Journal of information and communication convergence engineering 2022; 20(4): 303-308
Published online December 31, 2022
https://doi.org/10.56977/jicce.2022.20.4.303
© Korea Institute of Information and Communication Engineering
Correspondence to : Gwanghyun Jo (E-mail: gwanghyun@kunsan.ac.kr, Tel: +82-63-469-4542)
Department of Mathematics, Kunsan National University, Gunsan 54150, Republic of Korea
Seong-Yoon Shin (E-mail: s3397220@kunsan.ac.kr, Tel: +82-63-469-4860)
School of Computer Information & Communication Engineering, Kunsan National University, Gunsan 54150, Republic of Korea
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Simulating the heat transfer in a composite material is an important topic in material science. Difficulties arise from the fact that adjacent materials cannot match perfectly, resulting in discontinuity in the temperature variables. Although there have been several numerical methods for solving the heat-transfer problem in imperfect contact conditions, the methods known so far are complicated to implement, and the computational times are non-negligible. In this study, we developed a ResNet-type deep neural network for simulating a heat transfer model in a composite material. To train the neural network, we generated datasets by numerically solving the heat-transfer equations with Kapitza thermal resistance conditions. Because datasets involve various configurations of composite materials, our neural networks are robust to the shapes of material-material interfaces. Our algorithm can predict the thermal behavior in real time once the networks are trained. The performance of the proposed neural networks is documented, where the root mean square error (RMSE) and mean absolute error (MAE) are below 2.47E-6, and 7.00E-4, respectively.
Keywords Composite material, deep learning, heat transfer, Kapitza thermal resistance, ResNet
Heat transfer in composite media has been studied extensively because of its importance in material science (see [1,2] and the references therein). If the interfaces of the adjacent materials are in perfect contact, both the temperature and flux variables are continuously matched along the material interfaces. However, it is difficult to expect perfectly matched conditions in practice, that is, in the presence of gaps between the contacting domains, the temperatures are discontinuous across the material interfaces. Kapitza [3] modelled jump conditions for imperfect contact where temperature jumps appear across the interface, the amounts of which are affected by the Kapitza thermal resistance.
There have been various attempts to solve thermal transfer problems involving imperfect conditions using finite element methods (FEMs), see [4-6]. However, these methods are rather complicated because the Kapitza-type interface conditions must be carefully treated. In addition, simulations of the heat transfer in composite materials cannot be obtained in real time by FEM-based algorithms.
However, there have been huge developments in deep learning (DL) communities (see [7-12] and the references therein). One of the main advantages of DL-based methods is that once the networks are trained, they can produce predictions of the target variable in real time.
In this study, we developed a DL-based thermal simulation for a composite material. One good aspect of the artificial neural network (ANN)-based approach is that one can expect real-time thermal conduction simulation once the networks are trained. This is in contrast to conventional FEM-based methods, whose computation times are non-negligible. Another advantage of the proposed DL-based algorithms is that they are robust with respect to the geometry of the material-material interface. In fact, the inputs of the proposed DL architectures are material-material interfaces. Once the geometrical contribution of a composite material is substituted into the algorithm, the desired prediction of the target variable (i.e., temperature) can be obtained in real time. This method of obtaining solutions is more user-friendly compared to FEM-based approaches.
To train the neural networks, we produced datasets by numerically solving the heat equations for various configurations of composite material shapes. Here, the immersed finite element method (IFEM) [13] was used as the numerical method. For the architecture of the DL, we employed ResNet-type [7] structures, where so-called ResBlocks with identity maps are repeated several times. With the presence of identity-maps in ResBlocks, one can avoid the “gradient-vanishing” phenomenon even with a large number of layers.
The remainder of this paper is organized as follows. The model equation and the derivation of its weak form are presented in the next section. ResNet-based neural networks are developed in Section 3. The next section reports the performance of ResNet-based neural networks in predicting the solutions of heat-transfer models involving composite materials. Finally, we present our conclusions in the last section.
