Variational time integrators are derived in the context of discrete mechanical systems. In this area, the governing equations for the motion of the mechanical system are built following two steps: (a) Postulating a discrete action; (b) Computing the stationary value for the discrete action. The former is formulated by considering Lagrangian (or Hamiltonian) systems with the discrete action being constructed through numerical approximations of the action integral. The latter derives the discrete Euler-Lagrange equations whose solutions give the variational time integrator. In this paper, we build variational time integrators in the context of smoothed particle hydrodynamics (SPH). So, we start with a variational formulation of SPH for fluids. Then, we apply the generalized midpoint rule, which depends on a parameter @math , in order to generate the discrete action. Then, the step (b) yields a variational time integration scheme that reduces to a known explicit one if @math but it is implicit otherwise. Hence, we design a fixed point iterative method to approximate the solution and prove its convergence condition. Besides, we show that the obtained discrete Euler-Lagrange equations preserve linear momentum. In the experimental results, we consider artificial viscous as well as boundary interaction effects and simulate a dam breaking set up. We compare the explicit and implicit SPH solutions and analyze momentum conservation of the dam breaking simulations.
On the other hand, traditional SPH formulations for fluids rely on standard conservation equations and a particle framework to discretize the corresponding Navier-Stokes equations, generating models that treat the continuum fluid as a system of particles and recover continuous fields by using interpolation kernels @cite_27 . Variational formulations of SPH for fluid applications have been proposed, where the constitutive equation of the fluid is given by an internal energy term which is a function of the density @cite_17 @cite_22 . These formulations provide a basis to discuss momentum preserving properties of SPH approaches. In this paper, they are used to derive variational time integrators for SPH by computing the discrete action and its stationary point, as we shall see in the next sections.
@cite_27 @cite_22 @cite_17
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In this paper, we propose an interactive method based on Smoothed Particle Hydrodynamics (SPH) to simulate blood as a fluid with free surfaces. While SPH was originally designed to simulate astronomical objects, we gear the method towards fluid simulation by deriving the force density fields directly from the Navier-Stokes equation and by adding a term to model surface tension effects. In contrast to Eulerian grid-based approaches, the particle-based approach makes mass conservation equations and convection terms dispensable which reduces the complexity of the simulation. In addition, the particles can directly be used to render the surface of the fluid. Our method can be used in interactive surgical training systems with models of up to 3000 particles. The paper describes the variational formulation of smooth particle hydrodynamics for both fluids and solids applications. The resulting equations treat the continuum as a Hamiltonian system of particles where the constitutive equation of the continuum is represented via an internal energy term. For solids this internal energy is derived from the deformation gradient of the mapping in terms of a hyperelastic strain energy function. In the case of fluids, the internal energy term is a function of the density. Once the internal energy terms are established the equations of motion are developed as equations of Lagrange, where the Lagrangian coordinates are the current positions of the particles. Since the energy terms are independent of rigid body rotations and translations, this formulation ensures the preservation of physical constants of the motion such as linear and angular momentum. This paper presents a new variational framework for various existing Smooth Particle Hydrodynamic (SPH) techniques and presents a new corrected SPH formulation. The linear and angular momentum preserving properties of SPH formulations are also discussed. The paper will show that in general in order to preserve angular momentum, the SPH equations must correctly evaluate the gradient of a linear velocity field. A corrected algorithm that combines kernel correction with gradient correction is presented. The paper will illustrate the theory presented with several examples relating to simple free surface flows. In fluid simulation, enforcing incompressibility is crucial for realism; it is also computationally expensive. Recent work has improved efficiency, but still requires time-steps that are impractical for real-time applications. In this work we present an iterative density solver integrated into the Position Based Dynamics framework (PBD). By formulating and solving a set of positional constraints that enforce constant density, our method allows similar incompressibility and convergence to modern smoothed particle hydro-dynamic (SPH) solvers, but inherits the stability of the geometric, position based dynamics method, allowing large time steps suitable for real-time applications. We incorporate an artificial pressure term that improves particle distribution, creates surface tension, and lowers the neighborhood requirements of traditional SPH. Finally, we address the issue of energy loss by applying vorticity confinement as a velocity post process. This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler–Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. 