Learning Three-Dimensional Flow for Interactive Aerodynamic Design
NOBUYUKI UMETANI, Autodesk Research
BERND BICKEL, IST Austria
=0.27

=0.36

=0.33

=0.40

=0.27

=0.32

regression

prediction for new shapetraining datareal-time prediction
during interactive design

Fig. 1. Our interactive design system allows users to interactively design and optimize a free-form 3D shape while the pressure on the surface (color-coded)
and the velocity field (illustrated as stream lines) are accurately predicted in real-time. Our data-driven model is based on a novel representation for 3D shapes
and can be robustly trained from a set of exemplars.
We present a data-driven technique to instantly predict how fluid flows
around various three-dimensional objects. Such simulation is useful for computational fabrication and engineering, but is usually computationally expensive since it requires solving the Navier-Stokes equation for many time steps.
To accelerate the process, we propose a machine learning framework which
predicts aerodynamic forces and velocity and pressure fields given a threedimensional shape input. Handling detailed free-form three-dimensional
shapes in a data-driven framework is challenging because machine learning approaches usually require a consistent parametrization of input andoutput. We present a novel PolyCube maps-based parametrization that canbe computed for three-dimensional shapes at interactive rates. This allows
us to efficiently learn the nonlinear response of the flow using a Gaussian
process regression. We demonstrate the effectiveness of our approach forthe interactive design and optimization of a car body.
CCS Concepts: • Computing methodologies → Gaussian processes;
Real-time simulation; Physical simulation; • Applied computing → Engineering;
Additional Key Words and Phrases: machine learning, fluid simulation,
Gaussian process, parameterization
ACM Reference Format:
Nobuyuki Umetani and Bernd Bickel. 2018. Learning Three-Dimensional
Flow for Interactive Aerodynamic Design. ACM Trans. Graph. 37, 4, Article 89(August 2018), 10 pages. https://doi.org/10.1145/3197517.3201325

Authors’ addresses: Nobuyuki Umetani, Autodesk Research, Toronto, Canada, n.
umetani@gmail.com; Bernd Bickel, IST Austria, Vienna, Austria, bernd.bickel@ist.ac.at.

© 2018 Association for Computing Machinery.
This is the author’s version of the work. It is posted here for your personal use. Not forredistribution. The definitive Version of Record was published in ACM Transactions on
Graphics, https://doi.org/10.1145/3197517.3201325.

Submission ID: 285. 2018-05-19 12:58. Page 1 of 1–10.

1

INTRODUCTION

Computational Fluid Dynamics (CFD) has become an indispensable
component in the field of engineering and computational fabrication for design evaluation and optimization. Successful applications
include improving the aerodynamics of cars and airplanes, refining
heating, ventilation and air conditioning in architectural planning,or the optimization of casting and injection molding in manufacturing. Many of these applications would benefit from an interactive
design approach. However, analyzing a design is usually computationally expensive, as it requires repetitively solving a system withmany degrees of freedom over many time steps. Although modern
design tools allow the user to edit the shape interactively, computation times often prohibit the interactive visualization, exploration,and optimization of shapes and their surroundings based on CFD
simulations. This problem is extremely challenging, even whenreal-time simulation capabilities are available, i.e., the simulation ofindividual time-steps at real-time rates, because evaluating a designin simulation may require many time steps.
In this paper, we suggest a novel approach to achieve real-time
CFD performance based on a data-driven method. While CFD generally refers to fluid simulation for engineering purposes, we limit thescope to the evaluation of aerodynamic forces and flow around an
object given that it is moving at a constant speed in a single-phasefluid (e.g., air flow around cars or airplanes). Our approach enables
drastically faster evaluation by constructing a reduced model between input and output based on pre-existing data. Our modeltakes a 3D shape as the input, and outputs a drag coefficient, fluidpressure on the surface and velocity field around the shape. The
nonlinearly and globally coupled relationship between input andoutput is learned using the Gaussian Process (GP).
A key challenge in applying machine learning to CFD problems
is the parameterization of the input and output. As most machinelearning approaches take fixed dimensional vectors as input and
ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

89:2 • N. Umetani et. al.
output, we need to parameterize (i.e., represent as fixed dimensionalvariables) the input 3D shape and output pressure field on the shapeand velocity field around the shape. To address this problem, we
present a novel approach for parameterizing the 3D shape and volumetric field around it for machine learning based on the PolyCube
map [Tarini et al. 2004]. The PolyCube map represents arbitrary 3D
shapes by deforming a set of cubes. By deforming the space aroundthe cubes, we also achieve a natural parameterization of the fieldaround arbitrary 3D shapes. Our PolyCube-based shape and fieldparameterization is extremely efficient and can be done in real-timewith a small distortion.
Our proposed representation features a fixed-length parameterization of the shape and field, which allows a GP to easily learn andpredict their relationship. As a demonstration, we constructed aninteractive aerodynamic design tool where the user can edit a shapewhile getting real-time feedback on pressure distribution on thesurface and the air flow around the shape. Our tool also visualizes
reliability of the prediction as machine learning is unreliable when
input is very different from ones in the training data. The main
contributions of this work are:
• A novel, computationally efficient parameterization of 3D
shapes and volumetric fields based on the PolyCube map.
• A regression framework for three-dimensional flow aroundan object.
• An interactive interface for aerodynamically efficient design.

2

RELATED WORK

Reduction techniques for fluid simulation. Model reduction techniques perform simulations in a lower dimensional representationof the original simulation domain, thereby decoupling the grid resolution from the simulation complexity. In the computer graphics
community, Treuille et al. [2006] introduced model reduction for
the real-time simulation of incompressible fluids. Later, this technique was extended to modular tiles that can be assembled during
the runtime to adapt the domain and simulate novel fluid configurations [Wicke et al. 2009]. For moving solids, model reduction
can also be conducted on a moving grid [Cohen et al. 2010]. To
enable the reduced simulation of fluid flow around a deformableobject, Stanton et al. [2013] suggest a nonlinear Galerkin projection
approach to construct nonlinear bases. The subspace integration isaccelerated by the optimized cubature scheme [Kim and Delaney2013]. Eigenmodes of the Laplacian operator are used to constructbases of the reduced model [De Witt et al. 2012]. Since a model
reduction approach solves the Navier-Stokes equation step-by-step,
it requires the running of a simulation for some time to obtain theoverall fluid behavior. In contrast, our approach directly learns theresults, circumventing the use of a Navier-Stokes solver during theruntime. Stanton et al. [2014] use a state graph to synthesize animated fluid motion from examples, but it is unclear how this can beapplied to a high-dimensional input such as a free-form shape.
Machine learning for fluid simulation. Recently, accelerating fluid
simulations using machine learning has gained significant attention
Tompson et al. [2016] accelerate the pressure projection in Eulerian
fluid simulations using a neural network. Forces on Lagrangian
ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

