Smoothed Particle Hydrodynamics (SPH)

Smoothed Particle Hydrodynamics (SPH) is a particle-based Lagrangian Method in- vented by (Lucy, 1977) to simulate nonaxisymmetric phenomena in astrophysics. It is a method used for calculations involving self-gravitating fluids moving freely in 3D. It has been used to study many astrophysical systems, including large scale structure, galaxy formation, tidal disruption of stars by massive black holes and also coalescing com- pact binaries. Coalescing compact binaries are considered as the most promising sources of Gravitational Waves for detection from laser interferometers (LIGO, VIRGO, GEO, TAMA). The first gravitational wave detection (B.P. Abbott et al., 2016), provided a major new confirmation of Einstein’s theory of general relativity and the first direct proof of existence of black holes. Compact binaries could consist of binaries of neutron stars

Smoothed particle hydrodynamics (SPH) is a meshless particle method for simulation of fluid flows. It is especially suitable for simulating flows with rapid changes. Treating the solid boundaries, however, is not as straightforward as with finite element or finite volume based methods. This paper describes a non-discrete boundary with friction approach to model particle-boundary interaction. This approach is mathematically consistent with the solution for the particle-particle interaction, and it provides a continuous solution along the boundary. The proposed model was verified against the experiments of Martin & Moyce [1] and numerical simulations by other authors. The results showed at least as good overall agreement as the simulations of other models, while local behaviour at the boundaries was better.

Recently, smoothed particle hydrodynamics (SPH) method has become popular in computational fluid dynamic and heat transfer simulation. The simplicity offered by this method made some complex system in physics such as moving interface in multiphase flow, heat conductivity jumping in multiple material boundaries and many geometrical difficulties become relative easy to calculate. We will treat a relative easy example of melting process to test the method in solving fluid motion equation coupled by heat transfer process. The main heat transfer processes are caused by solid-liquid (medium to medium) heat diffusion and convection. System interaction with ambient temperature can be modeled by gas surrounding fluid-solid system. For the ambient temperature, we proposed surface particle heat transfer governed by convectional heat flux. Using local particle number density value as surface detection method, we applied cooling and heating to surface particle on the melting ice cube and water system. The simulation result is also verified by experiment.

Heat transfer in fluids and its effect upon motion is of interest in many areas within science and en- gineering including desalination plans, within reactor cores in power plants and in complex enhanced oil recovery techniques such as steam assisted gravity drainage. This is especially true when consider- ing complex multi-fluid or multi-phase interactions that are seen in these cases. With the introduction of heat conduction into the smoothed particle hydrodynamics (SPH) framework, it is necessary to consider what effect temperature will have upon the dynamics of a system.

A mesh-free 2-D numerical approach was examined for cold rolling process to gain a more simpler and faster way to simulate such complicated case. Regarding the wide range of gained reports about the simplicity and computationally efficient characters of the smoothed particle hydrodynamics (SPH) method, the computational simulation stands on SPH technique in current numerical study. In this paper, the computational efforts not only confirm the advantages of using the SPH, but also reveal some improvements in simulation efficiencies. The rolling test was performed for an aluminum strip: Al 6061. In this way, the rolls assumed to behave as rigid bodies and, the aluminum strip assumed to behave as an elastic-plastic continuum. In order to achieve the required assurance of the employed technique, the computed stress distribution patterns were compared with those reported from a finite element study. The comparison reveals good agreements of the two computed results. Moreover, as the final test case, the effect of some parameters that have been reported to play the main role in case; roll diameter, percentage of thickness reduction of the strip, and the rolling speed has been studied. Regarding all parts of the current work, it may be concluded that the SPH can be sufficient tool to gain a rapid and simple simulation for such complicated cases.

