Pattern-Guided Smoke Animation with Lagrangian Coherent Structure

Transcription

1 Pattern-Guided Smoke Animation with Lagrangian Coherent Structure Zhi Yuan Kent State University Fan Chen Kent State University Ye Zhao Kent State University (a) (b) (c) (d) Figure 1: Pattern-based fluid animation: high-quality animation is regulated by the flow dynamics pattern from low-resolution simulation. (a) Low-resolution simulation result; (b) High-resolution simulation with pattern regulation; (c) High-resolution simulation with partial pattern regulation on the smoke close to observer; (d) High-resolution simulation without regulation. Abstract Fluid animation practitioners face great challenges from the complexity of flow dynamics and the high cost of numerical simulation. A major hindrance is the uncertainty of fluid behavior after simulation resolution increases and extra turbulent effects are added. In this paper, we propose to regulate fluid animations with predesigned flow patterns. Animators can design their desired fluid behavior with fast, low-cost simulations. Flow patterns are then extracted from the results by the Lagrangian Coherent Structure (LCS) that represents major flow skeleton. Therefore, the final high-quality animation is confined towards the designed behavior by applying the patterns to drive high-resolution and turbulent simulations. The pattern regulation is easily computed and achieves controllable variance in the output. The method makes it easy to design special fluid effects, which increases the usability and scalability of various advanced fluid modeling technologies. 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; Keywords: Flow Pattern, Smoke Animation Design, Fluid Simulation, Lagrangian Coherent Structure, Finite Time Lyapunov Exponent 1 Introduction In computer graphics, extensive research has been conducted in fluid modeling for a long period, where direct simulation with the application of Computational Fluid Dynamics (CFD) methods has led to substantial success in creating realistic fluid phenomena. However, current fluid modeling technology still imposes great challenge on animators: the nonlinear flow dynamics is laborious to adjust for achieving desired fluid path and shape; and the expensive computational cost hinders their effort to design special effects in an interactive way. Therefore, advanced fluid design tools are direly needed by the animators which, ideally, can provide the functionality as a two-stage process: {zyuan,fchen,zhao}@cs.kent.edu Design Stage: Users design fluid behavior with multiple experiments of low-cost simulation. This procedure needs interactive adjustment of initial and boundary conditions, internal structures, and parameters; Output Stage: Once the expected fluid characteristics (such as major path and shape) are achieved, users can run highquality simulation to create final animation. However, unlike image or geometry objects, the transition from experimental design results to final outcome is not straightforward due to the inherent nonlinearity of fluids and numerical dissipation. Flow behavior greatly changes while simulation resolution increases or turbulence enhancement is exploited. The resultant fluid dynamics might not be what an animator prefers and has tested, which will greatly frustrate the animator considering his/her previous design efforts. Although researchers are actively presenting advanced techniques in fluid modeling and simulation, unfortunately, few efforts have been made to support such a two-stage design protocol which can provide great convenience for animators. Lack of such design tools also hamper the usability of the advanced fluid techniques for a wider audience. In this paper, we propose a novel pattern-based fluid animation approach for advancing fluid modeling in the two-stage animation scenario. Our method focuses on regulating high-quality animation with pre-computed patterns extracted from low-cost simulation results after design experiments. We employ a fast-emerging fluid analysis technique, in particular the Lagrangian Coherent Structure (LCS), to represent flow patterns. LCS defines the dynamic fluid skeletons of major flow trends. It provides a geometric instrument for revealing intrinsic property of complex fluids, which is computed as the locally maximum regions of the finite-time Lyapunov exponents (FTLE). LCS has evolved to become one of the most exciting avenues of research in dynamical systems [Peacock and Dabin 2010]. It can represent structural features and material separatrices which are often hidden when viewing the vector field or trajectories. Recently LCS has been successfully used in physics, meteorology, and oceanology to study real-world fluid dynamics. It has also been introduced in topology-based visualization for visually analyzing fluid flows. In our approach, we use LCS to define fluid patterns thanks to its capability of structural characterization of the main streams. Using the patterns to drive high-quality simulation, the final animation is confined to the design. To the best of our knowledge, this is the first time FTLE and LCS are used in 1

