Abstract

The authors present a generative adversarial network (GAN) model that demonstrates how to generate 3D models in their native format so that they can be either evaluated using complex simulation environments or realized using methods such as additive manufacturing. Once initially trained, the GAN can create additional training data itself by generating new designs, evaluating them in a physics-based virtual environment, and adding the high performing ones to the training set. A case study involving a GAN model that is initially trained on 4045 3D aircraft models is used for demonstration, where a training data set that has been updated with GAN-generated and evaluated designs results in enhanced model generation, in both the geometric feasibility and performance of the designs. Z-tests on the performance scores of the generated aircraft models indicate a statistically significant improvement in the functionality of the generated models after three iterations of the training-evaluation process. In the case study, a number of techniques are explored to structure the generate-evaluate process in order to balance the need to generate feasible designs with the need for innovative designs.

References

References
1.
Burnap
,
A.
,
Liu
,
Y.
,
Pan
,
Y.
,
Lee
,
H.
,
Gonzalez
,
R.
, and
Papalambros
,
P. Y.
,
2016
, “
Estimating and Exploring the Product Form Design Space Using Deep Generative Models
,”
42nd Design Automation Conference
,
Charlotte, NC
,
Aug. 21–24
, Vol.
2A
, p.
V02AT03A013
.
2.
Dering
,
M.
, and
Tucker
,
C.
,
2017
, “
Generative Adversarial Networks for Increasing the Veracity of Big Data
,”
2017 IEEE International Conference on Big Data
,
Boston, MA
,
Dec. 11–14
, pp.
2595
2602
.
3.
Dering
,
M. L.
, and
Tucker
,
C. S.
,
2017
,
Implications of Generative Models in Government
,
Reports of the 2017 AAAI Fall Symposium Series
,
Nov
.
9–11
, pp.
158
163
.
4.
Tran
,
D.
,
Bourdev
,
L.
,
Fergus
,
R.
,
Torresani
,
L.
, and
Paluri
,
M.
,
2015
, “
Learning Spatiotemporal Features With 3D Convolutional Networks
,”
IEEE International Conference on Computer Vision
,
Santiago, Chile
,
Dec. 13–16
, pp.
4489
4497
.
5.
Chan
,
T.-H.
,
Jia
,
K.
,
Gao
,
S.
,
Lu
,
J.
,
Zeng
,
Z.
, and
Ma
,
Y.
,
Dec. 2015
, “
PCANet: A Simple Deep Learning Baseline for Image Classification?
,”
IEEE Trans. Image Process.
,
24
(
12
), pp.
5017
5032
. 10.1109/TIP.2015.2475625
6.
Goodfellow
,
I.
,
Pouget-Abadie
,
J.
,
Mirza
,
M.
,
Xu
,
B.
,
Warde-Farley
,
D.
,
Ozair
,
S.
,
Courville
,
A.
, and
Bengio
,
Y.
,
2014
, “
Generative Adversarial Nets
,”
Advances in Neural Information Processing Systems 27
,
Montréal, Canada
,
Dec. 8–13
, pp.
2672
2680
.
7.
Denton
,
E. L.
,
Chintala
,
S.
, and
Fergus
,
R.
,
2015
, “
Deep Generative Image Models Using a Laplacian Pyramid of Adversarial Networks
,”
Advances in Neural Information Processing Systems
, pp.
1486
1494
.
8.
Venugopalan
,
S.
,
Xu
,
H.
,
Donahue
,
J.
,
Rohrbach
,
M.
,
Mooney
,
R.
, and
Saenko
,
K.
,
2015
, “
Translating Videos to Natural Language Using Deep Recurrent Neural Networks
,”
The 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies
,
Denver, CO
,
May 31–June 5
.
9.
Zheng
,
S.
,
Jayasumana
,
S.
,
Romera-Paredes
,
B.
,
Vineet
,
V.
,
Su
,
Z.
,
Du
,
D.
,
Huang
,
C.
, and
Torr
,
P. H.
,
2015
, “
Conditional Random Fields as Recurrent Neural Networks
,”
2015 IEEE International Conference on Computer Vision
,
Washington, DC
,
Dec. 7–13
, pp.
1529
1537
.
10.