As hydrodynamic simulations increase in scale and resolution, identifying structures with non- trivial geometries or regions of general interest becomes increasingly challenging. There is a growing need for algorithms that identify a variety of different features in a simulation without requiring a ‘by eye’ search. We present tensor classification as such a technique for smoothed particle hydrodynamics (SPH). These methods have already been used to great effect in N-Body cosmological simulations, which require smoothing defined as an input free parameter. We show that tensor classification successfully identifies a wide range of structures in SPH density fields using its native smoothing, removing a free parameter from the analysis and preventing the need for tessellation of the density field, as required by some classification algorithms. As examples, we show that tensor classification using the tidal tensor and the velocity shear tensor successfully identifies filaments, shells and sheet structures in giant molecular cloud simulations, as well as spiral arms in discs. The relationship between structures identified using different tensors illustrates how different forces compete and co-operate to produce the observed density field. We therefore advocate the use of multiple tensors to classify structure in SPH simulations, to shed light on the interplay of multiple physical processes.

Using lagrangian description for both fluid and solid domain is another remedy to the simulation of FSI problems, and Smoothed Particle Hydrodynamics (SPH) is one of the methods. In approaches mentioned above, mesh generation for the problem domain is a prerequisite for the numerical simulations. Therefore, the idea of eliminating the mesh has evolved naturally. SPH is a fully lagrangian mesh-less method that originally developed for astrophysical problems by Lucy [8], and separately by Gingold and Monaghan [9] and later extended to model a

The key design parameter for coastal defenses is the mean overtopping discharge or mean overtopping discharge per unit width of the structure q. Overtopping volume is used as one of the critical parameters in assessing the performance of defense structures. Whilst determining overtopping under different wave conditions could be easily achieved in laboratory, it is a challenging task to predict overtopping for real-world structures and assess the performance of existing structures with retrofits. Understanding the performance of defense structures and retrofits under extreme climatic conditions is very challenging and not even feasible in majority of experimental facilities. Hence, there is a need for an accurate wave overtopping prediction tool which could take into account various structural configurations and hydrodynamic conditions. The majority of existing overtopping prediction tools are based on empirical relations from lab-based data which are not very accurate as the overtopping dynamics rarely resemble the well-controlled expressions presented by empirical predictions. Wave activities and their interactions with defense structures create various responses to wave overtopping which are intricate. Understanding the underlying hydrodynamic mechanisms resulted from wave-structure interactions and ability to model these interactions could help to develop accurate simulation tools which are necessary for evaluating the coastal defenses and retrofit structures. This study adopts the capabilities of Lagrangian particle-based smoothed particle hydrodynamics (SPH) to develop a numerical tool for assessing wave overtopping from coastal retrofit structures. The numerical model is validated against the laboratory data collected at Warwick Water Laboratory and reported in Dong et al. (2018).

Smoothed particle hydrodynamics, is a truly meshfree and Lagrangian method in which the system is presented by a set of particles possessing material properties and move according to the governing equations. A vast variety of Lagrangian formulations have been proposed to calculate the local density, pressure, acceleration and velocity of the fluid. Since its first use was in astrophysical problem by Lucy (1977) and Gingold and Monaghan (1977), SPH has been comprehensively studied and extended to other fields including fluid flow with large nonlinear deformation [Liu and Liu, 2007]. SPH can be regarded as the oldest modern meshfree particle method [Liu et al., 2008 and Liu and Liu, 2010]. This method had been used for about ten years merely for astrophysical problems [Marrone, 2011]. Basics of SPH came from the probability theory, and statistical mechanics are widely employed for numerical estimation. Those initial formulations did not conserve linear and angular momentum. However, reasonably good results were obtained for many astrophysical. Several modifications have been proposed to improve the original formulation [Liu and Liu, 2010]. For instance, Monaghan and Gingold (1997) introduced a new SPH algorithm that conserves both linear and angular momentum, when they found non-conservative origin. A lot of efforts were put on the SPH method on the numerical aspects in accuracy, efficiency, stability and convergence.

Solution to the two-dimensional (2D) heat equation has been done by many researchers earlier. Numerical methods always being used in solving 2D heat equation. The usual numerical method being chosen is the finite difference techniques such as explicit finite difference method, Crank-Nicolson method and alternating direction implicit (ADI) method. However, in this research, the smoothed particle hydrodynamics (SPH) method is chosen and being studied to be applied in solving the 2D heat equation. The algorithm for SPH method is also being developed. As for making the comparisons to study on the accuracy of SPH method, 2D heat equation also being solved using ADI method. FORTRAN programming is used as a calculation medium for both the ADI and SPH method.