2 Velocity Field FTLE LCS Design Stage Output Stage Fluid Animation Design Guided Final Fluid Animation Low-cost Flow Simula on GPU Accelera on Controlled Thinning High-quality Flow Simula on Figure 2: Overview of our method. driving fluid simulation and animation. In detail, FTLE measures the rate of separation of very close particles after a given time interval inside a fluid. A sequence of FTLE fields, which are evolving scalar volumes, are computed from the velocity fields created over time in a low-cost flow simulation. Such velocity fields are the satisfied results for animators after a comprehensive design process. The FTLE ridges present geometries that divide the domain into coherent regions. Such ridges, i.e, LCS, play a role as material boundaries in the material s transport inside the flow and record the major flow trends. Our method regulates a highquality animation by enforcing its velocities on the LCS region to follow the pre-computed velocities. Therefore, the animation is guaranteed to follow the mainstream flow characteristics, e.g., major shape and direction, which have been designed and preserved on the extracted patterns. On remaining regions, the high-quality fluid dynamics is allowed to liberally develop thus leading to preferred realistic details. Based on this scheme, the high-quality animation is geometrically maneuvered by the predesigned results. Animators can adjust parameters to achieve different high-quality results while make them consistent with the design. In particular, the patterns are controllable to achieve results with variations, which are implemented by computing and representing the LCS regions with different sizes. A thick LCS skeleton will force a large portion of the flow to follow the pre-computed shape, while a thin skeleton gives the fluid more freedom to develop and leads to less shape-controlled results. We employ a skeleton thinning algorithm to achieve such control. These patterns are pre-computed in the design stage, and furthermore the parallel nature of their generation is utilized by GPU acceleration. In graphical modeling, the high-quality fluid simulation is typically accomplished by increasing the simulation resolution and using the turbulence enhancement methods. Our approach facilitates deterministic fluid animation by extracting and employing the dynamic signature of flows. It can be combined with these modeling methods in fluid design applications. Fig. 2 illustrates the overview of our method. In summary, our major contributions in this project are: Promoting the two-stage process in fluid animation, in particular fast design and high-quality outcome, which gives animators a very comfortable tool; Proposing geometric structures representing major trends of fluids, as a new operative instrument for graphical tasks involving fluid flows; Innovating the use of Lagrangian Coherent Structure in fluid animation beyond its traditional domain of visualizing and analyzing real-world or experimental fluid data; Developing techniques on using the flow patterns in guided fluid animation with acceleration and effect control. 2 Related Work Fluid dynamics is typically modeled by the incompressible Navier- Stokes (NS) equation in computer graphics. The equation is solved on discretized grids with numerical simulation preferably in unconditionally stable schemes [Stam 1999; Mullen et al. 2009]. Modular bases are used to solve the equation in a reduced space with constraints [Wicke et al. 2009]. The energy loss from numerical dissipation may impair the simulation quality, which is tackled by vorticity confinement [Fedkiw et al. 2001], vortex particle [Selle et al. 2005], fluid-implicit-particles [Zhu and Bridson 2005], circulation preservation [Elcott et al. 2007], or high-order advection schemes [Kim et al. 2007; Selle et al. 2008], etc. The flow results achieved on different grid resolutions vary largely due to the intrinsic complexity of energy transport among different scales. A variety of techniques have been proposed to enhance simulation with special turbulence effects. Synthetic noises are used together with low-cost simulators [Kim et al. 2008; Narain et al. 2008; Schechter and Bridson 2008; Pfaff et al. 2009], among which noises are integrated with simulated velocities considering Kolmogorov energy cascade theory. The random agitation is also added to simulations through special carrier particles [Pfaff et al. 2010; Chen et al. 2011]. These methods add good subscale details, whereas large-scale noises could induce strong deviation from the simulation results which may not be