Abstract

The demand for fast and accurate structural analysis is becoming increasingly more prevalent with the advance of generative design and topology optimization technologies. As one step toward accelerating structural analysis, this work explores a deep learning-based approach for predicting the stress fields in 2D linear elastic cantilevered structures subjected to external static loads at its free end using convolutional neural networks (CNNs). Two different architectures are implemented that take as input the structure geometry, external loads, and displacement boundary conditions, and output the predicted von Mises stress field. The first is a single input channel network called SCSNet as the baseline architecture, and the second is the multichannel input network called StressNet. Accuracy analysis shows that StressNet results in significantly lower prediction errors than SCSNet on three loss functions, with a mean relative error of 2.04% for testing. These results suggest that deep learning models may offer a promising alternative to classical methods in structural design and topology optimization. Code and dataset are available.2

References

1.
Roy
,
A. G.
,
Conjeti
,
S.
,
Karri
,
S. P. K.
,
Sheet
,
D.
,
Katouzian
,
A.
,
Wachinger
,
C.
, and
Navab
,
N.
,
2017
, “
Relaynet: Retinal Layer and Fluid Segmentation of Macular Optical Coherence Tomography Using Fully Convolutional Networks
,”
Biomed. Opt. Express
,
8
(
8
), pp.
3627
3642
. 10.1364/BOE.8.003627
2.
Mohan
,
A. T.
, and
Gaitonde
,
D. V.
,
2018
, “
A Deep Learning based Approach to Reduced Order Modeling for Turbulent Flow Control using LSTM Neural Networks
,” e-print arXiv:1804.09269.
3.
Farimani
,
A. B.
,
Gomes
,
J.
, and
Pande
,
V. S.
,
2017
, “
Deep Learning the Physics of Transport Phenomena
,” CoRR, abs/1709.02432.
4.
Kim
,
B.
,
Azevedo
,
V. C.
,
Thuerey
,
N.
,
Kim
,
T.
,
Gross
,
M. H.
, and
Solenthaler
,
B.
,
2018
, “
Deep Fluids: A Generative Network for Parameterized Fluid Simulations
,” CoRR, abs/1806.02071.
5.
Umetani
,
N.
,
2017
, “
Exploring Generative 3D Shapes Using Autoencoder Networks
,”
SIGGRAPH Asia 2017 Technical Briefs
, ACM, pp.
1
24
.
6.
Yu
,
Y.
,
Hur
,
T.
, and
Jung
,
J.
,
2018
, “
Deep Learning for Topology Optimization Design
,” CoRR, abs/1801.05463.
7.
Zhang
,
W.
,
Jiang
,
H.
,
Yang
,
Z.
,
Yamakawa
,
S.
,
Shimada
,
K.
, and
Kara
,
L. B.
,
2018
, “
Data-Driven Upsampling of Point Clouds
,” CoRR, abs/1807.02740.
8.
Ulu
,
E.
,
Zhang
,
R.
,
Yumer
,
M. E.
, and
Kara
,
L. B.
,
2014
, “
A Data-Driven Investigation and Estimation of Optimal Topologies Under Variable Loading Configurations
,”
Computational Modeling of Objects Presented in Images. Fundamentals, Methods, and Applications
,
Springer International Publishing
,
Berlin
, pp.
387
399
.
9.
Goh
,
G. B.
,
Hodas
,
N. O.
, and
Vishnu
,
A.
,
2017
, “
Deep Learning for Computational Chemistry
,”
J. Comput. Chem.
,
38
(
16
), pp.
1291
1307
. 10.1002/jcc.v38.16
10.
Mardt
,
A.
,
Pasquali
,
L.
,
Wu
,
H.
, and
Noé
,
F.
,
2018
, “
VAMPnets for Deep Learning of Molecular Kinetics
,”
Nat. Commun.
,
9
(
Jan
), p.
5
. 10.1038/s41467-017-02388-1
11.
