=Paper=
{{Paper
|id=Vol-2791/2020200001
|storemode=property
|title=Using Machine Learning to Predict the Stress-Strain State of a Rectangular Plate with a Circular Cut-Out
|pdfUrl=https://ceur-ws.org/Vol-2791/2020200001.pdf
|volume=Vol-2791
|authors=Oksana Choporova,Andrey Lisnyak
}}
==Using Machine Learning to Predict the Stress-Strain State of a Rectangular Plate with a Circular Cut-Out==
https://ceur-ws.org/Vol-2791/2020200001.pdf
Using Machine Learning to Predict the Stress-strain State
of a Rectangular Plate with a Circular Cut-out
Oksana Choporova1[0000-0003-3167-7869], Andrey Lisnyak2[0000-0001-9669-7858]
1Zaporizhzhia National University, Zhukovsky str., 66, Zaporizhzhia, 69600, Ukraine
o.choporova@gmail.com
2Zaporizhzhia National University, Zhukovsky str., 66, Zaporizhzhia, 69600, Ukraine
a.lisnyak@znu.edu.ua
Abstract. The paper describes the scheme of machine learning using for the
stress-strain state analysis of a rectangular plate with a circular cut-out. The
plate might be of arbitrary sizes and the cut-out might be of an arbitrary radius.
Each side of the plate is supposed to be free, supported or fixed. Additional in-
put parameters of the data set are following: size of plate’s side, thickness of the
plate, Young’s modulus, Poison’s coefficient, and pressure load. Initial parame-
ters have been random generated. The training set is generated by the finite el-
ement method. The artificial neural network merges numerical and one-hot in-
put layers. The developed regression model allows to predict von Mises stresses
for a rectangular plate with a circular cut-out.
Keywords: Machine Learning, Artificial Neural Network, Stress-Strain State,
Plate, Pre-diction, Regression.
1 Introduction
Machine learning, one of the six disciplines of Artificial Intelligence (AI) without
which the problems of having machines acting humanly could not be accomplished.
Machine learning allows us to ‘teach’ computers how to perform problems providing
examples of how they should be done [1]. Machine learning is a useful tool for abun-
dant data (also called examples or patterns) explaining a certain phenomenon. The
world is quietly being reshaped by machine learning, the Artificial Neural Network
(also referred in this manuscript as ANN or neural net) being its oldest [2] and most
powerful [3] technique. ANNs also lead the number of practical applications, virtually
covering any field of knowledge [4].
Applications of ANNs to engineering structures arises in a variety of industries
such as construction, automotive, space structures, etc. ANNs allow to develop mod-
els e.g. for the stress-train state estimation of some type of solids. Thus, the develop-
ment of machine learning methods for predicting the behavior of engineering struc-
tures is urgent.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
2
In aviation engineering and shipbuilding, the use of prismatic solids with a cut-out
in which one size (thickness) is much smaller than the other two is widespread. Such
solids could be modeled by plates. Models based on artificial neuron networks should
take into account the geometric and mechanical parameters of the body, as well as
boundary conditions.
1.1 Analysis of recent research and publications
The increasing popularity of artificial neural networks leads to an increasing amount
of researches devoted to the development of ANN models for modeling various fields.
Modelling of solids the stress-train state is also possible domain of ANNs applica-
tions. For example, [5–7] explores the possibilities of machine learning to solve the
problems of fracture mechanics. Particularly, in [5], the data of 64 computational
experiments and 3 full-scale experiments are used for the training of the neural net-
work to predict possible zones of beam destruction. In [6], a neural network based on
the Kalman filter is employed to predict the collapse of a highway on a bridge pro-
cessing temperature and oscillation data. In [7], a model based on the self-organizing
map of Kohonen is developed to detect the fracture using vibration data. In [8], the
possibilities of neural networks in the prediction the maximum displacements in rail
beams are investigated. The neural network model is constructed as a function of two
variables: the frictional parameter and load speed. 663 points are used for training,
which allowed to get maximum the finite element model error in 5.4%. In [9] a com-
bination of principal component analysis (PCA) and convolutional neural networks
(CNN) are used to predict the entire stress-strain behavior of binary composites
evauated over the entire failure path, motivated by the significantly faster inference
speed of empirical models. The authors show that PCA transforms the stress-strain
curves into an effective latent space by visualizing the eigenbasis of PCA. Despite
having a dataset of only 10-27% of possible microstructure configurations, the mean
absolute error of the prediction is <10% of the range of values in the dataset, when
measuring model performance based on derived material descriptors, such as modu-
lus, strength, and toughness. Their study demonstrates the potential to use machine
learning to accelerate material design, characterization, and optimization. In [10] the
proposed strategy demonstrates the effectiveness of machine learning to reduce exper-
imental efforts for damage characterization in composites.
