Abstract
In topology optimization using deep learning, the load and boundary conditions represented as vectors or sparse matrices often miss the opportunity to encode a rich view of the design problem, leading to less than ideal generalization results. We propose a new data-driven topology optimization model called TopologyGAN that takes advantage of various physical fields computed on the original, unoptimized material domain, as inputs to the generator of a conditional generative adversarial network (cGAN). Compared to a baseline cGAN, TopologyGAN achieves a nearly 3 × reduction in the mean squared error and a 2.5 × reduction in the mean absolute error on test problems involving previously unseen boundary conditions. Built on several existing network models, we also introduce a hybrid network called U-SE(Squeeze-and-Excitation)-ResNet for the generator that further increases the overall accuracy. We publicly share our full implementation and trained network.
Issue Section:
Design Automation
References
1.
Bendsøe
, M. P.
, and Kikuchi
, N.
, 1988
, “Generating Optimal Topologies in Structural Design Using a Homogenization Method
,” Comput. Methods Appl. Mech. Eng.
, 71
(2
), pp. 197
–224
. 10.1016/0045-7825(88)90086-22.
Suzuki
, K.
, and Kikuchi
, N.
, 1991
, “A Homogenization Method for Shape and Topology Optimization
,” Comput. Methods Appl. Mech. Eng.
, 93
(3
), pp. 291
–318
. 10.1016/0045-7825(91)90245-23.
Bendsøe
, M. P.
, 2009
, Topology Optimization
, Springer
, Berlin/Heidelberg
.4.
Diaz
, A.
, and Bendsøe
, M.
, 1992
, “Shape Optimization of Structures for Multiple Loading Conditions Using a Homogenization Method
,” Struct. Optim.
, 4
(1
), pp. 17
–22
. 10.1007/BF018940775.
Bendsøe
, M. P.
, 1989
, “Optimal Shape Design as a Material Distribution Problem
,” Struct. Optim.
, 1
(4
), pp. 193
–202
. 10.1007/BF016509496.
Bendsøe
, M. P.
, and Sigmund
, O.
, 1999
, “Material Interpolation Schemes in Topology Optimization
,” Arch. Appl. Mech.
, 69
(9–10
), pp. 635
–654
. 10.1007/s0041900502487.
Sigmund
, O.
, 2007
, “Morphology-Based Black and White Filters for Topology Optimization
,” Struct. Multidiscipl. Optim.
, 33
(4–5
), pp. 401
–424
. 10.1007/s00158-006-0087-x8.
Biyikli
, E.
, and To
, A. C.
, 2015
, “Proportional Topology Optimization: A New Non-sensitivity Method for Solving Stress Constrained and Minimum Compliance Problems and Its Implementation in Matlab
,” PLoS ONE
, 10
(12
), p. e0145041
. 10.1371/journal.pone.01450419.
Nie
, Z.
, Jung
, S.
, Kara
, L. B.
, and Whitefoot
, K.
, 2019
, “Optimization of Part Consolidation for Minimum Production Costs and Time Using Additive Manufacturing
,” ASME J. Mech. Des.
, 142
(7
), p. 072001
. 10.1115/1.404510610.
Suresh
, K.
, 2010
, “A 199-Line Matlab Code for Pareto-Optimal Tracing in Topology Optimization
,” Struct. Multidiscipl. Optim.
, 42
(5
), pp. 665
–679
. 10.1007/s00158-010-0534-611.
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-C12.
Chu
, D. N.
, Xie
, Y.
, Hira
, A.
, and Steven
, G.
, 1997
, “On Various Aspects of Evolutionary Structural Optimization for Problems With Stiffness Constraints
,” Finite Elements Anal. Des.
, 24
(4
), pp. 197
–212
. 10.1016/S0168-874X(96)00049-213.
Querin
, O.
, Steven
, G.
, and Xie
, Y.
, 1998
, “Evolutionary Structural Optimisation (ESO) Using a Bidirectional Algorithm
,” Eng. Comput.
, 15
(8
), pp. 1031
–1048
. 10.1108/0264440981024412914.
Young
, V.
, Querin
, O. M.
, Steven
, G.
, and Xie
, Y.
, 1999
, “3d and Multiple Load Case Bi-directional Evolutionary Structural Optimization (BESO)
,” Struct. Optim.
, 18
(2–3
), pp. 183
–192
. 10.1007/BF0119599315.
Reynolds
, D.
, McConnachie
, J.
, Bettess
, P.
, Christie
, W.
, and Bull
, J.
