Smoothed Particle Hydrodynamics on Triangle Meshes

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1 Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2011), pp. 1 9 A. Bargteil and M. van de Panne (Editors) Smoothed Particle Hydrodynamics on Triangle Meshes paper 158 Abstract We present a particle-based method for solving Navier-Stokes equations on the surface of objects of genus 0 and 1. When the mesh is conformally mapped to a domain of constant curvature, the problem is greatly simplified due to the regularity of the new domain and the uniform stretching property of conformal maps. We adapt the Smoothed Particle Hydrodynamics method to work in this framework and we reformulate the momentum equation so that it takes into account the geodesic path taken by the particles. A particle-based method allows the simulation time to be independent of the underlying mesh resolution and offers a natural way to compute free-surface flows. We therefore achieve real-time simulation without artifacts caused by the parameterization of the triangle mesh. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Three-Dimensional Graphics and Realism Animation 1. Introduction Fluid flows on the surface of an object can be observed at very different scales. Clouds moving across the surface of the earth can be well approximated by a fluid flowing on a sphere. At a smaller scale, soap films are another example of liquid constrained to move on a curved domain. But the complexity of such phenomena can be simplified by considering that the fluid is moving on a curved 2D domain instead of a small subset of a full 3D Euclidean domain. Such simplifications allow us to remove a degree of freedom, simplifying equations and saving computation time. Mesh methods for solving Navier-Stokes equations on an object represented by a triangular mesh have been developed, but they suffer from being very expensive to compute and depend heavily on the underlying mesh. We propose a particle approach, namely the Smoothed Particle Hydrodynamics method (SPH), to solve these equations in real time while keeping the dependency of the underlying mesh to a minumum. The resulting method is very simple and requires only minor modification of a standard 2D SPH simulation to be suitable for curved objects of genus 0 and 1. The simulation uses the uniform stretching property of conformal mapping to simplify equations in the SPH formulation. By first mapping the mesh conformally to a regular domain, the computation of the forces is entirely done in the parameter domain and the particles can be easily transposed to the original object space for displacement and visualization. Note that a simplified method that only projects particles to the tangent plane is very unstable when simulating flow on surfaces other than spheres. It also makes the number of particles dependent of the maximum curvature of the surface. Our contribution is the use of particle-based real-time simulation for fluid flow over a surface, allowing the simulation to be practically independent of the complexity of the mesh. By taking into account the geodesic path of a moving particle, we remove distortions resulting from the parameterization of the mesh. A particle-based simulation allows for simple surface tracking, which offers new possibilities for other forms of visualization, like free surface flows, all in a single unified method. More generally, the method proposes a way to solve differential equations defined on a surface where functions are defined only on a discrete set of points. Examples of such applications includes texture diffusion, de-noising and deformation, reaction-diffusion textures, mesh de-noising and deformation. As done in statistics to approximate probability density functions on a manifold [Pel05], this new formulation of the SPH equations can be simply used to interpolate values sampled on a surface. The paper is organized as follows : Section 2 reviews existing work on conformal parameterization, SPH and flow on surfaces. Section 3 explains the offline computations that need to be performed before doing the simulation. Section 4 recalls the basics of the SPH method that works on Euclidean domains. Then, Section 5 extends the SPH method to work on non-euclidean domains, which is our main contribution. Results are then presented in Section 6 showing the indepensubmitted to Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2011)