particles can be efficiently approximated using a regression forest [Ladický et al. 2015]. Detailed splashing effects can be added
to Lagrangian particles using neural networks to model the regression of splash formation [Um et al. 2017]. Chu et al. [2017] use a
Siamese network to learn a distance metric between fine and coarse
simulation to synthesize the details of smoke. Bonev et al. [2017]
suggest a data-driven technique to map changing simulation conditions such as user interactions to a reduced representation based on
space-time deformations. In addition, a network is trained to refine
the deformations and represent details present in the initial training
data. While these approaches either accelerate the computation ofindividual time steps or add detail as a post-process, our approachdirectly generates a time-averaged velocity field and pressure field.
Data-driven aerodynamics. There are few studies that use machine learning to predict aerodynamic forces for real-world applications. Pteromys [Umetani et al. 2014] constructs a relationshipbetween paper airplane shapes and aerodynamic forces based onrecorded flight trajectories of a large set of airplanes. The proposedmodel is limited to assemblies of free-form planar shapes. Addressing this limitation, OmniAD [Martin et al. 2015] models aerodynamic forces on an object from arbitrary directions using sphericalharmonics and learns model parameters from falling motions captured with a single camera. It also features the prediction of model
parameters of low-dimensional parameterized shapes. Schultz etal. [2017] present a sampling scheme to capture nonlinear simulation response by interpolating samples in the a low-dimensionalparameterized design space. Baqué et al. [2018] predict pressure
distribution on free-form 3D objects using graph convoluation. Our
approach learns aerodynamic forces of various free-form 3D objects,
in addition to the time-averaged velocity field around the object.
3D shapes in machine learning. When feeding a 3D shape intoa machine learning framework, a central question is how to parameterize the shape, i.e., how to express it as a set of variables.
Parameterization of free-form 3D shapes is a non-trivial task as
the shapes can have different resolutions, orientations, structures,and topologies. This is significantly different from other domains,such as natural language processing or vision, where text or image
datasets frequently come with a common parameterization [Xu et al.
2016]. For measuring shape similarity, many shape descriptors, suchas 3D shape histograms, were proposed. However, since the representation only parameterizes the shape as a distribution of features,
it loses a significant amount of information concerning the original
shape, making it unsuitable for our regression problem.
Recently, a number of deep neural network architectures havebeen proposed for 3D objects. These explore multiple views of a 3D
shape [Su et al. 2015] or voxel-based representations [Wang et al.
2017; Wu et al. 2014] as inputs. Instead of using descriptors, these approaches take a more direct representation of a three-dimensional
shape as an input. However, these implicit shape representation
approaches are not desirable for our regression problem since parameters change discontinuously with respect to the continuous
shape change. PointNet [Qi et al. 2016] explicitly encodes the 3D
geometry using a set of randomly distributed points on the surface.
However, it is difficult to discretize volumetric fields consistently
with such a grid-less point-based discretization.
Submission ID: 285. 2018-05-19 12:58. Page 2 of 1–10.

Learning Three-Dimensional Flow for Interactive Aerodynamic Design •89:3

accuracy
prediction

GAUSSIAN PROCESSREGRESSOR
PolyCube surface

parametric surfacepressure fielddrag coefficientvelocity field

velocity visualization

depth field

triangle meshthree-dimensional
conforming gridtraining data

Fig. 2. Overview of our method.

Addressing this problem, our shape representation is based on
deforming a template mesh. While Umetani [2017] deforms a cube
surface to parameterize a 3D shape, we further extend this approachby deforming a PolyCube. The PolyCube approach allows us to parameterize a wider variation of shapes (see Fig. 10) and to constructa hexahedral mesh outside the shape (see Fig. 5). As years of studies
show, PolyCube mapping is suitable to parameterize a wide rangeof three-dimensional shapes [Fang et al. 2016; Huang et al. 2014;
Tarini et al. 2004]. While these works focus on offline parameterization quality optimization, our focus is real-time computation forgenerating machine learning input shapes and output fields.

3

OVERVIEW

User interface. Our machine learning framework takes a shape S
and outputs drag coefficient cd , a velocity field v® around the shapeand a pressure field p on the shape. The drag coefficient shows themagnitude of drag force with respect to the projected frontal area
and is the most important indicator of aerodynamic efficiency. The
drag coefficient cd takes a value between 0.2 and 0.6 for typicalcars (e.g., 0.57 for a Hummer H2 and 0.30 for a Toyota Corolla).
The predicted drag coefficient is shown with a possible error range
because the prediction via machine learning is not perfect for unknown inputs (see Fig. 2-top rightmost). These predicted values aretime-averaged as they dynamically fluctuate over time. The pressure
field is visualized as a color contour on the object’s surface and thevelocity field is visualized with streamlines.
We do not have specific restrictions on the input shape representation, which we demonstrate using two popular shape representations: triangle meshes and boundary representation (B-Rep)
models, whose surfaces are defined by patches of parametric surfaces. These shapes can be edited freely with standard shape editing
operations (e.g., free-form deformation, changing the position ofcontrol points, etc.). Topology change is allowed in the input, i.e.,
the user can change the topology of the B-Rep by inserting an edge
inside a patch or by adding a control point to an edge. During theediting, the tool continuously updates the visualizations.
Computational pipeline. Figure 2 illustrates an overview of theruntime computation. We first convert the input shape into a set of
depth images that are taken from multiple orthogonal orientations.
Submission ID: 285. 2018-05-19 12:58. Page 3 of 1–10.

From the depth images, we construct a PolyCube-based representation (see Sec. 4), to parameterize the shape and velocity and pressure
fields. The pressure regressor estimates pressures at the grid points
on the shape and the velocity regressor estimates velocity at grid
points around the shape. We construct such regressors based on theGaussian Process (see Sec. 5).

4 PARAMETERIZATION OF 3D SHAPE AND FIELD
4.1 Shape Parameterization for the Regression Problem
In theory, state-of-the-art machine learning techniques such as
deep neural networks have the capability to approximate arbitrary
multivariable functions [Hornik 1991]. However, in practice, a hugeamount of training data is required to represent a high-dimensional
highly nonlinear function while avoiding over-fitting. How well
the regression predicts outputs for previously unseen data (i.e., thegeneralization error) depends heavily on the properties of the input
and output data. Three factors are usually desired:(fix) The input and output vectors for regression should have
fixed dimensions.
(compact) The dimension of the input and output vectors should
be as small as possible to avoid redundancy.
(linear) The relationship between input and output should be
as linear as possible to avoid complicated models, which are
difficult to train from a small amount of data.
Table 1. Comparison of shape and field parameterizations.

