Liquid Simulation on Lattice-Based Tetrahedral Meshes

Transcription

1 Liquid Simulation on Lattice-Based Tetrahedral Meshes Abstract This paper describes a simulation method for animating the behavior of incompressible liquids with complex free surfaces. The region occupied by the liquid is discretized with a boundary-conforming tetrahedral mesh that grades from fine resolution near the surface to coarser resolution on the interior. At each time-step, semi-lagrangian techniques are used to advect the fluid and its boundary forward, and a new conforming mesh is then constructed over the fluid-occupied region. The tetrahedral meshes are built using a variation of the body-centered cubic lattice structure that allows octree grading and deviation from the lattice-structure at boundaries. The semi-regular mesh structure can be generated rapidly and allows efficient computation and storage while still conforming well to boundaries and providing a meshquality guarantee. Pressure projection is performed using an algebraic multigrid method, and a thickening scheme is used to reduce volume loss when fluid features shrink below mesh resolution. Examples are provided to demonstrate that the resulting method can capture complex liquid motions that include fine detail on the free surfaces without suffering from excessive amounts volume loss or artificial damping. Keywords: Natural phenomena, physically based animation, computational fluid dynamics. 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; I.6.8 [Simulation and Modeling]: Types of Simulation Animation. 1 Introduction Numerical fluid simulation has proven to be an extremely effective technique for realistically animating the complex motions of liquids and gasses. Currently, the most commonly used methods employ either regular hexahedral grids or meshless collections of points. Recently however, a number of methods have been proposed that make use of unstructured tetrahedral meshes. (See, for example, [Feldman et al., 2005; Elcott et al., 2007; Klingner et al., 2006].) As with regular grids, these tetrahedral methods can easily model large volumes of incompressible fluid. However, tetrahedral meshes offer potential advantages because they can conform to complex boundaries and mesh grading can be used to focus computational effort where it is most beneficial. Tetrahedral methods have already been successfully demonstrated for situations where a fluid, typically a gas, fills the entire simulation domain and there are no free surfaces. However, animation of most scenarios involving liquids requires realistically capturing the behavior of the liquids free-surfaces. This work has been submitted for publication. Copyright may be transferred without further notice and the accepted version may then be posted by the publisher. Unauthorized viewing, duplication, and/or distribution are prohibited. Figure 1: These images show two views of a liquid pouring down a pair of chutes into a transparent container. The lower four images are a sequence of images evenly spaced in time. In this paper, we describe a method for animating incompressible liquids with free surfaces using tetrahedral meshes. The mesh tessellates the region occupied by the fluid, and the mesh s exterior surface conforms both to the fluid s fixed and free-surface boundaries. To avoid difficulties with meshtangling or self-intersection, we do not attempt to advect the mesh structure forward with the fluid. Instead, we use semi-lagrangian contouring [Bargteil et al., 2005] to determine the fluid s new boundary surface, and then construct a new mesh that tessellates the updated region. Once the new mesh has been constructed, physical properties are transfered from the old mesh to the new using a generalization of semi-lagrangian advection. The method we use for generating our meshes does not create a completely unstructured tetrahedral mesh. Instead, it builds a tiling using the nodes of a body-centered cubic lattice structure corresponding to an octree that encompasses the fluid region. The octree is refined based on distance to the fluid surface resulting in a fine-to-coarse grading from 1

2 surface to interior. Additionally, the nodes and tetrahedra near the surface are modified so that the mesh s exterior surface conforms to the fluid s surface. The mesh generation algorithm is fast, and includes quality guarantees on the resulting semi-structured mesh. The regular structure on the interior of the mesh facilitates efficient storage and computation because the majority of elements in the mesh share a small set of geometric templates. Although semi-lagrangian contouring generally does well at tracking movement of the fluid s free surface, as with any tracking method it still loses sheets and filaments of fluid that thin below the simulation method s finest resolution. This behavior can lead to noticeable volume loss and distracting artifacts. To combat the problem we use a thickening method that attempts to prevent features from thinning to the point where they can no longer be captured by the simulation. It attempts to locate regions where the surface is about to thin below the finest mesh resolution and then slightly thickens those areas. To prevent run-away volume gain as the thickened structures continue to stretch, an artificial effect is added to break the structures apart. The