2 2 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes to solve two linear systems. The resulting set of one-forms can then be optimized for some desired properties using the method described by Jin et al. [JWYG04]. Cutting the mesh along the homology basis, it can then be unfolded to a plane homeomorphic to a disk by integrating one-forms Smoothed Particle Hydrodynamics Smoothed Particle Hydrodynamics was introduced by Gingold and Monaghan [GM77] and Lucy [Luc77] but the first application of this method in the computer graphics community was made by Desbrun and Gascuel [DG96] for simulation of deformable bodies. Stora et al. [SAC 99] then used it to simulate lava flows, including heat transfer and nonconstant viscosity. Müller et al. [MCG03] used SPH to simulate fluid interactive time, including surface tension force. We have chosen to use this last formulation of SPH due to the simplicity of the formulas and the stability of the overall simulation. Figure 1: Free-surface flow over the skull and the Pensatore model. c) shows a quad-dominant mesh based on the particle distribution obtained with an anisotropic simulation and (d) shows a triangle mesh obtained with an isotropic simulation. dence of the mesh resolution and free-surface flows on mesh. Our conclusions are given in Section Related Work 2.1. Conformal parameterization Discrete conformal parameterization has been extensively studied in the past decade. We will concentrate on methods that map genus-0 objects to the unit sphere and genus-1 objects to the plane. Let us first enumerate sphere parameterization methods. Angenent et al. [AHTK99] and Haker et al. [HAT 00] describe a technique in which the object is first mapped to the plane, then stereographically projected to the sphere to obtain the conformal mapping. Gu and Yau [GY02], Gu [Gu03] and Gu et al. [GWC 03] describe a technique in which the Gauss map is taken as the initial approximation, then updated with a diffusion process to minimize an energy functional. The resulting map is then conformal. Li and Hartley [LH07] upgrade this method with a new initial map and a modified diffusion process that speeds up convergence. Global conformal parameterization of genus-1 objects is done by Gu and Yau [GY02, GY03] and Gu [Gu03] using differential one-forms defined on each edge of the mesh (d) 2.3. Surface flows A simulation of the Jupiter atmosphere was done by Yaeger et al. [YUM86] who used some heuristic to solve Navier- Stokes equations on a part of the planet. The simulation was done on a non-linear grid in the texture space to compensate for the deformation introduced by the mapping. They assumed that the space is nearly flat and that there is no vertical motion, thus rendering the method locked to this specific application. The first true fluid flow on surfaces was introduced to the computer graphics community by Stam [Sta03], who computed a local parameterization for each quadrilateral patch on which he solved the Navier-Stokes equations. Transition functions were defined between each pair of neighboring patches to transfer fluid properties across the entire mesh. Distortions are created by a misinterpretation of the advection term, resulting in a fluid that does not follow a geodesic path. Shi and Yu [SY04] did the simulation directly on the mesh using an adaptation of the semi-lagrangian method introduced by Stam [Sta99], eliminating distortion with parallel transport of the velocity vector in the backtracing step. Their method simulates incompressible and inviscid fluid flow by solving the Poisson equation and advecting fluid properties at each iteration. Lui et al. [LWC05] conformally parameterized the mesh to solve the Navier-Stokes equations on the parameter domain. The same misinterpretation of the advection term as in [Sta03] deviates the fluid from the geodesic path. Hegeman et al. [HMW 09] implemented a similar method on GPU by mapping the mesh to a conformal cube map. Elcott et al. [ETK 07] used the simplicial complex and the chain complex paradigm to translate equations into chain operations. Fan et al. [FZKH05] adapted the Lattice-Boltzmann

3 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes 3 method by using the tangent space at each vertex to project neighboring lattices on it. In computer animation with particles, Djado and Egli [DE09] used a particle system to visualize a velocity field resulting from a fluid simulation. The particles need some velocity correction to avoid interpenetration and two different steps are needed to do the simulation and the visualization. The simulation step is Eulerian and therefore also depends on the resolution of the mesh. The computational time of these simulations depends heavily on the complexity of the mesh, which makes them difficult to use for real-time applications. Particle simulation removes this dependency. In addition, it allows for new visualization techniques that would require additional processing steps using mesh simulation methods. We present a particle-based method for surface flows that can be used to solve more general PDEs on surfaces. The advection term is intrinsically considered as it is a langrangian method and the particles correctly follow geodesic paths. This