, 1999
, “Reverse Adaptivity–a New Evolutionary Tool for Structural Optimization
,” Int. J. Numer. Methods Eng.
, 45
(5
), pp. 529
–552
. 10.1002/(SICI)1097-0207(19990620)45:5<529::AID-NME599>3.0.CO;2-216.
Liu
, J.
, Parks
, G.
, and Clarkson
, P.
, 2000
, “Metamorphic Development: A New Topology Optimization Method for Continuum Structures
,” Struct. Multidiscipl. Optim.
, 20
(4
), pp. 288
–300
. 10.1007/s00158005015917.
Eschenauer
, H. A.
, Kobelev
, V. V.
, and Schumacher
, A.
, 1994
, “Bubble Method for Topology and Shape Optimization of Structures
,” Struct. Optim.
, 8
(1
), pp. 42
–51
. 10.1007/BF0174293318.
Eschenauer
, H.
, and Schumacher
, A.
, 1997
, “Topology and Shape Optimization Procedures Using Hole Positioning Criteria,” Topology Optimization in Structural Mechanics
, G. I. N.
Rozvany
, ed., Springer
, Vienna
, pp. 135
–196
.19.
Allaire
, G.
, Jouve
, F.
, and Toader
, A.-M.
, 2002
, “A Level-Set Method for Shape Optimization
,” Comp. Rend. Math.
, 334
(12
), pp. 1125
–1130
. 10.1016/S1631-073X(02)02412-320.
Allaire
, G.
, Jouve
, F.
, and Toader
, A.-M.
, 2004
, “Structural Optimization Using Sensitivity Analysis and a Level-Set Method
,” J. Comput. Phys.
, 194
(1
), pp. 363
–393
. 10.1016/j.jcp.2003.09.03221.
Allaire
, G.
, De Gournay
, F.
, Jouve
, F.
, and Toader
, A.-M.
, 2005
, “Structural Optimization Using Topological and Shape Sensitivity Via a Level Set Method
,” Control Cyber.
, 34
(1
), p. 59
.22.
Wang
, M. Y.
, Wang
, X.
, and Guo
, D.
, 2003
, “A Level Set Method for Structural Topology Optimization
,” Comput. Methods Appl. Mech. Eng.
, 192
(1–2
), pp. 227
–246
. 10.1016/S0045-7825(02)00559-523.
Challis
, V. J.
, and Guest
, J. K.
, 2009
, “Level Set Topology Optimization of Fluids in Stokes Flow
,” Int. J. Numer. Methods Eng.
, 79
(10
), pp. 1284
–1308
. 10.1002/nme.261624.
Suresh
, K.
, and Takalloozadeh
, M.
, 2013
, “Stress-Constrained Topology Optimization: a Topological Level-Set Approach
,” Struct. Multidiscipl. Optim.
, 48
(2
), pp. 295
–309
. 10.1007/s00158-013-0899-425.
Ulu
, E.
, Mccann
, J.
, and Kara
, L. B.
, 2017
, “Lightweight Structure Design Under Force Location Uncertainty
,” ACM Trans. Graph.
, 36
(4
), pp. 1
–13
. 10.1145/3072959.307362626.
Kelly
, D.
, and Elsley
, M.
, 1995
, “A Procedure for Determining Load Paths in Elastic Continua
,” Eng. Comput.
, 12
(5
), pp. 415
–424
. 10.1108/0264440951079972127.
Kelly
, D.
, Hsu
, P.
, and Asudullah
, M.
, 2001
, “Load Paths and Load Flow in Finite Element Analysis
,” Eng. Comput.
, 18
(1/2
), pp. 304
–313
. 10.1108/0264440011036592328.
Kelly
, D.
, Reidsema
, C.
, Bassandeh
, A.
, Pearce
, G.
, and Lee
, M.
, 2011
, “On Interpreting Load Paths and Identifying a Load Bearing Topology From Finite Element Analysis
,” Finite Elements Anal. Des.
, 47
(8
), pp. 867
–876
. 10.1016/j.finel.2011.03.00729.
Sobieszczanski-Sobieski
, J.
, 1999
, “Multidisciplinary Design Optimisation (MDO) Methods: Their Synergy With Computer Technology in the Design Process
,” Aeronaut. J.
, 103
(1026
), pp. 373
–382
. 10.1017/S000192400006459930.
Allison
, J. T.
, Kokkolaras
, M.
, and Papalambros
, P. Y.
, 2009
, “Optimal Partitioning and Coordination Decisions in Decomposition-Based Design Optimization
,” ASME J. Mech. Des.
