Thermal Conductivity Identification in Functionally Graded Materials via a Machine Learning Strategy Based on Singular Boundary Method

Abstract

A machine learning strategy based on the semi-analytical singular boundary method (SBM) is presented for the thermal conductivity identification of functionally graded materials (FGMs). In this study, only the temperature or heat flux on the surface or interior of FGMs can be measured by the thermal sensors, and the SBM is used to construct the database of the relationship between the thermal conductivity and the temperature distribution of the functionally graded structure. Based on the aforementioned constructed database, the artificial neural network-based machine learning strategy was implemented to identify the thermal conductivity of FGMs. Finally, several benchmark examples are presented to verify the feasibility, robustness, and applicability of the proposed machine learning strategy.

1. Introduction

Functionally graded materials [1,2] are a brand of novel composite materials whose volume fraction constituents vary with a specified gradient. This feature makes it possess a smooth variation of material properties. In many examples, FGMs can provide a better performance than traditional composite materials [3,4]; due to the superior thermal performance, the FGMs are generally used as important structures that are subjected to extreme thermal loadings, and the analyses of their thermal behaviors are necessary. The thermal conductivities are essential for the heat conduction analysis, which is of great significance for the design, optimization, and application of FGM structures [4,5,6], but in many practical engineering cases, the thermal conductivities of FGMs are usually unknown or change with long-term use. Therefore, a feasible and efficient inversion scheme of thermal conductivity for FGMs is necessary in the field of practical engineering.

The identification of thermal conductivity for FGMs faces enormous challenges. Unlike isotropic and anisotropic thermal materials, FGMs possess more thermal conductivity parameters, and the relative heat conduction governing equation is more complex and nonlinear, moreover, the identification task faces the troublesome ill-posedness of the inverse problem [7]. Therefore, it is necessary to propose a feasible, efficient, and robust inversion scheme. Some scholars have studied the parameter identification problems of heat conduction. Yueng and Lam [8] employed a second-order finite difference procedure for the identification of the unknown thermal conductivity in a one-dimensional domain. Huang and Chin [9] identified the unknown thermal conductivity and heat capacity by the conjugate gradient method. Dowding et al. [10] characterized the thermal conductivity of orthotropic carbon–carbon composite material by the finite element method. Divo et al. [11] applied an inverse boundary element method (BEM) to identify the unknown spatially dependent conductivity of heterogeneous materials through minimizing a regularized functional by genetic algorithm. Mera et al. [12] simultaneously predicted the unknown thermal conductivity and the unknown boundary data for a Cauchy steady-state heat conduction problem in a two-dimensional anisotropic thermal medium. Mierzwiczak and Kołodziej [13] investigated the non-iterative inverse determination of the temperature-dependent thermal conductivity for a steady-state heat conduction problem in a two-dimensional homogeneous thermal medium based on the method of fundamental solution (MFS). Chen et al. [14] characterized the unknown thermal conductivity for the steady-state heat conduction problem in two-dimensional FGMs by the combination of MFS and the Nelder–Mead simplex (NMS) method. Then, Chen et al. [15] identified the thermal conductivity coefficients of three-dimensional anisotropic media based on the SBM and Levenberg–Marquard (LM) algorithm. Nedin et al. [16] simultaneously retrieved the thermal conductivity and volumetric heat capacity of FGMs. Zhou et al. [17] retrieved the unknown thermal conductivity for orthotropic FGMs by the differential transformation dual reciprocity boundary element method and LM algorithm; and He and Yang [18] applied the element free Galerkin method [19] and the level-set method to identify the geometric boundary configurations in heat transfer problems.

The references above-mentioned are mostly based on traditional iterative gradient algorithms or intelligence heuristic algorithms. The traditional iterative gradient algorithms are based on the gradient information, so they converge rapidly, but the global convergence is not satisfactory. The intelligence heuristic algorithms have better global convergence, but they are usually time-consuming and unstable when the system contains noise error. Additionally, the historical data generated in the optimization process are not well utilized, which leads to repeated calculations while solving similar problems. In recent years, with the development of computer science and the great improvement in computer performance, the development of machine learning has been highly motivated [20]. The data-driven [21] and the physics informed [22] are the two general approaches of machine learning. The data-driven model can make full use of data information, which can decrease the intelligence and time costs associated with the constitutive model, and it possesses strong nonlinear fitting ability, which leads to the excellent global convergence in the inverse identification. In particular, for the artificial neuron network (ANN), if the network is trained well, it gives the output immediately when the input data are offered, and for similar problems, the network still responds immediately while the traditional optimization algorithms have to be calculated again. Due to its many advantages, the data-driven model has been successfully applied in various inverse heat conduction problems instead of the aforementioned traditional gradient or heuristic optimization algorithms. Chanda et al. [23] simultaneously retried the principal thermal conductivities for an anisotropic composite medium based on an artificial neural network and genetic algorithm. Shortly after, Chanda et al. [24] applied the ANN and genetic algorithm-based numerical scheme to estimate the principal thermal conductivities of layered honeycomb composites. Mohanraj et al. [25] reviewed the applications of ANN for the thermal analysis of heat exchangers. Rahman and Zhang [26] applied the ANN to predict the oscillatory heat transfer coefficient of a thermoacoustic heat exchanger. Chen et al. [27] identified the pipe’s inner radius by identifying the unknown thermal conductivity based on the deep learning network. There are also more applications of a machine learning strategy for other inverse problems, for example, Khodadadian et al. [28] proposed a parameter estimation framework for fracture propagation problems based on the Bayesian estimation method and phase field theory [29,30].