results in a method that is fast, simple and practically independent of the underlying mesh. 3. Parameterization Particle-based simulation would be very hard to perform directly on the mesh because of the curvature. The solution that we propose is to transform the mesh to a more regular shape, like a plane or a sphere, via a parameterization process. Many different mapping techniques exist. A subset of these techniques, called conformal maps, uniformly stretch an infinitesimal neighborhood of each point of a surface. This ensures angle preservation and avoids shearing transformation, which are very nice properties to have when working in the parameter domain. Conformal maps are preferred over exponential maps defining local isometries because the latter approach requires too many different maps to cover the mesh efficiently, whereas the former only needs one global map. When expressed in a conformal domain, Navier-Stokes equations only need to be slightly modified to remain valid in the new domain. The uniform stretching also ensures that a small circular neighborhood is kept circular in the new domain, which will be very useful for the neighbor search process of SPH. We will only use global mapping, since it removes the need to do a mesh segmentation and compute patch transition functions. Genus-0 surfaces are mapped to a sphere while genus-1 surfaces are mapped to a periodic plane. We will first describe what we get out of the mapping process, then compute some metric information about these mappings that is needed during the fluid simulation. It is to be noted that everything in this section is done offline. For what follows, M is the original mesh with vertices {v i }, edges {e i j } and triangles { f i jk }. M is the parameter domain mesh with vertices {v i}, edges {e i j} and triangles { f i jk } Genus-0 surfaces Since a genus-0 surface is homeomorphic to a sphere, a bijective map can be computed between the two surfaces. This is done by the method described in [GWC 03] and [LH07]. Given a genus-0 mesh as input, the method gives us a new mesh M such that every vertex v i is located on a unit sphere and such that the mapping is conformal. The new mesh topology remains unchanged since it is a bijective map Genus-1 surfaces For genus-1 surfaces, we used the method described in [GY03] to map the surface to the periodic plane. The mesh M will be cut in a way that allows it to be unfolded to a flat plane, obtaining the fundamental domain. Given a genus-1 mesh as input, the method gives us a new mesh M such that every vertex v i is on a plane. The topology of the new mesh is modified as some vertices and edges are added because of the fundamental domain cutting operation. By identifying the duplicated data, we reobtain a mesh that is topologically equivalent to the original mesh. This method also gives us the two generators for a homology basis B 0 and B 1 of the surface. The basis {B 0,B 1 } is simply a set of edges representing loops that cannot be contracted to a point on the surface. This basis will be used to represent the periodicity of the fundamental domain in section Fundamental parallelogram For genus-1 meshes, the unfolding of the fundamental domain can send two neighboring points very far apart. The periodicity of the fundamental domain is thus an important concept to take into account when computing distances on it, see section 5.2. For this purpose, we will define two vectors forming the fundamental parallelogram [GY03]. These are a pair of independent vectors representing the shortest nonnull translations needed to get back to an equivalent point on the parameter domain (see Figure 2). This equivalence relation is expressed by Equation 1: two points are equivalent if they are separated by an integral linear combination of the fundamental parallelogram vectors b 0 and b 1. p p+i b 0 + j b 1, i, j Z (1) These vectors are simply computed by integrating edge

4 4 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes 4. SPH Basics SPH is an approximation method that defines a continuous function from an unorganized set of points. To each point in the set is associated a mass m j and a density ρ j. To approximate the function at a position r, a convolution is done with the surrounding points inside a radius h, the smoothing length. The value of a property A contained at each point is weighted by a the kernel function W and summed over a local neighborhood. Having defined a continuous function over the whole space, it is now easy to define the gradient and the Laplacian of any property A. The relevant equations are the followings (see [MCG03]). Figure 2: Periodicity of the domain. a) The mapped mesh. b) The fundamental domain. c) The periodic structure of the domain with fundamental parallelogram vectors joining equivalent points. vectors of the parameter