B. Solenthaler ∗ University of Zurich R. Pajarola † University of Zurich
Figure 1: Three examples produced with our incompressible simulation: (Left) 2M particles splashing against the simulation boundaries. (Center) Close-up view of a wave tank. (Right) A ﬂuid represented by 700k particles colliding with cylinder obstacles.
We present a novel, incompressible ﬂuid simulation method based on the Lagrangian Smoothed Particle Hydrodynamics (SPH) model. In our method, incompressibility is enforced by using a predictioncorrection scheme to determine the particle pressures. For this, the information about density ﬂuctuations is actively propagated through the ﬂuid and pressure values are updated until the targeted density is satisﬁed. With this approach, we avoid the computational expenses of solving a pressure Poisson equation, while still being able to use large time steps in the simulation. The achieved results show that our predictive-corrective incompressible SPH (PCISPH) method clearly outperforms the commonly used weakly compressible SPH (WCSPH) model by more than an order of magnitude while the computations are in good agreement with the WCSPH results. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism— Animation. Keywords: ﬂuid simulation, SPH, incompressibility
process and thus renders particle methods less attractive for high quality and photorealistic water animations. In the context of Smoothed Particle Hydrodynamics (SPH), two different strategies have been pursued to model incompressibility. First, the weakly compressible SPH (WCSPH) method has been used where pressure is modeled using a stiff equation of state (EOS), and second, incompressibility has been achieved by solving a pressure Poisson equation. Although both methods satisfy incompressibility, the computational expenses of simulating high resolution ﬂuid animations are too large for practical use. In the standard SPH and WCSPH model the particle pressures are determined by an EOS. The characteristics of this equation and the stiffness parameter determine the speed of the acoustic waves in a medium. The EOS-based SPH with low stiffness according to [Desbrun and Cani 1996] was used in a series of papers to simulate water [M¨ ller et al. 2003; Adams et al. 2007], multiple u ﬂuids [M¨ ller et al. 2005; Solenthaler and Pajarola 2008], ﬂuidu solid coupling [M¨ ller et al. 2004b; Lenaerts et al. 2008], melting u solids [M¨ ller et al. 2004a; Keiser et al. 2005; Solenthaler et al. u 2007], and ﬂuid control [Th¨ rey et al. 2006]. In contrast to the u standard SPH formulation, WCSPH uses a stiff EOS [Monaghan 2005; Becker and Teschner 2007; Becker et al. 2009] resulting in acoustic waves traveling closer to their real speed through the medium. Typically, the stiffness value is chosen so large that the density ﬂuctuations do not exceed 1%. The required stiffness value to achieve this, however, is difﬁcult or even impossible to determine before running the simulation. Consequently, an animator cannot get around extensive testing and parameter tuning. Another drawback is that WCSPH imposes a severe time step restriction as the stiffness of the ﬂuid usually dominates the Courant-FriedrichsLevy (CFL) condition. Thus the computational cost increases with decreasing compressibility – since higher stiffness requires smaller time steps, making it infeasible to simulate high resolution ﬂuids within reasonable time. Rather than simulating acoustic waves, incompressibility in Lagrangian methods can be enforced by solving a pressure projection similar to Eulerian methods (e.g. [Enright et al. 2002]). These incompressible SPH (ISPH) methods ﬁrst integrate the velocity ﬁeld in time without enforcing incompressibility. Then, either the intermediate velocity ﬁeld [Cummins and Rudman 1999], the...