Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> Combining the kernel estimate with the Taylor series expansion is proposed to develop a Corrective Smoothed Particle Method (CSPM). This algorithm resolves the general problem of particle deficiency at boundaries, which is a shortcoming in Standard Smoothed Particle Hydrodynamics (SSPH). In addition, the method’s ability to model derivatives of any order could make it applicable for any time-dependent boundary value problems. An example of the applications studied in this paper is unsteady heat conduction, which is governed by second-order derivatives. Numerical results demonstrate that besides the capability of directly imposing boundary conditions, the present method enhances the solution accuracy not only near or on the boundary but also inside the domain. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States. <s> BIB001 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> A method for simulating two-phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high-order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid-based volume of fluid method for two-dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two-phase system and readily extend to three-dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free-surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd. <s> BIB002 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> An implementation of the smoothed particle hydrodynamics (SPH) method is presented to treat two-dimensional interfacial flows, that is, flow fields with different fluids separated by sharp interfaces. Test cases are presented to show that the present formulation remains stable for low density ratios. In particular, results are compared with those obtained by other solution techniques, showing a good agreement. The classical dam-break problem is studied by the present two-phase approach and the effects of density-ratio variations are discussed. The role of air entrapment on loads is discussed. <s> BIB003 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> In this paper we describe laboratory experiments and numerical simulations of the impact between a rigid body and water. The rigid body consists of a weighted rectangular box that slides down a curved ramp to enter the water at an angle to the horizontal of approximately 15°. As the box enters the water, a vigorous spray projects fluid beyond the box, then, as the water is heaved up, it falls back on the upper surface of the box and initiates a solitary wave. The experiments cover a wide range of impact velocities and three water depths in the main body of the tank. The experiments have been simulated using the Lagrangian particle method smoothed particle hydrodynamics (SPH). <s> BIB004 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> The interesting and complex behavior of fluids emerges mainly from interaction processes. While interactions of fluids with static or dynamic solids has caught some attention in computer graphics lately, the mutual interaction of different types of fluids such as air and water or water and wax has received much less attention although these types of interaction are the basis for a variety of important phenomena.In this paper we propose a new technique to model fluid-fluid interaction based on the Smoothed Particle Hydrodynamics (SPH) method. For the simulation of air-water interaction, air particles are generated on the fly only where needed. We also model dynamic phase changes and interface forces. Our technique makes possible the simulation of phenomena such as boiling water, trapped air and the dynamics of a lava lamp. <s> BIB005 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> We present a discrete particle based method capable of creating very realistic animations of bubbles in fluids. It allows for the generation (nucleation) of bubbles from gas dissolved in the fluid, the motion of the discrete bubbles including bubble collisions and drag interactions with the liquid which could be undergoing complex free surface motion, the formation and motion of coupled foams and the final dissipation of bubbles. This allows comprehensive simulations of dynamic bubble behavior. The underlying fluid simulation is based on the mesh-free Smoothed Particle Hydrodynamics method. Each particle representing the liquid contains an amount of dissolved gas. Gas is transferred from the continuum fluid model to the discrete bubble model at nucleation sites on the surface of solid bodies. The rate of gas transport to the nucleation sites controls the rate of bubble generation, producing very natural time variations in bubble numbers. Rising bubbles also grow by gathering more gas from the surrounding liquid as they move. This model contains significant bubble scale physics and allows, in principle, the capturing of many important processes that cannot be directly modeled by traditional methods. The method is used here to realistically animate the pouring of a glass of beer, starting with a stream of fresh beer entering the glass, the formation of a dense cloud of bubbles, which rise to create a good head as the beer reaches the top of the glass. <s> BIB006 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> To simulate multiple fluids realistically many important interaction effects have to be captured accurately. Smoothed Particle Hydrodynamics (SPH) has shown to be a simple, yet flexible method to cope with many fluid simulation problems in a robust way. Unfortunately, the results obtained when using SPH to simulate miscible fluids are severely affected, especially if density ratios become large. The undesirable effects reach from unphysical density and pressure variations to spurious and unnatural interface tensions, as well as severe numerical instabilities. In this work, we present a formulation based on SPH which can handle density discontinuities at interfaces between multiple fluids correctly without increasing the computational costs compared to standard SPH. The basic idea is to replace the density computation in SPH by a measure of particle densities and consequently derive new formulations for pressure and viscous forces. The new method enables the user to select the desired amount of interface tension according to the simulation problem at hand. We succeed to stably simulate multiple fluids with high density contrasts without the above described artifacts apparent in standard SPH simulations. <s> BIB007 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> We propose a hybrid method for simulating multiphase fluids such as bubbly water. The appearance of subgrid visual details is improved by incorporating a new bubble model based on