges but also intangible measures including education on disaster prevention and the development of hazard maps. The tsunami run-up analysis is expected to play a role as one of the countermeasures against tsunami. In this research, we aim to establish a tool to effectively analyze the tsunami run-up in urban areas based on the Smoothed particle hydrodynamics (SPH) method. And then, we propose a series of pre-process procedures to develop a detailed geography analysis model that reflects the geography, elevation, and exterior shapes of buildings by referring to 3D location information and digital elevation model data obtained from a geographical information system. Finally, we established a photorealistic visualization method so that citizen can understand the tsunami phenomenon intuitively.

Heat exchangers with small-diameter multi-path tubes have been recently used to improve the efficiency of air conditioners. The difficulty in using tubes with small diameters and multi-paths is the nonuniformity of refrigerant distribution in refrigerant distributors, which results in lower heat-exchange efficiency. Grid methods, such as the volume of fluid method, are now widely used to simulate detailed motions of gas–liquid interfaces. A weak point of grid methods is the numerical diffusion of interfaces that occurs if the scale of interfaces becomes close to the computational grid sizes. We previously developed a particle/grid hybrid method for simulating multi-scale free surfaces. For this study, we modified the hybrid method and applied it to gas–liquid flow simulations in a distributor. The liquid film behaviors in both the distributor and a bend pipe placed in the upstream of the distributor were simulated mainly using the particle method, and gas flows were simulated using the grid method. The predicted liquid film near the outer circumference of the curvature in the bend pipe was thicker than that of near inner circumference of the curvature, which qualitatively agreed with the measurement. The simulated distribution ratio under a steady-flow condition agreed well with the measurement; the predicted distribution ratio was 0.63 and the measured distribution ratio was 0.6.

The recently developed Consistent Particle Method (CPM) is used to model breaking waves in tsunami and violent sloshing waves in a moving tank. Solving the Navier-Stokes equations in a semi-implicit time stepping scheme, the CPM eliminates the use of kernel function which is somewhat arbitrarily defined and used in other particle methods. It is demonstrated that the method is applicable to large amplitude free surface wave problems that involve breaking phenomenon. Tsunami wave impact on a fixed structure is modeled using CPM. The simulated results show fairly good agreement to the actual nonlinear wave motions including overturning and breaking of waves. Large amplitude sloshing waves in a moving tank are investigated with CPM. Experiment was conducted in the laboratory to verify the CPM solutions. The hydrodynamic pressure computed by the CPM agrees well with the experimental results.

The fluid–soil interactions play a significant role in coastal and ocean engineering applications. However, there are still some complex mechanical problems with large deformations of water–soil interfaces to be solved. As a particle-based Lagrangian method, Smoothed Particle Hydrodynamics (SPH) is good at solving multiphase problems with large deformations of boundaries or interfaces. Therefore, in this work, the $\delta$-SPH method is extended for the simulation of fluid–soil interacting problems. First, based on the weakly compressible assumption, the water is modeled as a viscous fluid while the soil is considered as a material with elastic–perfectly plastic behaviors. The $\delta$-SPH method is implemented on the two phases separately, while the stress diffusive term only acts on the soil. The seepage force is introduced to model the interaction between two phases. After that, several numerical test cases with small to large interface deformations are presented. It is shown that the fluid–soil interacting model based on the $\delta$-SPH model gives satisfying results compared with experimental data. Finally, the model is further extended for the simulation of vertical or oblique water jet scouring problems which demonstrates the potential applications of the SPH model for complex engineering problems.

The paper presents improved MPS methods for the prediction of wave impact pressure on a coastal structure. By focusing on the momentum conservation properties of original MPS formulations, a new pressure gradient term is proposed in a momentum conservative form. As the first modification the original MPS formulation for pressure gradient is replaced by the new term which conserves both linear and angular momentum. Second modification is made by introducing a new source term for Poisson Pressure Equation (PPE). By revisiting the derivation of the PPE in MPS method, a higher-order source term is derived based on calculating the time differentiation of particle number density. Both first and second modifications are shown to significantly reduce the spurious fluctuations in particle number density (and thus pressure) field. The improved performance of the improved methods is demonstrated through the simulations of: a static fluid, a dam break with impact, a flip-through impact, and a slightly-breaking wave impact.

The paper presents a three-dimensional Corrected MPS (3D-CMPS) method for improvement of water surface tracking in breaking waves. The Corrected MPS (CMPS; Khayyer and Gotoh, 2008) has been extended to three dimensions and a 3D-CMPS method has been developed on the basis of the 3D-MPS method by Gotoh et al. (2005b). The improved performance of the 3D-CMPS method with respect to the 3D-MPS method has been shown by simulating a plunging breaking wave and resultant splash-up on a plane slope. Furthermore, the parallelization of 3D-CMPS method with two different solvers of simultaneous linear equations, namely, namely, the PICCG-RP (Parallelized ICCG with Renumbering Process; Iwashita and Shimasaki, 2000) and SCG (Scaled Conjugate Gradient; Jennings and Malik, 1978) techniques, has been performed to enhance the computational efficiency of the calculations. This study also applies a simple dynamic domain decomposition for an optimized load balancing among the processors.

This paper investigates the concept of modeling cliff stability and collapse during extreme erosion events using a hybrid model method combining Weakly Compresible Smoothed Particle Hydrodynamics (WCSPH) with a particle based geotechnical stability module (GeoSPH). The WCSPH model has been developed to include a sediment transport and morphological model to predict erosion of a beach slope under a wave climate. The GeoSPH model is based on a displacement stepping method and due to its particle nature it sidesteps the mesh straining and distortion that hinders many traditional models. The model concept and future research direction is investigated and the initial results of erosion and collapse in a storm event are presented. Conventional methods of bluff collapse are often probabilistic, based on extrapolation bluff retreat rates, or reliant on an equilibrium profile, and as such struggle to adapt the predicted movement when changes occur to either the bluff or to the wave climate. Using a hybrid model of wave-driven erosion and bluff stability allows for site by site analysis and can be run without any data of past events.

When a mooring chain is broken due to a high wave, a moored ship can be drifted and stranded. In order to attenuate damages to facilities in a port, it is important to predict the movement of a moored floating object driven by high waves. In some previous numerical methods for this kind of prediction, it was difficult to simulate a floating object appropriately under high waves including a fragmentation and a coalescence of water like a breaking wave. On the other hand, the particle method is suitable for analysis under such a complicated water surface change. Therefore, in this study, a simulation model of moored buoy based on a particle method is developed.