Smoothed Particle Hydrodynamics (SPH) is the oldest mesh-free method, having originally been introduced in the late 1970’s to simulate unbounded three-dimensional problems in astrophysics [9, 10]. It is a truly mesh- free Lagrangian technique, which has been successfully applied to a broad range of problems such as free sur- face flows [11], underwater explosions [12], problems of heat conduction [13], dynamic response with material strength [14] and many other fluid and mechanical ap- plications [15]. In spite of its wide application, it has two inherent weaknesses, namely the boundary deficiency problem and tensile instability, which have motivated the development of improved techniques. For example, the reproduced kernel particle method [16] was developed to improve the consistency of the SPH approach. Also, the Corrected Smoothed Particle Method (CSPM) [13] has addressed both the tensile instability and the inconsis- tency problems. More recently, Modified Smoothed Par- ticle Hydrodynamics (MSPH) has been introduced as a further enhancement over the CSPM. This method was developed simultaneously and independently by Zhang and Batra [17] and by Liu and Liu [18], whose works should be consulted for further details.

On the other hand, Haehnelt, Steinmetz & Rauch (1998) exam- ined a small number of dark matter haloes in a high-resolution (subkpc) smoothed particle hydrodynamics (SPH) simulation, and showed that such observational signatures can also be explained by a mixture of rotation, random motions, infall and mergers of protogalactic clumps. There are some observational indications (Le Brun et al. 1997; Rao & Turnshek 1998; Kulkarni et al. 2000, 2001) from direct imaging studies that luminous disc galaxies may not represent the dominant population of DLA galaxies (i.e. galaxies that host DLAs). Although the possibility of artefacts due to point spread function effects cannot be fully excluded, these observations suggest that some DLA galaxies could be compact, clumpy objects or low surface brightness galaxies, rather than large, well-formed protogalactic discs or spheroids.

Copyright © 2012 J. H. Pu and S. Shao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This research paper presents an incompressible smoothed particle hydrodynamics (ISPH) technique to investigate a regular wave overtopping on the coastal structure of different types. The SPH method is a mesh-free particle modeling approach that can efficiently treat the large deformation of free surface. The incompressible SPH approach employs a true hydrodynamic formulation to solve the fluid pressure that has less pressure fluctuations. The generation of flow turbulence during the wave breaking and overtopping is modeled by a subparticle scale (SPS) turbulence model. Here the ISPH model is used to investigate the wave overtopping over a coastal structure with and without the porous material. The computations disclosed the features of flow velocity, turbulence, and pressure distributions for different structure types and indicated that the existence of a layer of porous material can effectively reduce the wave impact pressure and overtopping rate. The proposed numerical model is expected to provide a promising practical tool to investigate the complicated wave-structure interactions.

From this research, we know that Smoothed particle hydrodynamics (SPH) is possess individual material properties and move according to the governing conservation equation where the state of a system represented by a set of particles. Smoothed particle hydrodynamics, as a meshfree, Lgrangian, particle method , has its particular characteristics. SPH has been extensively studied and extended to dynamics response with material strength as well as dynamic fluid flows with large deformations. A simulation using the SPH method involves particle approximation. The particle approximation is an issue related to only the initial creation of the particle and it can be solved using the existing software packages commercially available.

We propose viscoelastic smoothed particle hydrodynamics (SPH) with extended boundary conditions as a new method to model the extracellular matrix (ECM) in contact with a migrating cell. By drop out of the inertial terms in the SPH equations of motion, the new SPH formulation allows to solve problems in a low Reynolds environment with a timestep independent of the particle spacing, permitting to model processes at the cellular scale (i.e. µm-scale). The contact mechanics between a cell and ECM is modeled based on an existing boundary method in SPH that corrects for the well-known missing kernel support problem in Fluid Structure Interactions (FSI). This boundary method is here extended to allow the modeling of moving boundaries in contact with a viscoelastic solid. To validate the method, simulations are performed of tractions applied to a viscoelastic solid, Stokes flow around an array of square pillars, and indentation of a viscoelastic material with a circular indenter. The potential of the method to capture cell-ECM interactions is demonstrated by simulation of a self propelling object that locally degrades the ECM by fluidizing it to permit migration. This

