Fluid Animation from Simulation on Tetrahedral Meshes

Transcription

1 Fluid Animation from Simulation on Tetrahedral Meshes Bryan Eric Feldman Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS December 17, 2007

2 Copyright 2007, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission.

3 Fluid Animation from Simulation on Tetrahedral Meshes by Bryan Eric Feldman B.S. (University of California, Davis) 2000 M.S. (University of California, Berkeley) 2002 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor James F. O Brien, Chair Professor Jonathan Shewchuk Professor Panayiotis Papadopoulos Fall 2007

4 Fluid Animation from Simulation on Tetrahedral Meshes Copyright 2007 by Bryan Eric Feldman

5 1 Abstract Fluid Animation from Simulation on Tetrahedral Meshes by Bryan Eric Feldman Doctor of Philosophy in Computer Science University of California, Berkeley Professor James F. O Brien, Chair This thesis presents a simulation method for creating animations of gases and liquids that enhances the adaptability over current simulators within the computer graphics community. The method achieves adaptability in element size and shape by discretizing the domain with tetrahedra rather than regular hexahedra, the standard element shape in computer graphics. I also describe a method which allows the discretization to adapt arbitrarily from time step to time step without computational or numerical smoothing penalty. Additionally, I demonstrate a method to augment the fluid simulator with a rigid body simulator such that the fluid and rigid body simultaneously effect one another. Together these capabilities allow for complex scenarios to be simulated with a high level of detail while maintaining practical computation time, memory use, and ease of implementation.

6 Professor James F. O Brien Dissertation Committee Chair 2

7 i Contents List of Figures iii 1 Introduction Background Accuracy and Adaptivity Reference Frames Prior Methods Contributions of This Dissertation Previous Work Two-Dimensional Approximations Basic Three-Dimensional Simulator Alternative Discretizations Maintaining and Enhancing Detail Extensions Fluid Control Fluid-Solid Coupling Moving Meshes Particle Based Methods Surface Tracking and Rendering Meshing Gas Meshing Variational Tetrahedral Meshing Overview Modifications Comments Liquid Meshing Isosurface Stuffing Overview Modifications Comments... 36

8 ii 4 Simulation Overview Equations of Motion Where to Store Things Capabilities Required for Simulation Velocity Interpolation Derivative Operators Semi-Lagrangian Velocity Advection Generalized Semi-Lagrangian Advection External Forces Mass Conservation An Alternative Perspective Boundary Conditions Coupling Between Fluids and Rigid Bodies Multigrid Multigrid Background Algebraic Multigrid Background Details of our Implementation Multigrid Discussion and Results Results and Discussion Gas and Non Free Surface Liquid Animations Two-Way Coupling Free Surface Liquid Animations Overall Conclusions Bibliography 97

9 iii List of Figures 1.1 A one dimensional representation of discretization and simulation. On the left a continuous function (dashed line) is represented by discrete values (dots) s and an interpolation function (solid line). The spatial derivative, x is estimated. On the right are the discrete values at a later time. The values are changed by the partial differential equations of motion and the estimated spatial derivative Figure showing the advantage of spatial adaptivity. The representation on the right uses just as many data point but better approximates the continuous function A comparison of a regular grid (left) and tetrahedral (right) discretization of the same shape. Despite the fact that the tetrahedral mesh uses fewer elements, it better represents the input boundary Two frames from different animations that took equivalent times to simulate. On the left, a single fixed grid was used throughout the simulation. On the right, the mesh is dynamically updated to place small elements near the smoke and regions of high vorticity. The right animation features substantially less numerical smoothing despite taking the same time to simulate (including mesh generation) as the left animation Left: a visualization of the sizing field for a rectangular domain with an irregular obstacle at the top and a plume of smoke at the bottom. Right: the resulting simulation mesh. Obstacle faces are colored green This figure illustrates the two-dimensional version of isosurface stuffing. Yellow points are the vertices of the octree lattice. The green line is the fluid s surface. Blue points represent the warped locations of lattice vertices that were near the surface. Red points are vertices created where the surface cuts the lattice with no nearby vertices Two-dimensional representation of velocity interpolation. A velocity vector is computed for each tetrahedron from the face normal velocities. These velocities are at the vertices of the Voronoi cell. The velocity at some arbitrary location x can be computed by interpolating the Voronoi cell