In this section, the governing equation of the model and its derivation of a weak form are described. Consider a composite material Ω ⊂ R^{2} having two parts Ω = Ω_{1} ∪ Ω_{2} where Ω_{2} is imbedded in Ω. Here, denote Γ to be the material interface dividing Ω_{1} and Ω_{2}. The governing equation for the heat transfer model involving the Kapitza interface condition with a Kapitza thermal resistance α is as follows:
where
The weak problem for (1)-(5) can be derived as follows: For convenience, we assume that
By multiplying a function to (1) and (2) and by applying integration by parts in each subdomain, we have
Here, using Kapitza interface conditions (3) and (4), the second term of the above equation is written as follows:
Summarizing the above equations, a weak problem for (1)-(5) is written as follows: find such that it satisfies
Based on the weak problem (6), the IFEM is employed to generate datasets for training the ANN.
In this section, we propose ANN-based simulation methods for heat-transfer models in composite materials. In particular, ResNet-based neural networks produce an approximation solution for the temperature variable in Eqs. (1),-(5). The remainder of this section is organized as follows. The IFEM for model Eqs. (1),-(5) is described in the first subsection. The ResNet-based networks for heat transfer are described in the following subsections.
In this subsection, we propose the IFEM for heat equations involving imperfect contact conditions, based on [13]. Our version of the IFEM is similar to that of [13], but a different local space is employed. We briefly describe the methods as follows. (details of the derivations of discrete weak problems can be found in [13]).
Let 𝓣_{h} be a uniform triangulation of Ω by right triangles having an edge size h and let 𝓕_{h} be the set of edges of elements in 𝓣_{h}. Let E be a typical element in 𝓣_{h} and let
and coefficients (
where
We are now in a position to state IFEM for model equations (1)-(5): find satisfying
for all
It was reported in [13] that numerical solutions obtained by the IFEM for robo-interface condition elliptic partial differential equations have optimal error convergences, that is, as the mesh size is halved
In this subsection, ResNet-based neural networks are developed to predict the solutions (temperature variables) of heat transfer in a composite material. First, we explain the generation process of the datasets. We intend to develop ANNs that are robust with respect to the geometry of embedded materials. For this purpose, different interface shapes were considered, as shown in Fig. 1. Here, interfaces were created by perturbating circles using randomly chosen parameters
Here,Γ is chosen by level sets for
The formatting of datasets of type (
A two-dimensional input
Next, by IFEM in (7), the numerical solution 𝓣_{h} is obtained.
Finally, by matching the point-wise values of
The remainder of this subsection is devoted to the development of ANNs for a heat-transfer model in a composite material. Because the imperfect contact Kapitza resistance model has discontinuous solutions, it requires a large number of network parameters. To avoid the so-called gradient-vanishing phenomenon for deep neural networks, ResNet-type networks [7], which accompany the concept of skip connections, were employed in this study.
We denote
where
Finally, a ResNet type neural network is proposed as below:
ResNet(
Suppose that input X is given. ResNet( |
We note that at the down-sampling stage in the above algorithm, the resolution of the data is reduced while the number of kernels increases. Here, the total CPU time at the lower level does not increase even when the number of filters is doubled. In this manner, features at different resolutions can be effectively extracted. At the lowest level, RESBLOCK was applied
In this section, we report the performance of ResNet(
A dataset containing 10,000 IFEM generated solutions for the heat-transfer equations involving composite materials was used. Among these, 7000 samples were used for the training set, 1000 samples were used for the validation set, and 2000 samples were assigned to the test set. To train Res- Net, an ADAM optimizer with a learning-rate parameter of 0.0002 was used for 200 epochs. ResNet(
Table 1 reports CPU time and accuracies of solutions obtained by ResNet(
Table 1 . A comparison of ResNet(m,d) with respect to parameters (m,d) in terms of accuracies, total number of parameters, and CPU time
(m,d) | RMSE | MAE | CPU time (s) | Parameters |
---|---|---|---|---|
(5,5) | 1.71E-6 | 6.99E-4 | 1,258 | 163,901 |
(5,10) | 2.46E-6 | 6.08E-4 | 1,696 | 308,301 |
(5,15) | 2.28E-6 | 6.18E-4 | 2,050 | 452,701 |
(10,5) | 8.18E-7 | 4.77E-4 | 1,1378 | 653,601 |
(10,10) | 1.39E-6 | 6.57E-4 | 2,216 | 1,230,401 |
(10,15) | 1.91E-6 | 5.63E-4 | 2,755 | 1,807,201 |
The temperature variables predicted by ResNet(10,5) for heat-transfer problems with different composite material shapes are plotted in Fig. 3. Here, it is observed that the temperatures are discontinuous near the material-material interfaces. Indeed, owing to gaps between the two materials, there are relatively small temperature drops in the embedded material, which coincides with the phenomena observed in [5].