result is that thinning structures tend to fragment just before they reach the point where they would have become unresolvable. Finally, we show how an algebraic multigrid solver can be used to efficiently perform the pressure projection required to ensure that the fluid behaves incompressibly. We have found that this type of solver is easy to implement even with complex boundary geometry and that it significantly out performs the commonly used alternative of preconditioned conjugate gradient solvers. Our results, such as the example shown in Figure 1, demonstrate that tetrahedral meshes can be used to animate liquids without excessive amounts of volume loss or artificial damping. Because tetrahedra can adapt their shape and furthermore as the tetrahedral mesh grades to lower resolution away from the surface we can afford to use very fine tessellation near the surface that captures small surface details that would be difficult to model otherwise. 2 Related Work There are a number of challenges in attaining the level of realism, control, and efficiency required for generating highquality fluid animations. Simulation based methods have been widely adopted by the visual effect industry and the resulting demand has motivated a significant amount of work in this area. Some of the earliest work which established the foundation for subsequent research can be found in [Foster and Metaxas, 1996], [Stam, 1999], [Foster and Fedkiw, 2001]. While those papers generated results for smoke and water, the methods have since been extended to apply to more exotic materials for example, fire, [Nguyen et al., 2002], viscoelastic materials [Goktekin et al., 2004] bubbles, [Zheng et al., 2006], and explosions [Feldman et al., 2003]. Methods for interaction of fluids with rigid, thin, and deformable bodies ( [Carlson et al., 2004], [Guendelman et al., 2005] and [Chentanez et al., 2006] respectively), have also been investigated while [Losasso et al., 2006] presents a method to allow for the interaction of multiple fluids. Recently, [Treuille et al., 2006] introduced an exciting new direction of modelreduction techniques for real-time simulation. One of the fundamental difficulties for simulation is attaining detailed fluid motion and realistic liquid surfaces while maintaining stability and efficiency. One solution is to artificially enhance the vorticity as in [Fedkiw et al., 2001] and [Selle et al., 2005], while [Threy et al., 2006] looks a methods of maintaining the detail when applying control to the simulation. While the majority of papers use regular hexahedral grids to discretize the simulation some have investigated the use of more adaptable computational structures. For example, [Losasso et al., 2004] uses an octree data structure gasses and liquids, and [Klingner et al., 2006] uses an unstructured tetrahedral mesh for gas simulation. Others approaches have moved away from Eulerian grids entirely and use meshless collections of Lagrangian particles to discretize the fluid. A few examples include [Terzopoulos et al., 1989], [Desbrun and Cani, 1996], [Cani and Desbrun, 1997], [Premože et al., 2003], and [Müller et al., 2004]. We use tetrahedral grids to discretize the evolving liquid domain and therefore require a method to economically generate tetrahedral meshes of suitable quality. Mesh generation is a widely investigated field and [Owen, 1998] and [Teng and Wong, 2000] provide good surveys. We have adapted the meshing method by [Labelle and Shewchuk ] which uses a body-centered cubical lattice structure to tetrahedralize a domain defined by an implicit function. This type of meshing algorithm also appears in [Bridson et al., 2005] and it features a number of properties that make it particularly well suited for our fluid simulator. Surface tracking is an important part of a liquid simulator both for determining the location of the liquid to simulate and for generating a surface to render. The particle level set method presented in [Enright et al., 2002] is an effective technique that enjoys widespread use. A number of alternative methods exist and in our work we use a combination of semi-lagrangian contouring from [Bargteil et al., 2005] and the particle based method of presented in [Zhu and Bridson, 2005]. Grid-based level set methods are generally very successful at creating high quality liquid surfaces. The main problem these methods face is the loss of thin features that fall below the resolution of the grid. Solutions to this problem which detect the lost mass and replace it with particles appear in a number of papers for example, [Guendelman et al., 2005] and [Kim et al., 2006]. 3 Methods The basic steps of our simulation method are conceptually simple: 1. A tetrahedral mesh is generated that conforms to the liquid volume. 2. The fluid s governing equations are discretized onto this mesh and are used to determine how the fluid will move. 3. A mesh corresponding to the new configuration is generated, the fluid s state projected onto the new mesh, and the process repeats. Unfortunately, implementing these steps in a manner that produces useful results can be somewhat difficult. Inappropriate discretization of the fluid s governing equations can lead to nonsense solutions. Flowing liquids tend to produce highly complex shapes that are difficult to mesh with reasonable quality elements. Tracking the fluid s free surface forward in time without creating highly objectionable artifacts is a notoriously difficult problem. In the following sections we discuss these issues and explain how we have addressed them. As one would expect, the solutions we use are mainly based on prior work so we focus the discussion on how we have integrated the components and the changes necessary for them to work together. 2