10 iv 4.2 Divergence (left) is estimated by summing the outward-facing face normal velocities. In the figure s 2 is negative because n 2 points inward; all other s s are positive. The gradient (right) is estimated by taking the difference between the pressures in the tetrahedra adjoining the face Top left: If pressures are stored at circumcenters, the line connecting two adjacent pressures passes through the face circumcenter (the edge midpoint in 2D) and is parallel to the face s normal. Top right: Circumcenters may lie outside of an element such that the line segment between adjacent circumcenters does not intersect the face. Bottom left: In 3D the face circumcenter (dark triangle) is not necessarily in the face, making it a dubious place to locate the face normal velocity for divergence calculation. Bottom right: Storing quantities at barycenters. The line segment connecting adjacent tetrahedra barycenters does not necessarily pass through the face barycenter and is not generally parallel to the face normal A 2D representation of our generalized semi-lagrangian method. We trace back from the position where a velocity is stored in the new mesh x i =(x, y), interpolate the velocity using the old mesh and update the velocity in the new mesh A comparison of simulation with the same initial conditions on a static (top) and deforming (bottom) mesh. The deforming mesh is being stretched and squashed in the vertical direction. As shown the simulations are qualitatively very similar, demonstrating that mesh deformation does not effect the simulation Left) Pressure forces on rigid object create a force/torque couple. Right) The acceleration on the boundary (and hence the fluid simulation mesh faces at this boundary) of the rigid body can be computed from the linear and angular acceleration of the body An example of variable classification where c indicates that a triangle is labeled as coarse and f indicates a fine label. We have shown a geometry so that neighbor information is clear. In practice, the non-zero entries in the system matrix are used to determine neighborhood information. Subscripts indicate the order in which the variables are classified. For example, c 0 is added first (selected at random), then its neighbors are added as f 1 s. Next c 2 is added and its neighbors are labeled as f 3. Note that due to the classification algorithm each fine variable has at least one coarse neighbor and that there are relatively few coarse variable (8 of a possible 21) A paddle translates and rotates with scripted motion, stirring up the smoke in a box shape. On the right, the mesh is shown for three frames for this example. The domain is remeshed every time step so that it s boundary conforms to the paddle. The sizing function changes from time step to time step so that smaller tetrahedra are created near the paddle, as this is where the most interesting motion occurs

11 v 6.2 The leftmost image shows smoke inside the Stanford Buddha. The remaining images show the progression of green smoke as it is exhaled out. Smoke motion is induced by the motion of Buddha s belly, which determines the boundary conditions of the simulation domain This image shows smoke in a tube spinning and rising around a rotating blade A series of frames from a high resolution example with a jet shooting smoke up towards several obstacle rods Two animations where jets of air are shot towards a rigid bunny. The top sequence features a light bunny; the bottom features a heavier bunny. As expected, the light bunny is more affected by the cannon than the heavier one The central bulb expands and contracts with causes the red particles to move from the left tank to the right tank. The blue valves are coupled to the fluid simulation and prevent backflow Liquid initially in the shape of an angel falls under gravity, creating a splash Two nozzles spray liquid onto a dish with a hump shape in the middle Water, initially in the shape of a block, falls under the force of gravity. A half-hemisphere obstacle on the floor interacts with the moving liquid, as do the boundaries of the domain The meshes used to generate the animation in Figure 6.9. In the upper corner of the second, third, and forth panels a zoom in view is shown. Note that the elements become larger as the distance to the surface increases and that the mesh conforms well to that surface yet is highly structured Liquid is sprayed inside a transparent box. A thin sheet folds over after the spray hits the back wall Red Liquid erupts from a mountain (left). On the right a close up of the pool at the bottom where the liquid accumulates

12 vi Acknowledgments I d first like to thank my parents for supporting me through the many years of my education. I know they ve had to put up with countless questions of So...Bryan is STILL in school?. Hopefully, the rewards I have gained from this experience have been worth the annoyances I have created for you. Of course without my advisor James O Brien finishing the Ph.D wouldn t be possible. I mean that not only because without his signature, the university won t let me finish but of course because of the countless hours working with me on the details of projects and more importantly for the big picture things that I think are the most important part of what I ve learned, for guiding me in how to think about solving problems. I also will always appreciate him taking the risk to work with a Mechanical Engineering student that didn t (doesn t?) know that much about computers. I am indebted for his guidance in my transformation from an ignorant young grad student to a slightly less ignorant and substantially older graduate. All of those in Berkeley Computer Graphics and Vision group have been an absolute joy to work with and to know. Without them the grad school experience wouldn t be the great one that it has been. I especially want to thank all those who have a co-authored papers or worked on projects with me, I apologize if working on those deadlines with me shaved years off your life, hopefully those were the bad years anyways.

