A Multi-Resolution Interpolation Scheme for Path-Line Based Lagrangian Flow Representations

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1 Vision, Modeling, and Visualization (2014) J. Bender, A. Kuijper, T. von Landesberger, and P. Urban (Eds.) A Multi-Resolution Interpolation Scheme for Path-Line Based Lagrangian Flow Representations Paper ID: 1014 Abstract The computation of trajectories in vector fields is typically performed in an iterative fashion through numerical interpolation. If the field is given as a set of trajectories the Lagrangian representation this integration can be replaced by interpolation. We present a novel re-organization of such Lagrangian representations by subsampling a flow map into multiple levels of resolution, maintaining a bound over the amount of memory mapped by the file system. The base resolution facilitates a real-time, low memory cost, interactive environment for flow field exploration. Once an area of interest is selected, additional trajectories from other resolution levels are dynamically loaded in the vicinity of the desired feature, providing for a low memory footprint. Through path- and time-surface extraction, we show that interpolation outperforms traditional advection and that dynamic trajectory refinement leads to results whose accuracy converges to that of features extracted through numerical integration. 1. Introduction Flow fields are the output of some of the most widely used simulations to date. In the context of Computational Fluid Dynamics (CFD) simulations, they are used to represent material transport in a wide range of application fields such as airplane engineering or climate simulations. The ability to trace individual particles present in the flow domain allows the visualization of material transport during post-processing. Given the classic representation of flow fields as static meshes with fluid velocities given per time step, the so-called Eulerian representation, this tracing requires the computation of solutions to the underlying discretized differential equation known as numerical integration. Due to the unpredictability of particle movement, it is difficult to determine which sections of the vector field data will be needed as the particle is advected through the field. Solutions that load the entire mesh, or load data on demand, have to take special care not to exceed memory (large amounts of data) and time (large amounts of I/O operations) constraints [CGC 11]. Alternatively, flow fields can be represented in a Lagrangian way, where the discretization is no longer governed by a stationary mesh. This representation corresponds to a dense sampling of the flow map, which encodes the advection of massless particles under the influence of the vector field over a given time interval. Such data can be, for example, output of modern CFD simulation tools (cf. Smooth Particle Hydrodynamics [Mon88]), or the output of another post-processing technique such as Finite-Time Lyapunov Exponent (FTLE) computations [Hal01]. The availability of a full Lagrangian representation in the form of a set of dense particle trajectories opens up new possibilities for integration-based flow visualization. In order to trace an arbitrary particle through the flow field, one no longer has to perform numerical integration, but can interpolate within the given set of flow trajectories. The predictive nature of interpolation, combined with the spatial coherency properties of continuous flow, allows for the use of a local interpolation neighborhood to extract the trajectory over all time steps. This locality assures a bound on the information needed to perform interpolation, with the advantage of reducing I/O to only those trajectories needed for the interpolation neighborhood. We demonstrate how such a particle trajectory representation can be used to construct adaptive integral surface representations, and compare the results to traditional integrationbased surface generation with respect to matched accuracy, improved speed, and a reduced memory footprint. The advantages of a Lagrangian representation of fluid flow are further exemplified by a real-time, interactive data set exploration technique. By segmenting the flow map representation into multiple resolutions, the coarsest resolution allows for real-time surface reconstruction, while finer resolutions refine a specific surface with on-demand tracer loading. This technique gives users the ability to explore the entire flow data set for a desired surface, and once located, the surface is

2 refined to the best possible accuracy (defined by the total set of trajectories). Only those trajectories needed to reconstruct any given surface accurately are loaded in, thereby lowering the memory footprint of surface reconstruction. In summary, the main contributions of this work to the field of visualization are: A multi-resolution flow map organization for dense particle tracer sets. Efficient interpolation of arbitrarily seeded trajectories. Accurate and efficient extraction of integration-based flow geometry at multiple levels of detail, allowing real-time flow field exploration. Location specific I/O for a reduction in memory cost during integral surface representation. 2. Related Work Integral surfaces, as created from sets of trajectories or particle positions, are a tool capable of visualizing complex flow behaviors. Typically, the computation of underlying trajectories makes heavy use of numerical integration techniques such as Euler integration or higher-order adaptive Runge- Kutta Methods [CK90]. Recent surface generation methods [MLP 10] have the advantage of adaptive resolution and accuracy control [Hul92, GKT 08, KGJ09] but require high numbers of computationally expensive and iterative evaluation of vector fields. Additionally, surfaces with comparatively low accuracy and constrained adaptivity may also be computed on the GPU [BFTW09]. To increase computational efficiency, different approaches such as stream function generation with subsequent isosurface extraction [vw93, WJE00], or path-volume construction from integrated time-surfaces [OHBH13], or hierarchical construction of the integral curves [HSW11] have