Maier
,
J. R.
,
Fadel
,
G. M.
, and
Battisto
,
D. G.
,
2009
, “
An Affordance-Based Approach to Architectural Theory, Design, and Practice
,”
Desi. Stud.
,
30
(
4
), pp.
393
414
. 10.1016/j.destud.2009.01.002
11.
Ferguson
,
S.
,
Siddiqi
,
A.
,
Lewis
,
K.
, and
de Weck
,
O. L.
,
2007
, “
Flexible and Reconfigurable Systems: Nomenclature and Review
,”
ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
,
Las Vegas, NV
,
Sept. 4–7
, Vol.
6
, pp.
249
263
.
12.
Umeda
,
Y.
,
Ishii
,
M.
,
Yoshioka
,
M.
,
Shimomura
,
Y.
, and
Tomiyama
,
T.
,
1996
, “
Supporting Conceptual Design Based on the Function-Behavior-State Modeler
,”
Ai Edam
,
10
(
4
), pp.
275
288
. 10.1017/s0890060400001621
13.
Kang
,
S. W.
, and
Tucker
,
C.
,
2016
, “
An Automated Approach to Quantifying Functional Interactions by Mining Large-Scale Product Specification Data
,”
J. Eng. Des.
,
27
(
1–3
), pp.
1
24
. 10.1080/09544828.2015.1083539
14.
Christensen
,
B. T.
, and
Ball
,
L. J.
,
2016
, “
Dimensions of Creative Evaluation: Distinct Design and Reasoning Strategies for Aesthetic, Functional and Originality Judgments
,”
Desi. Stud.
,
45
(
Special Issue: Design Review Conversations
), pp.
116
136
.
15.
Bohm
,
M. R.
,
Stone
,
R. B.
,
Simpson
,
T. W.
, and
Steva
,
E. D.
,
2008
, “
Introduction of a Data Schema to Support a Design Repository
,”
Comput.-Aided Des.
,
40
(
7
), pp.
801
811
. 10.1016/j.cad.2007.09.003
16.
Dering
,
M.
,
Cunningham
,
J.
,
Desai
,
R.
,
Yukish
,
M. A.
,
Simpson
,
T. W.
, and
Tucker
,
C. S.
,
2018
, “
A Physics-Based Virtual Environment for Enhancing the Quality of Deep Generative Designs
,”
ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Quebec City, Canada
,
Aug. 26–29
.
17.
Bengio
,
Y.
,
Mesnil
,
G.
,
Dauphin
,
Y.
, and
Rifai
,
S.
,
2013
, “
Better Mixing Via Deep Representations
,”
The 30th International Conference on Machine Learning
,
Atlanta, GA
,
June 16–21
, pp.
552
560
.
18.
Gurumurthy
,
S.
,
Kiran Sarvadevabhatla
,
R.
, and
Venkatesh Babu
,
R.
,
2017
, “
Deligan: Generative Adversarial Networks for Diverse and Limited Data
,”
The IEEE Conference on Computer Vision and Pattern Recognition
,
Honolulu, HI
,
July 21–26
, pp.
166
174
.
19.
Ulu
,
N. G.
, and
Kara
,
L. B.
,
2015
, “
DMS2015-33: Generative Interface Structure Design for Supporting Existing Objects
,”
J. Vis. Lang. Comput.
,
31
(
Special Issue on DMS2015
), pp.
171
183
.
20.
Andrade
,
D.
,
Harada
,
M.
, and
Shimada
,
K.
,
2017
, “
Framework for Automatic Generation of Facades on Free-Form Surfaces
,”
Front. Archit. Res.
,
6
(
3
), pp.
273
289
. 10.1016/j.foar.2017.04.003
21.
Whiting
,
M.
,
Cagan
,
J.
, and
LeDuc
,
P.
,
2017
, “Automated Induction of General Grammars for Design,”
Design Computing and Cognition '16
,
J. S.
Gero
, ed.,
Springer
,
Cham, Switzerland
, pp.
267
278
.
22.
Pu
,
Y.
,
Gan
,
Z.
,
Henao
,
R.
,
Yuan
,
X.
,
Li
,
C.
,
Stevens
,
A.
, and
Carin
,
L.
,
2016
, “
Variational Autoencoder for Deep Learning of Images, Labels and Captions
,”
Advances in Neural Information Processing Systems 29
,
Barcelona, Spain
,
Dec. 5–10
, pp.
