## Abstract

The increase of the spatial resolution in numerical computation always leads to a decrease in computing efficiency with respect to the constraint of mesh density. In response to this problem of the inability to perform numerical computation, we propose a novel method to boost the mesh-density in the finite element method (FEM) within 2D domains. Running on the von Mises stress fields of the 2D plane-strain problems computed by FEM, the proposed method utilizes a deep neural network named SMNet to learn a nonlinear mapping from low mesh-density to high mesh-density in stress fields and realizes the improvement of numerical computation accuracy and efficiency simultaneously. By introducing residual density blocks into SMNet, we can extract abundant local features and improve prediction capacity. The result indicates that SMNet can effectively increase the spatial resolution of stress fields under multiple scaling factors in mesh-density: 2 ×, 3 ×, and 4 ×. Compared with the targets, the relative error of SMNet is $1.67%$, showing better performance than many other methods. SMNet can be generically used as an enhanced mesh-density boosting model of 2D physical fields for mesh-based numerical methods.

## References

1.
Bright
,
J. A.
, and
Rayfield
,
E. J.
,
2011
, “
The Response of Cranial Biomechanical Finite Element Models to Variations in Mesh Density
,”
Anat. Rec.: Adv. Integr. Anat. Evolution. Biol.
,
294
(
4
), pp.
610
620
.
2.
Ikramullah
,
I.
,
Afrizal
,
A.
,
Huzni
,
S.
,
Thalib
,
S.
,
Abdul Khalil
,
H.
,
Rizal
,
S.
,
2020
, “
Effect of Mesh Sensitivity and Cohesive Properties on Simulation of Typha Fiber/Epoxy Microbond Test
,”
Computation
,
8
(
1
), p.
2
.
3.
Chung
,
C. H.
,
Lee
,
J.
, and
Gil
,
J. H.
,
2014
, “
Structural Performance Evaluation of a Precast Prefabricated Bridge Column Under Vehicle Impact Loading
,”
Struct. Infrastruct. Eng.
,
10
(
6
), pp.
777
791
.
4.
Berger
,
M. J.
, and
Oliger
,
J.
,
1984
, “
Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations
,”
J. Comput. Phys.
,
53
(
3
), pp.
484
512
.
5.
Areias
,
P.
,
Reinoso
,
J.
,
Camanho
,
P.
,
De Sá
,
J. C.
, and
Rabczuk
,
T.
,
2018
, “
Effective 2D and 3D Crack Propagation With Local Mesh Refinement and the Screened Poisson Equation
,”
Eng. Fract. Mech.
,
189
(
22
), pp.
339
360
.
6.
Burstedde
,
C.
,
Wilcox
,
L. C.
, and
Ghattas
,
O.
,
2011
, “
p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees
,”
SIAM J. Sci. Comput.
,
33
(
3
), pp.
1103
1133
.
7.
Egan
,
R.
, and
Gibou
,
F.
,
2018
, “
Fast and Scalable Algorithms for Constructing Solvent-Excluded Surfaces of Large Biomolecules
,”
J. Comput. Phys.
,
374
(
4
), pp.
91
120
.
8.
Kossaczký
,
I.
,
1994
, “
A Recursive Approach to Local Mesh Refinement in Two and Three Dimensions
,”
J. Comput. Appl. Math.
,
55
(
3
), pp.
275
288
.
9.
MacNeice
,
P.
,
Olson
,
K. M.
,
Mobarry
,
C.
,
De Fainchtein
,
R.
, and
Packer
,
C.
,
2000
, “
Paramesh: A Parallel Adaptive Mesh Refinement Community Toolkit
,”
Comput. Phys. Commun.
,
126
(
3
), pp.
330
354
.
10.
Zhang
,
W.
,
Almgren
,
A.
,
Beckner
,
V.
,
Bell
,
J.
,
Blaschke
,
J.
,
Chan
,
C.
,
Day
,
M.
,
Friesen
,
B.
,
Gott
,
K.
, and
Graves
,
D.
,
2019
, “
Amrex: A Framework for Block-Structured Adaptive Mesh Refinement
,”
J. Open Source Softw.
,
4
(
37
), pp.
1370
1370
.
11.
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.
84
93
.
12.
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
.
13.
Bhatnagar
,
S.
,
Afshar
,
Y.
,
Pan
,
S.
,
Duraisamy
,
K.
, and
Kaushik
,
S.
,
2019
, “
Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks
,”
Comput. Mech.
,
64
(
2
), pp.
525
545
.
14.
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
.
15.
Pfaff
,
T.
,
Fortunato
,
M.
,
Sanchez-Gonzalez
,
A.
, and
Battaglia
,
P. W.
,
2020
, “
Learning Mesh-Based Simulation With Graph Networks
,” arXiv:2010.03409.
16.
,
L.
,
Jeong
,
S.
,
Solenthaler