In this study, we developed new algorithms to predict the solutions of heat-transfer models in composite materials. First, datasets were generated by solving heat-transfer equations numerically, and the IFEM was adopted for the implementation. To develop neural networks that are robust to the shapes of material subdomains, various geometries of material-material interfaces were considered in the datasets. Once the datasets were generated, we trained the ResNet-type neural networks, which we named ResNet(m,d). The results showed that the RMSEs were below 3E-6 and the MAEs were below 7E-4. In addition, the discontinuity of temperature variables were described by ResNet(m,d) in a reasonable manner.
The second author (G. Jo) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2020R1C1C1A01005396).
We would like to thank Editage (www.editage.co.kr) for English language editing.
received his M.S. degree in Computer Application Technology from Huazhong University of Science and Technology, Wuhan, China in 2009. From 2016 to the present, he has been an associate professor in the Information Technology Center of Jiujiang University in China. received his M.S. and Ph.D. degrees from the Dept. of Computer Information Engineering of Kunsan National University, Gunsan, Korea, in 2018 and 2021, respectively. F His research interests Deep Learning, Image Processing, Diagnosis, etc.
received his M.S. and Ph. D. degree in department of mathematical scient, KAIST in 2013 and 2018 respectively. From 2019 to the present, he has been a faculty member of the Department of Mathematics in Kunsan National University, Korea. His research interests include numerical analysis, computational fluid dynamics, machine learning.
received his M.S. and Ph.D. degrees from the Dept. of Computer Information Engineering of Kunsan National University, Gunsan, Korea, in 1997 and 2003, respectively. From 2006 to the present, he has been a professor in the same department. His research interests include image processing, computer vision, and virtual reality.
Journal of information and communication convergence engineering 2022; 20(4): 303-308
Published online December 31, 2022 https://doi.org/10.56977/jicce.2022.20.4.303
Copyright © Korea Institute of Information and Communication Engineering.
Guangxing Wang ^{1}, Gwanghyun Jo ^{2*}, and Seong-Yoon Shin^{3* }, Member, KIICE
^{1}Computer and Big Data Science, JiuJiang University, Jiujiang 332005, China
^{2}Department of Mathematics, Kunsan National University, Gunsan 54150, Republic of Korea
^{3}School of Computer Information & Communication Engineering, Kunsan National University, Gunsan 54150, Republic of Korea
Correspondence to:Gwanghyun Jo (E-mail: gwanghyun@kunsan.ac.kr, Tel: +82-63-469-4542)
Department of Mathematics, Kunsan National University, Gunsan 54150, Republic of Korea
Seong-Yoon Shin (E-mail: s3397220@kunsan.ac.kr, Tel: +82-63-469-4860)
School of Computer Information & Communication Engineering, Kunsan National University, Gunsan 54150, Republic of Korea
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Simulating the heat transfer in a composite material is an important topic in material science. Difficulties arise from the fact that adjacent materials cannot match perfectly, resulting in discontinuity in the temperature variables. Although there have been several numerical methods for solving the heat-transfer problem in imperfect contact conditions, the methods known so far are complicated to implement, and the computational times are non-negligible. In this study, we developed a ResNet-type deep neural network for simulating a heat transfer model in a composite material. To train the neural network, we generated datasets by numerically solving the heat-transfer equations with Kapitza thermal resistance conditions. Because datasets involve various configurations of composite materials, our neural networks are robust to the shapes of material-material interfaces. Our algorithm can predict the thermal behavior in real time once the networks are trained. The performance of the proposed neural networks is documented, where the root mean square error (RMSE) and mean absolute error (MAE) are below 2.47E-6, and 7.00E-4, respectively.