Montavon
,
G.
,
Rupp
,
M.
,
Gobre
,
V.
,
Vazquez-Mayagoitia
,
A.
,
Hansen
,
K.
,
Tkatchenko
,
A. R.
,
Müller
,
K.-R.
, and
Anatole von Lilienfeld
,
O.
,
2013
, “
Machine Learning of Molecular Electronic Properties in Chemical Compound Space
,”
New. J. Phys.
,
15
(
Sep
), p.
095003
. 10.1088/1367-2630/15/9/095003
12.
Tribello
,
G. A.
,
Ceriotti
,
M.
, and
Parrinello
,
M.
,
2010
, “
A Self-learning Algorithm for Biased Molecular Dynamics
,”
Proc. Natl. Acad. Sci. U. S. A.
,
107
(
41
), pp.
17509
17514
. 10.1073/pnas.1011511107
13.
Raissi
,
M.
,
Perdikaris
,
P.
, and
Karniadakis
,
G.
,
2018
, “
Multistep Neural Networks for Data-Driven Discovery of Nonlinear Dynamical Systems
.” arXiv preprint arXiv:1801.01236.
14.
Broecker
,
P.
,
Carrasquilla
,
J.
,
Melko
,
R. G.
, and
Trebst
,
S.
,
2017
, “
Machine Learning Quantum Phases of Matter Beyond the Fermion Sign Problem
,” Scientific Reports.
15.
Schütt
,
K. T.
,
Arbabzadah
,
F.
,
Chmiela
,
S.
,
Müller
,
K. R.
, and
Tkatchenko
,
A.
,
2017
, “
Quantum-chemical Insights From Deep Tensor Neural Networks
,”
Nat. Commun.
,
8
(
Jan
), p.
13890
. 10.1038/ncomms13890
16.
Barati Farimani
,
A.
,
Gomes
,
J.
,
Sharma
,
R.
,
Lee
,
F. L.
, and
Pande
,
V. S.
,
2018
, “
Deep Learning Phase Segregation
,” e-print arXiv:1803.08993.
17.
Zhang
,
W.
,
Jiang
,
H.
,
Yang
,
Z.
,
Yamakawa
,
S.
,
Shimada
,
K.
, and
Kara
,
L. B.
,
2018
, “
Data-Driven Upsampling of Point Clouds
,” CoRR, abs/1807.02740.
18.
Levin
,
R.
, and
Lieven
,
N.
,
1998
, “
Dynamic Finite Element Model Updating Using Neural Networks
,”
J. Sound Vib.
,
210
(
5
), pp.
593
607
. 10.1006/jsvi.1997.1364
19.
Atalla
,
M.
, and
Inman
,
D.
,
1998
, “
On Model Updating Using Neural Networks
,”
Mech. Syst. Signal Process.
,
12
(
1
), pp.
135
161
. 10.1006/mssp.1997.0138
20.
Javadi
,
A.
, and
Tan
,
P.T.
,
2003
, “
Neural Network for Constitutive Modelling in Finite Element Analysis
,”
Compute. Assist. Mech. Eng. Sci.
,
10
(
4
), pp.
523
530
.
21.
Oishi
,
A.
, and
Yagawa
,
G.
,
2017
, “
Computational Mechanics Enhanced by Deep Learning
,”
Comput. Methods Appl. Mech. Eng.
,
327
, pp.
327
351
.
22.
Spruegel
,
T.
,
Schröppel
,
T.
, and
Wartzack
,
S.
,
2017
, “
Generic Approach to Plausibility Checks for Structural Mechanics with Deep Learning
.”
Proceedings of the 21st International Conference on Engineering Design (ICED 17) Vol 1: Resource Sensitive Design, Design Research Applications and Case Studies
,
Vancouver, Canada
,
Aug. 21–25
, pp.
299
308
23.
Liang
,
L.
,
Liu
,
M.
,
Martin
,
C.