smoothed particle hydrodynamics (SPH) into an Eulerian grid-based simulation that handles background flows of large bodies of water and air. To overcome the difficulty in simulating small bubbles in the context of the multiphase flows on a coarse grid, we heuristically model the interphase properties of water and air by means of the interactions between bubble particles. As a result, we can animate lively motion of bubbly water with small scale details efficiently. <s> BIB008 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> In the present work a new SPH model for simulating interface and free surface flows is presented. This formulation is an extension of the one discussed in Colagrossi and Landrini (2003) and is related to the one proposed by Hu and Adams (2006) to study multi-fluid flows. The new SPH scheme allows an accurate treatment of the discontinuity of quantities at the interface (such as the density), and permits to model flows where both interfaces and a free surface are present. The governing equations are derived following a Lagrangian variational principle leading to an Hamiltonian system of particles. The proposed formulation is validated on test cases for which reference solutions are available in the literature. <s> BIB009 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> A new boundary force is proposed to solve rigid body interaction in SPH (Smoothed Particle Hydrodynamics). The new boundary force is impulse-based rather than potential-based. The impulse is converted to a boundary force with a fundamental observation that the impulse is a momentum change which is a result of a force acting on an object during a time interval. Our new impulse-based boundary force gives more stable and accurate results in all kinds of rigid body interactions, especially in rigid—rigid interaction since our boundary force faithfully reflects the dynamic features of rigid body. With our impulse-based boundary force, SPH fluid solver is capable of full two way interaction of fluid and rigid body and it turns to be a particle-based rigid body solver if only rigid bodies are simulated without fluid. Copyright © 2009 John Wiley & Sons, Ltd. ::: ::: A new method is presented to solve rigid body interaction in SPH. The new method is impulse-based and it gives more stable and plausible results than the conventional method in fluid-rigid and rigid-rigid interactions. Our SPH solver with this method may be regarded as a particle-based rigid body solver capable of two-way coupling to fluid and rigid body. <s> BIB010 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> We propose a novel boundary handling algorithm for particle-based fluids. Based on a predictor-corrector scheme for both velocity and position, one- and two-way coupling with rigid bodies can be realized. The proposed algorithm offers significant improvements over existing penalty-based approaches. Different slip conditions can be realized and non-penetration is enforced. Direct forcing is employed to meet the desired boundary conditions and to ensure valid states after each simulation step. We have performed various experiments in 2D and 3D. They illustrate one- and two-way coupling of rigid bodies and fluids, the effects of hydrostatic and dynamic forces on a rigid body as well as different slip conditions. Numerical experiments and performance measurements are provided. <s> BIB011 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> We propose a new ghost fluid approach for free surface and solid boundary conditions in Smoothed Particle Hydrodynamics (SPH) liquid simulations. Prior methods either suffer from a spurious numerical surface tension artifact or drift away from the mass conservation constraint, and do not capture realistic cohesion of liquid to solids. Our Ghost SPH scheme resolves this with a new particle sampling algorithm to create a narrow layer of ghost particles in the surrounding air and solid, with careful extrapolation and treatment of fluid variables to reflect the boundary conditions. We also provide a new, simpler form of artificial viscosity based on XSPH. Examples demonstrate how the new approach captures real liquid behaviour previously unattainable by SPH with very little extra cost. <s> BIB012 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> As a fully mesh-less,Lagrangian numerical method,smoothed particle hydrodynamics(SPH) has notable advantages in problems of complex flows with free surface and/or moveable interface.With the improvement in its accuracy and stability,the SPH method has been extensively used in different areas of science and engineering.The recent advances in SPH fundamentals are presented,with emphasis on interfacial flows,fluid-structure interaction,and non-Newtonian fluids.Finally,future development of the SPH method is prospected. <s> BIB013 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> We present a unified dynamics framework for real-time visual effects. Using particles connected by constraints as our fundamental building block allows us to treat contact and collisions in a unified manner, and we show how this representation is flexible enough to model gases, liquids, deformable solids, rigid bodies and cloth with two-way interactions. We address some common problems with traditional particle-based methods and describe a parallel constraint solver based on position-based dynamics that is efficient enough for real-time applications. <s> BIB014 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> The solid boundary handling has been a research focus in physically based fluid animation. In this paper, we propose a novel stable and fast particle method to couple predictive-corrective incompressible smoothed particle hydrodynamics and geometric lattice shape matching LSM, which animates the visually realistic interaction of fluids and deformable solids allowing larger time steps or velocity differences. By combining the boundary particles sampled from solids with a momentum-conserving velocity-position correction scheme, our approach can alleviate the particle deficiency issues and prevent the penetration artefacts at the fluid-solid interfaces simultaneously. We further simulate the stable deformation and melting of solid objects coupled to smoothed particle hydrodynamics fluids based on a highly extended LSM model. In order to improve the time performance of each time step, we entirely implement the unified particle framework on GPUs using compute unified device architecture. The advantages of our two-way fluid-solid coupling method in computer animation are demonstrated via several virtual scenarios. <s> BIB015 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> C. INTERFACIAL FLOW <s> This work extends existing multiphase-fluid SPH frameworks to cover solid phases, including deformable bodies and granular materials. In our extended multiphase SPH framework, the distribution and shapes of all phases, both fluids and solids, are uniformly represented by their volume fraction functions. The dynamics of the multiphase system is governed by conservation of mass and momentum within different phases. The behavior of individual phases and the interactions between them are represented by corresponding constitutive laws, which are functions of the volume fraction fields and the velocity fields. Our generalized multiphase SPH framework does not require separate equations for specific phases or tedious interface tracking. As the distribution, shape and motion of each phase is represented and resolved in the same way, the proposed approach is robust, efficient and easy to implement. Various simulation results are presented to demonstrate the capabilities of our new multiphase SPH framework, including deformable bodies, granular materials, interaction between multiple fluids and deformable solids, flow in porous media, and dissolution of deformable solids. <s> BIB016
Interfacial flow usually refers to a liquid-liquid and gas-liquid two-phase flow system with a variable phase interface. Fluid and fluid interaction, likewise known as multiphase flow, is highly common. In reality, similar to water and oil mixing, the clear majority of grid-based simulation methods are difficult to manage as systems that contain two or more fluids, and may not even be feasible. For SPH, the addition of a new fluid only means the introduction of a new type of particle, which is relatively simple. Compared with the mesh-based Euler method, particle representation could clearly distinguish the different phase fluid interfaces. Thus, the particle method is suitable for the treatment of the multiphase flow. Colagrossi et al. BIB003 implemented the SPH method to manage two-dimensional interfacial flows. The results are consistent with those obtained by other solution techniques. Chen et al. BIB001 proposed a particle-based solution to simulate the interaction between blood flow and vessel wall for virtual surgery. By coupling two particle-based techniques, SPH and mass-spring model (MSM), the author seamlessly simulated the blood flow and vessel deformation. Experimental results demonstrate the potential value of the proposed method in providing real-time and realistic interactions for virtual vascular surgery systems. For the first time, Morris achieved the liquid two-phase flow SPH simulation BIB002 . However, this model is highly sensitive to the non-uniformity of the particle distribution, and a large error may exist in the correlation calculation. Müller et al. BIB005 were able to simulate the multiphase flow with a low-density ratio by directly applying the SPH formula to different fluid particles. Solenthaler and Pajarola BIB007 introduced an improved density formula into the Navier-Stokes equations and the calculation formula of other forces, which can be used to solve the problem of false interface effect and to realize the simulation of a stable higher density than incompatible fluid. Figure 10 shows a schematic of the liquid vortex in a glass under direct particle use and different surface extraction methods. Compared with the liquid two-phase flow, the gas-liquid two-phase flow has a greater density ratio. Grenier et al. BIB009 established a Hamiltonian description for interface and free surface flows, simulating the rise of bubbles in water. Clerary et al. BIB006 used gas dissipation to simulate the bubbles, dealing with the gas and the liquid separately. The bubbles were simulated with a discrete solid of fixed shape, the fluid was simulated using the SPH method, and the effect of the bubble on the fluid was ignored. Hong et al. BIB008 proposed a coupling of meshless and grid methods, simulating gas using a meshless SPH method, and liquid with a grid-based method. The copling between the two methods is achieved by the velocity field. Schechter and Bridson BIB012 achieved a one-way coupling of gas and water, where the gas particles interacted only with the density field and not with the liquid, except for a passive interaction with the velocity field of the fluid. Ihmsen et al.  implemented a two-way coupling of gas and liquid, which can be used to simulate the effect of resistance and buoyancy. An important feature of the fluid-solid coupling problem is the interaction between the two-phase media; the solid moves under the action of the fluid, and the movement of the solid in turn affects the flow field of the fluid BIB013 . In the fluid-solid coupling problem, the properties of solid materials can usually be divided into two types, rigidity and elasticity. However, serious local errors exist in the Arbitrary Lagrangian-Eulerian (ALE) method, and the Immersed Boundary Method (IBM) does not accurately express the solid shape. Compared with the traditional ALE and IBM, an application of SPH method to deal with the fluid-solid coupling problem is more flexible and convenient. Monaghan et al. BIB004 first examined the process by which a rectangular object slid down along the slope and collided with the water surface. Oh et al. BIB010 combined the SPH method with the impulse-based rigid body dynamics to simulate the fluid-solid coupling phenomenon, but the method is not based on hydrodynamics and cannot guarantee non-penetrability. Becker et al. BIB011 used a predictive-corrected method to achieve bi-directional coupling between the rigid body and the fluid, ensuring that the fluid does not cross the solid boundary. However, the conservation of momentum is not guaranteed; thus, the authenticity of results is affected. Shao et al. BIB015 simulated the fluid-solid coupling phenomenon using a method based on velocity-position correction, which ensures momentum conservation and allows the use of a large time step to simulate. Macklin et al. BIB014 proposed a unified particle dynamics simulation framework, based on the location of the dynamics, the interaction between various substances into a unified particle processing, and the flexibility to achieve various object interaction simulation. However, in the above simulation, the particles can only be treated as pure liquid and solid. Yan et al. BIB016 extended the existing multiphase flow framework to achieve the interaction between deformed bodies, solids, particles, multiple fluids, and variable solids, as well as the dissolution of deformable solids. Figure 11 shows the process in which instant coffee and candy pieces are gradually dissolved in water under the action of agitation. 