# Prediction of Effective Elastic and Thermal Properties of Heterogeneous Materials Using Convolutional Neural Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dataset Collection

#### 2.1. Generation of Virtual Microstructure

- -
- Fix the volume fraction by fixing the dimensions of the matrix and the inclusion.
- -
- Define the random position of the inclusion by using the random function of the Numpy library.
- -
- Define the shape of the inclusion by changing the plotting function (Circle, Ellipse, Rectangle…)
- -
- Binarize the image using the THRESH_BINARY function.
- -
- Save the drawn figure.

#### 2.2. Finite Element Calculations

- -
- convert the binary images generated by the first code into “.ras”.
- -
- create the multi-phase mesh.
- -
- launch the calculation to obtain the “.post” file

## 3. Convolutional Neural Network

#### 3.1. Loading and Pre-Processing Data

#### 3.2. CNN Model

#### 3.2.1. Convolutional Layer

- Less error in learning because the model does not learn from images but from features.
- More accuracy in detection, because the model must recognize features and patterns.

#### 3.2.2. Maxpolling Layer

- -
- gain in accuracy by keeping only relevant data.
- -
- gain in speed: the learning of the model is done much faster because the data is getting progressively smaller.

#### 3.2.3. Flatten Layer

#### 3.2.4. Activation Layer

- The ReLU Function

- The Linear Function

#### 3.3. Compiling and Training

- m
_{t}: aggregate of gradients at time t [current] (initially, m_{t}= 0) - m
_{t − 1}: aggregate of gradients at time t − 1 [previous] - w
_{t}: weights at time t - w
_{t + 1}: weights at time + 1 - $\alpha $
_{t}: learning rate at time t - δL: derivative of Loss Function
- δw
_{t}: derivative of weights at time t - $\beta $: Moving average parameter (const, 0.9).

- ${w}_{t}$: weights at time t
- ${w}_{t\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1}$: weights at time $t+1$
- ${\alpha}_{t}$: learning rate at time t
- δL: derivative of Loss Function
- δ${w}_{t}$: derivative of weights at time t
- ${v}_{t}$: sum of square of past gradients. [i.e., sum (δL/δ${w}_{t-1}$)] (initially, v${}_{t}$ = 0)
- $\beta $: Moving average parameter (const, 0.9)
- $\u03f5$: A small positive constant (${10}^{-8}$).

- n is the number of fitted points.
- ${T}_{i}$ is the actual value.
- ${P}_{i}$ is the predicted value.

- n is the number of fitted points.
- ${T}_{j}$ is the actual value.
- ${P}_{j}$ is the predicted value.

- -
- The training data (train_X), the target data (train_y).
- -
- The validation data.
- -
- Epochs: the number of times the model will run the data. The more epochs we run, the more the model will improve up to a certain point. After this point, the model will stop improving at each epoch.

#### 3.4. Evaluation and Prediction

- ${y}_{i}$ is the actual value.
- ${\widehat{y}}_{i}$ is the predicted value.