, 131
(8
), p. 081008
. 10.1115/1.317872931.
Allison
, J. T.
, and Herber
, D. R.
, 2014
, “Special Section on Multidisciplinary Design Optimization: Multidisciplinary Design Optimization of Dynamic Engineering Systems
,” AIAA J.
, 52
(4
), pp. 691
–710
. 10.2514/1.J05218232.
Ulu
, E.
, McCann
, J.
, and Kara
, L. B.
, 2019
, “Structural Design Using Laplacian Shells,” Computer Graphics Forum
, H.
Hauser
and B.
Benes
, eds., Vol. 38
, Wiley Online Library
, New York
, pp. 85
–98
.33.
Sigmund
, O.
, and Maute
, K.
, 2013
, “Topology Optimization Approaches
,” Struct. Multidiscipl. Optim.
, 48
(6
), pp. 1031
–1055
. 10.1007/s00158-013-0978-634.
Guo
, T.
, Lohan
, D. J.
, Cang
, R.
, Ren
, M. Y.
, and Allison
, J. T.
, 2018
, “An Indirect Design Representation for Topology Optimization Using Variational Autoencoder and Style Transfer
,” 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
, Kissimmee, FL
, p. 0804
.35.
Yu
, Y.
, Hur
, T.
, Jung
, J.
, and Jang
, I. G.
, 2019
, “Deep Learning for Determining a Near-Optimal Topological Design Without Any Iteration
,” Struct. Multidiscipl. Optim.
, 59
(3
), pp. 787
–799
. 10.1007/s00158-018-2101-536.
Rawat
, S.
, and Shen
, M. H.
, 2019
, “Application of Adversarial Networks for 3d Structural Topology Optimization
,” Technical Report, SAE Technical Paper.37.
Sharpe
, C.
, and Seepersad
, C. C.
, 2005
, “Topology Design With Conditional Generative Adversarial Networks
,” ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Anaheim, CA
, Aug. 18–21
, American Society of Mechanical Engineers Digital Collection
, pp. 1
–9
.38.
Ulu
, E.
, Zhang
, R.
, and Kara
, L. B.
, 2016
, “A Data-Driven Investigation and Estimation of Optimal Topologies Under Variable Loading Configurations
,” Comput. Methods Biomech. Biomed. Eng.: Imag. Vis.
, 4
(2
), pp. 61
–72
. 10.1080/21681163.2015.103077539.
Raina
, A.
, McComb
, C.
, and Cagan
, J.
, 2019
, “Learning to Design From Humans: Imitating Human Designers Through Deep Learning
,” ASME J. Mech. Des.
, 141
(11
), p. 111102
. 10.1115/1.404425640.
Zhang
, W.
, Yang
, Z.
, Jiang
, H.
, Nigam
, S.
, Yamakawa
, S.
, Furuhata
, T.
, Shimada
, K.
, and Kara
, L. B.
, 2019
, “3d Shape Synthesis for Conceptual Design and Optimization Using Variational Autoencoders
,” ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Anaheim, CA
, Aug. 18–21
, American Society of Mechanical Engineers Digital Collection
, pp. 1
–10
.41.
Banga
, S.
, Gehani
, H.
, Bhilare
, S.
, Patel
, S.
, and Kara
, L.
, 2018
, “3d Topology Optimization Using Convolutional Neural Networks
,” preprint arXiv:1808.07440.42.
Cang
, R.
, Yao
, H.
, and Ren
, Y.
, 2019
, “One-Shot Generation of Near-Optimal Topology Through Theory-Driven Machine Learning
,” Comput. Aided Des.
, 109
(1
), 12
–21
. 10.1016/j.cad.2018.12.00843.
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
. 10.1115/1.404131944.
Kumar
, T.
, and Suresh
, K.
, 2019
, “A Density-and-Strain-Based K-clustering Approach to Microstructural Topology Optimization
,” Struct. Multidiscipl. Optim.
, 61
(4
), pp. 1
–17
. 10.1007/s00158-019-02422-445.
Rozvany
, G. I.
, Zhou
, M.
, and Birker
, T.
, 1992
, “Generalized Shape Optimization Without Homogenization
,” Struct. Optim.
, 4
(3–4
), pp. 250
–252
. 10.1007/BF0174275446.
Eschenauer
, H. A.
, and Olhoff
, N.
, 2001
, “Topology Optimization of Continuum Structures: A Review
,” ASME Appl. Mech. Rev.
, 54
(4
), pp. 331
–390
. 10.1115/1.138807547.