been taken in the past. Other flow structure construction techniques [SWS07] make direct use of sample trajectories as well, but do not aim at the creation of adaptive integral surface structures. These approaches do not, however, easily extend to 3D space + time. In similar contexts, other research [WT10] formulates time-varying integral surfaces as tangent curves in high-dimensional spaces [TS03], significantly increasing pre-computation and memory requirements. Memory reduction has been achieved in the form of vector field sub-sampling and compression [LRR00] with topology conservation but it is difficult to estimate the impact on trajectory advection. A class of visualization techniques that makes heavy use of pre-sampled vector fields is the field of Lagrangian flow analysis with the help of Finite-Time Lyapunov Exponent maps [Hal01]. Here, densely sampled flow fields are the basis for the approximation of flow stretching [SP07, GGTH07]. These approaches are suitable for direct execution on massively parallel machines due to locality of trajectory integration. To a lesser degree, this locality can assist curve integration in parallel. Massively parallel stream line integration covering seeding strategies has been implemented and analyzed [CGC 11], with a strong focus on I/O, citing computational hardship due to block loading of the vector field because of the inability to predict a particle s position over all time steps prior to advection. Focusing directly on pre-sampled vector fields, our method restructures trajectory information to allow for dynamic loading of pre-determined blocks, eliminating I/O due to the position unpredictability of particle advection. With I/O limited to desired areas of interest, we are therefore able to present a method for path and time surface extraction that has real-time, interactive character even for large data sets. 3. Vector Field Representations A vector field is a continuous, vector-valued function v over a four-dimensional (space-time) domain of definition Ω [0,T ] R 3 that maps a point (t,x) to a vector v(t,x) R 3. A vector field is called stationary if v does not depend on t, and is time-dependent otherwise. Eulerian The classic representation of a vector field as a stationary mesh that holds vector data at its nodes for every time step of a simulation is called Eulerian. In this representation, the spatial discretization of the flow domain is generally either static in time, or does not reflect fluid transport. Lagrangian In a Lagrangian representation, frequently used in SPH and other mesh-free methods, the flow field consists of a collection of dynamically moving flow particles (tracers). In this representation, the frame of reference is locally moving with the flow field, i.e., the spatial discretization of the flow field is attached to participating material in the fluid transport, following the trajectories of individual particles. A comparison of both views is shown in Figure 1. Flow Trajectories The trajectory of a massless particle is identified by a point (t 0,x 0 ) it contains, commonly called its seed point. All subsequent positions, as obtained through advection, then form a pathline or trajectory. Mathematically, trajectories are solutions to the differential equation d I(t) =v(t,i(t)) dt (1) I(t 0 )=x 0 (t 0,x 0 ) Ω (2) Like the vector fields over which they are defined, trajectories typically possess a straightforward connection to physical aspects of a given application: e.g., in the case of fluid flow, integral curves describe the trajectories of massless particles embedded in a flow. Arbitrary flow trajectories may be reconstructed from both Eulerian and Lagrangian representations of the flow field. The former representation requires the use of numerical integration schemes, whereas the

3 φ t Eulerian: v( ) = Lagrangian: φ t( ) = Figure 1: Eulerian vs. Lagrangian descriptions. The Eulerian view describes a flow field in terms of the instantaneous velocity v of particles, while the Lagrangian view considers the evolution of particle positions. latter performs interpolation within a set of given trajectories. The work presented investgates and utilizes Lagrangian flow representations for efficient flow feature extraction. 4. A System for Efficient Path Line Interpolation The accurate reconstruction of trajectories in flow fields is heavily influenced by two key factors: spatial resolution of the underlying data and appropriate choice of interpolation scheme. We develop a system that investigates and utilizes the strengths of Lagrangian flow representations towards efficient multi-resolution reconstruction of pathlines. System Outline The Lagrangian nature of our data allows us to design a system that efficiently reconstructs pathlines through interpolation. The gained efficiency allows for an interactive flow field exploration environment, instantly presenting initial results while progressively improving them over time. The initial feature reconstruction relies on pathlines interpolated from a low resolution sampling of the flow map. Once a feature of interest is located, additional tracers are loaded to improve reconstruction accuracy. By organizing the available Lagrangian data into varying levels of resolution, only relevant tracers are located and used by the reconstruction algorithm. This process may be repeated any number of times during data set exploration, loading trajectory information only when requested. The central steps of our algorithm are captured in a flow diagram in Figure 2 and are as follows. 1. Pre-Processing: The Lagrangian representation is processed and written to files as sub-sampled flow maps consisting of multiple levels of detail. This file representation is designed to allow efficient local and on-demand streaming of trajectory data, following a hierarchy based on grid resolution. 2. Base Resolution: The flow map with the lowest available resolution is loaded and kept in memory for fast trajectory exploration and interpolation. 