domain over each generator of the homology basis using Equation Metrics ( bi = v k v ) j e jk Bi A mapping generally induces local deformations that need to be taken into account in doing computations on the parameter domain. A metric is a numerical mesure of the deformation that lets us perform computation as if we were directly on the surface. The metric is generally represented as a matrix. But since a conformal map only locally induces a uniform stretch, the metric can be represented by a constant of proportionality. This constant is called the conformal factor and is denoted by λ 2. It is generally different for every point on the object and it is the only value that will be needed to compute lengths and areas in the parameter domain. In the discrete setting of an object represented by a triangle mesh, a conformal factor will be computed for every vertex v i of M. Concretely, λ represents the ratio of a unit of length on the mesh to a unit of length on the parameter domain. They are computed once the parameterization procedure is done using the formula in Equation 3 from [JWYG04]. (2) λ i = 1 e i j n i e i j M e i j (3) where n i is the number of incident edges to the vertex v i D kernels ρ i = m j W( r i r j,h) j N( r i) A( r) = A( r) = A( r) = m j A j W( r r j,h) ρ j m j A j W( r r j,h) ρ j m j A j W( r r j,h) ρ j The kernel functions used in this paper are the same as in [MCG03], except for the viscosity kernel who is adapted to the two dimensional case. These kernels are chosen for their simplicity, speed of computation and stability. Density computation uses W poly6 while the pressure and viscosity computations use W spiky and W visc, respectively. { W poly6 ( r,h)= 4 ( h 2 r 2) 3, r < h πh 8 0, otherwise { (h r) 3, r < h 0, otherwise W spiky ( r,h)= 10 πh 5 W visc ( r,h)= 40 πh 5 where r = r. r3 9 ( + hr2 4 + ) h 3 6 lnh lnr 5 6, r < h 0, otherwise The coefficients in front of each kernel are necessary to ensure the unity property. They are only valid in 2D and thus are valid for our application on surfaces. The complex form of the viscosity kernel is only to ensure that the laplacian will have the following simple form. W visc ( r,h)= 40 πh 5 (h r)

5 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes 5 by some factor. Noting that a unit of length in the parameter domain is stretched by a factor of λ to be valid on the surface, we need to define the new radius by Equation 5. h j = h λ j (5) Figure 3: Particle distribution. A uniform particle distribution on the surface induces a non-uniform distribution in the parameter domain due to the distortions caused by the parameterization. 5. SPH on a surface The first thing to do in order to simulate fluid flow on the surface of an object is to adapt the Navier-Stokes equations. Since the particles will be moving on the parameter domain, the distortions caused by the parameterization must be considered (see Figure 3). They appear in the computation of the gradient and the Laplacian, and of the path taken by particles. From now on, we need to be careful with the properties we are manipulating, since there are two different spaces in which they can "live": the surface and the parameter domain. Properties that live on the parameter domain will be written A and those that live on the surface will be written à to avoid any confusion. The new set of equations (see [Ari89]) benefits from the nice properties of the global conformal mapping. ρ D u Dt = 1 λ2 p +µ 1 }{{} λ 2 u + }{{} F }{{} external pressure viscosity D Note that the operator means that the particles need to Dt be moved over the surface and not in the parameter domain. We see that the only differences with the original Navier- Stokes equations are the use of the conformal factors λ 2 and the consideration of the geodesic path. Information on metrics is only known at vertices of the mesh. Since a particle is necessarily contained in a face of the mesh, these properties can be found by doing barycentric interpolation of the values stored at each vertex Corrected formulas First, we have already pointed out that a conformal parameterization maps circles to circles of different radius. This means that we can still use a circular neighborhood in the parameter domain, but we need to change the smoothing length (4) where h is the global smoothing length, λ j is the stretch factor at the position of particle j and h j is the local smoothing length of particle j. We can also point out that if lengths are stretched by a factor λ, areas will be stretched by a factor λ 2. For this reason, the area occupied by a particle in the parameter domain is computed by Equation 6. V j = m j ρ j λ 2 j As we wish to do every computation in only one domain and Equation 4 is written in such a way that the differential operators are to be applied in the parameter domain, the new