3 Online Submission ID: papers 0509 a. c. b. d. To interpolate velocity within the tetrahedral mesh, we first compute the full velocity vector at each tetrahedra center using its four face normal velocities as described in [Elcott et al., 2007]. Then to interpolate at an arbitrary point we use generalized barycentric interpolation of [Warren et al., 2004] to interpolate over the meshes dual cells. In steps where a small amount of additional smoothing will not produce objectionable artifacts, such as advecting marker particles or the trace-back steps in a semi-lagrangian method, we can optimize interpolating velocity to the primal mesh nodes and then using the barycentric interpolation of those values. This optimization should not be used for computing the values carried forward by the semi-lagrangian advection because the cumulative smoothing would produce excessive damping. We differ from [Klingner et al., 2006] in that we store quantities at the barycenters of tetrahedra and faces instead of at circumcenters. We prefer barycenters as their locations are better behaved. For example, in the interior of our mesh adjacent tetrahedra may have collocated circumcenters which would create difficulties for the gradient operator. While the gradient and interpolation operators were designed with the geometric properties of circumcenters in mind, we find that storing quantities at the barycenters works well in practice. Figure 2: The upper-left image shows a wave of liquid flowing over a hemispherical obstacle. The upper-right image corresponds to the area outlined in green, and it shows a cutaway view of the underlying tetrahedral simulation mesh. The two lower images correspond the small area outlined in yellow. The lower-left shows the surface triangulation used by the surface tracker and the lower-right shows the surface of the simulation mesh Governing Equation and Discretization The incompressible liquids we wish to model are governed by the standard Navier-Stokes equations: u f p = (u ) u + t ρ ρ (1) subject to the volume conserving constraint that: u = 0. (2) In these equations u is the fluid velocity, t time, p pressure, ρ density, and f external forces. The symbol denotes the vector of differential operators = [ / x, / y, / z]t. The basic simulation method follows that of [Stam, 1999]. The fluid state is evolved in time using operator splitting: integrating the state forward by each term of Equation (1) separately. The advection term is solved using the semilagrangian technique. Mass conservation is enforced by integrating the pressure term last, where the pressure is determined by solving a linear system that ensures the final integrated velocity field obeys Equation (2). We use the derivative operators and the velocity interpolation method for tetrahedra as described in [Klingner et al., 2006]. The interested reader should refer to this work for details but we will describe them briefly. Pressures are located at the centers of each tetrahedral cell and velocity is stored at each face center with the component of the velocity field that is normal to that face. To compute the gradient of pressure p at face f in the normal direction, Nf, one take the difference between adjacent pressures divided by the perpendicular distance Df between them. The divergence within a tetrahedra is the sum of the face-normal velocity components zi for each face of the cell weighted by the face area, Ai, multiplied by a ±1 value, s, that orients the face normals outward. ( p) Nf = 1 Df (p2 p1 ) u= 1 Vt P4 i=1 Meshing As show in Figure 2, the tetrahedral meshes we use for simulation conform to the volume occupied by the fluid. This approach has two main advantages. First, the computational cost is only dependent on the region occupied by fluid. Second, it greatly simplifies the handling of boundary conditions as the fluid s boundaries correspond to element boundaries. An additional benefit is that because the mesh is constructed at each frame to match the fluid s current configuration we are able to grade the mesh resolution so that it is fine near the surface where we want detail but coarse on the interior. We assume that the surface is specified by an implicit function that is positive outside the fluid region and negative inside. Initially this function can be specified using conventional modeling tools, and once the simulation has begun it will be updated using the semi-lagrangian contouring method from [Bargteil et al., 2005]. Given the implicit functions that define the interior of the liquid, and the obstacle geometry, the meshing algorithm returns tetrahedra which fill the liquid region, while conforming well to the liquid surface and the obstacles the liquid is in contact with. The meshing strategy that we employ is based on the provably robust algorithm presented by [Labelle and Shewchuk ]. It builds a semi-regular tetrahedral mesh according to body-centered cubical lattice structure that is defined