3. Efficient Interpolation: a. Interactive Extraction: During exploration, a preview version of the desired path- or time-surface is extracted in real time from the coarse level flow map rep- Figure 2: The process starts with an interactive exploration of a coarse resolution flow map. Once a feature of interest is located, the algorithm enters the refinement pipeline, loading additional tracers from finer resolutions. At this time, additional time step information may be loaded for the selected tracers. Once the seeding points are moved from the refined location, the process begins anew. resentation. For this matter, individual pathlines are interpolated from the existing set of flow map tracers. b. Spatial Refinement: On demand or after sufficient idle time has passed, higher resolution trajectory data is streamed from the file system. The existing surface representation is refined by taking novel trajectory data into account during pathline interpolation. The following section describes these steps in detail. 5. System Implementation We begin with an in-depth description of the data representation created by the pre-processing step, with a look at how tracer information is organized on the hard drive Organizing the Lagrangian Flow Representation If a small number of tracers is sufficient for reconstructing arbitrary pathlines under given accuracy constraints, loading the millions of tracers within a full resolution Lagrangian representation is an unnecessary burden. Therefore, the given Lagrangian representation is first pre-processed to place the tracers into multiple levels of resolution of varying density. Then, each resolution is segmented with a regular grid to allow for fast spatial location. Essentially, the entire Lagrangian representation is sub-sampled into smaller, separate, more computationally manageable representations. Each level of resolution breaks the data set up into a number of square voxels, or cells, of varying size, resulting in uniform grids of increasing resolution. An arbitrary time step τ of the vector field is chosen to represent the entire set of tracers. This means that a tracer s 3D position at τ is used to locate its designated cell. Beginning from the coarsest resolution, tracers are binned according to their tendency for

4 divergence, calculated as their FTLE value. This helps prioritize flow structures and material lines that play a key role in capturing representative structures. Binning continues until all tracers are exhausted and all resolution levels are filled. Resolution Specification The Lagrangian representation is broken up into multiple levels of resolution based on the data set dimensions, DIM xyz and the number of trajectories given, N total. The first resolution level to be created is the coarsest, providing a general subsampling over the entire domain. Each dimension is sampled by 100 cells, defining the cell width, CW c. If the data set does not have equal dimensions, the longest axis is represented by 100 cells, setting the resolution s cell width. The number of points within each cell of the coarse resolution, n c, is chosen to be 10, ensuring enough points in one cell to form a neighborhood for interpolation. To limit the amount of information mapped by the I/O, a value M is chosen where M represents the maximum number of points that should be mapped by the file system during pathline refinement. Setting a limit to the amount of information mapped at once assures that all resolutions store the same maximum amount of tracers. The number of levels, L, is defined by: N L = total N c (3) M where the numerator holds the amount of trajectories left to be stored outside of the coarsest resolution (N c being the total number of tracers in the coarsest resolution). Next, the cell size of each resolution is determined as CW i = CW c i where i is the current resolution level. With cell size known, the algorithm now determines how many tracers will be binned into each cell subject to the constraint, n i M (DIM x /CW i ) (DIM y /CW i ) (DIM z /CW i ) where i is the current resolution. The values n i will be used to sort tracers into the various resolution levels. There are two main objectives behind using multiple levels of resolution. First, sectioning the tracers into different levels lightens the load of the file system as resolutions are read in by the algorithm. Second, the varying density between resolutions helps ensure that loaded tracers may be used in the interpolation neighborhood of more than one seeded particle if a group of particles is seeded near one another. Binning of Trajectories For the purpose of locating a tracer in 3D, the input set of trajectories must be binned into the defined multi-resolution grids. Binning begins from the coarsest resolution to the finest, with the 3D position of a tracer at time τ used to determine which cell of the current resolution contains the tracer. A list of contained tracers is created for each cell, from which a subset is then selected. However, simply binning the n number of tracers can lead to an unbal- (4) Figure 3: Resolutions file layout. Resolutions are written in order with each resolution writing out its cells by collapsing the 3D representation into a linear one. anced cell. Therefore, a trajectory s spatial orientation determines whether or not it is chosen. Each cell has a designated radius r = 3 3 CW 3 i 8 n i that is used for spacing binned tracers, where CW i 3 n i is the cell dimensions divided by the number of trajectories to be binned. An empirically derived factor of 3 8 is applied to calculate the radius within the allotted space. If the list is exhausted before the n c criteria is met, trajectories furthest from those already selected are chosen to fill the cell. Any remaining tracers continue to the next resolution, where they are once again sorted into empty cells. Generating Resolution-Dependent Trajectory Files At the conclusion of the binning process, cells within all levels of resolution contain indices of their enclosed tracers. We write this data to files following resolution and cell order, one file for each time step. The order in which the trajectories appear in the file is based on their resolution and cell placement. Resolution levels are written in order from coarse to fine, while cells are laid out in the traditional xyz fashion with the x axis running