satisfied. These pursuits of advanced simulation and turbulence enhancement create realistic fluids with abundant details. However, the advanced simulation methods typically require high-cost computation and the turbulence integration involves challenging parameter adjustment. It is hard for animators to use them in a repetitive design task. Our method facilitates using these methods after easy animation planning with low-cost experiments. Fluid control has been studied in the literature. Extra forcing is a direct mechanism to perform the task, which drives fluids towards a predefined guiding object or simple path [Fattal and Lischinski 2004; Shi and Yu 2005; Thürey et al. 2006; Kim et al. 2006]. Advected radial basis function carried by moving particles provides an editable medium to modify the flow [Pighin et al. 2004]. Potential field is distributed on a shape to apply guiding forces, while the geometric shape is simply userdefined [Hong and Kim 2004]. Keyframe control and adjoint methods are used to drive the simulation at a sequence of keyframes with user-defined path/shapes [?;?]. Moreover, vortex filament and ring are used to control and edit artistic smoke behavior [Angelidis and Neyret 2005; Weißmann and Pinkall 2010]. Nielsen et al. [Nielsen et al. 2009] propose the optimization that allows a high-resolution simulation to follow low-resolution veloc- 2

3 (a) Velocity streamline (b) Forward FTLE (c) Backward FTLE (d) LCS with Hessian (e) LCS with threshold Figure 3: Flow pattern with FTLE and LCS. (a) Red: upward velocity. Green: downward velocity; (b)(c) Red: high FTLE value; Blue: low FTLE value; (d)(e) LCS region from (c). ity fields, which is enforced in the whole domain. They extend the original algorithm and use density erosion in the proportional control to provide efficient time-varying simulation and increase fluctuation [?]. The local density erosion adds variations to the simulation results but the paper states that a large stencil of density erosion will impair the control ability. The method does not consider the distinct importance of locations to adaptively add turbulence. This is different from our approach which utilizes critical geometric regions of major flow patterns for inhomogeneous control. Our method makes sure that the smoke follows major flow paths, and in other areas turbulence effects can be flexibly achieved. In our examples, we use the most straightforward forcing method to guide flow velocities. Our approach does not compete with these control techniques, since our focus is the extraction and representation of flow patterns, instead of control algorithms. Indeed, the guiding effects of our patterns can be applied together with the various control algorithms such as the density-based control. The application of FTLE and LCS has been used for studying time-dependent dynamical flow systems [Haller 2001; Peacock and Dabin 2010]. The technique has been successfully applied in topology-based flow visualization [Sadlo and Peikert 2007; Garth et al. 2007; Ferstl et al. 2010], where the structures are discovered to identify critical features of experimental or real flow data sets, giving insights of the dynamic processes. Based on these previous researches, our project initiates using FTLE and LCS in fluid animation missions. We anticipate the new technique can further develop into a powerful fluid modeling tool in computer graphics. 3 Flow Pattern A key challenge in fluid study is to find the representative pattern of a fluid flow so that such a pattern acts as an operative vehicle for flow manipulation, such as guiding high-quality animations. One major impediment is that the flow is dynamically-evolving. Another one is the very complex momentum variation on a wide range of spatial scales. While the Kolmogorov theory has been used to describe the dynamic energy transport, its statistical description is not eligible for unveiling geometric structures over time. We use the structural extraction methods to discover flow patterns. 3.1 Finite-Time Lyapunov Exponent (FTLE) Lyapunov exponent has its roots in the theory of dynamical systems, which characterizes the rate of separation of infinitesimally close trajectories. Haller [2001] used finite-time Lyapunov exponent for identifying LCS over a finite time interval. Considering the motion of a Lagrangian particle inside a fluid domain, its trajectory can be described by an ordinary differential equation: dp(t) dt = u(p(t),t), (1) wherepis the position of the particle at timet, anduis the velocity. In the parlance of dynamical systems, the trajectory that takes a particle forward T units in time from its initial position defines a flow map: Φ t 0+T t 0 (p) := p(t 0 +T). (2) It depends on the initial time,t 0, and the integration period,t. The flow map provides a way to compute the amount of local stretching, which is measured by the