Graphical Abstract Figure
Graphical Abstract Figure
Close modal

Abstract

Scalar fields, such as stress or temperature fields, are often calculated in shape optimization and design problems in engineering. For complex problems where shapes have varying topology and cannot be parametrized, data-driven scalar field prediction can be faster than traditional finite element methods. However, current data-driven techniques to predict scalar fields are limited to a fixed grid domain, instead of arbitrary mesh structures. In this work, we propose a method to predict scalar fields on arbitrary meshes. It uses a convolutional neural network whose feature maps at multiple resolutions are interpolated to node positions before being fed into a multilayer perceptron to predict solutions to partial differential equations at mesh nodes. The model is trained on finite element von Mises stress fields, and once trained, it can estimate stress values at each node on any input mesh. Two shape datasets are investigated, and the model has strong performance on both, with a median R2 value of 0.91. We also demonstrate the model on a temperature field in a heat conduction problem, where its predictions have a median R2 value of 0.99. Our method provides a potential flexible alternative to finite element analysis in engineering design contexts. Code and datasets are available online.

References

1.
Rankin
,
J. J.
, and
Ott
,
D. A.
,
1992
, “
The Open Approach to Fea Integration in the Design Process
,”
Mech. Eng.
,
114
(
9
), p.
70
. link.gale.com/apps/doc/A12800693/AONE
2.
Torczon
,
V.
, and
Trosset
,
M.
,
1998
, “
Using Approximations to Accelerate Engineering Design Optimization
,”
7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization
,
St. Louis, MO
,
Sept. 2-4, p. 4800
.
3.
Eriksson
,
M.
, and
Burman
,
Å.
,
2005
, “
Improving the Design Process by Integrating Design Analysis
,” DS 35: Proceedings ICED 05, the 15th International Conference on Engineering Design, Melbourne, Australia, Aug. 15–18, pp.
555
556
.
4.
Nourbakhsh
,
M.
,
Irizarry
,
J.
, and
Haymaker
,
J.
,
2018
, “
Generalizable Surrogate Model Features to Approximate Stress in 3D Trusses
,”
Eng. Appl. Artif. Intell.
,
71
(
01
), pp.
15
27
.
5.
Barmada
,
S.
,
Fontana
,
N.
,
Formisano
,
A.
,
Thomopulos
,
D.
, and
Tucci
,
M.
,
2021
, “
A Deep Learning Surrogate Model for Topology Optimization
,”
IEEE. Trans. Magn.
,
57
(
6
), pp.
1
4
.
6.
Nie
,
Z.
,
Jiang
,
H.
, and
Kara
,
L. B.
,
2019
, “
Stress Field Prediction in Cantilevered Structures Using Convolutional Neural Networks
,”
ASME J. Comput. Inf. Sci. Eng.
,
20
(
1
), p.
011002
.
7.
Jiang
,
H.
,
Nie
,
Z.
,
Yeo
,
R.
,
Farimani
,
A. B.
, and
Kara
,
L. B.
,
2021
, “
Stressgan: A Generative Deep Learning Model for Two-Dimensional Stress Distribution Prediction
,”
ASME J. Appl. Mech.
,
88
(
5
), p.
051005
.
8.
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
), pp.
2555
2569
.
9.
Whalen
,
E.
, and
Mueller
,
C.
,
2022
, “
Toward Reusable Surrogate Models: Graph-Based Transfer Learning on Trusses
,”
ASME J. Mech. Des.
,
144
(
2
), p.
021704
.
10.
Qi
,
C. R.
,
Su
,
H.
,
Mo
,
K.
, and
Guibas
,
L. J.
,
2017
, “
Pointnet: Deep Learning on Point Sets for 3D Classification and Segmentation
,”
Computer Vision and Pattern Recognition
,
Honolulu, HI
,
July 22–25
.
11.
Qi
,
C. R.
,
Yi
,
L.
,
Su
,
H.
, and
Guibas
,
L. J.
,
2017
, “
Pointnet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space
,”
Conference on Neural Information Processing Systems
,
Long Beach, CA
,
Dec. 4–9
.
12.
Kashefi
,
A.
,
Rempe
,
D.