by a balanced octree. Cells of the octree that lie within the liquid surface are filled with tetrahedra that have nodes at the corners and center of the cell. If a neighboring cell is of a finer level then nodes will appear at the face and edge centers of the larger cell. The tetrahedra are built from templates for the various relative refinement level of the cells neighbors. Tetrahedra in cells that are sliced by the liquid surface are modified to conform to the liquid surface. First, the locations are found where the surface intersects the edges of these tetrahedra as defined by the implicit function. If the distance between a lattice node and the edge crossing is a small fraction of the edge length, the node is moved to its closest crossing. Otherwise new nodes are added at the edge crossing location and the portion of the sliced tetrahedra zi Ai si 3

4 Figure 3: This figure illustrates the two-dimensional version of the meshing scheme. Yellow points are the nodes of the octree lattice and the green line shows the location of the fluid s surface. Blue points represent the warped locations of lattice nodes that were near the surface. Red points are nodes created where the surface cut the lattice with no nearby nodes. that is inside the surface is filled with smaller tetrahedra. These tetrahedra can also be built from templates that depend on the number of nodes of the base tetrahedra that have been warped to the surface and how many edge crossings there are. Space does not permit the description each of the possible templates here and we instead refer the reader to [Labelle and Shewchuk ]. Figure 3 shows an example of how this approach can be applied to a two-dimensional triangulation. We also refer the reader to [Labelle and Shewchuk ] for detailed discussion on how this meshing algorithm is guaranteed to produce good-quality tetrahedra and how the edge fraction threshold for node warping should be selected. Very briefly, the strategy generates good quality tetrahedra because in the interior it uses high quality tetrahedra generated by the lattice. Near the boundary it only moves the lattice nodes if the distance is sufficiently small to prevent creating bad tetrahedra or inversion, otherwise it adds a node to an edge that is far enough away from existing nodes to avoid creating new thin tetrahedra Obstacle Conformance In general, the zero-contour of the implicit function that defines the liquid body does not conform perfectly to the obstacle boundary due to small errors in its advection and because of finite resolution. To avoid cumulative error that could lead to meshing inside the obstacle or artificial gaps between the obstacle and fluid, we build a negative signed distanced function for the obstacle then intersect this with the signed distance function for the liquid and use the result as the input of the meshing algorithm. After the mesh is built, nodes that are within some small tolerance (we use 1/8 the edge length of the finest octree level) of an obstacle are moved along their normal direction to snap to the obstacle. We have found that moving by this small distance does not introduce bad quality tetrahedra. A boundary face is classified as liquid-obstacle interface if all of its nodes lie on the obstacle. All other boundary faces are classified as liquid-air interface. Liquid-obstacle faces are treated as closed, non-slip, and liquid-air faces are open Extrapolation Both the semi-lagrangian advection of fluid properties and semi-lagrangian contouring to determine the new fluid sur- Figure 5: Points approximately on the medial axis are generated by comparing cord-lengths to the distance between the cord-center and the nearest point on the surface. In this example, m 1 will be used but m 2 rejected. face will require velocity values outside the simulation domain. The mesher s octree structure provides an excellent structure for this task. The corner nodes of the interior cells are also nodes of the tetrahedral mesh and hence their velocities have already been computed. For the nodes outside the mesh whose velocities are not known, we use the efficient extrapolation technique in [Losasso et al., 2004]. 3.3 Surface The surface tracking algorithm determines liquid s location as it is advected each time step. It takes the current implicit function defining the liquid domain, L t 1 and the current velocity field, v t 1 and determines the liquid s new implicit function, L t, for the next time step. To track our surfaces we use the semi-lagrangian contouring method of [Strain, 2001] and [Bargteil et al., 2005]. This contouring method maintains an explicit polygonal surface representation and near the surface, function values are the exact signed distance to this surface. Away from the surface, function values are interpolated from distance values stored on an octree grid. The new surface is built by contouring the function defined by the composition of backward tracing with the current distance function. We refer the reader to [Bargteil et al., 2005] for more details and access to a publicly available, opensource implementation. As with other surface tracking methods, semi-lagrangian contouring will be unable to resolve fluid regions that thin below the finest resolution of the octree it uses for constructing the new surface. The visual result of this phenomena can include