13 1 Chapter 1 Introduction Animating fluids is an important capability to those in the computer graphics industry. Examples of fluid motions that might be animated include common, everyday, phenomena such as pouring a glass of water or exhaling of cigarette smoke, and (thankfully) extraordinary occurrences such as large scale explosions or tidal waves crashing into metropolises. Animating the motion of fluids by hand would often require much time and labor by a skilled artist and ad-hoc procedural methods for generating the motion do not generalize well. The main reason for these methods inadequacies is that fluid motion is extraordinarily complex. The complexity of a fluids motion contrasts with the simplicity of the equations that describe the fluid behavior (see Equations (4.1) and (4.2)). As a result, simulation, which is driven by these equations, has proven to be an extremely effective method to generate animations. This being the case, simulation methods have been widely investigated in academia and are commonly used in production.

14 2 Simulations that provide realistic and detailed motion are extremely resource intensive. Even as computers increase in speed and memory, fluid simulations continue to push the limits of the available computational power. Accordingly, those who use the simulators are in need of simulation methods that can increase the level of realism and detail for a given computational budget. In this work I present techniques that increase the efficiency of simulating fluids for the application of generating realistic animations. The method achieves this goal while maintaining the stability and ease of use and ease of implementation that is desired by practitioners of computer graphics. To accomplish these goals I investigate the use of tetrahedra as the discretizing element. Tetrahedral discretizations offer important advantages over regular grids, the commonly used discretization within computer graphics. First, because the size of the tetrahedra can vary over the domain, computational resources can be allocated efficiently by placing many small tetrahedra in regions where they are needed to resolve features of the flow, and few larger tetrahedra can be placed in the less important regions. Second, the shape of a tetrahedron is more adaptable than a cube. This shape adaptability means that for a fixed amount of computation resources, the discretization mesh can more accurately match a simulation domain. and therefore can create a more realistic animation. For these advantages to persist as the simulation progresses the discretization needs to adapt to the changing shape of the domain and adapt the regions where high resolution elements are located. To accomplish this adaptation, I present a method that allows the discretization to change over time without performance degradation.

15 3 Creating a realistic-appearing animation of gas or water is the primary goal of a simulator designed for use in computer graphics. This is a different goal than simulation in an engineering context, where the goal of simulation is predictive accuracy of real world fluid properties. The goals within computer graphics and those in engineering are not disjoint, and it is no coincidence that many of the methods used in computer graphics were first developed for use in the engineering field. That said, ultimately the purpose of a computer graphics fluid simulator is to generate an animation that is visually plausible, this goal and others such as speed, stability, and ease of implementation take precedence over strict guarantees of numerical accuracy. 1.1 Background To give perspective on the differences between previous methods and the methods presented in this thesis, I briefly review fluid simulation from a high level. Much of this review spells out the desirable properties of a simulation method and motivates the novel methods presented in this thesis. In a computer simulation the fluid state s (velocity, pressure, temperature, etc.) is discretized, meaning that the continuous properties of a real fluid are approximated by a discrete set of state values located at selected points throughout the simulation domain. The discretized fluid state serves as a description of these relevant fluid properties at a discrete moment in time t. The state at an arbitrary point in the domain can be computed by interpolating the discrete values. The simulator computes the states at the discrete data points for a series of times,

16 4 s(t 1 ), s(t 2 ),...,s(t N ) such that temporal evolution of the discretized fluid properties follow the equations of motion for a fluid. The equations of motion relate the temporal derivative s t of fluid properties to (among other things) the spatial derivative s x Mathematically this can be expressed as s t = f of fluid properties. ( ) s. (1.1) x Spatial derivatives can be estimated from the discrete values by, for example, taking derivatives of the interpolation function or, in our case, by utilizing Stokes Theorem. In plain English, Equation (1.1) expresses the idea that the way the fluid velocity, pressure, and temperature change over time depends on how these properties vary in space. The significance of this is that the fluid properties can be simulated over time given some initial state by executing the following steps: 1. Calculate spatial derivatives. 2. Use the equations of motion to determine a temporal derivative. 3. Apply this change to the initial properties to yield new fluid properties. The fluid state over time can be determined by repeating the steps using the updated fluid properties from one step as the new initial state in the next step Accuracy and Adaptivity There are several factors that affect the accuracy of a fluid simulation. One of the most important factors is the number of discrete values used to estimate the fluid properties. If more discrete values are available to estimate the fluid, the continuous properties of a

17 5 s s x s s t x x Figure 1.1: A one dimensional representation of discretization and simulation. On the left a continuous function (dashed line) is represented by discrete values (dots) and an s interpolation function (solid line). The spatial derivative, x is estimated. On the right are the discrete values at a later time. The values are changed by the partial differential equations of motion and the estimated spatial derivative. real fluid can be better approximated by the interpolation function. This results in better spatial derivative estimates, yielding more accurate simulations. Of course the downside of using more discrete values is that it requires more work to store and compute these values. One way to improve performance in this inherent trade-off is to put more discrete values where the fluid properties vary the most, and place fewer discrete values in areas where the fluid does not require as many discrete values to accurately approximate. An example of a beneficial use of spatial adaptivity is a discretization of the ocean. It is sufficient to describe the velocity a body of water with the velocity at a few locations if that water is moving with near constant velocity, such as would be found in the deep ocean. On the other hand, the velocity at many locations are needed to accurately describe the fluid velocity in the turbulent motion of a crashing wave. The motion of an ocean can be best approximated given a fixed budget of data values by placing a high density of points near the shore and a low density out in the deep ocean where the motion is less interesting. The location of desired high/low density points are often not as simple as the ocean example where dense points are generally always near the shore and sparse points far from