fastest and the z axis slowest, shown in Figure 3. Trajectory information for the current time step is written out strictly according to this layout, a size of 16 bytes per tracer, or one 32-bit integer and 3 floats. The integer holds the index of the tracer and the 3 floats hold its 3D position. An additional integer holds the number of tracers within the current cell. The memory layout is shown in Figure 4. The first file to be written is the time step τ. Recall that all tracers were binned into the resolution cells using their spatial position at time τ. Even though a tracer s 3D position may be different at a future time step, the tracer s location within any time step file remains static, as specified by the binning process. It is possible that a tracer with a valid position at τ may have ceased to exist during the fluid simulation Figure 4: Cell contents within a resolution for one time step file. Each cell begins with an integer which holds the number of trajectories within the cell, after which each trajectory is represented by its index and 3D position.

5 and therefore, may not have position information for an arbitrary future time step t i. To preserve the uniform layout between time step files, dummy values are inserted to make up for the missing 16 bytes of information. With prior knowledge of initial tracer location, specific tracers can quickly be found in these files with one seek operation for the enclosing cell. This knowledge comes from a reference file that is output simultaneously with the first time step file τ. The reference file is structured similarly to the time step files, ordered by resolution and cell number. Unlike the other files, the reference file holds the number of bytes to be read into a time step file to locate a given cell. This allows for the flexibility of reading and storing only specified tracers, reducing the overall memory footprint and I/O computation time. The aforementioned process is part of a pre-processing step that does not run in real-time due to intense data sorting and writing to disk. Generating the complete set of resolution-dependent trajectory files takes approximately 8 minutes for 25 million tracers, with 44 seconds taken by resolution creation and binning. Reading and writing the multiple time step files (40 in this example case) is the majority of the computational effort. Fetching Trajectories Our interactive flow exploration environment relies heavily on the files created during preprocessing. The coarsest resolution provides the trajectory sub-sampling used for fully interactive pathline interpolation; it is loaded upon initialization. Once a suitable location for the seeding curve or rake is found, refinement begins by loading additional tracers in the vicinity of the rake. These additional tracers are located in finer resolution levels, making it necessary to determine which grid cells are intersected. The reference file provided holds the dimensions and cell sizes for all resolutions. By combining this information with the 3D positions of rake seed points, the cells enclosing the rake can be found and sorted into a list for each resolution. Additionally, the adjacent neighbors of these cells are also sorted into the list by cell index. Using the beginning and end index of the cell list, the memory mapper maps only just enough of the resolution to cover the desired cells, exemplified in Figure 6. While a chunk of memory is mapped, only the desired cells are loaded since the reference file provides exact byte offsets for the file pointer to follow. As new trajectories are added to the pool of known tracers, pathlines are re-interpolated and a new visualization is created. The non-iterative nature of using interpolation for pathreconstruction is readily configured for parallelism (multithreading) during this process. Loading and interpolating position information for every rake seed point is done in parallel. To begin, all known tracers from the coarsest resolution are sorted into a spatial data structure and each seed point locates its nearest neighbors, building its neighborhood. Instead of restructuring the data structure as new tracers are loaded, tracers are added directly to the corresponding neighborhoods and each neighborhood is filtered by a k-nearest neighbor approach. By the end of the refinement process, the rake seed points are surrounded by the k-nearest trajectories over the entire input set of tracers, providing the best possible accuracy for scattered data interpolation Pathline Interpolation For complete flow field exploration, pathlines from arbitrarily seeded particles must be extracted throughout the domain. Such pathlines are interpolated from the input set of tracers provided by the Lagrangian representation. To begin interpolation, the seed location x is found within the current set of tracers at time τ, which form its neighborhood. A weighing kernel is applied to express the seed point as a function of its neighbors, x = f (x j (τ)), where x j are trajectories in the neighborhood. We use the following spline weight function [BMXK06], W(ρ)=1 6ρ 2 + 8ρ 3 3ρ 4 (5) where ρ is the Euclidean distance between x and x j divided by the neighborhood radius. The same function is then applied at a later time step, conditioned by the neighborhood at τ, to interpolate the seed point s new location, x(τ + i)= f (x j (τ + i)). Therefore, given a particle position and its neighborhood of tracers, scattered data interpolation can interpolate the particle s position at any time step during which the tracers exist. For the results presented in this paper, we have chosen to use the Moving Least Squares (MLS) [Lev98] method because of its continuity and local interpolation ability. Figure 5: Reference file. Ordered by resolutions, one integer is written out per cell to indicate how many bytes into the time step files must be skipped to reach the cell s contents. Figure 6: Memory mapped section of file. The rake, shown as a black line, is located within the blue cells. While information is only read from these cells, the entire area in red must be mapped for the file pointer to jump cell to cell, accessing the needed trajectory information.