(fix) (compact) (linear)

3D SHAPE

VOLUMETRIC

FIELD

our parameterization
B-Rep
triangle meshmulti-view projection
voxel
SDF on a Cartesian grid

○
×
×
○
○
○

○
○
○
×
×
×

○
○
○
×
×
○

our parameterization
tetrahedra meshvalues on a Cartesian grid

○
×
○

○
○
○

○
○
×

ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

89:4 • N. Umetani et. al.
Unfortunately, these desirable features are difficult to satisfy simultaneously. Table 1 compares common parameterization (representation) approaches of a 3D shape and field. Most shape editing
tools operate with boundary representation (B-Rep) models or triangle meshes. However, the number of parameters changes as the
topology of these representations is modified; for example, by inserting new edges and points or by subdividing or reconnecting thetriangle mesh. Representing shapes using density values on multiview projections or a three-dimensional Cartesian grid approach
alleviates this problem, but induces significant nonlinearity, making
the regression difficult. As illustrated in Figure 3-top, a continuous change in the shape results in abrupt changes of the velocity
and density at grid points which are passed by the boundary. The
volumetric and surface field representations suffer from a similar
nonlinearity problem when parameterized on a Cartesian grid. Parameterizing the shape using the signed distance field (SDF) insteadof a density function reduces this problem. However, the dimensionof the parameters is still large as we express the shape using a 3D
field, even though the input shape is locally two-dimensional. Representing the volumetric field with a tetrahedral mesh for various
shapes is difficult without changing its topology.

TYPICAL
PARAMETERIZATIONON A CARTESIAN GRID

density: 0
velocity:

discontinuity

density: 1
velocity: 0

OUR
PARAMETERIZATIONON A DEFORMING GRID

continuous velocity change
Fig. 3. Top: Small changes to a parameterized shape and field on a Cartesian
grid induce discontinuities. Bottom: We deform the grid continuously to
reduce nonlinearities.

Our solution is to deform the grid continuously on and aroundthe shape (see Figure 3-bottom). The deformed grid shares the interface with the input shape. Hence, the boundary does not crossgrid points, reducing the nonlinearity in the parameterization. The
deformation of the entire volumetric grid is uniquely determined by
the surface grid. Hence, if we specify the surface grid, we can define
the volumetric field just by giving the values at the grid points. In
the following section, we explain how we construct such a gridbased on the PolyCube map.
ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

4.2

Construction of the PolyCube Grid

The parametrization of the 3D shape is based on the work [Umetani
2017] in which we convert the input shape into GPU-computed
depth surfaces and we then project the template mesh in the static
normal directions to the depth surface. The key difference is the useof PolyCube instead of a single cube for the template mesh to parameterize a wider range of shapes by choosing the PolyCube close
to the inputs. In this paper, we define a PolyCube as a set of axisaligned unit cubes that are connected face-to-face. Such a PolyCube
is manually modeled with an interface similar to MineCraft [Mojang 2009]. Aside from continuous shape and field representation,
another important advantage of the cube representation is that we
can easily construct a grid to parameterize the field outside theinput shape by filling additional cubes outside the input cubes (seeFigure 4-a). Different to the original PolyCube map [Tarini et al.
2004], we hierarchically construct the mapping by projecting themesh in the normal direction similar to the Normal Mesh [Guskov
et al. 2000]. This allows us parameterizing a concave shape with alower number of variables — instead of storing three-dimensional
coordinate values for all the vertices we store a scalar height map —and fewer parameters come in handy for machine learning. We also
show our hierarchical construction of the parameterization benefits
the reguralization of the machine learning (see Sec. 5.3).
Point classification. We start with a single, user-provided PolyCube model as an approximation of the various input shapes. We
name the grid generated from an input and exterior cube as a level-0
grid. The level-0 grid has three types of points: corner points Pc ,surface points Ps and exterior points Pe . These points are classified based on how many input cubes belong to the points (i.e.,the valence). If the valence is an odd number, the point is a corner
point that is located at the corner of the input PolyCube. On theother hand, the exterior points have a valence of zero as they do
not belong to the input cube. If the valence is non-zero and an evennumber, since we do not construct a grid inside the shape, the pointis a surface point that is located on an edge or face of the input PolyCube. For corner and surface points, we define the normal directions
by averaging the normals of the input PolyCube faces where thesepoints belong.
Corner point placement. We first project the corner points onto
the object’s surface. We start initializing the location of corner
points by translating and scaling the input PolyCube in the axes
directions such that the bounding boxes become the same. Then,
we project the corner points onto the input shape by finding thenearest intersection point.
Surface and exterior point placement. Given the positions of thecorner points, we move the surface points and exterior points to thelocations linearly interpolated from the corner points (see Figure 4d). For the surface points, which are located on the faces or theedges of the input PolyCube, we apply two-dimensional Mean Value
Coordinates (MVC) [Floater 2003] to move them together with thecorner points. As a consequence, the surface points on the edge of
the PolyCube move as the linear interpolation of the two endpoints
with the ratio of their distances. A surface on a face of the input
PolyCube moves linearly with respect to the corner points that
Submission ID: 285. 2018-05-19 12:58. Page 4 of 1–10.

Learning Three-Dimensional Flow for Interactive Aerodynamic Design •(a)input cube
exterior cube
corner point
surface point
exterior point

(b)

(d)

(c)

(e)

(a) input shapelevel-0

level-0(b) PolyCube

level-1

89:5(c) bounding box fitting

level-2

level-3(d) PolyCube subdivision and projectionlevel-1(e) volumetric gridlevel-2

Fig. 5. Three-dimensional example of the subdivision of a level-0 grid. The
color visualizes the accumulated normal projection heights up to the current
level.

(f)surfacesubdivision point
exterior
subdivision point

Fig. 4. Our construction scheme of a PolyCube parameterization.

outline the face. For exterior points, we use the 3D extension of
MVC [Ju et al. 2005] to move the points linearly with respect to theposition of all the corner points. The MVC determines the weightthat transfers the movement of corner point pc to a movement of an
exterior point and the surface points. These weights are dependent
only on the initial PolyCube construction, thus we only need tocompute them once. After the linear interpolation, we project thesurface point onto the object surface. The set of projection heights
Hs for all the surface points specifies how much these points are
moved toward the normal direction, encoding the shape at the®c .
coarsest level together with the positions of the corner points P
Subdivision. So far, we have constructed a coarse grid such thatthe boundary between the input and exterior cubes is shared with
the input shape. We subdivide the exterior cube to construct further
detailed parameterization of the input shape and the output field(see Fig. 4-f). Although we can selectively choose which exterior
cubes are subdivided, in this paper, we uniformly subdivide theexterior cubes such that all cubes have eight children. New points
are introduced at the middle points of the edges, the center of thefour corners of the faces, and the cell center of the parent grid. If aface or edge subdivision point is on the boundary facing the interior,
we project it to the input shape in its normal direction. The normal
direction is computed as the normal on the input PolyCube at thecorresponding location. We denote the normal projection heights
at the surface subdivision points as Hiss , where i is the subdivisionlevel. Fig. 5 shows an example of the subdivision.
Submission ID: 285. 2018-05-19 12:58. Page 5 of 1–10.