2352
2360
.
23.
Genevay
,
A.
,
Peyré
,
G.
and
Cuturi
,
M.
,
2017
, “
GAN and VAE from an Optimal Transport Point of View
,” http://arxiv.org/abs/1706.01807.
24.
Kingma
,
D. P.
, and
Welling
,
M.
,
2014
, “
Auto-Encoding Variational Bayes
,”
2nd International Conference on Learning Representations
,
Banff, Canada
,
Apr. 14–16
.
25.
Arjovsky
,
M.
,
Chintala
,
S.
, and
Bottou
,
L.
,
2017
, “
Wasserstein Generative Adversarial Networks
,”
The 34th International Conference on Machine Learning
,
Sydney, Australia
,
Aug. 6–11
.
26.
Chen
,
W.
,
Fuge
,
M.
, and
Chazan
,
J.
,
2017
, “
Design Manifolds Capture the Intrinsic Complexity and Dimension of Design Spaces
,”
ASME J. Mech. Des.
,
139
(
5
), p.
051102
. 10.1115/1.4036134
27.
Dosovitskiy
,
A.
,
Springenberg
,
J. T.
,
Tatarchenko
,
M.
, and
Brox
,
T.
,
2017
, “
Learning to Generate Chairs, Tables and Cars With Convolutional Networks
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
39
(
4
), pp.
692
705
. 10.1109/tpami.2016.2567384
28.
Cang
,
R.
,
Vipradas
,
A.
, and
Ren
,
Y.
,
2017
, “
Scalable Microstructure Reconstruction with Multi-Scale Pattern Preservation
,”
ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Cleveland, OH
,
Aug. 6–9
,
American Society of Mechanical Engineers
,
New York
, p.
V02BT03A010
.
29.
Cang
,
R.
,
Xu
,
Y.
,
Chen
,
S.
,
Liu
,
Y.
,
Jiao
,
Y.
, and
Ren
,
M. Y.
,
2017
, “
Microstructure Representation and Reconstruction of Heterogeneous Materials Via Deep Belief Network for Computational Material Design
,”
ASME J. Mech. Des.
,
139
(
7
), p.
071404
. 10.1115/1.4036649
30.
Wu
,
J.
,
Zhang
,
C.
,
Xue
,
T.
,
Freeman
,
B.
, and
Tenenbaum
,
J.
,
2016
, “
Learning a Probabilistic Latent Space of Object Shapes Via 3d Generative-Adversarial Modeling
,”
Advances in Neural Information Processing Systems 29
,
Barcelona, Spain
,
Dec. 5–10
, pp.
82
90
.
31.
Ben-Hamu
,
H.
,
Maron
,
H.
,
Kezurer
,
I.
,
Avineri
,
G.
, and
Lipman
,
Y.
,
2018
, “
Multi-Chart Generative Surface Modeling
,”
ACM Transactions on Graphics
,
37
(
6
).
Article No. 215
.
32.
Tan
,
Q.
,
Gao
,
L.
,
Lai
,
Y. K.
, and
Xia
,
S.
,
2018
, “
Variational Autoencoders for Deforming 3d Mesh Models
,”
The IEEE Conference on Computer Vision and Pattern Recognition
,
Salt Lake City, UT
,
June 18–22
, pp.
5841
5850
.
33.
Gao
,
L.
,
Lai
,
Y. K.
,
Liang
,
D.
,
Chen
,
S. Y.
, and
Xia
,
S.
,
2016
, “
Efficient and Flexible Deformation Representation for Data-Driven Surface Modeling
,”
ACM Trans. Graphics (TOG)
,
35
(
5
), p.
158
. 10.1145/2908736
34.
Groueix
,
T.
,
Fisher
,
M.
,
Kim
,
V. G.
,
Russell
,
B. C.
, and
Aubry
,
M.
,
2018
, “
A Papier-Mâché Approach to Learning 3D Surface Generation
,”
The IEEE Conference on Computer Vision and Pattern Recognition
,
Salt Lake City, UT
,
June 18–22
, pp.
216
224
.
35.
Nash
,
C.
, and
Williams
,
C. K.
,
2017
, “
The Shape Variational Autoencoder: A Deep Generative Model of Part-Segmented 3D Objects
,”
Comput. Graphics Forum
,
36
(
5
), pp.