, and
Sun
,
W.
,
2018
, “
A Deep Learning Approach to Estimate Stress Distribution: a Fast and Accurate Surrogate of Finite-Element Analysis
,”
J. R. Soc. Interface
,
15
(
138
), pp.
20170844
.
24.
LeCun
,
Y.
,
Bengio
,
Y.
, and
Hinton
,
G.
,
2015
, “
Deep Learning
,”
Nature
,
521
(
05
), pp.
436
444
. 10.1038/nature14539
25.
Simonyan
,
K.
, and
Zisserman
,
A.
,
2015
, “
Very Deep Convolutional Networks for Large-Scale Image Recognition
,”
International Conference on Learning Representations
,
San Diego, CA
,
May 7–9
.
26.
He
,
K.
,
Zhang
,
X.
,
Ren
,
S.
, and
Sun
,
J.
,
2015
, “
Deep Residual Learning for Image Recognition
,” CoRR, abs/1512.03385.
27.
Jenkins
,
W.
,
1995
, “Neural Network-Based Approximations for Structural Analysis,”
Developments in Neural Networks and Evolutionary Computing for Civil and Structural Engineering
,
Civil-Comp Press
,
Edinburgh
, pp.
25
35
.
28.
Waszczyszyn
,
Z.
, and
Ziemiański
,
L.
,
2001
, “
Neural Networks in Mechanics of Structures and Materials–new Results and Prospects of Applications
,”
Comput. Struct.
,
79
(
22–25
), pp.
2261
2276
. 10.1016/S0045-7949(01)00083-9
29.
Goh
,
G. B.
,
Hodas
,
N. O..
, and
Vishnu
,
A.
,
2017
, “
Deep Learning for Computational Chemistry
,”
J. Comput. Chem.
,
38
(
16
), pp.
1291
1307
.
30.
Abendroth
,
M.
, and
Kuna
,
M.
,
2003
, “
Determination of Deformation and Failure Properties of Ductile Materials by Means of the Small Punch Test and Neural Networks
,”
Comput. Mater. Sci.
,
28
(
3–4
), pp.
633
644
. 10.1016/j.commatsci.2003.08.031
31.
Wu
,
X.
,
Ghaboussi
,
J.
, and
Garrett
Jr,
J.
,
1992
, “
Use of Neural Networks in Detection of Structural Damage
,”
Comput. Struct.
,
42
(
4
), pp.
649
659
. 10.1016/0045-7949(92)90132-J
32.
Zang
,
C.
, and
Imregun
,
M.
,
2001
, “
Structural Damage Detection Using Artificial Neural Networks and Measured Frf Data Reduced Via Principal Component Projection
,”
J. Sound Vib.
,
242
(
5
), pp.
813
827
. 10.1006/jsvi.2000.3390
33.
Tsou
,
P.
, and
Shen
,
M.-H.
,
1994
, “
Structural Damage Detection and Identification Using Neural Networks
,”
AIAA J.
,
32
(
1
), pp.
176
183
. 10.2514/3.11964
34.
Huber
,
N.
, and
Tsakmakis
,
C.
,
1999
, “
Determination of Constitutive Properties From Spherical Indentation Data Using Neural Networks. Part Ii: Plasticity With Nonlinear Isotropic and Kinematic Hardening
,”
J. Mech. Phys. Solids.
,
47
(
7
), pp.
1589
1607
. 10.1016/S0022-5096(98)00110-0
35.
Zhang
,
Z.
, and
Friedrich
,
K.
,
2003
, “
Artificial Neural Networks Applied to Polymer Composites: a Review
,”
Compos. Sci. Technol.
,
63
(
14
), pp.
2029
2044
. 10.1016/S0266-3538(03)00106-4
36.
Settgast
,
C.
,
Abendroth
,
M.
, and
Kuna
,
M.
,
2019
, “
Constitutive Modeling of Plastic Deformation Behavior of Open-cell Foam Structures Using Neural Networks
,”
Mech. Mater.