Thus, the analysis of last researches and publications allows to conclude that prob-
lems of developing models based on neural networks for predictions stress-stain state
are actual.
1.2 The Problem Statement
Computer-aided design requires the development of methods and software for stress
components fast estimation. The classical methods of mathematical modeling (e.g. the
finite element method) allow to evaluate the stress-strain state with a good accuracy.
Moreover, the preparation of adequate mathematical models and the corresponding
computational experiments could be time-consuming. A possible alternative is ma-
3
chine learning methods. Artificial neural networks (ANN) are frequently used in ma-
chine learning. ANN could be trained over a data set of an object states and then
employed as interactive assistants in the design process.
The problem of predicting the parameters of the state of an object by its geometric
and mechanical parameters could be classified as a regression problem.
1.3 Purpose
The purpose of this analysis is to make a prediction model which be able to predict
the stress-strain state of a rectangular isotropic plate with a circular cut out. A plate
with width 𝑤 , height ℎ , radius of cut out 𝑟, the center of a circle (𝑥0 , 𝑦0 ) , uniform
thickness ℎ, Young's modulus 𝐸, Poisson's ratio 𝜗 . A plate is loaded transversely
by a distributed load 𝑞 per unit area. Edges of a plate may be clamped, simply sup-
ported, or free.
Research objectives such as:
1. Develop an algorithm for training and testing models.
2. Explore the capabilities of ANN for prediction of maximum plate deflection.
3. Develop a neural network to predict the maximum deflection of the plate.
2 Research methods
The plate might be of arbitrary sizes and the cut-out might be of an arbitrary radius.
Each side of the plate is supposed to be free, supported or fixed. Additional input
parameters of the data set are following: size of plate’s side, thickness of the plate,
Young’s modulus, Poison’s coefficient, and pressure load.
A dataset is generated using the finite element method. Parameters of a plate are
randomly generated with following restrictions:
𝑤 ∈ [0.1; 10] (meters);
ℎ ∈ [0.1; 10] (meters);
1
𝑟 ∈ [3𝑠; 𝑎 − 3𝑠] (meters), where s is a size of a background cell in a meshing rou-
2
max(𝑤,ℎ)
tine (for an uniform mesh we could use 𝑠 = , 𝑛2 is a number of cells in the
𝑛
mesh), 𝑎 = min(𝑤, ℎ);
1
𝑥0 ∈ [0; 𝑎 − 𝑟 − 3𝑠] (meters);
2
1
𝑦0 ∈ [ − 𝑟 − 3𝑠] (meters);
2𝑎
1 1
ℎ ∈ [ 𝑎; 𝑎] (meters);
80 5
𝐸 ∈ [50000; 300000] (MPa);
4
𝜗 ∈ [0; 0,45];
𝑞 ∈ [0,01; 0,1](MPa).
Boundary conditions are also randomly generated. These values are categorical da-
ta. Possible boundary conditions are following: a free edge, a supported edge, a
clamped (fixed) edge. Any combination of boundary conditions is possible excluding
situations with four free edges or one supported edge and three free edges. Denote
𝑤 ℎ
boundary conditions at the edge 𝑥 = − by 𝑐0 , at the edge 𝑦 = − by 𝑐1 , at the edge
2 2
𝑤 ℎ
𝑥 = by 𝑐2 , at the edge 𝑦 = by 𝑐3 . If we also enumerate a free edge by 0, a sup-
2 2
ported edge by 1, a clamped edge by 2, then we get the following restriction:
𝑐1 + 𝑐2 + 𝑐3 + 𝑐4 ≥ 2
Categorical data one hot encoded by vector of 76 values.