,
B.
,
Pollefeys
,
M.
, and
Gross
,
M.
,
2015
, “
Data-Driven Fluid Simulations Using Regression Forests
,”
ACM Trans. Graph.
,
34
(
6
), pp.
1
9
.
17.
Nie
,
Z.
,
Lin
,
T.
,
Jiang
,
H.
, and
Kara
,
L. B.
,
2021
, “
Topologygan: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain
,”
ASME J. Mech. Des.
,
143
(
3
), p.
031715
.
18.
Xie
,
Y.
,
Franz
,
E.
,
Chu
,
M.
, and
Thuerey
,
N.
,
2018
, “
tempoGAN: A Temporally Coherent, Volumetric GAN for Super-Resolution Fluid Flow
,”
ACM Trans. Graph.
,
37
(
4
), pp.
1
15
.
19.
Belbute-Peres
,
F. d. A.
,
Economon
,
T.
, and
Kolter
,
Z.
,
2020
, “
Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction
,”
International Conference on Machine Learning
,
Virtual
,
July 13–18
, PMLR, pp.
2402
2411
.
20.
Dong
,
C.
,
Loy
,
C. C.
,
He
,
K.
, and
Tang
,
X.
,
2014
, “
Learning a Deep Convolutional Network for Image Super-Resolution
,”
European Conference on Computer Vision
,
Zurich, Switzerland
,
Sept. 6–12
, Springer, pp.
184
199
.
21.
Shi
,
W.
,
Caballero
,
J.
,
Huszár
,
F.
,
Totz
,
J.
,
Aitken
,
A. P.
,
Bishop
,
R.
,
Rueckert
,
D.
, and
Wang
,
Z.
,
2016
, “
Real-Time Single Image and Video Super-Resolution Using an Efficient Subpixel Convolutional Neural Network
,”
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
,
Las Vegas, NV
,
June 26–July 1
, pp.
1874
1883
.
22.
Lim
,
B.
,
Son
,
S.
,
Kim
,
H.
,
Nah
,
S.
, and
Mu Lee
,
K.
,
2017
, “
Enhanced Deep Residual Networks for Single Image Super-Resolution
,”
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops
,
Honolulu, HI
,
July 21–26
, pp.
136
144
.
23.
Kim
,
J.
,
Lee
,
J. K.
, and
Lee
,
K. M.
,
2016
, “
Deeply-Recursive Convolutional Network for Image Super-Resolution
,”
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
,
Las Vegas, NV
,
June 26–July 1
, pp.
1637
1645
.
24.
Tong
,
T.
,
Li
,
G.
,
Liu
,
X.
, and
Gao
,
Q.
,
2017
, “
Image Super-Resolution Using Dense Skip Connections
,”
Proceedings of the IEEE International Conference on Computer Vision
,
Venice, Italy
,
Oct. 22–29
, pp.
4799
4807
.
25.
Tai
,
Y.
,
Yang
,
J.
,
Liu
,
X.
, and
Xu
,
C.
,
2017
, “
MemNet: A Persistent Memory Network for Image Restoration
,”
Proceedings of the IEEE International Conference on Computer Vision
,
Venice, Italy
,
Oct. 22–29
, pp.
4539
4547
.
26.
Zhang
,
Y.
,
Li
,
K.
,
Li
,
K.
,
Wang
,
L.
,
Zhong
,
B.
, and
Fu
,
Y.
,
2018
, “
Image Super-Resolution Using Very Deep Residual Channel Attention Networks
,”
Proceedings of the European Conference on Computer Vision (ECCV)
,
Munich, Germany
,
Sept. 8–14
, pp.
286
301
.
27.
Hu
,
X.
,
Mu
,
H.
,
Zhang
,
X.
,
Wang
,
Z.
,
Tan
,
T.
, and
Sun
,
J.
,
2019
, “
Meta-SR: A Magnification-Arbitrary Network for Super-Resolution
,”
Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition
,
Long Beach, CA
,
June 15–20
, pp.
1575
1584
.
28.
Yang
,
F.
,
Yang
,
H.
,
Fu
,
J.
,
Lu
,
H.
, and
Guo
,
B.
,
2020
, “
Learning Texture Transformer Network for Image Super-Resolution
,”
Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition
,
Virtual
,
June 14–19
, pp.
5791
5800
.
29.
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
,
June 26–July 1
, pp.
770
778
.
30.
Zhang
,
Y.
,
Tian
,
Y.
,
Kong
,
Y.
,
Zhong
,
B.
, and
Fu
,
Y.
,
2018
, “
Residual Dense Network for Image Super-Resolution
,”
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
,
Salt Lake City, UT
,
June 18–22
, pp.
2472
2481
.
31.
Szegedy
,
C.
,
Liu
,
W.
,
Jia
,
Y.
,
Sermanet
,
P.
,
Reed
,
S.
,
Anguelov
,
D.
,
Erhan
,
D.
,
Vanhoucke
,
V.
, and
Rabinovich
,
A.
,
2015
, “
Going Deeper With Convolutions
,”
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
,
Boston, MA
,
June 7–12
, pp.
1
9
.
32.
Liang
,
J.
,
Cao
,
J.
,
Sun
,
G.
,
Zhang
,
K.
,
Van Gool
,
L.
, and
Timofte
,
R.
,
2021
, “
SwinIR: Image Restoration Using Swin Transformer
,”
IEEE International Conference on Computer Vision Workshops.
,
Virtual
,
Oct. 11–17
.