Plocher
, J.
, and Panesar
, A.
, 2019
, “Review on Design and Structural Optimisation in Additive Manufacturing: Towards Next-Generation Lightweight Structures
,” Mater. Des.
, 183
(5
), p. 108164
. 10.1016/j.matdes.2019.10816448.
Ambrosio
, L.
, and Buttazzo
, G.
, 1993
, “An Optimal Design Problem With Perimeter Penalization
,” Calc. Var. Partial Differ. Equ.
, 1
(1
), pp. 55
–69
. 10.1007/BF0216326449.
Petersson
, J.
, 1999
, “Some Convergence Results in Perimeter-Controlled Topology Optimization
,” Comput. Methods Appl. Mech. Eng.
, 171
(1–2
), pp. 123
–140
. 10.1016/S0045-7825(98)00248-550.
Sharpe
, C.
, Wiest
, T.
, Wang
, P.
, and Seepersad
, C. C.
, 2019
, “A Comparative Evaluation of Supervised Machine Learning Classification Techniques for Engineering Design Applications
,” ASME J. Mech. Des.
, 141
(12
), p. 121404
. 10.1115/1.404452451.
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
. 10.1115/1.404422952.
Lin
, Q.
, Hong
, J.
, Liu
, Z.
, Li
, B.
, and Wang
, J.
, 2018
, “Investigation Into the Topology Optimization for Conductive Heat Transfer Based on Deep Learning Approach
,” Int. Commun. Heat Mass Transfer
, 97
(1
), 103
–109
. 10.1016/j.icheatmasstransfer.2018.07.00153.
Sosnovik
, I.
, and Oseledets
, I.
, 2019
, “Neural Networks for Topology Optimization
,” Russ. J. Numer Anal. Math. Model.
, 34
(4
), pp. 215
–223
. 10.1515/rnam-2019-001854.
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
, Montreal, Canada
, pp. 2672
–2680
.55.
Mirza
, M.
, and Osindero
, S.
, 2014
, “Conditional Generative Adversarial Nets
,” preprint arXiv:1411.1784.56.
Isola
, P.
, Zhu
, J.-Y.
, Zhou
, T.
, and Efros
, A. A.
, 2017
, “Image-to-Image Translation With Conditional Adversarial Networks
,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
, Honolulu, HI
, pp. 1125
–1134
.57.
Long
, J.
, Shelhamer
, E.
, and Darrell
, T.
, 2015
, “Fully Convolutional Networks for Semantic Segmentation
,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
, Boston, MA
, pp. 3431
–3440
.58.
Ronneberger
, O.
, Fischer
, P.
, and Brox
, T.
, 2015
, “U-net: Convolutional Networks for Biomedical Image Segmentation
,” International Conference on Medical Image Computing and Computer-Assisted Intervention
, Munich, Germany
, Springer
, pp. 234
–241
.59.
He
, K.
, Zhang
, X.
, Ren
, S.
, and Sun
, J.
, 2016
, “Deep Residual Learning for Image Recognition
,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
, Las Vegas, NV
, pp. 770
–778
.60.
Hu
, J.
, Shen
, L.
, and Sun
, G.
, 2018
, “Squeeze-and-Excitation Networks
,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
, Salt Lake City, UT
, pp. 7132
–7141
.61.
Nie
, Z.
, Jiang
, H.
, and Kara
, L. B.
, 2020
, “Stress Field Prediction in Cantilevered Structures Using Convolutional Neural Networks
,” ASME J. Comput. Inf. Sci. Eng.
, 20
(1
), p. 011002
. https://doi.org/10.1115/1.404409762.
63.
Gómez
, J.
, and Guarín-Zapata
, N.
, 2018
, “Solidspy: 2d-Finite Element Analysis With Python.” https://github.com/AppliedMechanics-EAFIT/SolidsPy64.
Kingma
, D. P.
, and Ba
, J.
, 2014
, “Adam: A Method for Stochastic Optimization
,” preprint arXiv:1412.6980.65.
Duchi
, J.
, Hazan
, E.
, and Singer
, Y.
, 2011
, “Adaptive Subgradient Methods for Online Learning and Stochastic Optimization
,” J. Mach. Learn. Res.
, 12
(7
), pp. 2121
–2159
.66.
Tieleman
, T.
, and Hinton
, G.
, 2012
, “Lecture 6.5-rmsprop: Divide the Gradient by a Running Average of Its Recent Magnitude
,” COURSERA: Neural Netw. Mach. Learn.
, 4
(2
), pp. 26
–31
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