Citation: 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
Received: 3 February 2023
Revised: 21 February 2023
Accepted: 23 February 2023
Published: 27 February 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
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conditions of the Creative Commons
Attribution (CC BY) license (https://
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4.0/).
Article
Prediction of Effective Elastic and Thermal Properties of
Heterogeneous Materials Using Convolutional
Neural Networks
Hamdi Béji , Toufik Kanit * and Tanguy Messager
Univ. Lille, ULR7512—Unité de Mécanique de Lille—Joseph Boussinesq (UML), F-59000 Lille, France;
hamdi.beji@univ-lille.fr (H.B.); tanguy.messager@polytech-lille.fr (T.M.)
* Correspondence: tkanit@univ-lille.fr
Abstract:
The aim of this study is to develop a new method to predict the effective elastic and
thermal behavior of heterogeneous materials using Convolutional Neural Networks CNN. This
work consists first of all in building a large database containing microstructures of two phases
of heterogeneous material with different shapes (circular, elliptical, square, rectangular), volume
fractions of the inclusion (20%, 25%, 30%), and different contrasts between the two phases in term of
Young modulus and also thermal conductivity. The contrast expresses the degree of heterogeneity
in the heterogeneous material, when the value of C is quite important (C » 1) or quite low (C « 1),
it means that the material is extremely heterogeneous, while C= 1, the material becomes totally
homogeneous. In the case of elastic properties, the contrast is expressed as the ratio between Young’s
modulus of the inclusion and that of the matrix ( C =
E
i
E
m
), while for thermal properties, this ratio is
expressed as a function of the thermal conductivity of both phases (C =
λ
i
λ
m
). In our work, the model
will be tested on two values of contrast (10 and 100). These microstructures will be used to estimate
the elastic and thermal behavior by calculating the effective bulk, shear, and thermal conductivity
values using a finite element method. The collected databases will be trained and tested on a deep
learning model composed of a first convolutional network capable of extracting features and a second
fully connected network that allows, through these parameters, the adjustment of the error between
the found output and the expected one. The model was verified using a Mean Absolute Percentage
Error (MAPE) loss function. The prediction results were excellent, with a prediction score between
92% and 98%, which justifies the good choice of the model parameters.
Keywords:
numerical homogenization; convolutional neural network; deep learning; regression model
1. Introduction
Heterogeneous and composite materials [
1
–
3
] are becoming increasingly popular in
many industrial sectors, including the aeronautic and automotive. However, their potential
cannot be fully exploited nowadays because of their variability and the complexity of their
microstructural morphology.
In order to predict the performance (e.g., thermal, mechanical, . . .) of these composites
and heterogeneous media for arrangements and morphologies of heterogeneities as varied
as in reality, it would be too costly in time and means to rely directly on experimental
and numerical homogenization approaches [4–6] based on the use of real [7] or virtual [8]
microstructures. However, it is possible to use the experimental data to generate many
random but statistically equivalent virtual microstructures. The goal is the systematic
prediction of macroscopic behavior patterns of these materials. The large range of materials
to be treated represents a very complete database for the use of artificial intelligence and
neural networks to build homogenized and especially well-optimized behavioral models [
9
]
on the level of determining parameters such as shape, volume fraction, contrast. ..
Appl. Mech. 2023, 4, 287–303. https://doi.org/10.3390/applmech4010016 https://www.mdpi.com/journal/applmech
