My thesis presents a physics-based simulation method for animating sand. To allow for efficiently scaling up to large volumes of sand, we abstract away the individual grains and think of the sand as a continuum. In particular we show that an existing water simulator can be turned into a sand simulator within frictional regime with only a few small additions to account for inter-grain and boundary friction, yet with visually acceptable result. We also propose an alternative method for simulating fluids. Our core representation is a cloud of particles, which allows for accurate and flexible surface tracking and advection, but we use an auxiliary grid to efficiently enforce boundary conditions and incompressibility. We further address the issue of reconstructing a surface from particle data to render each frame. ii Contents ii

Figure 1: A silver block catapulting some wooden blocks into an oncoming wall of water. We present the Rigid Fluid method, a technique for animating the interplay between rigid bodies and viscous incompressible fluid with free surfaces. We use distributed Lagrange multipliers to ensure two-way coupling that generates realistic motion for both the solid objects and the fluid as they interact with one another. We call our method the rigid fluid method because the simulator treats the rigid objects as if they were made of fluid. The rigidity of such an object is maintained by identifying the region of the velocity field that is inside the object and constraining those velocities to be rigid body motion. The rigid fluid method is straightforward to implement, incurs very little computational overhead, and can be added as a bridge between current fluid simulators and rigid body solvers. Many solid objects of different densities (e.g., wood or lead) can be combined in the same animation.

Figure 1: Left: A solid stirring smoke runs at interactive rates, two orders of magnitude faster than previously. Middle: Fully coupled rigid bodies of widely varying density, with flow visualized by marker particles. Right: Interactive manipulation of immersed rigid bodies. Physical simulation has emerged as a compelling animation tech-nique, yet current approaches to coupling simulations of fluids and solids with irregular boundary geometry are inefficient or cannot handle some relevant scenarios robustly. We propose a new varia-tional approach which allows robust and accurate solution on rela-tively coarse Cartesian grids, allowing possibly orders of magnitude faster simulation. By rephrasing the classical pressure projection step as a kinetic energy minimization, broadly similar to modern approaches to rigid body contact, we permit a robust coupling be-tween fluid and arbitrary solid simulations that always gives a well-posed symmetric positive semi-definite linear system. We provide several examples of efficient fluid-solid interaction and rigid body coupling with sub-grid cell flow. In addition, we extend the frame-work with a new boundary condition for free-surface flow, allowing fluid to separate naturally from solids.

A distance field is a representation where, at each point within the field, we know the distance from that point to the closest point on any object within the domain. In addition to distance, other properties may be derived from the distance field, such as the direction to the surface, and when the distance field is signed, we may also determine if the point is internal or external to objects within the domain. The distance field has been found to be a useful construction within the areas of computer vision, physics, and computer graphics. This paper serves as an exposition of methods for the production of distance fields, and a review of alternative representations and applications of distance fields. In the course of this paper, we present various methods from all three of the above areas, and we answer pertinent questions such as How accurate are these methods compared to each other? How simple are they to implement?, and What is the complexity and runtime of such methods?

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This article introduces the Hierarchical Run-Length Encoded (H-RLE) Level Set data structure. This novel data structure combines the best features of the DT-Grid (of Nielsen and Museth [2004]) and the RLE Sparse Level Set (of Houston et al. [2004]) to provide both optimal efficiency and extreme versatility. In brief, the H-RLE level set employs an RLE in a dimensionally recursive fashion. The RLE scheme allows the compact storage of sequential nonnarrowband regions while the dimensionally recursive encoding along each axis efficiently compacts nonnarrowband planes and volumes. Consequently, this new structure can store and process level sets with effective voxel resolutions exceeding 500030003000 (45 billion voxels) on commodity PCs with only 1 GB of memory. This article, besides introducing the H-RLE level set data structure and its efficient core algorithms, also describes numerous applications that have benefited from our use of this structure: our unified implicit object representation, efficient and robust mesh to level set conversion, rapid ray tracing, level set metamorphosis, collision detection, and fully sparse fluid simulation (including RLE vector and matrix representations.) Our comparisons of the popular octree level set and Peng level set structures to the H-RLE level set indicate that the latter is superior in both narrowband sequential access speed and overall memory usage

We propose a novel approach for dynamically simulating articulated rigid bodies undergoing frequent and unpredictable contact and collision. In order to leverage existing algorithms for nonconvex bodies, multiple collisions, large contact groups, stacking, etc., we use maximal rather than generalized coordinates and take an impulse based approach that allows us to treat articulation, contact and collision in a unified manner. Traditional constraint handling methods are subject to drift, and we propose a novel pre-stabilization method that does not require tunable potentially stiff parameters as does Baumgarte stabilization. This differs from poststabilization in that we compute allowable trajectories before moving the rigid bodies to their new positions, instead of correcting them after the fact when it can be difficult to incorporate the effects of contact and collision. A post-stabilization technique is used for momentum and angular momentum. Our approach works with any black box method for specifying valid joint constraints, and no special considerations are required for arbitrary closed loops or branching. Moreover, our implementation is linear both in the number of bodies and in the number of auxiliary contact and collision constraints, unlike many other methods that are linear in the number of bodies but not in the number of auxiliary constraints.

The computation of Delaunay triangulations from static point sets has been extensively studied in computational geometry. When the points move with known trajectories, kinetic data structures can be used to maintain the triangulation. However, there has been little work so far on how to maintain the triangulation when the points move without explicit motion plans, as in the case of a physical simulation. In this paper we examine how to update Delaunay triangulations after small displacements of the defining points, as might be provided by a physics-based integrator. We have implemented a variety of update algorithms, many new, toward this purpose. We ran these algorithms on a corpus of data sets to provide running time comparisons and determined that updating Delaunay can be significantly faster than recomputing.

We propose a novel technique for melting and burning solid materials including the simulation of the resulting liquid and gas. The solid is simulated with traditional mesh-based techniques (triangles or tetrahedra) which enable robust handling of both deformable and rigid objects, collision and self-collision, rolling, friction, stacking, etc. The subsequently created liquid or gas is simulated with modern grid-based techniques including vorticity confinement and the particle level set method. The main advantage of our method is that state of the art techniques are used for both the solid and the fluid without compromising simulation quality when coupling them together or converting one into the other. For example, we avoid modeling solids as Eulerian grid-based fluids with high viscosity or viscoelasticity, which would preclude the handling of thin shells, self-collision, rolling, etc. Thus, our method allows one to achieve new effects while still using their favorite algorithms (and implementations) for simulating both solids and fluids, whereas other coupling algorithms require major algorithm and implementation overhauls and still fail to produce rich coupling effects (e.g. melting and burning solids).

Figure 1: Sample frames captured from our real-time simulation system (approximately 100,000 wave particles) We present a new method for the real-time simulation of fluid surface waves and their interactions with floating objects. The method is based on the new concept of wave particles, which offers a simple, fast, and unconditionally stable approach to wave simulation. We show how graphics hardware can be used to convert wave particles to a height field surface, which is warped horizontally to account for local wave-induced flow. The method is appropriate for most fluid simulation situations that do not involve significant global flow. It is demonstrated to work well in constrained areas, including wave reflections off of boundaries, and in unconstrained areas, such as an ocean surface. Interactions with floating objects are easily integrated by including wave forces on the objects and wave generation due to object motion. Theoretical foundations and implementation details are provided, and experiments demonstrate that we achieve plausible realism. Timing studies show that the method is scalable to allow simulation of wave interaction with several hundreds of objects at real-time rates.