Keywords: Composite material, deep learning, heat transfer, Kapitza thermal resistance, ResNet
Heat transfer in composite media has been studied extensively because of its importance in material science (see [1,2] and the references therein). If the interfaces of the adjacent materials are in perfect contact, both the temperature and flux variables are continuously matched along the material interfaces. However, it is difficult to expect perfectly matched conditions in practice, that is, in the presence of gaps between the contacting domains, the temperatures are discontinuous across the material interfaces. Kapitza [3] modelled jump conditions for imperfect contact where temperature jumps appear across the interface, the amounts of which are affected by the Kapitza thermal resistance.
There have been various attempts to solve thermal transfer problems involving imperfect conditions using finite element methods (FEMs), see [4-6]. However, these methods are rather complicated because the Kapitza-type interface conditions must be carefully treated. In addition, simulations of the heat transfer in composite materials cannot be obtained in real time by FEM-based algorithms.
However, there have been huge developments in deep learning (DL) communities (see [7-12] and the references therein). One of the main advantages of DL-based methods is that once the networks are trained, they can produce predictions of the target variable in real time.
In this study, we developed a DL-based thermal simulation for a composite material. One good aspect of the artificial neural network (ANN)-based approach is that one can expect real-time thermal conduction simulation once the networks are trained. This is in contrast to conventional FEM-based methods, whose computation times are non-negligible. Another advantage of the proposed DL-based algorithms is that they are robust with respect to the geometry of the material-material interface. In fact, the inputs of the proposed DL architectures are material-material interfaces. Once the geometrical contribution of a composite material is substituted into the algorithm, the desired prediction of the target variable (i.e., temperature) can be obtained in real time. This method of obtaining solutions is more user-friendly compared to FEM-based approaches.
To train the neural networks, we produced datasets by numerically solving the heat equations for various configurations of composite material shapes. Here, the immersed finite element method (IFEM) [13] was used as the numerical method. For the architecture of the DL, we employed ResNet-type [7] structures, where so-called ResBlocks with identity maps are repeated several times. With the presence of identity-maps in ResBlocks, one can avoid the “gradient-vanishing” phenomenon even with a large number of layers.
The remainder of this paper is organized as follows. The model equation and the derivation of its weak form are presented in the next section. ResNet-based neural networks are developed in Section 3. The next section reports the performance of ResNet-based neural networks in predicting the solutions of heat-transfer models involving composite materials. Finally, we present our conclusions in the last section.
In this section, the governing equation of the model and its derivation of a weak form are described. Consider a composite material Ω ⊂ R^{2} having two parts Ω = Ω_{1} ∪ Ω_{2} where Ω_{2} is imbedded in Ω. Here, denote Γ to be the material interface dividing Ω_{1} and Ω_{2}. The governing equation for the heat transfer model involving the Kapitza interface condition with a Kapitza thermal resistance α is as follows:
where
The weak problem for (1)-(5) can be derived as follows: For convenience, we assume that
By multiplying a function to (1) and (2) and by applying integration by parts in each subdomain, we have
Here, using Kapitza interface conditions (3) and (4), the second term of the above equation is written as follows:
Summarizing the above equations, a weak problem for (1)-(5) is written as follows: find such that it satisfies
Based on the weak problem (6), the IFEM is employed to generate datasets for training the ANN.
In this section, we propose ANN-based simulation methods for heat-transfer models in composite materials. In particular, ResNet-based neural networks produce an approximation solution for the temperature variable in Eqs. (1),-(5). The remainder of this section is organized as follows. The IFEM for model Eqs. (1),-(5) is described in the first subsection. The ResNet-based networks for heat transfer are described in the following subsections.
In this subsection, we propose the IFEM for heat equations involving imperfect contact conditions, based on [13]. Our version of the IFEM is similar to that of [13], but a different local space is employed. We briefly describe the methods as follows. (details of the derivations of discrete weak problems can be found in [13]).
Let 𝓣_{h} be a uniform triangulation of Ω by right triangles having an edge size h and let 𝓕_{h} be the set of edges of elements in 𝓣_{h}. Let E be a typical element in 𝓣_{h} and let
and coefficients (
where
We are now in a position to state IFEM for model equations (1)-(5): find satisfying
for all
It was reported in [13] that numerical solutions obtained by the IFEM for robo-interface condition elliptic partial differential equations have optimal error convergences, that is, as the mesh size is halved
In this subsection, ResNet-based neural networks are developed to predict the solutions (temperature variables) of heat transfer in a composite material. First, we explain the generation process of the datasets. We intend to develop ANNs that are robust with respect to the geometry of embedded materials. For this purpose, different interface shapes were considered, as shown in Fig. 1. Here, interfaces were created by perturbating circles using randomly chosen parameters
Here,Γ is chosen by level sets for
The formatting of datasets of type (
A two-dimensional input
Next, by IFEM in (7), the numerical solution 𝓣_{h} is obtained.