1
12
. 10.1111/cgf.13240
36.
Achlioptas
,
P.
,
Diamanti
,
O.
,
Mitliagkas
,
I.
, and
Guibas
,
L.
,
2017
, “
Learning Representations and Generative Models for 3d Point Clouds
,”
Thirty-fifth International Conference on Machine Learning
,
Stockholm, Sweden
,
July 10–15
.
37.
Li
,
C. L.
,
Zaheer
,
M.
,
Zhang
,
Y.
,
Poczos
,
B.
, and
Salakhutdinov
,
R.
,
2019
, “
Point Cloud Gan
,”
ICLR Workshop on Deep Generative Models for Highly Structured Data
,
New Orleans, LA
,
May 6–9
.
38.
Clayton
,
M. J.
,
Teicholz
,
P.
,
Fischer
,
M.
, and
Kunz
,
J.
,
1999
, “
Virtual Components Consisting of Form, Function and Behavior
,”
Autom. Constr.
,
8
(
3
), pp.
351
367
. 10.1016/S0926-5805(98)00082-X
39.
Umeda
,
Y.
,
Kondoh
,
S.
,
Shimomura
,
Y.
, and
Tomiyama
,
T.
,
2005
, “
Development of Design Methodology for Upgradable Products Based on Function–Behavior–State Modeling
,”
Ai Edam
,
19
(
3
), pp.
161
182
. 10.1017/s0890060405050122
40.
Crilly
,
N.
,
Moultrie
,
J.
, and
Clarkson
,
P. J.
,
2004
, “
Seeing Things: Consumer Response to the Visual Domain in Product Design
,”
Desi. Stud.
,
25
(
6
), pp.
547
577
. 10.1016/j.destud.2004.03.001
41.
Balduzzi
,
F.
,
Ferrara
,
G.
,
Babbini
,
A.
, and
Pratelli
,
G.
,
2012
, “
CFD Evaluation of the Pressure Losses in a Reciprocating Compressor: A Flexible Approach
,”
ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis
,
Nantes, France
,
July 2–4
,
American Society of Mechanical Engineers
,
New York
, pp.
63
72
.
42.
Jeong
,
H. Y.
,
Ha
,
K. S.
,
Kwon
,
Y. M.
,
Lee
,
Y. B.
,
Hahn
,
D.
,
Cahalan
,
J. E.
, and
Dunn
,
F. E.
,
2007
, “
Evaluation of the Conduction Shape Factor With a CFD Code for a Liquid–Metal Heat Transfer in Heated Triangular Rod Bundles
,”
Nucl. Eng. Des.
,
237
(
6
), pp.
648
654
. 10.1016/j.nucengdes.2006.09.030
43.
Conner
,
M. E.
,
Baglietto
,
E.
, and
Elmahdi
,
A. M.
,
2010
, “
CFD Methodology and Validation for Single-Phase Flow in PWR Fuel Assemblies
,”
Nucl. Eng. Des.
,
240
(
9
), pp.
2088
2095
. 10.1016/j.nucengdes.2009.11.031
44.
Marchesse
,
Y.
,
Changenet
,
C.
,
Ville
,
F.
, and
Velex
,
P.
,
2011
, “
Investigations on CFD Simulations for Predicting Windage Power Losses in Spur Gears
,”
ASME J. Mech. Des.
,
133
(
2
), p.
024501
. 10.1115/1.4003357
45.
Krepper
,
E.
,
Končar
,
B.
, and
Egorov
,
Y.
,
2007
, “
CFD Modelling of Subcooled Boiling—Concept, Validation and Application to Fuel Assembly Design
,”
Nucl. Eng. Des.
,
237
(
7
), pp.
716
731
. 10.1016/j.nucengdes.2006.10.023
46.
Montazeri
,
H.
, and
Blocken
,
B.
,
2013
, “
CFD Simulation of Wind-Induced Pressure Coefficients on Buildings With and Without Balconies: Validation and Sensitivity Analysis
,”
Building and Environment
,
60
, pp.
137
149
.
47.