large holes opening up in thin sheets, large sections of spray spontaneously vanishing, and noticeable loss of overall fluid volume. Other researchers have observed that Lagrangian particles can be useful for correcting these problems. In [Enright et al., 2002], for example, the particles play a similar role as the triangle mesh in the semi-lagrangian contouring, namely to correct interpolation values within grid cells near the surface. In our thickening approach we place particles only on the medial axis of thin regions that are in danger of falling below resolution. These particles are then used to explicitly thicken the region so that it can be resolved. This thickening will lead to run-away volume gain so our algorithm also forces the thickened sheets or filaments to break apart. This method is inspired by surface tension effects where thin regions break apart and the liquid is forced away from the break. The scale of the actual surface tension effects are substantially smaller than we can feasibly simulate, but the method s approximation of this behavior captures a roughly similar effect. Or simulations still exhibit some volume loss, but as shown in Figure 4 thickening greatly reduces volume- 4

5 Online Submission ID: papers 0509 Figure 4: These image sequences compare results generated with and without our thickening scheme. The top row uses thickening while the bottom does not. The example with no thickening suffers from visible holes and other artifacts where thin structures have been lost. The volume loss for the non-thickened example is also noticeable in the last frame were the container has failed to fill properly. loss artifacts. We approximate sample locations of medial axis using the following simple strategy that is illustrated by Figure 5. At a vertex pi of the contouring method s polygonal surface, we shoot a ray in the negative normal direction ni and check if the ray hits another surface within distance γslc. If so, we compute the point mi, the midway between pi and the first ray intersection. The location mi is a potential estimate sample position of the medial axis of the thin region. We then compute the minimal distance to the surface, di, using the contouring method s distance function and throw away the sample points where di disagree substantially with the distance from pi to mi. The particles are tagged with their local thickness radius and are advected forward using the velocity vi 1. Referencing the contouring method s advected distance function, we test for particles that fall in grid cells of all positive values. These particles represent mass that will not be contoured by ˆ t, the implicit function before our thickening extension is L added. We refer to these particles as outside particles while those that are in cells with at least one negative value are called inside particles as they are either in or very near the ˆt. location of a surface that would be contoured by L We make an approximate redistribution of the lost mass of the outside particles to their neighboring inside particles. For each outside particle we use its thickness radius to increment the thickness of the inside particles within a small neighborhood radius r1. Currently, we simply distribute the thickness evenly to its neighboring inside particles. While more sophisticated procedures can be developed, we find that this simple procedure suffices. With the updated radii, ˆ t and an implicit the final liquid domain, Lt is the union of L function obtained from the inside particles which we obtain using the method described in [Zhu and Bridson, 2005]. Because thin regions that contain inside particles that neighbor outside particles are thickened a bit they are less likely to be lost in subsequent frames. Optionally, inside particles that are in regions thinner than a threshold can be artificially incremented to be thicker. While this corresponds to a slight increase in mass, often this is preferable to risking losing the thin feature. It is very difficult to perceive a slight increase in the thickness of a thin sheet while the perception of a thin sheet vanishing can be quite jarring. Excessive artificial thickening can, however, create problems in certain situations. For example, if liquid is allowed to spread under gravity, incrementing the thickness too aggressively will allow the liquid to expand outwards without limit. The thickening parameter used in this situation does not correspond to a physical quantity and should be picked to suit aesthetic preferences. Outside particles that have no inside neighbors correspond to larger regions that have thinned within a single time-step. We increment their thickness by a small amount such that the they can contribute directly to Lt. They could also be rendered as particle spray as in [Guendelman et al., 2005] Mesh Thickening The resolution we use for generating the simulation mesh is typically a factor of two coarser than the resolution used for surface tracking. This sampling strategy helps to ensure that we do not smooth surface features during advection, but it can potentially create a situation were no tetrahedra are created in some thin part of the volume returned by the contouring method. To avoid having unstimulated regions of fluid hanging outside the mesh, we perform a second thickening process that only effects the meshing generator, not the surface tracking. We attach an offset function to the particles that were used for the previous surface thickening. The function passed to the mesh generator is the sum of the implicit function defined by the contouring method and the particle offsets. The offsets are computed as: P offset(x) = i W (x pi, h)(γmesh di ) P W (x pi, h) i (3) where γmesh is the size of the mesh generator s finest octree level multiplied by a constant a little greater than one. Our 5