18 6 s s x x Figure 1.2: Figure showing the advantage of spatial adaptivity. The representation on the right uses just as many data point but better approximates the continuous function.. shore. More often the location of regions where computation effort is desired varies over the course of the simulation. This points to the importance of being able to alter the location and even number of simulation data points over the simulation in order to maintain the benefits of spatial adaptivity Reference Frames If the discrete values that are used to estimate the fluid state are placed at locations in space that don t change over time the simulation is said to take an Eulerian view. A second alternative is the Lagrangian view where the discrete values are associated with a particular lump of fluid mass and therefore move with the fluid. Lastly, the locations of fluid values can be placed and moved arbitrarily. This reference frame is known as the Arbitrary Lagrangian-Eulerian (ALE) view. As an analogy, the flow of a river can be described with a fixed number of data points in a number of different ways. In the Eulerian way, assistants wade out into the river, stand fixed on the river bottom and report the fluid velocity that is flowing past them over time. In the Lagrangian way, they float down the river on innertubes. Each assistant describes the velocity that they are moving at over time. Lastly, in an ALE view

19 7 the assistants could swim around in the river, perhaps moving to points of the flow that are interesting, and report how quickly the fluid was moving past them, and how fast they were swimming. Probably the most widely used perspective for performing fluid simulation is the Eulerian approach. The primary advantage of the Eulerian approach is that this method is amenable to use of a simulation mesh. A simulation mesh is a geometric structure which partitions the simulation domain into simple elements, typically squares or triangles in 2D and cubes or tetrahedra in 3D. The mesh functions as an organizational structure for the discrete fluid values. It is used to define convenient interpolation functions of the discrete values, and provides a clear specification for the region of space that each discrete value is responsible for. As a result, the mesh simplifies the operations required for performing a simulation such as interpolating the discrete data points, and estimating derivatives. It is difficult to use a mesh with a Lagrangian view because, in a Lagrangian method the data points follow the complex motion of the fluid. As a result, the mesh is altered into a configuration that would make simulation impossible. Elements are twisted into shapes that are numerically unsuited to represent a function or to compute a derivative. Furthermore, mesh elements can become inverted, creating the nonsensical situation that these inverted elements represent a negative volumes of fluid. To avoid these problems when simulating a fluid with Lagrangian view, meshless methods can be used. These method build no explicit partitioning of the space into elements nor do they force elements to maintain neighborhood relations over the simulation. Rather points are allowed to move freely and neighborhood relations are determined for each time step. Interpolation is performed by

20 8 taking weighted averages of a time steps current local neighboring data Prior Methods The majority of simulation methods developed for computer graphics application use an Eulerian approach and discretize the domain with cube elements. An all cube element discretization is also referred to as a regular grid. The advantage of a regular grid is that the discretization is easy to generate, interpolation is straightforward, and accurate derivatives can be estimated using a simple finite difference method. The main disadvantage is that this discretization is very inflexible because all elements are the same size and shape. As a consequence many cubical elements are required to accurately represent the shape of the fluid being simulated, in particular at irregular boundaries such as at interface of fluid and an irregularly shaped obstacle or at the free surface of liquids. Inaccuracies in the shape of the discretization can result in noticeable artifacts such as stair-stepping. Another result of a regular grid discretization s lack of adaptability is that with fixed element size the data points distribution is uniform and can not take advantage of spatial adaptivity to focus computation where the fluid motion is most interesting. 1.2 Contributions of This Dissertation This dissertation presents methods that increase the adaptability of simulation methods used for computer graphics. Care is taken to maintain, as much as possible, the desirable properties of many of the prior graphics simulation methods; speed, stability, and ease of implimentation/use. The methods presented in this thesis increase adaptability over

