Rapid, stable fluid dynamics for computer graphics

By M. Kass, Gavin S. P. Miller. 1990

We present a new method for animating water based on a simple, rapid and stable solution of a set of partial differential equations resulting from an approximation to the shallow water equations. The approximation gives rise to a version of the wave equation on a height-field where the wave velocity is proportional to the square root of the depth of the water. The resulting wave equation is then solved with an alternating-direction implicit method on a uniform finite-difference grid. The computational work required for an iteration consists mainly of solving a simple tridiagonal linear system for each row and column of the height field. A single iteration per frame suffices in most cases for convincing animation.Like previous computer-graphics models of wave motion, the new method can generate the effects of wave refraction with depth. Unlike previous models, it also handles wave reflections, net transport of water and boundary conditions with changing topology. As a consequence, the model is suitable for animating phenomena such as flowing rivers, raindrops hitting surfaces and waves in a fish tank as well as the classic phenomenon of waves lapping on a beach. The height-field representation prevents it from easily simulating phenomena such as breaking waves, except perhaps in combination with particle-based fluid models. The water is rendered using a form of caustic shading which simulates the refraction of illuminating rays at the water surface. A wetness map is also used to compute the wetting and drying of sand as the water passes over it.

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Citations in the corpus

Latent Space Physics: Towards Learning the Temporal Evolution of Fluid Flow (2018, methodology): “In the area of visual effects, Kass and Miller were the ﬁrst to employ height ﬁeld models [KM90], while Foster and Metaxas employed a ﬁrst three-dimensional NS solver [FM96].”
GPGPU: general purpose computation on graphics hardware (2004, methodology): “In contrast regular grids, used in graphics applications such as [Kass and Miller 1990; Stam 1999], lead to highly structured sparse matrices which map more directly onto the GPU since they are akin to pixel (2D) or voxel (3D) structures.”
Coupling water and smoke to thin deformable and rigid shells (2005, background): “[Kass and Miller 1990] solved a linearized form of the three dimensional Navier-Stokes equations for water that interacted with a variable height terrain.”
Simulating water and smoke with an octree data structure (2004, methodology): “[Kass and Miller 1990] solved a linearized form of the three dimensional Navier-Stokes equations, and [Chen and Lobo 1994] solved the two dimensional Navier-Stokes equations using the pressure to define a height field.”
Practical animation of liquids (2001, background): “O’Brien and Hodgins [20] simulated splashing liquids by combining a particle system and height field, while Miller and Pearce [19] used viscous springs between particles to achieve dynamic flow in 3D. Terzopoulos, Platt and Fleischer [27] simulated melting deformable solids using a molecular dynamic”
Animation and rendering of complex water surfaces (2002, methodology): “[Kass and Miller 1990] use a linearized form of the 2D shallow water equations to obtain a height field representation of the water surface.”
NeuroAnimator: fast neural network emulation and control of physics-based models (1998, methodology): “Physics-based animation has also been applied to the simulation some more speci c domains such as uids (Kass and Miller, 1990), gases (Stam and Fiume, 1993; Foster and Metaxas, 1997), chains (Barzel and Barr, 1988), and tree leaves (Wejchert and Haumann, 1991).”
Linear algebra operators for GPU implementation of numerical algorithms (2003, methodology): “These techniques have a variety of applications in physics based simulation and modelling, and they have been frequently employed in computer graphics to provide realistic simulation of real-world phenomena [Kaas and Miller 1990; Chen and da Vitoria Lobo 1995; Foster and Metaxas 1996; Stam 1999; Fos”
Interactive localized liquid motion editing (2013, methodology): “Animating fluid flows goes back to the early procedural method [Kajiya and Von Herzen 1984] and the simple linear water flow model [Kass and Miller 1990].”
Subdivision Methods for Geometric Design: A Constructive Approach (2001, methodology): “On the computer graphics side, Kass and Miller [78] simulate surface waves in water by approximately solving a two-dimensional shallow water problem.”