For the parameter identification problem, it is essential to first solve the direct problem. The SBM was proposed by Chen and collaborators [31,32,33,34,35] and can be viewed as an improvement in MFS [36,37] to solve some boundary value problems of partial differential equations. It can also be regarded as a modified method of BEM, as it inherited their advantages but possesses its unique innovation. It is a semi-analytical boundary collocation method that only needs the collocation points on the boundary, no grid is required, does not involve integration, which is sometimes troublesome, expensive, and complicated; and it overcomes the perplexing fictitious boundary issue associated with the MFS. In SBM, the source points and the collocation points are the same set of points, and the interpolating basis function is the fundamental solution, but the SBM introduced the original intensity factors (OIFs) to eliminate the singularity of the fundamental solution, which appears when the source point and the collocation point coincide. Prior to this study, the SBM has been successfully used to solve various physical problems such as potential problems [38], heat conduction problems [39,40], wave equations [41,42] and elastic problems [43,44].

Prior to this study, the SBM has been successfully applied to simulate the thermal behavior of FGMs [45] and the thermal conductivity identification based on traditional gradients and heuristic optimization algorithms [14,15]. To overcome the drawbacks of the traditional optimization algorithms above-mentioned, the more superior data-driven ANN model is introduced to identify the unknown thermal conductivity of FGMs. In this study, the thermal conductivity of FGMs is assumed to be a function that changed with spatial coordinates with unspecified parameters [46], the thermal conduction behavior of functionally graded structure is simulated by the SBM, the effective thermal conductivity is predicted by the ANN, and the identification of the thermal conductivity is achieved merely based on the measured temperature in the material or on the boundary.

The novelty of this paper is that it is the first to combine the SBM with machine learning strategy to identify the unknown thermal conductivity of FGMs; the numerical results show that the proposed inverse scheme is feasible and owns good applicability; and the ANN is more stable than traditional iterative LM algorithm.

The rest of this paper is organized as follows. The SBM for the heat conduction problem of FGMs is shown in Section 2. The inversion scheme of the thermal conductivity of FGMs based on ANN and SBM is shown in Section 3. Section 4 examines the efficiency, accuracy, and robustness of the proposed approach through some benchmark examples. Finally, some conclusions are summarized in Section 5.

2. SBM for Heat Conduction Problem of FGM

This section shows the SBM procedure to solve the steady-state heat conduction equation in FGMs and presents the latest techniques to calculate the origin intensity factors for the heat conduction problems in FGMs on the Neumann and Dirichlet boundaries.

2.1. Heat Conduction Problem in FGMs

Suppose a two-dimensional exponentially FGM medium occupying the bounded domain $\mathsf{\Omega }$ with the boundary $\mathsf{\Gamma }$; $\mathsf{\Gamma }={\mathsf{\Gamma }}_D\cup {\mathsf{\Gamma }}_N, {\mathsf{\Gamma }}_D\cap {\mathsf{\Gamma }}_N=\varnothing$; ${\mathsf{\Gamma }}_D$ represents the Dirichlet boundary (essential boundary); and ${\mathsf{\Gamma }}_N$ represents the Neumann boundary (nature boundary). The thermal conductivity can be donated as follows:

$$\widehat{K}{(\mathit{x})}_{ij}=K_{ij}{e}^{2{\beta }_i·x_i}\mathit{x}=(x_{1,}{x}_2)\in \mathsf{\Omega }(i,j=1,2),$$

(1)

where $K_{ij}$ is assumed to be symmetric ($K_{12}=K_{21}$) and positive-definite (${\Delta }_k=\det (K)=K_{11}{K}_{22}-K_{12}{}^2>0$), while ${\beta }_i$ represents the constants of material property characteristic.