21 9 the prior methods in two ways. First, we demonstrate use of tetrahedra for discretizing the fluid simulation domain. Secondly, we allow the mesh discretization to change arbitrarily from time step to time step. The following paragraphs elaborate on these points and their effects on the animations. This dissertation describes simulation methods where tetrahedra are used to discretize the domain as opposed to the standard computer graphic discretization, regular cubical elements. Tetrahedra are more adaptable in their shape, and unlike a regular grid of cubical elements, their size can vary over the domain. These factors together allow a tetrahedral mesh to conform well to the desired discretization boundaries, and to take advantage of spatial adaptivity. Conforming well to boundaries is important because if the elements do not match the intended boundary well, noticeable artifacts can be seen in the motion of the fluid. With regular grids, accurate matching of the boundary is achieved by using many small elements, increasing the cost of simulation. With tetrahedra, larger elements can be used to match the boundary with the same accuracy as a much larger number of regular grid elements. In Figure 1.3 the same shape has been discretized in one case using cubes and in another using tetrahedra. As shown, even though more elements are used in the cubical discretization, the tetrahedral discretization does a much better job of representing the object s boundary. Spatial adaptivity allows efficient use of available computations resources. It allows effort to be focused where it is most needed. Small tetrahedra can be placed near the boundary, where the fluid exhibits turbulent motion, or generally wherever the fluid exhibits features that a user thinks necessitate more detail. Following the recommendation of the

22 10 Figure 1.3: A comparison of a regular grid (left) and tetrahedral (right) discretization of the same shape. Despite the fact that the tetrahedral mesh uses fewer elements, it better represents the input boundary. work by Losasso et al. [43] I defining regions of interest based on where the visible smoke is, regions of high vorticity, or near boundaries. Away from these regions the size of tetrahedra can grade to larger sizes, thus saving computational effort and memory. In Figure 1.4 we show an example comparing a simulation using a static mesh to one that adapts to use small elements where smoke and vorticity are. As shown in this example, for the same computation time, the adaptive discretization creates an animation with substantially more detail and less numerical smoothing. The second way that the methods presented in this thesis increase the adaptability of the simulation is by allowing the mesh to change from time step to time step. Changing the mesh allows it to adapt to changes in the simulation such as the location of the boundaries or the regions where simulation effort is desired. This capability is important to maintain the advantages that tetrahedra offer over the progression of the simulation.

23 11 The approach I take does not fall neatly into the above three reference frames, Eulerian, Lagrangian, or ALE. It perhaps most closely resembles an ALE view as the mesh is neither fixed in world or material coordinates, but the method we use to accomplish this differs substantially from previous ALE approaches. Furthermore, since the topology and the number of discretization points may change arbitrarily between time steps, our method is more general than typical ALE methods, where some limited mesh movement is used, and topological changes and changes in the number of discrete points are not handled. The adaptability of size and shape that comes with of tetrahedra does not, unfortunately, come without cost. Most of the costs can be attributed to the loss of the structure that is inherent to a regular grid discretization. The structured nature of a regular grid allows for optimizations of both execution time and memory. Furthermore, generation of a regular grid is trivial. Generation of unstructured tetrahedral meshes with suitable quality elements is a difficult problem and so even the initial step of creating the mesh is an obstacle using a tetrahedral based simulator. In this work I show that by choosing appropriate meshing methods and by taking advantage of properties of the resulting meshes, many of the problems typically associated with tetrahedral meshes are mitigated and I obtain a viable simulator. The method I use to alter the mesh between time steps, on the other hand, incurs little cost. A standard part of fluid simulation for computer graphics is performing semi- Lagrangian advection. I show in Section that by making a simple generalization of the procedure, a mesh can be changed essentially arbitrarily as part of this step. Furthermore, the generalization does not degrade the simulation performance. The method is applicable

24 12 to both tetrahedral grids or regular grids, although a tetrahedral mesh is needed to fully exploit the adaptability that the method offers. Fluid simulation, in particular on tetrahedral meshes, has been widely studied outside of the computer graphics in the computational fluid dynamics (CFD) community. However, the goals of practitioners within the CFD community are fundamentally different from the ultimate goal of this thesis. Generally, researchers outside of computer graphics are concerned with creating simulations that accurately predict numerical behavior of a fluid. In this work I present a simulation method that uses a tetrahedral discretization to create animations of liquids and gases that is designed to meet the needs and desires of those in computer graphics. In particular the method is meant to be stable, efficient in execution and memory usage, and fairly straightforward to implement. It achieves these goals by applying methods that are well established in computer graphics to tetrahedral meshes. Furthermore, when applying the simulator to free-surface liquids I chose a meshing technique, [41], that has properties that can be exploited to maintain some of the computational and memory efficiency of a regular grid, yet retains much of the adaptability of general tetrahedra.