Dye
,
J. R.
,
Tay
,
Y. Y.
, and
Lankarani
,
H. M.
,
2015
, “
Development and Application of Planar Computational General-Purpose Constrained Multibody Simulations in Matlab with Simple Graphical/Visualization Capability
,”
ASME 2015 International Mechanical Engineering Congress and Exposition
,
Houston, TX
,
Nov. 13–19
,
American Society of Mechanical Engineers
,
New York
, p.
V04BT04A002
.
48.
Sam
,
R.
,
Arrifin
,
K.
, and
Buniyamin
,
N.
,
2012
, “
Simulation of Pick and Place Robotics System Using Solidworks Softmotion
,”
2012 International Conference on System Engineering and Technology
,
Bandung, Indonesia
,
Sept. 11–12
,
IEEE
, pp.
1
6
.
49.
Turrell
,
M. D.
,
Stopford
,
P. J.
,
Syed
,
K. J.
, and
Buchanan
,
E.
,
2004
, “
CFD Simulation of the Flow Within and Downstream of a High-Swirl Lean Premixed gas Turbine Combustor
,”
ASME Turbo Expo 2004: Power for Land, Sea, and Air
,
Vienna, Austria
,
June 14–17
,
American Society of Mechanical Engineers
,
New York
, pp.
31
38
.
50.
Field
,
D. A.
,
2004
, “
Education and Training for CAD in the Auto Industry
,”
Comput.-Aided Des.
,
36
(
14
), pp.
1431
1437
. 10.1016/j.cad.2003.10.007
51.
Rozvany
,
G. I.
,
2014
,
Topology Optimization in Structural Mechanics
,
Springer
,
New York
,
Vol. 374
.
52.
Zhu
,
J. H.
,
Zhang
,
W. H.
, and
Xia
,
L.
,
2016
, “
Topology Optimization in Aircraft and Aerospace Structures Design
,”
Arch. Comput. Meth. Eng.
,
23
(
4
), pp.
595
622
. 10.1007/s11831-015-9151-2
53.
Xia
,
L.
, and
Breitkopf
,
P.
,
2017
, “
Recent Advances on Topology Optimization of Multiscale Nonlinear Structures
,”
Arch. Comput. Meth. Eng.
,
24
(
2
), pp.
227
249
. 10.1007/s11831-016-9170-7
54.
Kanno
,
Y.
,
2016
, “
Redundancy Optimization of Finite-Dimensional Structures: Concept and Derivative-Free Algorithm
,”
J. Struct. Eng.
,
143
(
1
), p.
04016151
. 10.1061/(ASCE)ST.1943-541X.0001630
55.
Mohammadi
,
B.
, and
Pironneau
,
O.
,
2010
,
Applied Shape Optimization for Fluids
,
2nd ed
.,
Oxford University Press
,
New York
.
56.
Borrvall
,
T.
, and
Petersson
,
J.
,
2003
, “
Topology Optimization of Fluids in Stokes Flow
,”
Int. J. Numer. Methods Fluids
,
41
(
1
), pp.
77
107
. 10.1002/fld.426
57.
Zhou
,
S.
, and
Li
,
Q.
,
2008
, “
A Variational Level Set Method for the Topology Optimization of Steady-State Navier–Stokes Flow
,”
J. Comput. Phys.
,
227
(
24
), pp.
10178
10195
. 10.1016/j.jcp.2008.08.022
58.
Othmer
,
C.
,
Manosalvas-Kjono
,
D. E.
,
Jameson
,
A.
, and
Alonso
,
J. J.
,
2017
, “
Aerodynamic Topology Optimization: Some Observations on Hysteresis in Separated Flows
,”
23rd AIAA Computational Fluid Dynamics Conference
,
Denver, CO
,
June 5–9
, p.
4413
.
59.
Zegard
,
T.
, and
Paulino
,
G. H.
,
2016
, “
Bridging Topology Optimization and Additive Manufacturing
,”
Struct. Multi. Optim.
,
53
(
1
), pp.
175
192
. 10.1007/s00158-015-1274-4
60.
Langelaar
,
M.
,
2016
, “
Topology Optimization of 3D Self-Supporting Structures for Additive Manufacturing
,”
Additive Manufacturing
,
12
, pp.
60
70
. 10.1016/j.addma.2016.06.010
61.
Guo
,
X.
,
Zhou
,
J.
,
Zhang
,
W.
,
Du
,
Z.