6 The advantage of interpolation over integration in regards to pathline construction is in the speed of position calculation. For the Eulerian framework, advecting a pathline to a time τ + i involves computationally expensive numerical integration for each previous time step. The position of the particle at times τ +[1,2,...,i 2,i 1] must be calculated before its position at τ +i can be computed. In contrast, only one scattered data reconstruction using the Lagrangian representation is needed to achieve the same position information, thereby lowering the computational cost. We can further reduce the computational cost of pathline representation by choosing the frequency with which position information is calculated. That is, if the goal is to draw a pathline between times τ and τ +i, it may not be necessary to connect the positions of all intermediate time steps to get a sense of the fluid flow in the region, but only a subset of positions. A value κ defines the time interval at which position values are calculated. Therefore, the number of interpolations needed to visually represent a pathline decreases from i to κ i per seeded particle Interactive Data Exploration We demonstrate the advantage of fast position calculation on the CPU by offering an interactive seeding environment to explore the data set. The environment allows flow behavior at any data set position to be viewed in real time, and enables notable flow features such as salient structures to be sought out. To reduce the memory footprint of the exploration environment and to promote real time rake manipulation, only tracers from the coarse resolution are loaded by the algorithm. Because only a subset of tracers are initially loaded, additional trajectories within the rake neighborhood exist in the finer resolutions. Once an area of interest is located, these trajectories are loaded in the vicinity of the rake to improve the interpolation quality of the pathline. In other words, the pathlines are interpolated from the closest possible trajectories to the seed point, providing the best possible accuracy of the reconstruction. The dynamic loading of tracers significantly reduces the amount of I/O when compared to the Eulerian approach. 6. Results and Analysis Let C be a univariate space curve parametrized in s R. A path-surface is the union of all pathlines passing through such a user-defined seeding curve, the rake, where s designates a specific pathline seeded at C(s). Creating visual connections between these curves along the s and t directions creates the path-surface [GKT 08]. If the rake forms a surface instead of a curve, it becomes a two-dimensional family of pathlines with a seed surface S(u,v) R 2, otherwise referred to as a time-surface. Points seeded in this fashion jointly traverse the flow, thus presenting a snapshot of the surface s deformation at a time t i. Stitching these coordinates along the (u, v) axis forms a visual surface through time [KGJ09]. Both types of surfaces can be computed using the trajectory interpolation presented in this paper. For the presented results, the interpolation neighborhood was set to the 12 nearest neighbors and curves created with a value of κ = 4. All computations were done on an Intel Core i7 processor with 7200 RPM and 8GB of RAM. The colors chosen are for illustration purposes only and serve to simply differentiate between interpolated and integrated surfaces Karman Vortex Street We begin by exploring a data set exhibiting a Karman Vortex Street. The given trajectories have a total time step length of 36 and the data set is scaled to fit into a unit cube and 1 contains 1,656 trajectories per 100 of a unit cube, split into 6 levels of resolution. The time step τ chosen for the preprocessing step is randomly selected from the middle of the simulation, giving a snapshot of the flow after a significant amount of time. The Karman data set has one inflow boundary on the left hand side and one outflow boundary on the right where the first quarter of the data set, from left to right, holds a cylindrical object lodged in the flow. As the fluid flows over the cylinder, it pushes downward towards the middle which in turn pushes particles on the side outward and up. Figure 7 exemplifies this trend through a time surface representation. The rake begins as a rectangle, densely packed with points, which is then interpolated through time and visualized at intervals of κ. Here, the rake is positioned above the obstacle showing time surfaces heading downwards. For a close look directly behind the cylinder, an area exhibiting extremely high turbulence, the top of Figure 9 shows three different images of the moving plane in the same location. The first two are reconstructed at the coarse and refined resolutions using our approach, while the final surface is an Eulerian-based advection using adaptive Runge-Kutta 4/5. The first image shows the path-surface reconstruction during interactive exploration, revealing its general shape. The center image shows the surface reconstructed after additional trajectories in the vicinity of the rake are loaded from Figure 7: Time surfaces behind the cylinder. Surfaces are extracted at an interval of κ = 4, showing the downward trend of trajectories from the top towards the bottom.