5 MACHINE LEARNING
5.1 Data for Regression
So far, we have explained our parameterization framework that
encodes shape and field in fixed dimensional vectors while also
avoiding excessive nonlinearities between them. In this section, we
explain how we construct the regressor to approximate the output
parameter vector from the input parameter vector.
Dimensional analysis. We start with four parameters as inputs:
shape S, incoming wind velocity V∞ , mass density ρ, and viscosityµ. We first reduce these four parameters into two using dimensional analysis. The dimensional analysis determines how physicalphenomena can be parameterized with a minimum number of independent non-dimensional variables. Non-dimensionalization isconvenient for machine learning of simulations because it reduces
the number of parameters without loss of accuracy. Furthermore,
we can reduce the amount of training data by avoiding redundant,i.e., dynamically similar, examples. OmniAD [Martin et al. 2015]
uses the Buckingham Π theory to construct a model of aerodynamic
forces on 3D objects. Similarly, our problem is parameterized by
the Reynolds number Re = ρLV∞ /µ and the non-dimensionalized(i.e., normalized) shape S ′ which is uniformly scaled such that the
representative length becomes one. The scaling of the shape canbe done in the parameter space by scaling corner point positions® c and projection heights H. Here, we define the representativeP
length as the diagonal length of the bounding box of the input
shape S. From these non-dimensionalized input parameters, we obtain non-dimensionalized velocity and pressure values as v®′ = v/V® ∞2 ). We can easily compute the final velocity andand p ′ = p/(ρV∞
2 , respectively.
pressure values by scaling them with V∞ and ρV∞
Input and output vectors. In Section 4, we explained our polycube
parameterization. Since our problem setting (i.e, a single object inside a constant flow) is invariant to translation and the grid moves
together with the object, we can remove the translation component from the input parameters. We move the corner points such
ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

89:6 • N. Umetani et. al.
that the averaged position of the corner points rests at the origin.
The input parameter vector becomes a concatenation of the scaled
and translated corner point positions, and projection heights of the
1 , H2 , . . .}. We have three regres® c , Hs , Hsssurface points x = {Psssors: for drag coefficient, velocity, and pressure. Thus, the level ofsubdivision can be chosen differently for each of them. The machine learning outputs a vector concatenating the drag coefficientand the nondimentionalizeds values at all grid points for velocity
v = {v®0′ , v®1′ , . . .} and pressure p = {p0′ , p1′ , . . .}.

5.2

Gaussian white noise with variance θ n > 0 assumed in the output.
These hyperparameters θ s , θ, θ n are optimized to agree with thedataset (see Sec. 5.3).
Given the covariance matrix K, which defines how much eachelements are relevant, we can compute the conditional probabilityof y ∗ under the training data D is given asp(y ∗ |D) = N (w∗ y, k(x∗ , x∗ ) − w∗ k∗ ),w = k∗ K−1
D
∗
1 ∗

(2)

∗

(3)
N

∗ T

k = {k(x , x ), . . . , k(x , x )}Gaussian Process

We choose Gaussian Process (GP) regression for inferring the CFD
simulation data. The advantages of a GP in our context is four folds:(i) The GP can express a highly nonlinear relationship between input
and output from a small number of training samples compared tothe dimension of inputs (T ≪ M). (ii) The GP is a probabilistic modelthat interpolates values from the training data probabilistically, i.e.,
we can obtain the confidence for the regression result aside from theregressed output itself. (iii) We can optimize its hyperparameters
directory from the training data to increase the accuracy. (iv) We canhandle noise in the training data. Thus, it is robust to the over-fitting
problem.
Leaving the comprehensive details of GP to a textbook [Rasmussen and Williams 2005], we only briefly explain its key concepts.
Here, we rely on common notations in statistic such as E(·) forthe expected value and p(·) for the probability. Suppose we would
like to predict a scalar value y ∗ from a high-dimensional input
N , wherex∗ ∈ RM given the set of training pairs D = {(y n , xn )}n=1
the expected value of the output is zero E[y] = 0. We denote theoutputs in the training data concatenated with the unknown output
as Y = {y 1 , . . . , y N , y ∗ }. The GP assumes that a set of outputs Y
follows a multivariate Gaussian distribution p(Y) = N (0, K), where
K denotes the (N + 1) × (N + 1) covariance matrix. The desired
solution is inferred from the conditional probability distribution of
the output y ∗ given the known training data p(y ∗ |D).
Each element of the covariance matrix K is defined as the expected value of the multiplication of two outputs K i j = E[y i y j ],where 1 ≤ i, j ≤ (N + 1). Intuitively, the diagonal entry K ii provides
information on the variance of the output y i , while the off-diagonal
elements of the covariance matrix K i j gives the magnitude of correlation between the output y i and y j (i.e., the similarity of the y i
and y j ). The GP regression models an element of the covariancematrix from the corresponding inputs using the kernel function
K i j = k(xi , xj ). Our choice of the kernel function is
( M
)
Õ
j
k(xi , xj ) = θ s exp −θ k (x ki − x k )2 + θ n δ i j .
(1)
k=1

The first term in (1) defines the similarity of the two inputs xi
and xj in terms of the similarity of the outputs y i and y j . The
hyperparameter θ s is a positive uniform scaling factor, while θ ={θ 1 , . . . , θ M } > 0 are positive scaling factors for the each dimensionof the input. This squared exponential covariance function is one of
the most favored choices of kernel functions in statistics because it
can be shown such a kernel corresponds to linear regression with
infinite number of basis functions. The second term of (1) models
ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

(4)ij

The K D is the N × N covariance matrix in the dataset K D =k(xi , xj ).
We apply GP regression to the regression of the drag coefficient
cd . As the GP requires the output value to have a zero mean, we
offset the drag coefficients with their average from the training
data y = cd − c̃d , where c̃d means the average of the cd from thetraining examples. How much the regression result is trustworthy
is visualized to the user as the range of µ ± σ where the probability
to fall in that range is about 69% based on a normal distribution.
cd (x∗ ) = w∗ cd + ω ∗c̃d , where ω ∗ = 1 −NÕw n∗ .

