Learning Three-dimensional Flow
for Interactive Aerodynamic Design

Nobuyuki Umetani1, Bernd Bickel2
12Motivation• Predicting aerodynamics at an interactive rate

Aerodynamics is Important

Images from Shutterstock and PixaBay

Computational Fluid Dynamics• Traditional workflow is expensive & slow
outputinput

3D shape

meshgeneration

NS equation• Cd value• pressure field• velocity fieldReplacing CFD with MLparameterization uncertainty parameterization
input

outputparameter

regressor………………………….

………….

3D shape

input
parameter

output• Cd value• pressure field• velocity fieldParameterization Problem• Shape need to be represented by fixed
dimensional vector/tensor
3D shape
2D Image

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A
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3D field0153.3
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Parameterization Problem• Triangle mesh / NURBS are not suitable for ML– Topology / #points are not constant

Triangle mesh

NURBS

Related Work: Voxel Model• Difficult to handle detailed 3D shape[FPNN, Li et al., 2017][3D GAN, 2016] [Girdhar et al., 2016] [O-CNN, Wang et al., 2017][liu et al., 2017] [OctNet, Riegler et al., 2017]

Related Work: Point-based Model• Ordering of the point is not consistent[PointNet, Qi et al, 2017][PointNet++, Qi et al, 2017] [Gadelha et al.,2017][DeformNet, Kurenkov et.al, 2017] [Fan et al, 2017]

Our Approach: Mesh Representation• PolyCube as a template quad mesh
• Deforming the quad mesh into the input shapesubdivide
designing
PolyCube

deformation

3D Shape as “Height Field”• “Height field” from a PolyCube in normal directions[Umetani 2018], Normal Meshes [Guskov et al. 2000]

height2.5 dimensional

3 dimensional

Hierarchical Projection• We repeat subdivision and projection• Key observation: concave shape is locally convex

initialization

subdivision & projection

Parameterization Example

Parameterization Example

Parameterization Example

Parameterization Example

Parameterization Example

Parameterization Example

Parameterization Example

Parameterization Example

Parameterization Example

Overview of Our Approach
input

parameterization

inputparameteroutput
parameterregressor………………………….

………….

3D shape

parameterization

output• pressure field• velocity field• Cd value

Challenge for Velocity Field Learning• Velocity changes rapidly at the boundary

V ' V1

V '0

Challenge for Velocity Field Learning• Velocity changes rapidly at the boundary

widenshapeChallenge for Velocity Field Learning• Velocity changes rapidly at the boundary

V ' V1

V '0

Challenge for Velocity Field Learning• Regular grid has too much nonlinearity

V ' V1V '0

Parameterization of the Velocity Field
We cannot use fixed gridParameterization of the Velocity Field
We cannot use fixed grid

We use conformable gridParameterization of the Velocity Field• Continuous representation of velocity field J
3D Mean Value Coordinate [Tao et al, 2005]

extendshape

Parameterization Result

Parameterization Result

Challenges
parameterization
input

input
parameterparameterization
outputoutputparameter

regressor………………………….

………….

3D shape

uncertainty• Cd value• pressure field• velocity fieldChallenges
input

inputC/F/Adinput………….

3D shape3D shape
F : drag force

A: arearegressor………………………….

parameter

outputparameter

output• Cd value• pressure field• velocity fieldCd Value is Important• Car design is strongly influenced by the Cd value
1951

1980

1995

2005

2017

Citroen 2CV

Ford Mercury

Ford Falcon

Ford Fusion

Tesla Model3

Cd=0.51

Cd=0.36

Cd=0.31

Cd=0.275

Cd=0.23

Preview of Result

Preview of Result
predicted error range

Regression: Gaussian Process(GP)• GP: non-parametric Bayesian regression model– Handles nonlinearity well J– Learning from few samples without over–fitting J– Prediction of error J

Regression: Gaussian Process(GP)• GP: non-parametric Bayesian regression modelInterpolation weight is determined by distances
Cd = 2.5
interpolation

Cd = 2.7

Cd = 2.7

Cd =?
Cd = 2.8

Regression: Gaussian Process(GP)• GP: non-parametric Bayesian regression model
Covariance (similarity of the distributions) is determined by distances
Cd = 2.5
interpolation

Cdd = 2.7

Cd = 2.7

CCd d==?
Cd = 2.8

Regression: Gaussian Process(GP)• Standard deviation gives how “confident” theprediction isless confident

Cd =confident

Cd =Distance Metric Optimization• Distance metric is optimized for maximizing likelihood of
the training data
bad distance metricdifferent

Cd =

Cd =

Cd =

good distance metricsimilar

Cd =

Cd =

Cd =

Cd =

Cd =
Cd =

Cd =

The Weights in the Distance Metricheight difference
separation location

distributed weightfor heights

stagnant
weight low

separated flow
weight highTraining Data
• 889 CFD simulation for car
shapes from ShapeNet• Each simulation takes 3 hourscomputed on Amazon AWS• All data available

http://www.nobuyuki-umetani.com/

-0.05

number of samples
0-0.02

-0.03

-0.04

error in
0.05

0.04

0.03

0.02

0.01

0.00

-0.01

Accuracy= 0.01

50

4030

2010

Live Demo!

Shape with Genus One Topology
PolyCube3D Printed Hood Ornament

Limitations• Prediction limited by training data
• Highly concave shapeside mirror

Future Work• Convolution operation on the PolyCube• Application to more complex phenomena– thermal convection, acoustics

spoilerAcknowledgement• Anonymous reviewers• Alec Jacobson, Ryan Schmidt, Eitan Grinspun
Funding:• European Research Council (ERC) No 715767 -- MATERIALIZABLE

Learning Three-dimensional Flowfor Interactive Aerodynamic Design

Nobuyuki Umetani, Bernd Bickel