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. We propose a momentum-conserving two-way coupling method of SPH fluids and arbitrary rigid objects based on hydrodynamic forces. Our approach samples the surface of rigid bodies with boundary particles that interact with the fluid, preventing deficiency issues and both spatial and temporal discontinuities. The problem of inhomogeneous boundary sampling is addressed by considering the relative contribution of a boundary particle to a physical quantity. This facilitates not only the initialization process but also allows the simulation of multiple dynamic objects. Thin structures consisting of only one layer or one line of boundary particles, and also non-manifold geometries can be handled without any additional treatment. We have integrated our approach into WCSPH and PCISPH, and demonstrate its stability and flexibility with several scenarios including multiphase flow. Traditional fluid simulations require large computational resources even for an average sized scene with the main bottleneck being a very small time step size, required to guarantee the stability of the solution. Despite a large progress in parallel computing and efficient algorithms for pressure computation in the recent years, realtime fluid simulations have been possible only under very restricted conditions. In this paper we propose a novel machine learning based approach, that formulates physics-based fluid simulation as a regression problem, estimating the acceleration of every particle for each frame. We designed a feature vector, directly modelling individual forces and constraints from the Navier-Stokes equations, giving the method strong generalization properties to reliably predict positions and velocities of particles in a large time step setting on yet unseen test videos. We used a regression forest to approximate the behaviour of particles observed in the large training set of simulations obtained using a traditional solver. Our GPU implementation led to a speed-up of one to three orders of magnitude compared to the state-of-the-art position-based fluid solver and runs in real-time for systems with up to 2 million particles. The vessel wall and the blood flow interact and influence each other, and real-time coupling between them is of great importance to the virtual surgery as well as the research and diagnosis of vascular disease. On the basis of smoothed particle hydrodynamics SPH, we present a new approach to solve non-Newtonian viscous force of blood and a parallel mixed particles-based coupling method for blood flow and vessel wall. Meanwhile, we also design a proxy particle-based vessel wall force visualization method. Our method is as follows. Firstly, we solve the non-Newtonian viscous forces of blood through the SPH method to discretize the Casson equation. Secondly, in each time step, we combine blood particles and sampling proxy particles on the blood vessel wall to form mixed particles and calculate the interaction forces through the SPH method between every pair of the neighboring mixed particles inside the graphics processing unit. Thirdly, the forces of the proxy particles will be mapped to the color display of the proxy particle. Experimental results demonstrate that our method is able to implement real-time sizeable coupling of blood flow and vessel wall while mainly ensuring physical authenticity and it can also provide real-time and obvious information about vessel wall force distribution. Copyright © 2015 John Wiley & Sons, Ltd. Abstract We study an incompressible ideal fluid with a free surface that is subject to surface tension; it is not assumed that the fluid is irrotational. We derive a priori estimates for smooth solutions and prove a short-time existence result. The bounds are obtained by combining estimates of energy type with estimates of vorticity type and rely on a careful study of the regularity properties of the pressure function. An adequate artificial coordinate system is used instead of the standard Lagrangian coordinates. Under an assumption on the vorticity, a solution to the Euler equations is obtained as a vanishing viscosity limit of solutions to appropriate Navier–Stokes systems. We present a new algorithm to compute a collision-free trajectory for a robot manipulator to pour liquid from one container to the other. Our formulation uses a physical fluid model to simulate its highly deformable motion. We present a simulation guided and optimization based method to automatically compute the transferring trajectory. We use the full-featured and accurate Navier-Stokes model that provides the fine-grained information of velocity distribution inside the liquid body. Moreover, this information is used as an additional guiding energy term for the planner. Our approach includes a tight integration between the fine-grained fluid simulator, liquid transfer controller, and the optimization-based planner. We have implemented the method using a hybrid particle-mesh fluid simulator (FLIP) and demonstrated its performance on 4 benchmarks with different cup shapes and viscosity coefficients. In this paper we present a simple and rapid implementation of a fluid dynamics solver for game engines. Our tools can greatly enhance games by providing realistic fluid-like effects such as swirling smoke past a moving character. The potential applications are endless. Our algorithms are based on the physical equations of fluid flow, namely the Navier-Stokes equations. These equations are notoriously hard to solve when strict physical accuracy is of prime importance. Our solvers on the other hand are geared towards visual quality. Our emphasis is on stability and speed, which means that our simulations can be advanced with arbitrary time steps. We also demonstrate that our solvers are easy to code by providing a complete C code implementation in this paper. Our algorithms run in real-time for reasonable grid sizes in both two and three dimensions on standard PC hardware, as demonstrated during the presentation of this paper at the conference. Abstract Motivated by the study on the uniqueness problem of the coupled model, in this paper, we revisit 2d incompressible Navier–Stokes equations in bounded domains. In fact, we establish some new smoothing estimates to the Leray solution based on the spectral analysis of Stokes operator. To understand well these estimates, on one hand, we establish some new Brezis–Waigner type inequalities in general domain and in any dimension and disclose the connection between both of them. On the other hand, we show that these new estimates can be applied to prove the existence and uniqueness of the weak solutions for two coupled models: Boussinesq system with partial viscosity (no dissipation for the temperature) and Fluid Particle system, in two dimension and in bounded domains. Image inpainting involves filling in part of an image or video using information from the surrounding area. Applications include the restoration of damaged photographs and movies and the removal of selected objects. We introduce a class of automated methods for digital inpainting. The approach uses ideas from classical fluid dynamics to propagate isophote lines continuously from the exterior into the region to be inpainted. The main idea is to think of the image intensity as a 'stream function for a two-dimensional incompressible flow. The Laplacian of the image intensity plays the role of the vorticity of the fluid; it is transported into the region to be inpainted by a vector field defined by the stream function. The resulting algorithm is designed to continue isophotes while matching gradient vectors at the boundary of the inpainting region. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis. In this paper we introduce a novel parallel and interactive SPH simulation and rendering method on the GPU using CUDA which allows for high quality visualization. The crucial particle neighborhood search is based on Z-indexing and parallel sorting which eliminates GPU memory overhead due to grid or hierarchical data structures. Furthermore, it overcomes limitations imposed by shading languages allowing it to be very flexible and approaching the practical limits of modern graphics hardware. For visualizing the SPH simulation we introduce a new rendering pipeline. In the first step, all surface particles are efficiently extracted from the SPH particle cloud exploiting the simulation data. Subsequently, a partial and therefore fast distance field volume is rasterized from the surface particles. In the last step, the distance field volume is directly rendered using state-of-the-art GPU raycasting. This rendering pipeline allows for high quality visualization at very high frame rates. A submersible unmanned aerial vehicle (UAV) is proposed firstly, which is capable of operating in both air and water. One of the outstanding characteristics of the UAV is that the air-water transition imitates that of a gannet, i.e., plunge-diving. In this paper, the plunge-diving process of this UAV is simplified as a water-entry problem with a certain initial velocity, and the impact force is calculated by the method of the computational fluid dynamics (CFD). The Volume of Fluid is coupled with the 3-D Navier-Stokes equations to establish the model of the flow field, and the equations are solved in Fluent 6.3. The phase distribution and the pressure distribution during water-entry are presented and analyzed. Furthermore, the effects of the dropping height and the wing's sweptback angle on the impact force are investigated and discussed. Renewal theory is used to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non‐Markovian) fluid chord length distributions. One conclusion (for unimodal distributions) that is drawn is that the mean and variance of the chord length distributions through each fluid is sufficient knowledge of the statistics to give a reasonably accurate description of the ensemble averaged intensity. Expressions for effective cross sections and an effective source to be used in the usual deterministic transport equation are also obtained. The use of these effective quantities allows statistical information to be introduced very simply into a standard transport equation. An analysis is given which shows how the transport description, including scattering, in a Markovian mixture can be modified to yield an approximate description of transport in a non‐Markovian mixture. Numerical results are given to assess the accuracy of this model... This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier?Stokes equations. We extend a discontinuous finite element discretization originally considered for hyperbolic systems such as the Euler equations to the case of the Navier?Stokes equations by treating the viscous terms with a mixed formulation. The method combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means of Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. As a consequence the method is ideally suited to compute high-order accurate solution of the Navier?Stokes equations on unstructured grids. The performance of the proposed method is illustrated by computing the compressible viscous flow on a flat plate and around a NACA0012 airfoil for several flow regimes using constant, linear, quadratic, and cubic elements. The Navier-Stokes equations are utilized to derive upper bounds on the turbulent energy dissipation rate for an incompressible Newtonian fluid confined between parallel comoving plates. These estimates provide a rigorous foundation for one of the basic scaling ideas of turbulence theory, namely, the independence of the dissipation rate and the viscosity at high Reynolds number. The bounds are compared to experiments on turbulence in the Couette-Taylor system Abstract This paper reports large-scale direct numerical simulations of homogeneous-isotropic fluid turbulence, achieving sustained performance of 1.08 petaflop s on gpu hardware using single precision. The simulations use a vortex particle method to solve the Navier–Stokes equations, with a highly parallel fast multipole method ( fmm ) as numerical engine, and match the current record in mesh size for this application, a cube of 4096 3 computational points solved with a spectral method. The standard numerical approach used in this field is the pseudo-spectral method, relying on the fft algorithm as the numerical engine. The particle-based simulations presented in this paper quantitatively match the kinetic energy spectrum obtained with a pseudo-spectral method, using a trusted code. In terms of parallel performance, weak scaling results show the fmm -based vortex method achieving 74 parallel efficiency on 4096 processes (one gpu per mpi process, 3 gpu s per node of the tsubame -2.0 system). The fft -based spectral method is able to achieve just 14 parallel efficiency on the same number of mpi processes (using only cpu cores), due to the all-to-all communication pattern of the fft algorithm. The calculation time for one time step was 108 s for the vortex method and 154 s for the spectral method, under these conditions. Computing with 69 billion particles, this work exceeds by an order of magnitude the largest vortex-method calculations to date.