In a companion paper we have shown how the equations describing gas and dust as two fluids coupled by a drag term can be re-formulated to describe the system as a single-fluid mixture. Here, we present a numerical implementation of the one-fluid dusty gas algorithm using smoothed particle hydrodynamics (SPH). The algorithm preserves the conservation properties of the SPH formalism. In particular, the total gas and dust mass, momentum, angular momentum and energy are all exactly conserved. Shock viscosity and conductivity terms are generalized to handle the two-phase mixture accordingly. The algorithm is benchmarked against a comprehensive suit of problems: DUSTYBOX , DUSTYWAVE , DUSTYSHOCK and DUSTYOSCILL ,

The impact of a laser pulse onto a liquid metal droplet is numerically investigated by utilising a weakly compressible single phase model; the thermodynamic closure is achieved by the Tait equation of state (EoS) for the liquid metal. The smoothed particle hydrodynamics (SPH) method, which has been employed in the arbitrary Lagrangian Eulerian (ALE) frame- work, offers numerical efficiency, compared to grid related discretization methods. The latter would require modelling not only of the liquid metal phase, but also of the vacuum, which would necessitate special numerical schemes, suitable for high density ratios. In addition, SPH-ALE allows for the easy deformation handling of the droplet, compared to interface tracking methods where strong mesh deformation and most likely degenerate cells occur. Then, the laser-induced deformation of the droplet is simulated and cavitation formation is predicted. The ablation pattern due to the emitted shock wave and the two low pressure lobes created in the middle of the droplet because of the rarefaction waves are demon- strated. The liquid metal droplet is subject to material rupture, when the shock wave, the rar- efaction wave and the free surface interact. Similar patterns regarding the wave dynamics and the hollow structure have been also noticed in prior experimental studies.

It is desired to resolve soil contamination with reduced costs. “Insoluble treatment” is a soil im- provement method for heavy metal containing soil, which uses soil mixers to mix soil and soil im- provement liquid agents. To reduce the costs of this method, soil mixers have to be optimized. However, it is not achieved due to the lack of theoretical knowledge on mixing solid with liquid. Therefore, a numerical model to simulate the dynamic behavior of solid and liquid is on the de- velopment in this study using Smoothed Particle Hydrodynamics (SPH) method. To validate the numerical model, several experiments were carried out and numerically reproduced. The com- parisons of the results showed that the numerical model replicated a liquid flow with an error rate of 2.1% and a seepage flow with an error rate up to 26.1%. Especially, the water distribution in the soil pores was highly improved with absolute gaps in volumetric water content up to 4.4% in the porosity range of 10% - 90%. For the water absorption into dry sand, the simulation result be- came more realistic by concerning soil suction.

The water wave generation by wave paddle and a freely falling rigid body are examined by using an Incompressible Smoothed Particle Hydrodynamics (ISPH). In the current ISPH method, the pressure was evaluated by solving pressure Poisson equation using a semi-implicit algorithm based on the projection scheme and the source term of pressure Poisson equation contains both of divergence free velocity field and density invariance condition. Here, the fluid-structure interaction is introduced in free surface flows and the structure is taken as a rigid body motion. In this study, we generated the water waves using the Scott Russell wave generator, in which the heavy box sinking vertically into water. Also, the solitary wave is generated by using the wave paddle and the generated solitary wave profiles are compared with the available results with a good agreement. Free falling of torpedo over the water in tank was simulated by using 3D-ISPH method.

development of adequate mathematical models of heat and mass transfer. The paper proposed a formulation of the problem of melt motion in the framework of the Lagrangian description. The mathematical statement includes the balance equations for mass, momentum and energy, and physical equations for describing heat and mass transfer. Methods: The smoothed particle hydrodynamics method was used for numerical simulation of the process of wire- based electron-beam surfacing on the substrate made from same materials (titanium or steel). A finite-difference analog of the equations is given and the algorithm for solving the problem is implemented. To integrate the