- -
- Outliers*: a data which does not ”fit in” with the rest of the data that we are analysing.
- -
- IQR*: the interquartile range, it’s the measure of statistical dispersion equal to the difference between 25% and 75% percentile.
- -
- Z-score*: a tool capable of re-scaling data, its value is between $-3$ and 3 in the most cases.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Moumen, A.E. Prévision du Comportement des Matériaux Hétérogènes Basée sur l’Homogénéisation Numérique: Modélisation, Visualisation et Étude Morphologique. Ph.D. Thesis, IBN Zohr University, Agadir, Marocco, 2014. [Google Scholar]
- Kováčik, J.; Simančík, F. Aluminium foam—Modulus of elasticity and electrical conductivity according to percolation theory. Scr. Mater.
**1998**, 39, 239–246. [Google Scholar] [CrossRef] - Ding, Y. Analyse Morphologique de la Microstructure 3D de Réfractaires Électrofondus à Très Haute Teneur en Zircone: Relations Avec les Propriétés Mécaniques, Chimiques et le Comportement Pendant la Transformation Quadratique-Monoclinique. Ph.D. Thesis, Ecole Nationale Supérieure des Mines de Paris, Paris, France, 2012. [Google Scholar]
- Zhou, Q.; Zhang, H.W.; Zheng, Y.G. A homogenization technique for heat transfer in periodic granular materials. Adv. Powder Technol.
**2012**, 23, 104–114. [Google Scholar] [CrossRef] - Chaboche, J. Le Concept de Contrainte Effective Appliqué à l’Élasticité et à la Viscoplasticité en Présence d’un Endommagement Anisotrope. In Mechanical Behavior of Aniotropic Solids; Springer: Dordrecht, The Netherlands, 1982. [Google Scholar] [CrossRef]
- Wu, T.; Temizer, I.; Wriggers, P. Computational thermal homogenization of concrete. Cem. Concr. Compos.
**2013**, 35, 59–70. [Google Scholar] [CrossRef] - Kanit, T.; N’guyen, F.; Forest, S.; Jeulin, D.; Reed, M.; Singleton, S. Apparent and effective physical properties of heterogeneous materials: Representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 3960–3982. [Google Scholar] [CrossRef] - González, C.; Segurado, J.; Llorca, J. Numerical simulation of elasto-plastic deformation of composites: Evolution of stress microfields and implications for homogenization models. J. Mech. Phys. Solids
**2004**, 52, 1573–1593. [Google Scholar] [CrossRef] - Liu, X.; Tian, S.; Tao, F.; Yu, W. A review of artificial neural networks in the constitutive modeling of composite materials. Compos. Part B Eng.
**2021**, 224, 109152. [Google Scholar] [CrossRef] - Mentges, N.; Dashtbozorg, B.; Mirkhalaf, S.M. A micromechanics-based artificial neural networks model for elastic properties of short fiber composites. Compos. Part B Eng.
**2021**, 213, 108736. [Google Scholar] [CrossRef] - Li, B.; Zhuang, X. Multiscale computation on feedforward neural network and recurrent neural network. Front. Struct. Civ. Eng.
**2020**, 14, 1285–1298. [Google Scholar] [CrossRef] - Ford, E.; Maneparambil, K.; Rajan, S.; Neithalath, N. Machine learning-based accelerated property prediction of two-phase materials using microstructural descriptors and finite element analysis. Comput. Mater. Sci.
**2021**, 191, 110328. [Google Scholar] [CrossRef] - Minfei, L.; Yidong, G.; Ze, C.; Zhi, W.; Erik, S.; Branko, Š. Microstructure-informed deep convolutional neural network for predicting short-term creep modulus of cement paste. Cem. Concr. Res.
**2022**, 152, 106681. [Google Scholar] [CrossRef] - Sun, Z.; Lei, Z.; Zou, J.; Bai, R.; Jiang, H.; Yan, C. Prediction of failure behavior of composite hat-stiffened panels under in-plane shear using artificial neural network. Compos. Struct.
**2021**, 272, 114238. [Google Scholar] [CrossRef] - Barbosa, A.; Upadhyaya, P.; Iype, E. Neural network for mechanical property estimation of multilayered laminate composite. Mater. Today Proc.
**2020**, 28, 982–985. [Google Scholar] [CrossRef] - Kim, D.W.; Lim, J.H.; Lee, S. Prediction and validation of the transverse mechanical behavior of unidirectional composites considering interfacial debonding through convolutional neural networks. Compos. Part B Eng.
**2021**, 225, 109314. [Google Scholar] [CrossRef] - Venkatesan, N. An Introduction to Making Scientific Publication Plots with Python. Available online: https://towardsdatascience.com/an-introduction-to-making-scientific-publication-plots-with-python-ea19dfa7f51e (accessed on 2 February 2023).
- Gregori, E. Introduction To Computer Vision Using OpenCV. Presented at the 2012 Embedded Systems Conference, San Jose, CA, USA, 26–29 March 2011. [Google Scholar]
- Tanner, G. Introduction to Deep Learning with Keras. 2019. Available online: https://gilberttanner.com/blog/introduction-to-deep-learning-withkeras/ (accessed on 2 February 2023).
- Abadi, M.; Barham, P.; Chen, J.; Chen, Z.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Irving, G.; Isard, M.; et al. TensorFlow: A System for Large-Scale Machine Learning. In Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI ‘16), Savannah, GA, USA, 2–4 November 2016. Section: GBlog. [Google Scholar]
- Kanit, T.; Forest, S.; Galliet, I.; Mounoury, V.; Jeulin, D. Determination of the size of the representative volume element for random composites: Statistical and numerical approach. Int. J. Solids Struct.
**2003**, 40, 3647–3679. [Google Scholar] [CrossRef] - Rogala, T.; Przystałka, P.; Katunin, A. Damage classification in composite structures based on X-ray computed tomography scans using features evaluation and deep neural networks. Procedia Struct. Integr.
**2022**, 37, 187–194. [Google Scholar] [CrossRef]