SPH formulation will be described by Equation 7. A( r) = m j A j ρ j λ 2 j ( W d( r, r j ), h ( 1 2 λ( r) + 1 )) λ j where d( r, r j ) means the vector representing the shortest straight path between r and r j. This subtlety will be fully discussed in Section 5.2. Given this new formulation, it is easy to define the density, gradient and Laplacian on the parameter domain using equations 8, 9 and 10. ρ i = 1 λ 2 i p( r i ) = 1 λ 2 i 1 λ 2 u( r i ) = 1 i λ 2 i (6) (7) m j λ 2 W( d i j,h i j ) (8) j m j p i j ρ j λ 2 W( d i j,h i j ) (9) j m j u i j ρ j λ 2 W( d i j,h i j ) (10) j where d i j = d( r i, r j ), u i j = ( u j u i ), h i j = (h i + h j )/2, p i j = ( p i + p j )/2 and p j = k ρ j, with k being a constant. Note that for liquid flow with free surface, the pressure is rather expressed by the Tait s equation 11. ( ( ) 7 ρ p j = B 1) ρ 0 (11)

6 6 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes Nb triangles Torus Kitten Rocker arm ms 6.80ms 5.95ms ms 6.45ms 5.81ms ms 6.41ms 5.74ms Table 1: Simulation time per step of 1000 particles on 3 models of different resolutions. Figure 4: Effect of the geodesic path. shows the paths taken by a particle with (in yellow) and without (in blue) considering geodesics while b) shows the same paths in the parameter domain Distance computation As explained in Section 3.2.1, the distance between two points on the global conformal domain needs to take into consideration the periodic structure of the domain. Due to this periodicity, multiple straight paths can be drawn between two points on the parameter domain, as illustrated in Figure 2. Only the shortest path is relevant to the computation and this is the path that will be used for the SPH formulation. For genus-0 surfaces, the mesh is mapped to a sphere. In this case, the Euclidean distance is a good approximation of the real distance and naturally gives the shortest distance. For genus-1 surfaces, the mesh is mapped to a periodic flat domain. We need to look for the closest translated copy of a particle p j around the particle position p i. This is done by translating p j by integer values of the fundamental parallelogram vectors defined in Section and taking the difference relative to the closest translated position of the particle. Mathematically, this path vector is defined by Equation 12. d( p 1, p 2 )= d i j, d i j = min d kl (12) k,l { 1,0,1} where d i j = p 2 p 1 + i b 1 + j b Geodesic path By definition, a geodesic is a curve that have no acceleration when projected in the tangent space of the manifold. So geodesic curves are the natural way to generalize straight lines on manifolds. The path of a free particle moving on a manifold without force acting on it will then follow a geodesic. This intuitive fact simply means that we need to transpose particles on the actual surface to move them. To transpose Nb Particles h=0.03 h=0.06 h= ms 2.73ms 4.08ms ms 8.54ms 14.28ms ms 30.30ms 52.63ms ms ms 250ms Table 2: Simulation time per step for 3 different smoothing radii h for a certain amount of particles on the fluid-filled torus. Simulations in bold were those who offered good stability and rapidity at the same time. the position of a particle and its velocity in the surface domain, we use the affine transformation induced by the triangle of the parameter domain in which the particle lies and the corresponding triangle of the mesh. Once the position and the velocity is expressed in the surface domain, we move the particle following the direction of the velocity. Each time the particle cross the edge of a triangle, the direction must be rotated about the edge axis so it is tangent to the incident triangle. Repeating this process until the path is completed, the particle is garanteed to remain on the surface and to follow the geodesic path. At this point, the new position and the transformed velocity must be transposed back in the parameter domain using the inverse of the affine transformation described above. Note that although this process introduced an explicit dependency of the underlying mesh, only a small percentage of the particles cross an edge of a triangle at each time step, resulting in a very small impact on the computation time. 6. Results The method is now applied to different situations, to confirm that simulation time is in fact independent of the mesh resolution, and to demonstrate free-surface flows. Various other examples will also be presented to complete the section. All results presented in this section were obtained with a 2.0GHz AMD Athlon 64 processor The resolution independence is demonstrated in Table 1 where the simulation time for a single frame is displayed for every resolution of a given mesh. Some simulation times are given in Table 2 for a given mesh for different number of particles and smoothing radii. Figure 5 demonstrates free-surface flows over a torus and