6 choice of W is a W poly6 kernel, { W poly6 (r, h) = 315 (h 2 r 2 ) 3 0 r h 64πh 9 0 otherwise as in [Müller et al., 2003]. We choose h to be 4 times the size of the smallest grid of the simulation mesh s octree. The reason we use different thickening approaches for the contoured surface and the simulation mesh is because they are used for different purposes. For surface tracking, we prefer to get smooth surface and potential lose a bit volume than to get bumpy surface. For meshing, we prefer to not to have any of the rendered surface left hanging outside the simulation mesh and bumps in mesh surface do not impact the rendered appearance. 3.4 Multigrid Solver Enforcing incompressibility for the fluid requires solving Poisson s equation for a pressure field that causes the resulting velocity field to be divergence free. For our discretized system, this amounts to solving a large, sparse, symmetric linear system. We do this efficiently using an algebraic multigrid method. A detailed description of the multigrid method is beyond the scope of this paper but we briefly describe the relevant high level concepts and refer the interested reader to [Mc- Cormick, 1987] and [Trottenberg et al., 2001] for more details. Multigrid is an iterative method that performs some number of steps of a local iterative method, for example Gauss-Seidel, on the original system, Ax = b, and then corrects the solution estimate using estimates from coarser versions of the problem, A k x k = b k, where k is the level of coarseness of the problem. Correction by a coarser system is applied recursively such that the original problem is corrected by an estimate from a coarsened version of the original system, which is obtained with a correction from an even coarser version of the system and so on until the system is small enough to be solved easily with a direct, exact method. The idea is similar to multiresolution methods which start from a solution on a very coarse system and then repeatedly use coarse systems to initialize solutions on refined levels until ultimately solving the problem on the original resolution. When solving a large linear system, it is more effective to make several cycles through all levels of resolution. The particular scheduling for levels of corrections we use is called the full multigrid method and we refer the interested reader to [McCormick, 1987] and [Trottenberg et al., 2001] for details. Often there is no obvious way to construct a coarse version of the problem by geometrically coarsening the mesh, then applying the discrete differential operators to the coarsened mesh. In our case, this is because grouping together tetrahedra will generally not create larger tetrahedra. A similar problem occurs for regular hexahedral grids in the common case when the simulation domain can not be represented using only cells of the largest size (the coarsest level). We therefore use algebraic multigrid, an extension of the multigrid method which is called so because it uses only the matrices to construct coarser versions of the problem instead of geometrically coarsening the problem. Algebraic multigrid works by considering variables i and j to be neighbors (similar to the geometric sense of the meaning) if entry A ij is non-zero. This neighbor information is used to form the prolongation, P, and restriction, R, operators which allow the transfer of values from one level of the problem to another. P interpolates variables from one system to the (4) Sim Mesh SLC Tets(M) Tris(M) Chutes Spray(T) Spray(NT) Angel Dam Break Volcano Table 1: This table shows a break down of simulation times for our examples. Sim denotes the time used for numerical integration and pressure correction. Mesh is the time required for mesh generation. SLC is the time used by the surface tracker. All times are number of second for one frame of animation. The last two columns are the average numbers, in millions, of tetrahedra in the simulation mesh and of triangles in the mesh used by the surface tracker. The (T) and (NT) annotations indicate the spray example with thickening and with no-thickening. next finer level and R downsamples variables to the next coarsest level. With these operators a coarse (k + 1) solution estimate can correct the finer (k) solution estimate by ˆx k = P k x k+1 and the coarser version of the problem is found by A k+1 = R k A k P k and b k+1 = R k b k. We use the popular Galerkin projection method which sets R = P T. To form P first we classify variables of level k as either coarse or fine where coarse variables are those that will exist in the k + 1 th level (the coarser system). Then P is a matrix that interpolates variables on the k th level by taking an average of its coarse labeled neighbors. Therefore, each variable need have at least one coarse labeled neighbor (including itself). The optimal labeling will classify as few coarse variables as possible so that the coarser system is as small as possible. This is the NP complete minimum dominating set or ice cream stands problem from graph theory where the variables are nodes of the graph and edges are determined by non-zero entries in A. We find that the following non-optimal greedy algorithm for variable labeling achieves a good balance between simplicity and effectiveness. 