,
Liu
,
C.
, and
Liu
,
Y.
,
2017
, “
Self-supporting Structure Design in Additive Manufacturing Through Explicit Topology Optimization
,”
Computer Methods in Applied Mechanics and Engineering
,
323
, pp.
27
63
.
62.
Brackett
,
D.
,
Ashcroft
,
I.
, and
Hague
,
R.
, “
Topology Optimization for Additive Manufacturing
,”
Proceedings of the Solid Freeform Fabrication Symposium
,
Aug. 2011
,
Austin, TX
,
Vol. 1
, pp.
348
362
.
63.
Bendsoe
,
M. P.
, and
Sigmund
,
O.
,
2004
,
Topology Optimization: Theory, Methods and Applications
,
2nd ed.
,
Springer-Verlag
,
Berlin, Heidelberg, New York
.
64.
Allaire
,
G.
,
Jouve
,
F.
, and
Toader
,
A. M.
,
2002
, “
A Level-Set Method for Shape Optimization
,”
C.R. Math.
,
334
(
12
), pp.
1125
1130
. 10.1016/S1631-073X(02)02412-3
65.
Xie
,
Y. M.
, and
Steven
,
G. P.
,
1993
, “
A Simple Evolutionary Procedure for Structural Optimization
,”
Comput. Struct.
,
49
(
5
), pp.
885
896
. 10.1016/0045-7949(93)90035-C
66.
Liu
,
J.
, and
Ma
,
Y.
,
2016
, “
A Survey of Manufacturing Oriented Topology Optimization Methods
,”
Advances in Engineering Software
,
100
, pp.
161
175
.
67.
Sigmund
,
O.
, and
Maute
,
K.
,
2013
, “
Topology Optimization Approaches
,”
Struct. Multi. Optim.
,
48
(
6
), pp.
1031
1055
. 10.1007/s00158-013-0978-6
68.
Guo
,
X.
,
Li
,
W.
, and
Iorio
,
F.
,
2016
, “
Convolutional Neural Networks for Steady Flow Approximation
,”
Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining
,
San Francisco, CA
,
Aug. 13–17
,
ACM
, pp.
481
490
.
69.
Oh
,
S.
,
Jung
,
Y.
,
Kim
,
S.
,
Lee
,
I.
, and
Kang
,
N.
,
2019
, “
Deep Generative Design: Integration of Topology Optimization and Generative Models
,”
ASME J. Mech., Des.
,
141
(
11
), p.
111405
. 10.1115/1.4044229
70.
Lei
,
X.
,
Liu
,
C.
,
Du
,
Z.
,
Zhang
,
W.
, and
Guo
,
X.
,
2019
, “
Machine Learning-Driven Real-Time Topology Optimization Under Moving Morphable Component-Based Framework
,”
ASME J. Appl. Mech.
,
86
(
1
), p.
011004
. 10.1115/1.4041319
71.
Montgomery
,
D. C.
,
2012
,
Design and Analysis of Experiments
,
8th ed
,
John Wiley & Sons Inc.
,
Hoboken, NJ
.
72.
Qi
,
C. R.
,
Su
,
H.
,
Mo
,
K.
, and
Guibas
,
L. J.
,
2017
, “
Pointnet: Deep Learning on Point Sets for 3d Classification and Segmentation
,”
The IEEE Conference on Computer Vision and Pattern Recognition
,
Honolulu, HI
,
July 21–26
.
73.
Chang
,
A. X.
,
Funkhouser
,
T.
,
Guibas
,
L.
,
Hanrahan
,
P.
,
Huang
,
Q.
,
Li
,
Z.
,
Savarese
,
S.
,
Savva
,
M.
,
Song
,
S.
,
Su
,
H.
and
Xiao
,
J.
,
2015
, “
Shapenet: An Information-Rich 3d Model Repository
,” eprint arXiv preprint .
74.
Kondoh
,
T.
,
Matsumori
,
T.
, and
Kawamoto
,
A.
,
2012
, “
Drag Minimization and Lift Maximization in Laminar Flows via Topology Optimization Employing Simple Objective Function Expressions Based on Body Force Integration
,”
Struct. Multi. Optim.
,
45
(
5
), pp.
693
701
. 10.1007/s00158-011-0730-z
You do not currently have access to this content.