25 Figure 1.4: Two frames from different animations that took equivalent times to simulate. On the left, a single fixed grid was used throughout the simulation. On the right, the mesh is dynamically updated to place small elements near the smoke and regions of high vorticity. The right animation features substantially less numerical smoothing despite taking the same time to simulate (including mesh generation) as the left animation. 13

26 14 Chapter 2 Previous Work Fluid simulation has been a popular subject in computer graphics for many years. Outside computer graphics, fluid simulation has a long history within engineering. I only mention the most relevant papers from these fields here as the literature is so vast. Even within computer graphics a massive number of papers have been published on the subject. The following sections discuss a selection of these papers that have been influential in the development of this thesis. 2.1 Two-Dimensional Approximations In computer graphics the end goal is to generate an animation, and for liquids the animation is of the liquid s surface. If the motion of the surface is simple enough that it can be represented by a height field than the simulation can be simplified substantially by representing the liquid as a height field over a 2D domain. This idea was first used in computer graphics by [52] who used sinusoidal functions

27 15 to describe ocean waves and simulated their evolution over time by computing the change due to gravity. Other early work by Kass and Miller [38] represent the fluid as a 2D height field and simulated the height field with a partial differential equation (PDE) developed from an approximation to the shallow water equations. The shallow water equations are a simplification of the 3D equations for a fluid to a 2D height field representation that a linear pressure variation in height columns of the fluid. The result was a simple, fast, stable system for generating wave motion. To allow for more dynamic splashes O Brien and Hodgins [50] present a hybrid 2D column of fluid and particle method. The first use of the Navier-Stokes equations in the graphics community appears in a 2D version of the equations [8] where the height of the liquid surface is determined from pressures computed in the simulation. 2.2 Basic Three-Dimensional Simulator The 2D methods discussed above describe a liquid by specifying the height of the liquid surface over a 2D plane. This description of the liquid is somewhat limited. For example, it can not model an overturning wave where the liquid surface exists at multiple heights. Additionally, the 2D simplification is not appropriate for gasses because the motion of the gas throughout the 3D domain is of visual interest. To handle general liquids, Foster and Metaxas [24] perform a full 3D Navier-Stokes simulation. This work is based on the pioneering work of Harlow and Welch [30] who present an effective and often used Eulerian grid based 3D simulator in the engineering literature. Foster and Metaxas not only prove the feasibility and usefulness of such an approach to fluid animation but also discuss a number

28 16 of useful ideas in their paper including a staggered storage scheme for the fluid velocity and pressure variables. They also introduce preliminary work on simulating situations where a fluid interacts with a solid and vice versa. Lastly, the authors present preliminary work on fluid control. Fluid control is the manipulation of natural fluid motion through control forces to achieve some desired properties of the motion. The authors followed up the next year with an application of the method to gases [25]. The paper Stable Fluids by Jos Stam [62] presents a number of important improvements to fluid simulation for graphics application. First, this paper presents the semi- Lagrangian advection technique. (The method was known in other fields and rediscovered by Stam.) Semi-Lagrangian advection is an unconditionally stable method for simulating the fluids advection. Explicit advection techniques, such as the one used in [24], are subject to the CFL (Courant, Friedrich, Levy) stability condition. This condition states that the time step for which an explicit method will remain stable is related to the velocity and element size. For stability, the time step must be small enough such that the velocity field could not propagate material more than an element size. The semi-lagrangian technique is not subject to any such condition on it s stability. The simulation time step may be arbitrarily large, and while the accuracy of the solution may suffer, the method is guaranteed not to go unstable. Robustness of the simulator is typically considered more important than accuracy in computer graphics applications and for this reason the method is popular within the field. Furthermore, the semi-lagrangian method is important to this work because the extension of the semi-lagrangian method that we describe allows us to change meshes between time steps efficiently and accurately. Secondly, Stam uses the projection

29 17 approach of Chorin [11] to conserve mass instead of the relaxation technique used by Foster and Metaxas [24]. With this approach mass conservation can be abstracted to the problem of solving a large sparse linear system. Years of work on developing fast linear system solvers can be leveraged to obtain an efficient mass conservation procedure. 2.3 Alternative Discretizations The efficiency of the early 2D methods has been revisited in more recent work. For explosions that do not interact with obstacles, the motion of the explosion is radially symmetric. Rasumssen et al. [55] exploit this symmetry and generate high resolution explosions by simulating with several high resolution, 2D domains that are rotated about the axis of symmetry. Irving et al. [36] present a hybrid shallow water and 3-D fluid simulator is used to model deep bodies of water by placing a layer of 3D voxels atop the column elements of a shallow water method. The method can simulate 3D effects like splashing and overturning waves at a cost somewhere between a 2D and 3D method. Stam [63] presents a method for animating flow on 2D manifolds in 3D space by projecting the simulation onto a subdivision surface. Most work within computer graphics has used regular hexahedral grids. Regular grids have the advantages that they are easy to generate and taking derivatives and interpolating the grid values is straightforward. The disadvantage of regular grids is that their elements lack adaptability in shape and size. Some authors have investigated alternative discretizations of the simulation domain. Lossaso and colleagues [43] develop a method that uses octree decomposition of

