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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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. 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. 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. 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.