Cauchy-Green deformation tensor: ( := Φ t 0+T t 0 ) T ( (p) Φ t 0+T t 0 ) (p), (3) where () T denotes transpose of a matrix. is a 2 2 matrix for 2D flow or 3 3 matrix for 3D flow, respectively. It is computed on each grid site of a discrete grid on the fluid domain. The implementation is described in Sec has positive eigenvalues since the matrix is positive definite. The eigenvalues measure the rate of separation of the underlying flow at a locationp. In particular, the FTLE value is defined as a time-dependent scalar using the maximum eigenvalue λ max: σ T(p,t) = 1 T log λ max( ). (4) 3.2 Forward and Backward FTLE In Eqns. 2-4, the time interval T indeed can be either positive or negative. The Lagrangian particle thus moves forward or backward over time along its trajectories, respectively. For T > 0, the FTLE measures forward trajectory separation and the associated LCS (i.e., local maxima of FTLE) represents repelling surfaces (stable manifold) in the flow. If T < 0, the separation is evaluated backward in time and the resulting LCS acts as attracting surfaces (unstable manifold). Fig. 3b and Fig. 3c display a forward FTLE and a backward FTLE field, respectively. Note that the FTLE fields are dynamically evolving and these are the snapshots at a moment. The velocity field at that moment is shown in Fig. 3a. Haller and Sapsis [2011] addressed that the attracting LCSs form the backbones of forward-evolving trajectory patterns over the time interval [t 0,t], 3

4 (a) (b) (c) (d) Figure 5: 2D Pattern-based fluid animation. (a) Low-resolution simulation result; (b) High-resolution simulation with regulation after 10 thinning passes; (c) High-resolution simulation with regulation after 20 thinning passes; (d) High-resolution simulation without regulation. From the computed FTLE field, the LCS is identified as the local maxima of the field. Fluids are represented as such invisible repelling/attracting structures that outline the boundaries between different regions and reveal material transport pathways. Ridge extraction is an established topic in differential geometry. Second order derivative of the FTLE field is evaluated by the Hessian matrix, Γ = d2 σ T (p), whose smallest eigenvalue π dp 2 min and its related eigenvector n satisfy π min < 0 and (σ) n = 0 for the LCS ridge. Unfortunately, such an accurate LCS computation is very sensitive to small fluctuations in the FTLE field, and thus results in scattered small patches/points which impair the ability to identify the desired skeleton. The fluctuation comes from the sensitive nature of the particle trajectories to small velocity variations, and also from the inaccuracy of numerical integration. A practical algorithm of extracting LCS from the Hessian was proposed by setting the largest eigenvalue, π max < 10 3, of Γ in extracting 2D LCS [Ferstl et al. 2010]. A 2D curve-thinning approach was applied after such extraction to find the exact skeleton. Fig. 3d shows the LCS created from Fig. 3c with the largest eigenvalue threshold. The approach is good at continuous ridge extraction, but still results in many separated ridges. The shattered regions are not appropriate in our application: firstly, they introduce irregular and unnatural aliases in flow simulation when we apply flow guiding forces on them; secondly, it is hard to control different regulation levels in the fluid animation results, which is an essential feature of our approach. (a) 5 thinnings (b) 15 thinnings Figure 4: Control domain with LCS thinning from Fig. 3(e). acting as central structures on which nearby trajectories accumulate. In our application we want to make high-quality simulations confine to pre-computed trends, in other words, to attract the simulations towards the skeletons. Therefore, the attracting surfaces and hence the backward FTLE are used in our implementation. Fig. 3c displays the major characteristic of the flow above the ball. The integration time T is chosen depending on the amount of details needed in the resulting FTLE: a large T achieves smooth and largescale structures, but it should not be too large in which necessary vortex is ignored. We use T = 1 second in our examples. 