7 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes 7 a sphere. The gravitational force is transposed to the parameter domain and is applied to each particle. Surface tension is added using the simple formulation by Becker and Teschner [BT07] adapted to the formalism presented in Section 5.1. In this example, the non-realistic visualization is done by sampling the density of the fluid on each vertex of the mesh and using a marching squares-like method to extract the interface of the fluid. The simulations were done with 3000 particles. Figure 7 demonstrates higher resolution free-surface flows on complex objects. Using the same formulation, we rendered the fluid as if it was mercury flowing between two thin glass layers. The simulations were done with 7000 particles. Figure 1 also demonstrates free-surface flows using the same number of particles with different materials. The skull model is filled with a copper-like metallic liquid while the Pensatore model is hidden to show only the liquid flowing inside. Two examples of fluid-filled space simulation are shown in Figure 6. External forces are added by dragging the mouse over the mesh or over the parameter domain. The simulations were done with 1800 particles. The same simulation can be used to advect texture on the mesh as demonstrated in Figure 8. Figure 9 demonstrates that the method can be applied to other computer graphics problems using the same formalism. We show this by performing a mesh smoothing algorithm using the simple diffusion equation defined by p t = α p. As an additional interesting application, Figure 1 (c,d) and Figure 10 show that the method can be used to remesh objects. Indeed, the particle distribution obtained when the simulation is in steady state is isotropic (Figure 1 ) and is easy to mesh to get a equal-area triangle mesh independent of the parameterization. Different particle alignment can be computed using a generalized L p distance computation [?]. This anisotropic distance computation favors particle alignment that can be used for quad-dominant remeshing. The anisotropic distance must be computed relative to a particular direction using a orientation matrix M so that the length of a vector v is computed by M v p = ( (M v) x p +(M v) y p ) 1 p. Figure 10 (a,b,c) have been computed to be aligned with the gradient of the conformal factor and Figure 1 has been computed to be aligned with the x-axis of the parameter domain. We found that integrating the force directly to the position leads to a smoother simulation and that the smoothing radius of the simulation must be small enough for the alignment to be done correctly. Figure 5: Non-realistic free-surface flow over a sphere (a,b) and a torus (c,d). 7. Conclusion In this paper, we have presented a particle-based method for simulating fluid flows on the surface of an object of genus 0 or 1. The idea is to conformally map the triangle mesh to a constant curvature domain and solve the Navier-Stokes equations on the parameter domain. The uniform stretching property of conformal maps ensures minimal changes to these equations. The reformulation of the advection term lets the particles follow the geodesic path without distortions caused by the parameterization. Results show that the simulation time is independent of the resolution of the mesh, so real-time performance is achieved even with high resolution triangle mesh. The method was used to simulate free-surface flows by taking advantage of the natural surface tracking that a particle-based simulation offers. References [AHTK99] ANGENENT S., HAKER S., TANNENBAUM A., KIKI- NIS R.: Conformal geometry and brain flattening. In Proceedings of MICCAI 99 (1999), Springer-Verlag, pp [Ari89] ARIS R.: Vectors, tensors and the basis equations of fluid mechanics. Dover, [BT07] BECKER M., TESCHNER M.: Weakly compressible sph for free surface flows. In Proceedings of SCA 07 (2007), Eurographics Association, pp [DE09] DJADO K., EGLI R.: Particle-based fluid flow visualization on meshes. In AFRIGRAPH 09: Proceedings of the 6th International Conference on Computer Graphics, Virtual Reality, (d)

8 8 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes (d) Figure 6: Fluid-filled space simulation over a genus-0 object (duck) and a genus-1 object (rocker arm). Figure 9: Laplacian mesh smoothing of a torus (a,b) and a skull model (c,d). Figure 7: Realistic free-surface flow over the Pensatore model. Figure 8: Texture advection using the velocity field defined by the particles over the kitten model. (d) Figure 10: Quad-dominant meshing based on an anisotropic L p -norm of a sphere, a torus and the kitten model. Distribution of aligned points obtained with an anisotropic simulation (d). Note that the quads have practically the same area and is independent on the parameterization.