The steady-state heat conduction equation in the absence of heat sources in FGMs is stated as follows:

$${({\widehat{K}}_{ij}(\mathit{x})u_{,j}(\mathit{x}))}_{,i}=0\mathit{x}=(x_{1,}{x}_2)\in \mathsf{\Omega },$$

(2)

with two types of boundary conditions

$$u(\mathit{x})=u_{\mathsf{\Gamma }}(\mathit{x})\mathit{x}=(x_{1,}{x}_2)\in {\mathsf{\Gamma }}_D,$$

(3)

$$q(\mathit{x})=-{\widehat{K}}_{ij}{u}_{,j}(\mathit{x}){\mathit{n}}_i=q_{\mathsf{\Gamma }}(\mathit{x})\mathit{x}=(x_{1,}{x}_2)\in \in {\mathsf{\Gamma }}_N,$$

(4)

where $u(\mathit{x})$ represents the temperature on position $\mathit{x}$, $u_{\mathsf{\Gamma }}(\mathit{x})$, and $q_{\mathsf{\Gamma }}(\mathit{x})$ represent the temperature on the Dirichlet boundary and the heat flux on the Neumann boundary, respectively. ${\mathit{n}}_i$ represents the i-th direction cosine of the unit outward normal vector to the boundary $\mathsf{\Gamma }$. It should be noted that the Einstein summation convention was employed in this study, and the comma indicates spatial derivatives, for example, $u_{,j}(\mathit{x})$ donates $\frac{\partial u(\mathit{x})}{\partial x_j}$.

Through the following transformation:

$$u=\mathsf{\Phi }{\mathrm{e}}^{-{\beta }_i·x_i},$$

(5)

the governing Equation (2) can be rewritten in the following form:

$$\left(K_{ij}\mathsf{\Phi }{,}_{ij}(\mathit{x})-{\lambda }^2\mathsf{\Phi }(\mathit{x})\right)=0, \mathit{x}=(x_1,x_2)\in \mathsf{\Omega },$$

(6)

where $\lambda =\sqrt{{\beta }_i{K}_{ij}{\beta }_j}$. Correspondingly, the boundary conditions (3) and (4) can be rewritten in the following forms:

$$\mathsf{\Phi }(\mathit{x})=u(\mathit{x}){\mathrm{e}}^{{\beta }_i·x_i}\mathit{x}=(x_{1,}{x}_2)\in {\mathsf{\Gamma }}_D,$$

(7)

$${\mathsf{\Phi }}_q(\mathit{x})=-{\widehat{K}}_{ij}{\mathsf{\Phi }}_{,j}(\mathit{x}){\mathit{n}}_i\mathit{x}=(x_{1,}{x}_2)\in \in {\mathsf{\Gamma }}_N$$

(8)

Next, we give the SBM formulation for solving Equations (6)–(8).

2.2. The SBM Formulation for Direct Heat Conduction

The SBM is a method corresponding to MFS. The same as MFS, SBM selects the singularity fundamental solution as the basic function of interpolation. It is well-known that due to the singularity of the fundamental solution, the MFS has to set a fictitious boundary outside the physical boundary, so the selection of the fictitious boundary limits its engineering applications and popularity. Unlike MFS, the source points ${\left\{{\mathit{s}}^j\right\}}_{j=1}^N$ and collocation points ${\left\{{\mathit{x}}^j\right\}}_{j=1}^N$ of SBM are the same set of points, which are all placed on the physical boundary. The notable feature of the SBM is to introduce the OIFs to eliminate the singularity of the fundamental solution. The SBM interpolation of Equations (6)–(8) can be expressed as

$$\mathsf{\Phi }({\mathit{x}}^i)=\left\{\begin{array}{l}{\displaystyle \sum _{j=1}^N{\alpha }_j{G}^{*}({\mathit{x}}^i,{\mathit{s}}^j){\mathit{x}}^i\in \mathsf{\Omega }}\\ {\displaystyle \sum _{j=1,i\ne j}^N{\alpha }_j{G}^{*}({\mathit{x}}^i,{\mathit{s}}^j)+{\alpha }_i{\vartheta }_{ii}^D{\mathit{x}}^i\in {\mathsf{\Gamma }}_D}\end{array}\right.,$$

(9)

$${\mathsf{\Phi }}_q({\mathit{x}}^i)={\displaystyle \sum _{j=1,i\ne j}^N{\displaystyle \sum _{k=1}^2{{\displaystyle \sum _{p=1}^2-K_{pk}{\alpha }_j\frac{\partial G^{*}({\mathit{x}}^i,{\mathit{s}}^j)}{\partial x_k^i}{\mathit{n}}_p^{x^i}+{\alpha }_i\vartheta }}_{ii}^N}}{\mathit{x}}^i\in {\mathsf{\Gamma }}_N,$$