3.3 Lagrangian Coherent Structure (LCS) 3.4 Thinning To address the problem, we apply the skeleton thinning method directly after a threshold-selection on the FTLE field without Hessian computation. First, we use a threshold (k) to find LCS region (σ T(p) > k). This method may miss some LCS ridges with a very small FTLE value which are related to small fluctuations. It is appropriate since the mainstream guidance usually does not focus on the small details, and very small fluctuations can introduce aliases in animation. The resulting LCS region is a portion of the simulation area, which defines the control domain where the guidance of animation takes effect. Fig. 3e shows the result with such threshold extraction from Fig. 3c. In comparison to Fig. 3d from the Hessian-based computation, it achieves smooth and integral LCS control domain. The threshold k plays a role in achieving different results. Using a small k leads to a large control domain and hence strong fluid guiding effects. k = 0.1 is used in our examples for a normalized FTLE range between [0, 1]. This initial control domain can be pared by applying a 3D thinning algorithm [Lee et al. 1994]. Skeleton thinning preserves the geometric feature which is better than directly using a large k. The thinning algorithm is iteratively applied with n thin passes which is selected by users, so as to achieve a smaller control domain and hence a desired level of animation regulation. Fig. 4 shows two control domains with different sizes after the iterative thinning from Fig. 3e. Thinning 15 times retains a smaller major flow region than thinning 5 times. 4

5 To appear in ACM TOG 30(6). (a) (b) (c) (d) Figure 6: Pattern-based fluid animation on a moving ball simulation. (a) Low-resolution simulation result; (b) High-resolution simulation with regulation; (c) High-resolution simulation with regulation after 8 passes of thinning; (d) High-resolution simulation without regulation Pattern-driven Fluid Animation Implementation From a sequence of low-cost simulation results on a coarse grid, the FTLE is computed numerically at each grid point at a time t. In a 2D domain, for each grid point (i, j), four particles are positioned at pl = (i τ, j), pr = (i + τ, j), pd = (i, j τ ), pu = (i, j + τ ) with a small perturbation τ (e.g. 0.1 is used in our experiments for a unit grid interval). The four particles are back traced in the velocity fields for a period of T, and their stopping positions p l, p r, p d, p u are used to numerically compute Eqn. 3 where x(p ) x(p ) x(p ) x(p ) r Φtt00 +T (p) = l u High-resolution Simulation: Dynamic flow patterns extracted from a low-cost fluid simulation, SL, are used to drive a highquality animation. At a time t, the pattern represented by the LCS control domain, Ω(t), determines where a high-quality animation should follow the predesigned velocities. A high-resolution flow simulation, SH, is influenced by applying special guiding forces in Ω, which confines its velocity to SL. The guiding force is computed at a point p as ( 1 c δt (u SL (t) ush (t)) if p Ω(t); F(t) = (6) 0 if p / Ω(t). d 2τ 2τ y(p r ) y(p l ) 2τ y(p u ) y(p d ) 2τ (5) Here, u SL is the upsampled velocity on the high resolution grid. A unitless constant c is a scaling parameter which defines the strength of the guiding force. It is chosen empirically to yield an appropriate control effect (e.g. c = 1.5 in our 3D examples and c = 2.0 in 2D examples). Note that simply adjusting c is not preferable to LCS thinning. Specifically, decreasing c might make the different control effects at different regions less distinguishable, while increasing thinning times will create smaller control regions and flows will be allowed to freely evolve in larger areas. Note that the two-stage design still requires the user to perform parameter modification iterations at the high resolution. Even so, our method reduces this effort to only adjusting the thinning times and the constant c. The FTLE value σt (p, t) is then achieved using Eqn. 4 by calculating the maximum eigenvalue. In a 3D domain, six particles should be positioned in a similar way. The trajectory tracing is imt plemented within δt steps, where δt is the time step size in an NS simulator. For example, in Fig. 3, δt = 0.1 seconds and T is 1 second. We adopt a fourth-order Runge-Kutta integration scheme in the tracing and use linear interpolations in the computation. Further enhancement using smaller particle moving step size or highorder integration and interpolation methods can increase the accuracy, which has been used in some approaches of real flow analysis and visualization. However, the computational cost also grows greatly. In our practice, the fluid control tasks in animation can be satisfied with the current computational scheme and hence we avoid the time-consuming computation. Turbulence Enhancement: Besides using high-resolution simulations for final animation, noises can be integrated with a pre-computed velocity field for turbulence enhancement in a post-processing stage without solving the NS equation (e.g., [Kim et al. 2008]). Such methods successfully add small-scale details to an existing flow, whereas a large vortex scale of noises will result in diverted and unnatural flow behavior. We also use the LCS control domain to adjust noise integration to further leverage the potential of these methods. Noise-based fluctuation is only added out of the control domain, so that the major flow trend is preserved when a noise field with larger scales is used. The FTLE fields are time-varying volume data in 3D. In