,
131
, pp.
1
10
.
37.
Liu
,
M.
,
Liang
,
L.
, and
Sun
,
W.
,
2019
, “
Estimation of in Vivo Constitutive Parameters of the Aortic Wall Using a Machine Learning Approach
,”
Comput. Methods Appl. Mech. Eng.
,
347
, pp.
201
217
. 10.1016/j.cma.2018.12.030
38.
Yildiz
,
A.
,
Öztürk
,
N.
,
Kaya
,
N.
, and
Öztürk
,
F.
,
2003
, “
Integrated Optimal Topology Design and Shape Optimization Using Neural Networks
,”
Struct. Multidiscipl. Optim.
,
25
(
4
), pp.
251
260
. 10.1007/s00158-003-0300-0
39.
Papadrakakis
,
M.
, and
Lagaros
,
N. D.
,
2002
, “
Reliability-based Structural Optimization Using Neural Networks and Monte Carlo Simulation
,”
Comput. Methods Appl. Mech. Eng.
,
191
(
32
), pp.
3491
3507
. 10.1016/S0045-7825(02)00287-6
40.
Sosnovik
,
I.
, and
Oseledets
,
I.
,
2017
, “
Neural Networks for Topology Optimization
,” e-print arXiv:1709.09578.
41.
Khadilkar
,
A.
,
Wang
,
J.
, and
Rai
,
R.
,
2019
, “
Deep Learning–based Stress Prediction for Bottom-up Sla 3d Printing Process
,”
Int. J. Adv. Manuf. Technol.
,
102
(
5–8
), pp.
2555
2569
.
42.
Gómez
,
J.
,
Guarín-Zapata
,
N.
,
2018
, “
SolidsPy: 2D-Finite Element Analysis With Python
.”
43.
Masci
,
J.
,
Meier
,
U.
,
Cireşan
,
D.
, and
Schmidhuber
,
J.
,
2011
, “Stacked Convolutional Auto-Encoders for Hierarchical Feature Extraction”,
International Conference on Artificial Neural Networks
,
Springer
, pp.
52
59
.
44.
Geng
,
J.
,
Fan
,
J.
,
Wang
,
H.
,
Ma
,
X.
,
Li
,
B.
, and
Chen
,
F.
,
2015
, “
High-Resolution Sar Image Classification Via Deep Convolutional Autoencoders
,”
IEEE Geosci. Remote Sens. Lett.
,
12
(
11
), pp.
2351
2355
. 10.1109/LGRS.2015.2478256
45.
Holden
,
D.
,
Saito
,
J.
,
Komura
,
T.
, and
Joyce
,
T.
,
2015
, “Learning Motion Manifolds with Convolutional Autoencoders,”
SIGGRAPH Asia 2015 Technical Briefs
,
ACM
,
New York
, p.
18
.
46.
LeCun
,
Y.
, and
Bengio
,
Y.
,
1998
, “Convolutional Networks for Images, Speech, and Time Series,”
The Handbook of Brain Theory and Neural Networks
,
M. A.
Arbib
, ed.,
MIT Press
,
Cambridge, MA
, pp.
255
258
.
47.
Krizhevsky
,
A.
,
Sutskever
,
I.
, and
Hinton
,
G. E.
,
2017
, “
Imagenet Classification with Deep Convolutional Neural Networks
,”
Commun. ACM
,
60
(
6
), pp.
84
90
. 10.1145/3098997
48.
Johnson
,
J.
,
Alahi
,
A.
, and
Li
,
F.
,
2016
, “
Perceptual Losses for Real-Time Style Transfer and Super-Resolution
,” CoRR, abs/1603.08155.
49.
Hu
,
J.
,
Shen
,
L.
, and
Sun
,
G.
,
2017
, “
Squeeze-and-Excitation Networks
,” CoRR, abs/1709.01507.
You do not currently have access to this content.