, and
Guibas
,
L. J.
,
2021
, “
A Point-Cloud Deep Learning Framework for Prediction of Fluid Flow Fields on Irregular Geometries
,”
Phys. Fluids.
,
33
(
2
), p.
027104
.
13.
Attene
,
M.
,
Katz
,
S.
,
Mortara
,
M.
,
Patane
,
G.
,
Spagnuolo
,
M.
, and
Tal
,
A.
,
2006
, “
Mesh Segmentation—A Comparative Study
,”
IEEE International Conference on Shape Modeling and Applications 2006 (SMI'06)
,
Matsushima, Japan
,
June 14-16, p. 7
.
14.
Kalogerakis
,
E.
,
Hertzmann
,
A.
, and
Singh
,
K.
,
2010
, “
Learning 3D Mesh Segmentation and Labeling
,”
ACM Trans. Graph.
,
29
(
3
), pp.
1
12
.
15.
Wu
,
Z.
,
Pan
,
S.
,
Chen
,
F.
,
Long
,
G.
,
Zhang
,
C.
, and
Yu
,
P. S.
,
2021
, “
A Comprehensive Survey on Graph Neural Networks
,”
IEEE Trans. Neural Netw. Learn. Syst.
,
32
(
1
), pp.
4
24
.
16.
Wang
,
Y.
,
Sun
,
Y.
,
Liu
,
Z.
,
Sarma
,
S. E.
,
Bronstein
,
M. M.
, and
Solomon
,
J. M.
,
2019
, “
Dynamic Graph CNN For Learning on Point Clouds
,”
ACM Transactions on Graphics (TOG)
,
38
(
5
), pp.
1
12
. https://dl.acm.org/doi/10.1145/3326362
17.
Sanchez-Gonzalez
,
A.
,
Godwin
,
J.
,
Pfaff
,
T.
,
Ying
,
R.
,
Leskovec
,
J.
, and
Battaglia
,
P. W.
,
2020
, “
Learning to Simulate Complex Physics with Graph Networks
,”
International Conference on Machine Learning
,
Virtual Only (formerly Vienna)
,
July 12–18
.
18.
Meyer
,
L.
,
Pottier
,
L.
,
Ribes
,
A.
, and
Raffin
,
B.
,
2021
, “
Deep Surrogate for Direct Time Fluid Dynamics
,”
NeurIPS 2021 - Thirty-fifth Workshop on Machine Learning and the Physical Sciences
,
Virtual Only
,
Dec. 6–14
.
19.
Maurizi
,
M.
,
Gao
,
C.
, and
Berto
,
F.
,
2022
, “
Predicting Stress, Strain and Deformation Fields in Materials and Structures With Graph Neural Networks
,”
Sci. Rep.
,
12
(
1
), p.
21834
.
20.
Li
,
G.
,
Müller
,
M.
,
Thabet
,
A.
, and
Ghanem
,
B.
,
2019
, “
DeepGCNs: Can GCNs Go as Deep as CNNs?
International Conference on Computer Vision (ICCV)
,
Seoul, South Korea
,
Oct. 27–Nov. 2
.
21.
Pfaff
,
T.
,
Fortunato
,
M.
,
Sanchez-Gonzalez
,
A.
, and
Battaglia
,
P. W.
,
2021
, “
Learning Mesh-Based Simulation With Graph Networks
,”
International Conference on Learning Representations (ICLR)
,
Virtual Only
,
May 3–7
.
22.
Zhao
,
L.
, and
Akoglu
,
L.
,
2020
, “
Pairnorm: Tackling Oversmoothing in GNNs
,”
ICLR 2020 ·The Eighth International Conference on Learning Representations
,
Virtual Only
,
Apr. 26–May 1
.
23.
Javadi
,
A.
, and
Tan
,
T.
,
2003
, “
Neural Network for Constitutive Modelling in Finite Element Analysis
,”
Comput. Assist. Mech. Eng. Sci.
,
10
(
4
), pp.
523
530
. https://api.semanticscholar.org/CorpusID:15777198
24.
de Avila Belbute-Peres
,
F.
,
Economon
,
T. D.
, and
Kolter
,
J. Z.
,
2020
, “
Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction
,”
Proceedings of the 37th International Conference on Machine Learning, PMLR
,
Virtual Only (formerly Vienna)
,
July 12–18
.