generation, multiple particle tracings are required for each grid point. Fortunately, this is a pre-processing computation on a low-resolution grid. Moreover, the FTLE computational algorithm is explicit and embarrassingly parallel, which is implemented on GPUs to further increase the performance. The computed FTLE fields are stored after the design stage. In practice, we only need to compute and store FTLE fields every β steps (e.g. β = 5), since they change gradually. In the output stage they are retrieved and linearly interpolated to the compatible high resolution domain and every time step. Then, LCS is computed with nthin passes of thinning for high-quality animation (Sec. 4). Such a pattern-driven simulation imposes fluid control on LCS pattern domains. This is the key difference of our approach from previous fluid control methods, which have focused on different methods of applying control to fluid simulation, such as optimization 5

6 To appear in ACM TOG 30(6). (a) (b) (c) Figure 7: Pattern visualization of the Fig. 1 example. (a) 4 thinnings; (b) 8 thinnings; (c) 12 thinnings. (a) [Nielsen et al. 2009;?], detail-preserving [Thu rey et al. 2006], etc. But they did not study the intrinsic structural features of fluids and the corresponding spatial regions to apply their control. Indeed, the new scheme complements these previous approaches. Our method can be combined with different control methods besides using Eqn (b) (c) Figure 8: Pattern-based fluid animation on vortex particles. (a) Low-resolution simulation result; (b) Adding vortex particles with regulation (4 thinnings); (c) Adding vortex particles without regulation. Results and Performance Several examples are implemented on a workstation with two Intel Xeon E GHz Quad-Core CPUs and 12 GB memory. We use a stable solver with MacCormack advection [Selle et al. 2008]. The length scale of the simulations is related to the grid resolution and the physical size of the domain. Here, smoke density volume is advected along the flow field. All these example animations are compared in the supplemental video. (a) Fig. 5 is a 2D example of fluid animation using our method. Fig. 5a is a low-resolution simulation snapshot with a grid. For better illustration, we assign two different colors to the smoke on the left and right side of the ball. In Fig. 5d, a high-resolution grid is used. However, this configuration of high-quality animation makes Fig. 5d totally different from Fig. 5a at the same time step. The high-resolution simulation has distinct characteristic shapes than the low-resolution one. To make the final output consistent to the low-quality simulation, we extract LCS patterns and apply regulation. Fig. 5b and Fig. 5c display the snapshots using 10 passes and 20 passes of thinning, respectively. Their shapes are similar to Fig. 5a with two different levels of details. (b) (c) Figure 9: Pattern-based fluid animation on turbulence enhancement with wavelet noise. (a) Low-resolution simulation result; (b) Adding noise with regulation (8 thinnings); (c) Adding noise without regulation. 1b illustrates detailed smoke but still follows the design. Furthermore, we choose to only apply guidance on half of the LCS control domain. Fig. 1c shows that the smoke close to the observer still follows the design shape, while the smoke far away to the observer evolves more freely. This example indicates that the geometric pattern can be further edited and adjusted for spatially maneuverable fluid animation. In Fig. 6, a 3D flow simulation with a moving ball is performed on a grid. Fig. 6a shows the basic flow shape in the low-cost design. A four times larger grid is then used for creating final animation and a small amount of vorticity confinement [Fedkiw et al. 2001] is added to achieve turbulent results shown in Fig. 6d. However, the smoke behavior deviates from our satisfied design. It also evolves faster in part due to the extra confinement energy. In the two stage design scenario, this transition difference hinders an animator to use the turbulence enhancement technique. Fig. 6b is the result applying pattern guidance with a large LCS control domain (0 thinning). The snapshot has detailed smoke behavior and the shape agrees with Fig. 6a. Applying 8 passes of thinning, Fig. 6c shows more turbulence and details. In Fig. 8 we show our regulation result used in a fluid simulation which is fluctuated by the vortex particles [Selle et al. 2005]. Vortex particles apply vorticity forces to a high-resolution simulation ( ) as in Fig. 8c. The enhanced flow shows scattered smoke rotations, which is not like the original simulation in Fig. 8a ( ). If the original shape is preferred, Fig. 8b is the confined result after applying the LCS patterns. We also apply our method for noise based turbulence. Kim et al. [2008] successfully added sub-scale wavelet noises to low-cost simulation