7 top of the jet. Figure 8 is a path-surface extracted near this plume. The rake is seeded as a circle that captures trajectories spreading out as they curl in on themselves. Figure 8: Path surfaces reconstructed using interpolation (left and middle) and integration (right). The left surface represents a real time interpolated reconstruction from trajectory information, which captures the general shape of the actual surface. The middle image shows how the interpolated reconstruction converges to the same surface when the flow map resolution is increased by dynamic loading of new sample trajectories. The same surface extracted with numerical integration is shown on the right. The color is for illustration purposes only and serves to differentiate the interpolated and integrated surfaces. all levels of resolution. The increased density of trajectories drastically improves the accuracy of the reconstruction. Below each image in Figure 9 are reconstruction metrics measuring accuracy, performance, and I/O. Accuracy is measured as the Euclidean distance between the trajectories that make up the surface reconstructed through interpolation and the surface created through numerical integration. We treat the advected surface as the ground truth and show that our method can achieve comparable accuracy. The left-most and center image show that as additional trajectories are loaded, the accuracy improves and the difference in surfaces, when compared to the ground truth, diminishes. While not as accurate, the coarse reconstruction allows for a surface preview with the benefit of real-time visualization. To match ground truth accuracy, loading additional tracers takes 1,143ms, which is slightly above half the time it takes for the Eulerian surface (1,936ms). Additional trajectories are only loaded in the vicinity of the plane, resulting in a total of 25.7MB necessary to perform the reconstruction. When compared to the 84.8MB of velocity field information needed to perform advection, our approach uses over 3 times less memory to create a near identical surface. Moreover, our method also takes less time to construct the surface, reaching comparable accuracy faster and with less memory than the Eulerian representation The Jet We continue with a look at a "Jet" data set. The given trajectories have a total time step length of 30 and the data set is scaled to fit into a unit cube and contains 1,656 trajectories 1 per 100 of a unit cube, split into 4 levels of resolution. An overall turbulent data set, the Jet has an inflow boundary at the bottom in the form of four jets at the center. A prominent feature of interest is the plume that forms towards the To explore the plume in further detail, we switch to a timesurface to better capture the curl of the structure. The bottom of Figure 9 shows three surfaces, two of which are extracted using interpolation and the final using numerical integration. All images look up at the underside of the forming plume, capturing it folding in on itself. As additional trajectories are loaded from finer resolutions, the interpolated surface matches up visually to the integrated surface. This is shown numerically through the average distance error metric below the images, which decreases with refinement. For a time surface, refinement loading is constant because position information is required at only one time step. With numerical integration however, all seed points must be advected to the representative time, thereby making computation time linear. The time-surface was extracted after 30 time steps, taking numerical integration 10,941ms, while completely refined interpolation took only 2,362ms. The results also shows that through the refinement step, the Lagrangian representation loaded 27.4MB of memory, compared to the 737MB loaded with numerical integration. Our method only needs to load 2 time steps when extracting time surfaces, which significantly reduces the computation time and I/O of creating the surface when compared with advection with the Eulerian representation. 7. Concluding Remarks The main contribution of this work is a novel file system together with interpolation techniques that provide significant advantages over classic time-varying integration, such as direct control over memory constraints. We show that tracer interpolation reduces computational effort in such a way as to provide a real-time, interactive tool for exploring fluid flow. While the overall process involves a pre-processing step for a given time step τ, this step only needs to be performed once to gain the benefits of a Lagrangian flow map representation over the entire data set. 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