25.
Müller
,
T.
,
Evans
,
A.
,
Schied
,
C.
, and
Keller
,
A.
,
2022
, “
Instant Neural Graphics Primitives With a Multiresolution Hash Encoding
,”
ACM Trans. Graph.
,
41
(
4
), pp.
1
15
.
26.
Pandey
,
R. K.
, and
Ramakrishnan
,
A. G.
,
2018
,
“A Hybrid Approach of Interpolations and CNN to Obtain Super-Resolution,“ arXiv preprint arXiv:1805.09400
.
27.
Brnic
,
J.
,
2018
,
Analysis of Engineering Structures and Material Behavior
,
John Wiley & Sons
,
Hoboken, NJ
.
28.
The MathWorks
,
I.
,
2021
,
Partial Differential Equation Toolbox
,
Natick
,
MA
.
29.
Kou
,
X.
, and
Tan
,
S.
,
2010
, “
A Simple and Effective Geometric Representation for Irregular Porous Structure Modeling
,”
Comput. Aid. Des.
,
42
(
10
), pp.
930
941
.
30.
Sorkine
,
O.
,
Cohen-Or
,
D.
,
Lipman
,
Y.
,
Alexa
,
M.
,
Rössl
,
C.
, and
Seidel
,
H.-P.
,
2004
, “
Laplacian Surface Editing
,”
SGP04: Symposium on Geometry Processing
,
Nice, France
,
July 8–10, pp. 175–184
.
31.
Ronneberger
,
O.
,
Fischer
,
P.
, and
Brox
,
T.
,
2015
, “
U-Net: Convolutional Networks for Biomedical Image Segmentation
,”
Medical image computing and computer-assisted intervention--MICCAI 2015: 18th international conference
,
Munich, Germany
,
Oct. 5–9
.
32.
Osher
,
S.
, and
Fedkiw
,
R.
,
2003
,
Level Set Methods and Dynamic Implicit Surfaces
,
Springer
,
New York City
, pp.
17
22
.
33.
Mauch
,
S.
,
2000
,
"A Fast Algorithm for Computing the Closest Point and Distance Transform," Caltech ASCI Technical Report 077.
.
34.
Sethian
,
J. A.
,
1996
, “
A Fast Marching Level Set Method for Monotonically Advancing Fronts
,”
Proc. Natl. Acad. Sci. U. S. A.
,
93
(
4
), pp.
1591
1595
.
35.
Strutz
,
T.
,
2021
,
"The Distance Transform and its Computation," arXiv preprint arXiv:2106.03503
.
36.
Paszke
,
A.
,
Gross
,
S.
,
Chintala
,
S.
,
Chanan
,
G.
,
Yang
,
E.
,
DeVito
,
Z.
,
Lin
,
Z.
,
Desmaison
,
A.
,
Antiga
,
L.
, and
Lerer
,
A.
,
2017
, “
Automatic Differentiation in PyTorch
,”
Conference on Neural Information Processing Systems
,
Long Beach, CA,
Dec. 4–9
.
37.
Thévenaz
,
P.
,
Blu
,
T.
, and
Unser
,
M.
,
2000
, “
Image Interpolation and Resampling
,”
Handbook of Medical Imaging, Processing and Analysis
,
1
(
1
), pp.
393
420
. https://dl.acm.org/doi/10.5555/374166.374424
38.
Kingma
,
D. P.
, and
Ba
,
J.
,
2017
, “
Adam: A Method for Stochastic Optimization
,”
International Conference on Learning Representations (ICLR 2015)
,
San Diego, CA
,
May 7–9
.
39.
Cotter
,
A.
,
Shamir
,
O.
,
Srebro
,
N.
, and
Sridharan
,
K.
,
2011
, “
Better Mini-Batch Algorithms Via Accelerated Gradient Methods
,”
Advances in Neural Information Processing Systems 24 (NIPS 2011)
,
Granada, Spain
,
Dec. 12–17
.
40.
Miles
,
J.
,
2005
,
R-Squared, Adjusted R-Squared
,
John Wiley and Sons, Ltd
,
Hoboken, NJ
.
You do not currently have access to this content.