Finally, by matching the point-wise values of
The remainder of this subsection is devoted to the development of ANNs for a heat-transfer model in a composite material. Because the imperfect contact Kapitza resistance model has discontinuous solutions, it requires a large number of network parameters. To avoid the so-called gradient-vanishing phenomenon for deep neural networks, ResNet-type networks [7], which accompany the concept of skip connections, were employed in this study.
We denote
where
Finally, a ResNet type neural network is proposed as below:
ResNet(
Suppose that input X is given. ResNet( |
We note that at the down-sampling stage in the above algorithm, the resolution of the data is reduced while the number of kernels increases. Here, the total CPU time at the lower level does not increase even when the number of filters is doubled. In this manner, features at different resolutions can be effectively extracted. At the lowest level, RESBLOCK was applied
In this section, we report the performance of ResNet(
A dataset containing 10,000 IFEM generated solutions for the heat-transfer equations involving composite materials was used. Among these, 7000 samples were used for the training set, 1000 samples were used for the validation set, and 2000 samples were assigned to the test set. To train Res- Net, an ADAM optimizer with a learning-rate parameter of 0.0002 was used for 200 epochs. ResNet(
Table 1 reports CPU time and accuracies of solutions obtained by ResNet(
Table 1 . A comparison of ResNet(m,d) with respect to parameters (m,d) in terms of accuracies, total number of parameters, and CPU time.
(m,d) | RMSE | MAE | CPU time (s) | Parameters |
---|---|---|---|---|
(5,5) | 1.71E-6 | 6.99E-4 | 1,258 | 163,901 |
(5,10) | 2.46E-6 | 6.08E-4 | 1,696 | 308,301 |
(5,15) | 2.28E-6 | 6.18E-4 | 2,050 | 452,701 |
(10,5) | 8.18E-7 | 4.77E-4 | 1,1378 | 653,601 |
(10,10) | 1.39E-6 | 6.57E-4 | 2,216 | 1,230,401 |
(10,15) | 1.91E-6 | 5.63E-4 | 2,755 | 1,807,201 |
The temperature variables predicted by ResNet(10,5) for heat-transfer problems with different composite material shapes are plotted in Fig. 3. Here, it is observed that the temperatures are discontinuous near the material-material interfaces. Indeed, owing to gaps between the two materials, there are relatively small temperature drops in the embedded material, which coincides with the phenomena observed in [5].
In this study, we developed new algorithms to predict the solutions of heat-transfer models in composite materials. First, datasets were generated by solving heat-transfer equations numerically, and the IFEM was adopted for the implementation. To develop neural networks that are robust to the shapes of material subdomains, various geometries of material-material interfaces were considered in the datasets. Once the datasets were generated, we trained the ResNet-type neural networks, which we named ResNet(m,d). The results showed that the RMSEs were below 3E-6 and the MAEs were below 7E-4. In addition, the discontinuity of temperature variables were described by ResNet(m,d) in a reasonable manner.
The second author (G. Jo) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2020R1C1C1A01005396).
We would like to thank Editage (www.editage.co.kr) for English language editing.
Table 1 . A comparison of ResNet(m,d) with respect to parameters (m,d) in terms of accuracies, total number of parameters, and CPU time.
(m,d) | RMSE | MAE | CPU time (s) | Parameters |
---|---|---|---|---|
(5,5) | 1.71E-6 | 6.99E-4 | 1,258 | 163,901 |
(5,10) | 2.46E-6 | 6.08E-4 | 1,696 | 308,301 |
(5,15) | 2.28E-6 | 6.18E-4 | 2,050 | 452,701 |
(10,5) | 8.18E-7 | 4.77E-4 | 1,1378 | 653,601 |
(10,10) | 1.39E-6 | 6.57E-4 | 2,216 | 1,230,401 |
(10,15) | 1.91E-6 | 5.63E-4 | 2,755 | 1,807,201 |