The randomly generated dataset includes approximately 11000 records. Approximate-
ly 7500 records are left after data cleaning because the maximum deflection must be
greater than 10−5 meters and less 0.2 of the plate thickness.
Fig. 1. The layers of neurons in the model.
The model of an artificial neural network is developed. This model includes few
dense layers of neurons (see Fig. 1). Separate input layers are used for numerical and
categorical data. Each input layer is connected with hidden dense layers. Then the
5
dense layer merges the output the hidden layers for numerical and categorical data
processing. The last layer is connected with the additional hidden dense layer. Finally,
the output layer is connected with the last hidden layer.
For machine learning, the original set of values is divided into two parts: training and
test. The training part contains the values of the input parameters of the results, which
use an algorithm for constructing trees. The test part is usually less than the training
part, its values are used to test (in the case of regression analysis) the accuracy of
training. The accuracy of the regression analysis is evaluated by calculating the root
mean square deviation of the values calculated for the test part as a result of machine
learning from the original values.
Input and hidden layers use hyperbolic tangent function as activation. The output
layer uses the linear activation.
Fig. 2. Mean square deviation
The number of neurons in the input layers are the same as the number of elements
in input arrays. I.e. 8 neurons are used to process float data and 76 neurons are used to
process categorical data. A number of neurons in hidden layers is variated to get the
minimum of the loss function. The mean squared error is employed is a loss function.
The Fig. 2 shows loss functions for both train and test sets. The mean absolute per-
centage error of the model is approximately 20%.
3 Conclusion
Developed ANN allows to predict the maximum deflection of a rectangular plate with
a circular cutout loaded transversely by a distributed load. The Fig. 2 shows that train-
ing has a good convergence. The mean error of ANN predictions is approximately
6
20%. Prospects for further studies are associated with genetic algorithms using in
ANN optimization.
The key advantage of an artificial neural network is the speed of prediction: the
plate deflection is predicted almost instantaneously (milliseconds) comparing the
finite element method. So, pretrained artificial neural networks could be employed as
interactive assistances in computer-aided design.
References
1. Hertzmann A, Fleet D. Machine Learning and Data Mining, Lecture Notes CSC 411/D11,
Computer Science Department, University of Toronto, Canada(2012).
2. McCulloch WS, Pitts W. A logical calculus of the ideas immanent in nervous activity. Bul-
letin of Mathematical Biophysics, 5(4):115–133 (1943).
3. Hern A. Google says machine learning is the future. Available at:
www.theguardian.com/technology/2016/jun/28/all, last accessed 2019/12/20.
4. Wilamowski BM, Irwin JD. The industrial electronics handbook: Intelligent Systems, CRC
Press, Boca Raton (2011).
5. Abambres M., Marcy M., Doz G. Potential of Neural Networks for Structural Damage Lo-
calization //engrXiv. 2018. URL: https://engrxiv.org/rghpf/. DOI: 10.31224/osf.io/rghpf.
6. Damage detection of a highway bridge under severe temperature changes using extended
Kalman filter trained neural network / C. Jin, S. Jang, X. Sun [et. al] // Journal of Civil
Structural Health Monitoring. 2016. Vol. 6,Iss. 3, P. 545–560.
7. Onur Avci P. O., Abdeljaber A. O. Self-Organizing Maps for Structural Damage Detec-
tion: A Novel Unsupervised Vibration-Based Algorithm // Journal of Performance of Con-
structed Facilities. 2016. Vol. 30, Iss. 3. P.1–11.
8. Neural Network-based formula for the buckling load prediction of I-section cellular steel
beams /M. Abambres, K. Rajana, K. Tsavdaridis, T. Ribeiro // engrXiv. 2018. URL:
https://engrxiv.org/wg7hd/.DOI: 10.31224/osf.io/wg7hd, last accessed 2020/01/20
9. Ch. Yang, Youngsoo Kim Seunghwa, Ryu Grace X. Gu. Prediction of composite micro-
structure stress-strain curves using convolutional neural networks // Materials and Design
189 (2020). URL: https://doi.org/10.1016/j.matdes.2020.108509
10. Navid Zobeirya, Johannes Reinerb, Reza Vaziric. Theory-guided machine learning for
damage characterization of composites // Composite Structures Volume 246, 15 August
2020, 112407. URL: https://doi.org/10.1016/j.compstruct.2020.112407