=Paper=
{{Paper
|id=None
|storemode=property
|title=Statistical Fluid Flow Synthesis
|pdfUrl=https://ceur-ws.org/Vol-588/114.pdf
|volume=Vol-588
}}
==Statistical Fluid Flow Synthesis==
https://ceur-ws.org/Vol-588/114.pdf
Statistical Fluid Flow Synthesis
Mauricio Vines* Won-Sook Lee†
University of Ottawa
fluid behavior in real-time. Composing fluid behavior from a set
ABSTRACT of simulation components or tiles has been applied with model
We present our ongoing work on developing techniques for real- reduction [6]. Here each component contains a base of possible
time simulation and visualization of fluids. We propose fluid velocities. In our scenario, we do not use model reduction
synthesizing fluid motion from simulation examples obtained with for our simulations, but synthesize motion from examples
a full numerical solver. We segment our simulation data in order obtained by numerically solving the Navier-Stokes equations.
to capture the properties of local fluid behavior and apply
statistical techniques to obtain the main modes of variation of the
flow interacting with solid obstacles. Most research in fluid
simulation has been focused on techniques for solving the Navier-
Stokes equations for fluid motion, obtaining physically accurate
simulations. We aim to obtain visually plausible flows by
combining the results of physically accurate simulations
preserving the general flow behavior. Our technique is suitable to
be applied for visual effects design in movies as well as real-time
applications such as videogames and virtual environments with
haptic interaction, where efficient computations are required.
KEYWORDS: Fluid simulation, principal component analysis,
visualization.
Figure 1. Schema of our approach
INDEX TERMS: I.3.7 [Computer Graphics]: Three-Dimensional
Graphics and Realism—Virtual Reality; I.6.8 [Simulation and 2 METHODOLOGY
Modeling]: Types of Simulation—Animation
In this section we discuss the two stages of our simulation
1 INTRODUCTION framework: the construction of the examples database and the
synthesis of new fluid motion from these examples. We represent
Simulating fluids remains one of the most challenging problems the fluid using an Eulerian approach, both in the preprocessing
in computer graphics. Recent advances in computer applications and motion synthesis stages. The simulation space is divided into
demand more realism and interactivity. Several techniques have a regular grid of cells. In each cell the fluid velocity and pressure
been developed to provide accurate and realistic fluid simulations, are stored. Solid objects are assumed to be static and they are
however, at a great computational cost. This is due to the interplay voxelized as in [2], conforming to the simulation space
of different complex phenomena involved, such as advection, configuration. In the preprocessing stage simulation examples are
diffusion and turbulence, which are difficult to simulate. Fluid produced and analyzed. In the synthesis stage we find velocity
behavior is characterized by the well known Navier-Stokes values for each grid cell to approximate fluid behavior by
equations which are a system of partial differential equations for combining the flow information from the simulation examples. A
which no analytical solution is known. schema of our approach is shown in Figure 1.
In computer graphics, the main goal is to achieve visual
plausibility, even at the cost of physical accuracy. Several 2.1 Database construction
approximations to fluid behavior have been developed, including Fluid flow examples are obtained from simulating the fluid on the
particle-based simulations [4] and height field approximations for grid in the pre-processing stage. For simulating the fluid we
liquid surfaces [3]. Applications of fluid simulations include employ a finite difference method based on operator splitting and
visual effects in movies and videogames, as well as real-time semi-Lagrangian advection [5]. We run several simulation rounds
interaction with virtual environments [1][7]. for different solid configurations in order to have a rich set of
We propose a different method to efficiently obtain visually fluid motion examples. In a simulation round the obstacle
plausible simulations using fluid motion examples. These geometry is defined, and the fluid is simulated under different
examples are obtained by running a standard fluid simulation flow conditions, varying the direction and magnitude of the fluid
under different flow conditions and the resulting motion data is velocity. Velocity values for each time step of simulation are
spatially partitioned into different fragments. We apply principal stored for segmenting and statistical analysis.
component analysis (PCA) on the fragments and we employ this Once the examples have been obtained, the results of each time
information to synthesize new fluid motion. Fluid motion step are segmented in fragments of equal size. A fragment
databases have been used in the past for computing fluid forces on consists on several grid cells, each storing velocity and pressure
immersed solid objects [1], however we focus on visualizing the values. For each fragment we keep track of its neighboring
fragments as well as the corresponding fragment in the next time
* e-mail: mvine059@uottawa.ca step. As we consider only the case of static solids, the velocities in
†
e-mail: wslee@uottawa.ca the grid cells occupied by a solid are zero. An example of the fluid
examples generated and its fragments is shown in Figure 2.
4
We apply PCA on the fragments to capture the main modes of definition of a free surface. A non-mass conserving fluid will
variation of the fluid velocity. We construct a covariance matrix appear as evaporating, hence losing mass. To properly simulate
from the fluid velocity data in each fragment and we obtain the liquids using our technique, examples of the free surface have to
eigenvectors and eigenvalues of this matrix. Then we transform be generated and statistically analyzed as well.
our simulation data into the PCA space, this is, we represent the
data in terms of the principal components, which are the
eigenvectors with highest eigenvalues associated. The fragments
spatial and temporal associations are stored, enabling to quickly
determine the local evolution of the fluid for a specific fragment.
2.2 Flow synthesis
Synthesizing new fluid motion in the simulation grid is performed
using the data from the examples and the current velocity in the
grid. A specific velocity field may be defined by the user as initial
conditions for the simulation, or as result of real-time interaction.
The new simulation data is segmented into fragments the same
way as the examples and the velocity data for each fragment is
translated into the PCA space. For fragments that contain solid
cells, or that share a boundary with a solid, a pattern matching Figure 2. Frames of fluid simulation examples and a detail of
step is performed in order to determine the best example fragments for each
fragments that resemble the specific local solid geometry.
Let f be a fragment in the new simulation and S the set of We note as well that the different visual effects that can be
example fragments that are geometrically compatible with f. The obtained by this technique are limited to the fluid motion
fluid velocity in f is determined by finding a combination of variations in the database. Extending the examples database
examples such that it matches the behavior observed in f. would allow simulating more diverse solid-fluid interactions.
Formally, let e1, e2,,…,en be the first n principal components
derived from the examples, and let B=[b1, b2,…bn]T be the vector 4 CONCLUSION
of coefficients corresponding to f in the PCA space. For each We present an approach to synthesize fluid motion from
example si∈S, let Ci=[ci1,…, cin]T be the vector of coefficients in examples. Statistical analysis is performed on the examples and
the PCA space. Then synthesizing new motion reduces to solving they are combined in order to approximate fluid behavior. This
the following system: approach is suitable to be used for real-time applications such as
videogames and interaction with virtual environments. In the
[C ]
future we aim to extend our method to properly simulate fluid
1
L Ck ⋅ x = B (1) interaction with moving solids. Also we aim to provide more
animation control by allowing the user to specify the velocity in
specific fragments in different locations and times, so fluid motion
The coefficients of vector x indicate how the samples must be must be synthesized to match the user constraints.
combined to match the current motion in the current time step. As
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