30 18 the domain allowing adaptive resolution. Tetrahedra allow for adaptation of both size and shape, and a number of works have investigated their use for computer graphics fluid simulation. Feldman et al. [21] simulates gases with a static discretization that uses cubical and a tetrahedral elements. Cubical elements are used in open regions and tetrahedral elements are used near the boundaries, so the mesh can conform to complex boundaries. Wendt et al. [69] use an all-tetrahedral mesh and a finite volume method to simulate both gases and liquids interacting with complex obstacles. Elcott et al. [16] simulated fluids using a vortex approach on tetrahedral meshes. Vortex methods simulate using the curl of the velocity field to describe the liquid s motion. When the velocity field is needed for smoke or liquid surface advection it can be solved for from the curl. Shi et al. [60] use a surface triangulation discretization to simulate fluid motion on the surface of objects. They describe interpolation and derivative operators using a triangle mesh and perform projections of velocity onto the surface manifold. 2.4 Maintaining and Enhancing Detail Generating fluid animations with detailed turbulent motion is very important to computer graphics practitioners. However, direct simulation of turbulent effects is prohibitively costly. The vorticity confinement method was introduced to the graphics community [19] to address this issue. The method enhances the vortices already present in the field which would otherwise be damped out from numerical smoothing. The vortex particle method, [56], presents a more flexible method which can add vortices to a fluid by associating vorticity with particles that are advected with the flow. Rasmussen and co-authors

31 19 [55] use Kolmogorov noise to enhance motion in explosions at a scale more finely resolved than the simulation scale. The detail of flames is enhanced in work by Hong et al. [33]. In this work the characteristic cellular patterns observed in flames are achieved by augmenting motion of the flame with equations from detonation shock dynamics. 2.5 Extensions The basic fluid simulator presented by Stam [62] has been extended in a number of ways to increase the type of fluids that can be modeled and to allow for more general interaction with other materials. The basic method is extended to apply to high viscosity fluids in Melting and Flowing [7]. By adding this capability, materials such as melting wax, wet sand, and squirting tubes of toothpaste may be simulated. Elastic materials exhibit forces that try to restore the material from a deformed configuration back to the original configuration. The forces can be computed from the deformation which is locally described by a strain tensor. For solids, the material can be simulated using a Lagranian method and strain tensors can be calculated directly. Fluid simulators typically use Eulerian methods. To simulate viscoelastic liquids, liquids that exhibit elastic behavior, Goktekin et al. [28], advects and integrates the strain rate tensor so that elastic behavior can be incorporated to fluid simulation on an Eulerian grid. Sand is simulated [71] as a fluid by adding sand friction forces to a standard fluid simulator. Nguyen and co-authors [49] simulate flames by using a level set to track the reaction front, and by modifying the semi-lagrangian advection to account for the reaction process. Fire animations are generated by using a mix of procedural and simulation methods [42]

32 20 with the goal of creating controllable flames. Explicit complex reaction processes have been modeled for graphics by Ihm et al. [35]. This allows for generation of the complex behavior of interacting fluids by adding some simple rules for the different fluids interaction. As an example one fluid making contact with another can generate fire and reactive product which are all three effected heat and expansion created by reaction. Explosions have been investigated in a couple of different works. The first such work by Yngve et al. [70] uses a compressible method instead of using the standard incompressible assumption and models the blast wave of the explosion. Since compressible simulations are very stiff and the blast wave moves very fast, this method requires very small time steps. Later work, by Feldman et al. [20] ignores the largely invisible blast wave and instead focuses on the fireball, the most apparent visual phenomena of the explosion. By ignoring the blast wave and focusing on the fireball an incompressible fluid simulator can be used by modifying the simulator to account for expansion of the fireball. Additionally, this work uses a hybrid particle/grid method is used to account for interaction of fuel and air. In some situations surface tension effects and the interaction of air with a liquid has a noticeable effect on the fluid motion. The paper Discontinuous Fluids [32] focused on simulating these cases by describing methods to properly simulate the large discontinutiy in material properties at the liquid-air interface. More recent work has looked at foam [39] and frothing liquids [12]