**Figure 1.**Descriptive flowchart of the data collection process through a python script based on the use of Matplotlib and OpenCV.

**Figure 2.**Binarization of the different microstructure images with a different shape and a fixed volume fraction (30%) using THRESH_BINARY function of OpenCV library.

**Figure 3.**The elastic (Young modulus) and thermal (Thermal conductivity) properties of two different material phases (the values in the table are not realistic, just used for the calculation).

**Figure 4.**Descriptive flowchart of the output collection process based on the use of finite element software.

**Figure 5.**Example of a finite element calculation of the bulk modulus based on a 2D morphological image of a composite.

**Figure 6.**Detection of the important features of the 2D microstructure using the different filters of convolutional layer.

**Figure 7.**Reduction of the image shape using polling operation consists at extracting the maximum value from each sub-matrix.

**Figure 8.**Transformation of the output matrix into a vector using flattening by recovering the pixels line by line and adding them to the final vector.

**Figure 9.**Mean Absolute Percentage Error curves using training and validation data consists in following the evolution of the training according to the number of epochs, (

**a**) presents scenario 4 while (

**b**) presents scenario 6.

**Figure 10.**Mean Absolute Error curves using training and validation data consists in following the evolution of the training according to the number of epochs, (

**a**) presents scenario 4 while (

**b**) presents scenario 6.

**Table 1.**The 6 different calculation scenarios to be tested, collected by modifying the contrast and the volume fraction.

Scenarios | Contrast | Volume Fraction |
---|---|---|

Scenario 1 | 20% | |

Scenario 2 | 10 | 25% |

Scenario 3 | 30% | |

Scenario 4 | 20% | |

Scenario 5 | 100 | 25% |

Scenario 6 | 30% |

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**MDPI and ACS Style**

Béji, H.; Kanit, T.; Messager, T.
Prediction of Effective Elastic and Thermal Properties of Heterogeneous Materials Using Convolutional Neural Networks. *Appl. Mech.* **2023**, *4*, 287-303.
https://doi.org/10.3390/applmech4010016

**AMA Style**

Béji H, Kanit T, Messager T.
Prediction of Effective Elastic and Thermal Properties of Heterogeneous Materials Using Convolutional Neural Networks. *Applied Mechanics*. 2023; 4(1):287-303.
https://doi.org/10.3390/applmech4010016

**Chicago/Turabian Style**

Béji, Hamdi, Toufik Kanit, and Tanguy Messager.
2023. "Prediction of Effective Elastic and Thermal Properties of Heterogeneous Materials Using Convolutional Neural Networks" *Applied Mechanics* 4, no. 1: 287-303.
https://doi.org/10.3390/applmech4010016