9 paper 158 / Smoothed Particle Hydrodynamics on Triangle Meshes 9 Visualisation and Interaction in Africa (2009), ACM, pp [DG96] DESBRUN M., GASCUEL M.-P.: Smoothed particles: A new paradigm for animating highly deformable bodies. In Proceedings of the Eurographics Workshop on Computer Animation and Simulation 96 (1996), Springer-Verlag New York, Inc., pp [ETK 07] ELCOTT S., TONG Y., KANSO E., SCHRÖDER P., DESBRUN M.: Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 1 (2007), 4. 2 [FZKH05] FAN Z., ZHAO Y., KAUFMAN A., HE Y.: Adapted unstructured LBM for flow simulation on curved surfaces. In SCA 05: Proceedings of the 2005 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (2005), ACM, pp [GM77] GINGOLD R. A., MONAGHAN J. J.: Smoothed particle hydrodynamics - Theory and application to non-spherical stars. Royal Astronomical Society, Monthly Notices 181 (1977), [Gu03] GU X.: Parametrization for surfaces with arbitrary topologies. PhD thesis, Harvard University, [GWC 03] GU X., WANG Y., CHAN T. F., THOMPSON P. M., YAU S.-T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Transactions on Medical Imaging 23 (2003), , 3 [GY02] GU X., YAU S.-T.: Computing conformal structures of surfaces. Communications in Information and Systems 2 (2002), [GY03] GU X., YAU S.-T.: Global conformal surface parameterization. In Proceedings of SGP 03 (2003), Eurographics, pp , 3 [HAT 00] HAKER S., ANGENENT S., TANNENBAUM A., KIKI- NIS R., SAPIRO G., HALLE M.: Conformal surface parameterization for texture mapping. IEEE Transactions on Visualization and Computer Graphics 6, 2 (2000), [HMW 09] HEGEMAN K., MICHAEL A., WANG H., QIN H.,, GU X.: Gpu-based conformal flow on surfaces. Communications in Information & Systems 9, 2 (2009), [JWYG04] JIN M., WANG Y., YAU S.-T., GU X.: Optimal global conformal surface parameterization. In Proceedings of VIS 04 (2004), IEEE Computer Society, pp , 4 [LH07] LI H., HARTLEY R.: Conformal spherical representation of 3d genus-zero meshes. Pattern Recogn. 40, 10 (2007), , 3 [Luc77] LUCY L. B.: A numerical approach to the testing of the fission hypothesis. Astronomical Journal 82 (1977), [LWC05] LUI L. M., WANG Y., CHAN T. F.: Solving PDEs on manifolds with global conformal parametrization. In VLSM 05 (2005), pp [MCG03] MÜLLER M., CHARYPAR D., GROSS M.: Particlebased fluid simulation for interactive applications. In Proceedings of SCA 03 (2003), Eurographics Association, pp , 4 [Pel05] PELLETIER B.: Kernel density estimation on riemannian manifolds. Statistics & Probability Letters 73, 3 (2005), [SAC 99] STORA D., AGLIATI P.-O., CANI M.-P., NEYRET F., GASCUEL J.-D.: Animating lava flows. In Graphics Interface, GI 99, June, 1999 (1999), Mackenzie I. S., Stewart J., (Eds.), Canadian Human-Computer Communications Society, pp [Sta99] STAM J.: Stable fluids. In SIGGRAPH 99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques (1999), ACM Press/Addison-Wesley Publishing Co., pp [Sta03] STAM J.: Flows on surfaces of arbitrary topology. In Proceedings of SIGGRAPH 03 (2003), ACM, pp [SY04] SHI L., YU Y.: Inviscid and incompressible fluid simulation on triangle meshes: Research articles. Comput. Animat. Virtual Worlds 15, 3-4 (2004), [YUM86] YAEGER L., UPSON C., MYERS R.: Combining physical and visual simulation creation of the planet jupiter for the film SIGGRAPH Comput. Graph. 20, 4 (1986),