results. However, when the spatial scale of the noises (i.e., the size of the added vortices) increases, the flow will be greatly diverted from the designed path and behavior, which will lead to unnatural final results. Fig. 9c shows the strong fluctuation result, in comparison to the original flow in Fig. 9a. In contrast, Fig. 9b displays the turbulent smoke with its shape similar to Fig. 9a. In Fig. 1, smoke flows past two internal objects. When a grid is used in the design stage, Fig. 1a shows the designed simulation snapshot. Once patterns are extracted, a grid is applied for high-quality output. Fig. 7 visualizes three thinning results of the patterns at a step. The finer simulation, enhanced by the vorticity confinement, creates good smoke effects (Fig. 1d). However, the result abandons the predefined smoke behavior. We thus apply our pattern based method after 4 thinning passes. Fig. Table 1 reports the system performance. When using a highresolution simulation, the average simulation time (including density advection) per frame greatly increases, which depicts the necessity of the two-stage animation design. Besides the memory usage by the fluid solver, the FTLE stores a scalar floating-point 6

7 Table 1: Performance Report (in seconds). Examples Low-cost Simulation Low-cost Ave. Simulation FTLE Ave. Time Per Frame Thinning Thinning High-quality Simulation High-quality Ave. Simulation Resolution Time Per Frame CPU GPU passes time Resolution Time Per Frame Fig Fig Fig Fig Fig value at each low-resolution grid site. It is up-sampled to the desired high-resolution and used to compute the LCS region represented as a grid with binary values. The total extra memory used by the FTLE/LCS computation is around 10% more than the classic fluid solver. Computation of 3D FTLEs using fourth-order Runge- Kutta integration and trilinear interpolation is slow on the CPU, but this computation is done only on the low-resolution grid and is highly parallel. We thus compute low-resolution FTLE fields with GPU acceleration on an nvidia Tesla C1060. The speedup factor of GPU/CPU is great due to the parallel nature of the algorithm. For example, on a grid, the GPU computation is 150x faster than the CPU version for Fig. 1 and 64x faster for Fig. 9. The discrepancy between the two factors is due to the different scenes and flow behaviors. Please note that the computation is only applied every β = 5 steps. The thinning algorithm is also very fast compared to the high-quality simulation. The guiding force computation and feedback are very trivial in computational requirement. The wavelet noise method uses noise integration instead of a NS simulator in creating the high-quality animation. 6 Conclusion and Future Work We have exploited the emerging fluid analysis techniques, FTLE and LCS, to promote easy fluid animation. After users design the fluid animation in low-cost, and hence fast and interactive, numerical experiments, the extracted flow patterns are integrated with high-quality simulations to provide final animation output that is consistent to the design. One limitation of the method is that if the LCS control is set too low on a long simulation, the results may appear unconstrained later in the simulation due to the accumulated fluctuation of energy. Thus, we plan to study how to adaptively adjust the control effects along time to further enhance the method. Also, we currently use the LCS region as a mask (defined by discrete grid cells), but using other geometric representations such as meshes or dynamic implicit surfaces may allow improved control techniques or increased analytic power. In addition, the use of runlength encoding (RLE) can also help to save memory of the LCS regions. Lastly, the FTLE and LCS parameters need empirical selection. The guiding force algorithm can also be deliberately adjusted. We plan to utilize other existing control methods to couple the patterns in simulations. Acknowledgment This work is partially supported by National Science Foundation grant IIS We thank the reviewers for their insightful reviews and suggestions. We also thank Michael Nielsen and Sean Reber for the help in preparing the paper. References ANGELIDIS, A., AND NEYRET, F Simulation of smoke based on vortex filament primitives. In Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation, ACM, New York, NY, USA, CHEN, F., ZHAO, Y., AND YUAN, Z Langevin particle: A self-adaptive lagrangian primitive for flow simulation enhancement. Computer Graphics Forum 30, ELCOTT, S., TONG, Y., KANSO, E., SCHRÖDER, P., AND DES- BRUN, M Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 1, 4. FATTAL, R., AND LISCHINSKI, D Target-driven smoke animation. In SIGGRAPH 04: ACM SIGGRAPH 2004 Papers, ACM, New York, NY, USA, FEDKIW, R., STAM, J., AND JENSEN, H Visual simulation of smoke. 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