(5)n=1

Here cd is a vector concatenating cd values in the training data.
The GP regression scheme can be seen as interpolation from thetraining outputs with weight w∗ . Here we used the same weight tointerpolate the training data for the pressure field and velocity field
regressionp(x∗ ) = w∗ P + ω ∗ p̃,∗∗∗v(x ) = w V + ω ṽ,

(6)
(7)

where P and V are matrices stacking training data in rows. The
weight w∗ is optimized for the drag coefficient and reusing themfor pressure and velocity regression is probably not the optimal.
However, it is much computationally efficient compared to optimizing the weight for every element of the velocity and pressure.
Moreover, the result shows the weight have much better accuracy
compared to the naïve approach (see Sec.6). We leave the rigorous
weight optimization for the pressure and velocity as a future work.

5.3

Hyperparameter Optimization

Given θ = {1, . . . , 1} and θ n = 0, the above interpolation scheme
becomes equivalent to a radial based function (RBF) network, whichmeasures the difference between inputs as a function of uniform
Euclidean distance. When the dimension of the input M is verylarge, the naïve Euclidean distance suffers the curse of dimensionality,
where the distances to the example inputs fail to give information of
the closeness of the outputs. To alleviate this problem, we optimize
θ to emphasize elements of x that is relevant to the output. We
cannot optimize all the elements of θ as its dimension is muchhigher than the number of the training sample without assuming
smoothly varying parameters (i.e., regularization). The smoothness
distribution of the parameter over the shape is reasonable because
the positions of the neighboring nodes have strong correlations andpossibly the similar influence to the outputs. Our proposed PolyCube
Submission ID: 285. 2018-05-19 12:58. Page 6 of 1–10.

Learning Three-Dimensional Flow for Interactive Aerodynamic Design •representation naturally supports this approach. We directly specify
the weights at the coarse subdivision level θ c and obtain the smoothweights for the remaining nodes in the finer subdivision level byinterpolation. The interpolation scheme is based on the subdivision
scheme in Sec. 4.2 – we interpolate the weight by averaging thetwo end points of an edge if the fine point is on the edge, and by
averaging four corner points of a quadrant if the fine points are in
the quadrant.
We denote Θ for all the hyperparameters – uniform kernel weightθ s , the weights for coarse nodes θ c and noise variance θ n . We
optimize these hyperparameters using the maximum likelihood
method such that we have maximum probability of the training
outputs under the optimized parameter asΘoptimum = arg max log p(y|Θ) = −L(Θ),

(8)

Θ

1
1
N
L(Θ) = log |K D (Θ)| + yT K−1
D (Θ)y + 2 log(2π ),22where θ s > 0, θ c > 0, θ n > 0.

(9)
(10)

As the gradient of the log-likelihood in eq. (9) can be analyticallycomputed [Rasmussen and Williams 2005], we solve this constraint
optimization problem using the projected gradient decent method.

6

RESULT

Training data generation. We start with describing the car aerodynamics example we created using our framework. We then prepare
the input shapes from the “car” category of ShapeNet [Chang et al.
2015] (see Figure 1-left). We manually modify the shape to remove
the side mirrors, spoilers and tires. Then, we solve the Navier-Stokes
equation in a highly detailed spatial grid for many time steps until
we accurately obtain the time-averaged velocity and pressure fields.
The degrees of freedom in these simulations range from 600 k to
700 k. We simulate 10 seconds of air flow and average the resultsof the last 4 seconds to obtain the training data. Each simulation
takes approximately 50 minutes. Note that our machine learning approach does not depend on the fluid solver. In our example, the fluid
flow is simulated using our in-house finite element Navier-Stokes
solver with the k-epsilon turbulence model and SUPG stabilization [Zienkiewicz et al. 2013]. The simulation runs on a tetrahedralmesh that conforms to the boundary and we adaptively refined themesh around the surface and the back of the car to resolve the boundary layer and separated vortices resulting from non-slip boundary
condition. We used realistic physics parameters for the car simulation with the car driving in air at 72 km/h speed (Re = 5 × 106 ).

coarse control points

0max

Fig. 6. Optimized weight θ in the computation of the kernel. We set thecontrol points at the nodes on the coarse level (left) and smoothly interpolate
their values (middle and right). The weight illustrates how relevant theunderlying shape is to the prediction.

Submission ID: 285. 2018-05-19 12:58. Page 7 of 1–10.

89:7

Figure 5 illustrates the PolyCube and the resulting grid. Thismodel is built on a PolyCube with three cubes assuming sedan orwagon-type cars. To encode the shape of the car, we choose level-4
subdivision for the input shape M = 3698, level-4 subdivision forthe surface pressure discretization (3682 degrees of freedom), andlevel-3 for the velocity on the spatial volumetric grid (88494 degrees
of freedom). The pressure and velocity values are sampled at the grid
points, whose location can be computed from the input parameters.
We prepare 889 pairs of input shapes, and the output simulation
data consists of the drag coefficient value cd , pressure and velocity
field data computed with the CFD simulation. All training data is
included in the supplemental material∗ .
Training. We optimize the hyperparameters according to the maximum likelihood method in (8). The hyperparameter optimizationis the most time-consuming training step, taking approximately
30 minutes. With the optimized hyperparameters, the GP does not
require expensive computations aside from the computation andinversion of the kernel matrix, which is done only once. We trained
and ran the real-time demonstration program, as shown in the accompanied video, on a MacBook Pro 2014 model with a 3.0 GHz Core
i7 CPU using the Intel MKL LibraryTM . Fig. 6 shows the color-coded
resulting optimized spatially-varying weights θ in the kernel computation. Intuitively, the weights reflect the importance of specificinput positions in predicting the output. In our example, the weightis high on the windshield and the A-pillar, while low around thefront grill and the rear side. This is reasonable because the boundary layer grows and separates around the windshield and the air
is stagnant at the front and back of the car. We also have analyzed
the impact of the hyperparameter optimization. Fig. 8-right shows
a comparison of the error with and without weight optimization.
The unoptimized weight scenario is equivalent to a naïve RBF interpolation where all the weights are the same. With hyperparameter
optimization, we can observe a significant improvement in accuracy,with an error decrease of about 68% (see Fig. 8).
Performance. We show our runtime aerodynamic design tool in
Figure 1 and the accompanying supplemental video. While the user
is interactively modifying the shape by moving the control pointsof B-Rep or the free-form deformation, the system consistently
maintains 10 > frames per second for all shape-editing operations. By showing this drag force in real-time, the user can design aerodynamically-efficient shapes based on interactive trialand-error shape exploration. The memory footprint for the threeregressors (cd , pressure and velocity) combined is 270MB.
Accuracy. To examine the accuracy of our model, we conducted
9-fold cross-validations, i.e., we split all the data into nine equalsized subgroups and executed the cross-validation five times. For
each cross-validation, we use one subgroup for validation and therest for training. Fig. 7-left shows the distribution of the magnitudeof errors with respect to the predicted standard distribution σ ∗ . Thisshows that when the confidence is low (i.e., the standard deviation ishigh) the error tends to be large. Fig. 7-right shows the distributionof the errors divided by the standard deviation. When the errors∗ http://www.nobuyuki-umetani.com/publication/mlcfd_data.zip

ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0

0.05

0.04

0.03

0.02

0.01

0.00

-0.01

-0.02

-0.03

-0.04

error in0

error innormalized with

Fig. 7. (Left) Distribution of the magnitude of errors with respect to thepredicted standard distribution. (Right) Histogram of the errors divided by
the predicted standard distribution.

are normalized by their standard deviation, they follow the normal
distribution N (0, 1).
Fig. 8 compares the accuracy of the drag coefficient with other
regression approaches including linear regression, random forest,
GP with optimized uniform weight, and GP without hyperparameter
optimization (θ = 1). Fig. 8 shows the standard deviation of theerrors of the drag coefficient with respect to the size of the training
set for various regression strategies including linear regression
with L 2 reguralizer, random forest, GP with optimized uniform
weight, GP without hyperparameter optimization (θ = 1) and neuralnetworks with and without the 50% dropout [Srivastava et al. 2014]. For the regression forest method, we used the implementationin the scikit-learn library with its default parameters. The neuralnetwork we used here is a perceptron with one hidden layer thathas ten neurons with the logistic activation function. We trained
the neural network until convergence (sum of squared error wasless than 10−6 times the initial error). We observed that the neuralnetwork regression significantly performed worse in validation thanthe others because of the overfitting problem as the number of the
training data (∼ 0.8k) is much smaller than the dimension of the
inputs (∼ 3.7k). The dropout alleviated the overfitting problem tosome extent, but the result was still worse than the other regressors.
We are surprised to find the linear model is very competitive. Thanks
to our PolyCube parameterization, we managed to make the input
and output relationship close to linear. We include a Python script
for linear least square fitting in the supplemental material∗ .

average accuracy caseone of the worst accuracy caseregression

10

10

ground truth

20

20

Compared to the other approaches, our regression achieves thehighest accuracy in predicting the cd value. The standard deviation of the error is about ±0.012, which relates to a relative errorof less than ±3.4% when normalized with an average drag coefficient of the example c̃d = 3.5. According to an extensive study by
SAE International, the standard deviation of the cd values obtained
from ten large different wind tunnel testing facilities throughout
the world is ±2.2% and the difference can be up to 5% [Buchheim
et al. 1983; Hucho 1998]. Allowing the assumption that the offline
simulation would perfectly reproduce reality, our learned flow accuracy would be comparable to that of a wind tunnel. However, inpractice the offline simulation is not perfectly accurate, leaving uswith an unknown error relative to the true real-world behaviour.
Our goal in this paper is to show that we can efficiently reproduce
the time-consuming offline simulation result. Because no dataset ofwind tunnel experiments with many different car shapes is publicly
available, we leave the physical validation of the accuracy as futurework.
We also evaluated the errors in the pressure and velocity fields.
Fig. 9 shows the distribution of errors in the velocity and pressure
fields obtained from the cross-validation. The horizontal axis shows
the mean squared errors for the velocity and pressure values, whilethe vertical axis shows the frequency of the errors. Even in thecase of the largest error in the validation, we did not observe asignificant difference in the velocity and pressure fields comparedto the ground truth. This graph also suggests that the distributed
weight optimized for cd improves the accuracy of the velocity andpressure fields prediction in comparison with the naïve weight θ = 1.

regression

30

30

average=0.48
after optimization

0.04

Neural Network (with dropout)

0.03

Gaussian Process (unoptimized
weights)
Random Forrest Regression

0.02

Least Square Fitting (linear model)
Gaussian Process (uniform
optimized weights)

0.01
0 100 200 300 400 500 600 700 800

number of training samples

Gaussian Proces (distributed
optimized weights)

Fig. 8. Convergence of the standard variation of the errors in the dragcoefficient prediction with respect to the training size for different regression
strategies.
ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

80

140

number of samples

Neural Network (without dropout)error in

0.06
0.05

number of samples

0

normal distribution

40

40

ground truth

50

number of samples

50

-0.05

number of samples

89:8 • N. Umetani et. al.

120

average=0.92
before optimization

60

20
0

0.5

1

1.5

2

velocity error

2.5

average=13.7
before optimization

100

40

0

average=8.1
after optimization

3

80
60
40
20
0
0

10

20

30

pressure error

40

Fig. 9. (Top) Comparisons between ground truth velocity and pressure fields
and regression results for one average accuracy case and one worst accuracy
case. (Bottom) Frequency of the squared norm error of the velocity field (left)
and pressure field (right).

Submission ID: 285. 2018-05-19 12:58. Page 8 of 1–10.

Learning Three-Dimensional Flow for Interactive Aerodynamic Design •More specifically, mean squared errors averaged over the example
decreased 47% for velocity and 41% for the pressure. In terms ofthe maximum absolute value of the errors in the velocity and thepressure fields, their averaged value over the examples are 8.80
m/s and 126 Pa, respectively, using the optimized weight. In the
supplemental material, we include about one hundred comparisons
between ground truth and machine learning results.

PLANE

DOLPHINE

parameterization for concave shapes. Note that for the torus example (Fig. 10-bottom top left), there is no way to continuously fit a
cube into this genus-1 shape, thus the previous technique [Umetani
2017] results in a failure.
For the torus model, we also construct a machine learning CFD
model based on 50 training examples. These example shapes are chosen randomly from the recording of the user’s modeling sequence.
For the torus model, we assume that the shape is used as a car-top
sign, so we set the dimension of the torus about 10 cm in diameterand 4 cm in thickness. We encourage the readers to look at the
supplemental material for the interactive modelling sequence ofthe car-top sign while getting the feedback from the real-time CFD
estimation. Fig. 10-(b) and -(c) show the CFD simulation results on
the initial torus shape and final torus shape after the editing. Using
our system, the torus has 15% less drag coefficient (from 0.31 to0.26) compared to the original undeformed torus shape based on
the simulation. The bottom rightmost image in Fig. 10-(a) shows
the resulting aerodynamically-efficient torus shape we 3D printed.