33 Fluid Control In a production environment it is very important to be able to generate an animation that meets some specified goals or has a particular look and feel. For example a film director might request that smoke form the shape of a dragon after it is blown. Finding initial conditions that happen to move smoke into the desired configuration is unlikely or impossible. So fluid control is used to help nudge the fluid with forces to influence its motion. One of the first works on fluid control in graphics, Controlling fluid animations, [26] manipulates the flow by altering boundary conditions and the fluid velocities and pressure. A keyframe approach to control was first introduced by Treuille et al. [66]. This paper uses an optimization procedure to solve for control forces which manipulate smoke into the keyframes. The optimization is quite expensive so in later work the authors utilize the adjoint method, a method from the control and optimization engineering literature, to improve efficiency [46]. A non-optimal keyframe method also has been introduced for smoke [18] and water [61]. A more subtle approach to fluid control, which uses user animated control particles was introduced by Rasmussen and colleagues [54]. 2.7 Fluid-Solid Coupling Early fluid simulation papers realized the need for interaction between fluids and moving objects. One-way coupling, from fluid to solid, is achieved [24] by using the fluid velocity to passively advect soda cans in the water. Two-way coupling between explosions and fracturing solids was presented by Yn-

34 22 gve et al. [70] in the context of compressible fluid simulation. To obtain two way coupling in an incompressible simulator, Rigid Fluid [6] simulates the solid as a fluid and then projects all velocities within the solid to act rigidly. This method generates convincing results in most situations and incurs relatively little overhead. One problem is that when the velocities are altered to act rigidly, the divergence-free condition is broken in the cells that surround the rigid body. This problem is addressed in this thesis by performing the coupling simultaneously. A method for simulating the interaction of fluid and deformable thin sheets is presented by Guendleman et al. by [29]. The interaction is achieved by alternating steps of simulating the fluid with fixed boundary conditions for the deformable body, then performing the deformable body simulation fixing the pressure from the fluid simulation. Alternatively, the coupling between fluid and deformable bodies and the fluid can be performed simultaneously as is done in Chantenez et al. [10]. 2.8 Moving Meshes Prior work on moving meshes has been investigated in graphics albeit in a substantially more limited fashion than what is presented in this thesis. The method by Rasmussen et al. [54] translates the grid by cell size units so that the locations where velocities are stored in the previous mesh overlap with the locations in the new mesh. This allows the mesh to be moved without interpolating velocities on the new mesh and therefore avoids the problem of numerical diffusion created by interpolation. The principle of Galilean Invariance is used by Shah et al. [58] to allow meshes to translate with some moving reference frame such that the grid can follow, for example, a plume of rising smoke. This prevents the need

35 23 for large simulation domains where much of the simulation domain contains velocity values that do not effect the smoke. The method I present in this thesis which first appeared in the paper Fluid in Deforming Meshes [22] allows for substantially more freedom in mesh changes. This first paper has examples with affine transformations and simple deformation of meshes while in later work with Klingner et al. [40] we demonstrate that the method is applicable to meshes that move, deform, and alter topologically. In a related work by Bargteil et al. [4] the idea of altering a mesh over the course of simulation is applied to Lagrangian simulation of highly deformable plastic solids. 2.9 Particle Based Methods The most common way fluids are simulated is by discretizing the domain with a mesh and representing properties of the fluid using discrete values on that mesh that remain fixed in world space, an Eulerian view. Alternatively, the fluid can be described by meshless particles that are advected with the flow where each particle represents some lump of fluid mass (Lagrangian). Smoothed Particle Hydrodynamics (SPH) defines continuous functions of fluid values from the discrete particles by taking a weighted average of surrounding particles. Weights are determined using densities that are attached to the particles and smoothing kernels which fall off with distance. From these functions derivatives of the state properties can be formulated. SPH was first introduced independently by Ginggold and Monaghan [27] and Lucy [44]. The SPH method was introduced for use in graphics by Desbrun and Cani, [14]. It has been used in a number of situations including lava, [64], boiling water, [48], and soft bodies [13]. Mueller et al. [47] perform an SPH method with a

36 24 small number of particles ( 5000) achieving real time results. Adams et al. [1] use adaptively sized particles in order to have a multi-resolution simulation. This allows computational effort to be focused in areas that most benefit from the increased resolution. The results of this work are visually compelling and are generated in times very competitive with regular grid methods. One draw back of the SPH method is that incompressibility is not enforced. As a result, visible compression can be noticed unless the fluid is simulated with a very high resistance to any compression. For the system to remain stable with this large compression resistance very small time steps are required. An alternative approach introduced by Premoze et al. [53] uses of the Moving Particle Semi-Implicit method which is a particle based method that explicitly enforces a divergence free velocity field. This work generated impressive results, and unlike previous particle methods, it rendered the surface using a level set method, instead of defining the surface using an implicit function by the simulation particles such as in Cani et al. [5]. Within academia at least, particle based implementations have yet to produce the visual realism of grid based methods. Despite this they promise to be an interesting avenue of future research. It is noteworthy that one of the most popular commercial software packages Real Flow 1 is SPH based and has been used in production movies a number of times although specific details of their implementation are unknown. 1