1. Pick a variable at random, label it as coarse. 2. Label the neighbors of the new coarse variable as fine. 3. For all unlabeled neighbors of the new fine variables increment a counter of their total number of fine neighbors. 4. Select the unlabeled neighbor of the new fine variables with the largest number of fine neighbors as the next coarse variable. 5. Repeat steps 2 4 until all variables have been labeled Although the multigrid scheme we are using is not novel, we are not aware of it s prior use for fluid animations. The simulations we used contained between 0.8 million and 1.3 million tetrahedra. For these systems our algebraic multigrid solver was between 1.8 and 2.1 faster than a preconditioned conjugate gradient solver. We also tested on a larger 5 million tetrahedra mesh, that was not used for one of our fluid simulations, and observed a 3 speed up over preconditioned conjugate gradient. We expect that this multigrid solver would also be useful for fluid simulations on other discretizations and that the performance benefit will grow for larger systems. 6

7 Online Submission ID: papers 0509 Figure 6: This sequence shows an initial fluid configuration in the shape of an angle on a square base. Once the simulation begins the structure collapses creating a complex splash pattern. Figure 7: In this sequence a block of liquid is released at one side of a closed container. The resulting wave of water flows over an obstacle and crashes against the far side of the container. 4 Results and Discussion We have tested this simulation method by using it to animate several example scenarios. These example animations appear in the supplemental material for this paper. Our implementation was done in C++, and the examples were run single-threaded on Intel-based workstations running at between 2.8 and 3.2 GHz with 3 or 4 GB of memory. All renderings were done using the open source render, pixie1. Table 1 summarizes timing results. In general, the cost of pressure projection and integration was roughly equal to the cost of mesh generation. In all cases surface tracking dominated the overall cost. The images in Figure 1 show liquid flowing from two suspended sources down a pair of chutes and into a transparent basin. Flow from the left source begins after the right source has already run for a short time. The basin has a bulging bottom that disrupts the liquid s motion. In Figure 4 we compare results generated with and without our thickening scheme. A jet of liquid is sprayed against the side of a container. As the spray hits the wall it forms a thin sheet that rises up against the wall and folds over onto itself. Without thickening, large parts of this sheet vanish. Thickening corrects this problem creating a more plausible result. Figures 6 and 7 both show initial fluid configurations that when released collapse downward creating complex splashes. Some volume loss can be observed in the angel sequence when fluid is ejected upward from the center of the tank. This is an example where our thickening scheme is not completely successful. For that example, increasing thickening to the point were no visible volume loss occurs results in an implausible animation where small ejected structures spontaneously grow and form strange-looking fluid sheets. Our final example, shown in Figure 8, shows fluid flowing down a rough surface and accumulating in a small depression. The fluid wells up from a recessed source and creates flow patterns typical of real fluids. Figure 8: This example shows liquid rising from a recessed source, flowing down over an irregular surface, and the collecting in a depression. The image on the right shows a rotated closeup view of the pool of liquid forming in the depression. In general our simulations exhibit a high degree of fine surface detail. This detail is made possible by the relatively fine resolution used for both surface tracking and for simulation near the fluid s surface. The ability of tetrahedral meshes to focus detail where desired in one of their strong points and a strong motivation for using them over regular grids. In our current implementation we only vary the mesh resolution based on distance from the fluid s surface. However as pointed out in both [Losasso et al., 2004] and [Klingner et al., 2006] there are other useful refinement strategies that could be employed. Previous work in [Klingner et al., 2006] and [Chentanez et al., 2006] looked at how tetrahedral based gas simulations could be simultaneously coupled to simulations of solid bodies. We believe that these techniques should also work with this method. Although some of our examples still exhibit a small amount of volume loss, we are not aware of a liquid simulation method for animation that does not exhibit some type of related artifact. Our thickening scheme provides an adjustable compensation mechanism that allows the user to adjust a simulation s behavior

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