7

SPACESHIPFLYING SAUCER

TORUS(a) 3D printed shape(b) initial shape(c) final shape

Fig. 10. Examples of our PolyCube 3D shape and field parameterization fora various shapes. The resulting aerodynamically efficient shape can be 3D
printed and attached to a car (bottom-rightmost).

Parameterization of more complex shapes. Our shape and fieldparameterization technique is not limited to car-like, almost concave shapes. We can parameterize a wide variety of shapes startingfrom a PolyCube that approximates the shapes. Figure 10 shows
examples of our shape parameterization including an airplane, adolphin, a flying saucer, a spaceship, and a torus. While the previous
technique [Umetani 2017] fits a cube into the shape, our PolyCubebased technique can achieve better quality (i.e., less bias) in the
Submission ID: 285. 2018-05-19 12:58. Page 9 of 1–10.

89:9

LIMITATIONS AND FUTURE WORK

Currently, the user needs to manually specify a PolyCube as a roughrepresentation of the input shape. Automatically generating sucha PolyCube representation would further reduce the user’s workload. We currently use a single regression model which is built on asingle PolyCube. An interesting avenue for future work would be
to explore the construction of multiple regression models for different PolyCube configurations and switch them during the user’s
interaction. This would require developing a reliable framework to
determine decision criteria and methods for choosing and switching
between PolyCube representations.
In this work, we constructed a parameterization by projecting
points of a subdivided PolyCube onto the input shape in predetermined directions. While this parameterization can handle a wide
range of shapes similar to the input PolyCube, it is still challenging
to parameterize a highly concave shape or very sharp and thin features (e.g., side mirrors) without excessive distortion of the grid. We
are planning to increase the representation capability by adaptively
changing the resolution of the input template PolyCube in regionsin which we know a priori that high curvatures might occur.
We believe our approach could be extended to a wide range ofother complex physical phenomena, which we plan to investigate
in the future. Since our approach parameterizes the shape and associated 3D field, we can parameterize boundary conditions andthe solution of many partial differential equations such as the thermal fluid equation, supersonic fluid equation, or electromagnetic
fluid equations. Finally, we think it would be exciting to extend
our method to include a state graph and/or a temporal frequency
decomposition of the field. As an application, we are particularlyinterested in biomechanical simulation (e.g., the simulation of flow
inside a heart or blood vessels), such that doctors can give morereliable diagnostics based on preexisting simulation data.

8

CONCLUSION

We have presented a technique to use to interactively predict how
fluid flows around a three-dimensional object given its shape. Our
ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

89:10 •N. Umetani et. al.

primary contribution is a novel interactive PolyCube representation
to parameterize the input shape and represent a field on and aroundthe shape. Using that parameterization, we successfully constructeda regression model using Gaussian Process. Our method is fastenough to be integrated into existing interactive shape modelingtools. With the real-time feedback, the user can manually optimize
the aerodynamics of an object. We evaluated the accuracy of our
method by cross-validation, confirming a good match of our results
to ground truth simulation.

ACKNOWLEDGEMENTS
We thank anonymous reviewers and Ryan Schmidt for their comments and advice. We appreciate the assistance from Rin Ishikawa
for the production of the supplemental video. This project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant agreement No 715767 – MATERIALIZABLE).

REFERENCES
Pierre Baqué, Edoardo Remelli, François Fleuret, and Pascal Fua. 2018. Geodesic Convolutional Shape Optimization. CoRR abs/1802.04016 (2018). arXiv:1802.04016
http://arxiv.org/abs/1802.04016
B. Bonev, L. Prantl, and N. Thuerey. 2017. Pre-computed Liquid Spaces with
Generative Neural Networks and Optical Flow. ArXiv e-prints (April 2017).
arXiv:cs.GR/1704.07854
R. Buchheim, R. Unger, P. Jousserandot, E. Mercker, F. K. Schenkel, Y. Nishimura,and D. J. Wilsden. 1983. Comparison Tests Between Major European and North
American Automotive Wind Tunnels. In SAE Technical Paper. SAE International.
https://doi.org/10.4271/830301
A. X. Chang, T. Funkhouser, L. Guibas, P. Hanrahan, Q. Huang, Z. Li, S. Savarese, M.
Savva, S. Song, H. Su, J. Xiao, L. Yi, and F. Yu. 2015. ShapeNet: An Information-Rich
3D Model Repository. ArXiv e-prints (Dec. 2015). arXiv:cs.GR/1512.03012
Mengyu Chu and Nils Thuerey. 2017. Data-Driven Synthesis of Smoke Flows with
CNN-based Feature Descriptors. Transaction on Graphics (SIGGRAPH) 36(4) (Apr
2017), 14.
Jonathan M Cohen, Sarah Tariq, and Simon Green. 2010. Interactive fluid-particle simulation using translating Eulerian grids. In Proceedings of the 2010 ACM SIGGRAPH
symposium on Interactive 3D Graphics and Games. ACM, 15–22.
Tyler De Witt, Christian Lessig, and Eugene Fiume. 2012. Fluid Simulation Using
Laplacian Eigenfunctions. ACM Trans. Graph. 31, 1, Article 10 (Feb. 2012), 11 pages.
https://doi.org/10.1145/2077341.2077351
Xianzhong Fang, Weiwei Xu, Hujun Bao, and Jin Huang. 2016. All-hex Meshing Using
Closed-form Induced Polycube. ACM Trans. Graph. 35, 4, Article 124 (July 2016),
9 pages. https://doi.org/10.1145/2897824.2925957
Michael S. Floater. 2003. Mean Value Coordinates. Comput. Aided Geom. Des. 20, 1(March 2003), 19–27. https://doi.org/10.1016/S0167-8396(02)00002-5
Igor Guskov, Kiril Vidimče, Wim Sweldens, and Peter Schröder. 2000. Normal Meshes.
In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive
Techniques (SIGGRAPH ’00). ACM Press/Addison-Wesley Publishing Co., New York,NY, USA, 95–102. https://doi.org/10.1145/344779.344831
Kurt Hornik. 1991. Approximation Capabilities of Multilayer Feedforward Networks.
Neural Netw. 4, 2 (March 1991), 251–257. https://doi.org/10.1016/0893-6080(91)90009-TJin Huang, Tengfei Jiang, Zeyun Shi, Yiying Tong, Hujun Bao, and Mathieu Desbrun.
2014. &#8467;1Basedd Construction of Polycube Maps from Complex Shapes. ACM
Trans. Graph. 33, 3, Article 25 (June 2014), 11 pages. https://doi.org/10.1145/2602141
Hucho. 1998. Aerodynamics of road vehicles : from fluid mechanics to vehicle engineering.
Society of Automotive Engineers, Warrendale, PA.
Tao Ju, Scott Schaefer, and Joe Warren. 2005. Mean Value Coordinates for Closed
Triangular Meshes. ACM Trans. Graph. 24, 3 (July 2005), 561–566. https://doi.org/
10.1145/1073204.1073229
Theodore Kim and John Delaney. 2013. Subspace Fluid Re-simulation. ACM Trans.
Graph. 32, 4, Article 62 (July 2013), 9 pages. https://doi.org/10.1145/2461912.2461987
L’ubor Ladický, SoHyeon Jeong, Barbara Solenthaler, Marc Pollefeys, and Markus Gross.
2015. Data-driven Fluid Simulations Using Regression Forests. ACM Trans. Graph.
34, 6, Article 199 (Oct. 2015), 9 pages. https://doi.org/10.1145/2816795.2818129
Tobias Martin, Nobuyuki Umetani, and Bernd Bickel. 2015. OmniAD: Data-driven
Omni-directional Aerodynamics. ACM Trans. Graph. 34, 4, Article 113 (July 2015),
12 pages. https://doi.org/10.1145/2766919

ACM Trans. Graph., Vol. 37, No. 4, Article 89. Publication date: August 2018.

Mojang. 2009. Minecraft. (2009).
Charles Ruizhongtai Qi, Hao Su, Kaichun Mo, and Leonidas J. Guibas. 2016. PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation. CoRR
abs/1612.00593 (2016). arXiv:1612.00593 http://arxiv.org/abs/1612.00593
Carl Edward Rasmussen and Christopher K. I. Williams. 2005. Gaussian Processes for
Machine Learning (Adaptive Computation and Machine Learning). The MIT Press.
Adriana Schulz, Jie Xu, Bo Zhu, Changxi Zheng, Eitan Grinspun, and Wojciech Matusik.
2017. Interactive Design Space Exploration and Optimization for CAD Models. ACM
Trans. Graph. 36, 4, Article 157 (July 2017), 14 pages. https://doi.org/10.1145/3072959.
3073688
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan
Salakhutdinov. 2014. Dropout: A Simple Way to Prevent Neural Networks from
Overfitting. Journal of Machine Learning Research 15 (2014), 1929–1958. http://jmlr.org/papers/v15/srivastava14a.html
Matt Stanton, Ben Humberston, Brandon Kase, James F. O’Brien, Kayvon Fatahalian,and Adrien Treuille. 2014. Self-refining Games Using Player Analytics. ACM Trans.
Graph. 33, 4, Article 73 (July 2014), 9 pages. https://doi.org/10.1145/2601097.2601196
Matt Stanton, Yu Sheng, Martin Wicke, Federico Perazzi, Amos Yuen, Srinivasa
Narasimhan, and Adrien Treuille. 2013. Non-polynomial Galerkin Projection on
Deforming Meshes. ACM Trans. Graph. 32, 4, Article 86 (July 2013), 14 pages.
https://doi.org/10.1145/2461912.2462006
Hang Su, Subhransu Maji, Evangelos Kalogerakis, and Erik Learned-Miller. 2015. Multiview Convolutional Neural Networks for 3D Shape Recognition. In Proceedings ofthe 2015 IEEE International Conference on Computer Vision (ICCV) (ICCV ’15). IEEE
Computer Society, Washington, DC, USA, 945–953. https://doi.org/10.1109/ICCV.
2015.114
Marco Tarini, Kai Hormann, Paolo Cignoni, and Claudio Montani. 2004. PolyCubeMaps. ACM Trans. Graph. 23, 3 (Aug. 2004), 853–860. https://doi.org/10.1145/
1015706.1015810
Jonathan Tompson, Kristofer Schlachter, Pablo Sprechmann, and Ken Perlin. 2016.
Accelerating Eulerian Fluid Simulation With Convolutional Networks. CoRR
abs/1607.03597 (2016). arXiv:1607.03597 http://arxiv.org/abs/1607.03597
Adrien Treuille, Andrew Lewis, and Zoran Popović. 2006. Model Reduction for Realtime Fluids. ACM Trans. Graph. 25, 3 (July 2006), 826–834. https://doi.org/10.1145/
1141911.1141962
K. Um, X. Hu, and N. Thuerey. 2017. Liquid Splash Modeling with Neural Networks.
ArXiv e-prints (April 2017). arXiv:cs.GR/1704.04456
Nobuyuki Umetani. 2017. Exploring Generative 3D Shapes Using Autoencoder Networks. In SIGGRAPH Asia 2017 Technical Briefs (SA ’17). ACM, New York, NY, USA,
Article 24, 4 pages. https://doi.org/10.1145/3145749.3145758
Nobuyuki Umetani, Yuki Koyama, Ryan Schmidt, and Takeo Igarashi. 2014. Pteromys:
Interactive Design and Optimization of Free-formed Free-flight Model Airplanes.
ACM Trans. Graph. 33, 4, Article 65 (July 2014), 10 pages. https://doi.org/10.1145/
2601097.2601129
Peng-Shuai Wang, Yang Liu, Yu-Xiao Guo, Chun-Yu Sun, and Xin Tong. 2017. O-CNN:
Octree-based Convolutional Neural Networks for 3D Shape Analysis. ACM Trans.
Graph. 36, 4, Article 72 (July 2017), 11 pages. https://doi.org/10.1145/3072959.3073608
Martin Wicke, Matt Stanton, and Adrien Treuille. 2009. Modular Bases for Fluid
Dynamics. ACM Trans. Graph. 28, 3, Article 39 (July 2009), 8 pages. https://doi.org/
10.1145/1531326.1531345
Zhirong Wu, Shuran Song, Aditya Khosla, Xiaoou Tang, and Jianxiong Xiao. 2014.
3D ShapeNets for 2.5D Object Recognition and Next-Best-View Prediction. CoRR
abs/1406.5670 (2014). http://arxiv.org/abs/1406.5670
Kai Xu, Vladimir G Kim, Qixing Huang, Niloy Mitra, and Evangelos Kalogerakis. 2016.
Data-driven shape analysis and processing. In SIGGRAPH ASIA 2016 Courses. ACM,
4.
Olek C Zienkiewicz, Robert L Taylor, and P. Nithiarasu. 2013. The Finite Element
Method for Fluid Dynamics, Seventh Edition (7 ed.). Butterworth-Heinemann. http://amazon.com/o/ASIN/1856176355/

Submission ID: 285. 2018-05-19 12:58. Page 10 of 1–10.