(10)

where

$$G^{*}(x^i,s^j)=-\frac{1}{2\pi \sqrt{{\Delta }_K}}{H}_0(R({\mathit{x}}^i,{\mathit{s}}^j)),{\mathit{x}}^i,{\mathit{s}}^j\in \mathsf{\Omega }$$

(11)

is the fundamental solution of Equation (6) in exponentially FGMs [47]. ${\alpha }_j$ stands for the unknown coefficients. ${\Delta }_K$ donates the determinant of $K$. $H_0$ donates the modified Bessel function of the second kind and zero-order. It should be noted that

$$R(\mathit{x},\mathit{s})=\sqrt{(x_i-s_i)P_{ij}(x_j-s_j)}\mathit{x},\mathit{s}\in \mathsf{\Omega },$$

(12)

where

$$P_{ij}=K_{ij}{}^{-1}=\frac{1}{\Delta K}\left(\begin{array}{cc}{K}_{22}& -K_{12}\\ -K_{12}& K_{11}\end{array}\right),$$

(13)

${\vartheta }_{ii}^D$ and ${\vartheta }_{ii}^N$ donate the OIFs for the Dirichlet boundary condition and Neumann boundary condition, respectively. They are used to replace the singular terms of the diagonal elements of the original interpolation matrix. Section 2.3. will show how to obtain the corresponding original intensity factors in heat conduction problems for FGMs.

2.3. Origin Intensity Factors in the SBM

This section introduces the latest techniques to calculate OIFs to solve two-dimensional heat conduction problems in FGMs on the Neumann and Dirichlet boundary conditions, respectively.

2.3.1. Origin Intensity Factors on Neumann Boundary Conditions

Through some transformations [48] the SBM formulation for Neumann boundary conditions (8) can be expressed as

$${\mathsf{\Phi }}_q({\mathit{x}}^i)={\displaystyle \sum _{j=1,i\ne j}^N{\displaystyle \sum _{p,k=1}^2-K_{pk}\left[{\alpha }_j\frac{\partial G^{*}({\mathit{x}}^i,{\mathit{s}}^j)}{\partial {\mathit{x}}_k^{x^i}}{n}_p^{x^i}-\frac{{\alpha }_i}{{\mathit{l}}_i}{\displaystyle \sum _{j=1,j\ne i}^N\frac{\partial G^{*}({\mathit{x}}^i,{\mathit{s}}^j)}{\partial {\mathit{x}}_k^{x^j}}}{\mathit{n}}_p^{x^j}\right]}},$$

(14)

where ${\mathit{l}}_i$ donates the half length of the curve $\stackrel{⏜}{{\mathit{s}}_{j-1}{\mathit{s}}_{j+1}}$ between the source point ${\mathit{s}}^{i-1}$ and ${\mathit{s}}^{i+1}$, as shown in Figure 1, and the OIFs on the Neumann boundary conditions can be donated as follows:

$${\vartheta }_{ii}^N=-\frac{1}{{\mathit{l}}_i}{\displaystyle \sum _{j=1,j\ne i}^N{\displaystyle \sum _{p,k}^2-K_{pk}{\mathit{l}}_j\frac{\partial G^{*}({\mathit{x}}^i,{\mathit{s}}^j)}{\partial s_k^{s^j}}{\mathit{n}}_k^{s^j}}}.$$

(15)

One can find the detailed derivations of the above Equation (15) in [48].

2.3.2. Origin Intensity Factors on Dirichlet Boundary Conditions

It should be noted that the fundamental solutions of the Helmholtz (modified Helmholtz) equation and Laplace equation have the same order of the singularities [49]. The OIFs for two-dimensional Laplace equation on Dirichlet boundary conditions can be donated as follows:

$$u_{ii}{}^L=\ln \left(\frac{{\mathit{l}}_i}{2\pi }\right).$$

(16)

Based on theoretical analysis and numerous numerical experiments [49], the OIFs for the two-dimensional modified Helmholtz equation on the Dirichlet boundary conditions can be written as

$$u_{ii}{}^H=\ln \left(\frac{{\mathit{l}}_i}{2\pi }\right)+\kappa ,$$

(17)

where the constant $\kappa =0.5772156649015328606065120900824024310421$, and through two-step variable transformations in [47] the governing equation of heat conduction in FGMs can be transformed to a modified Helmholtz equation, and the OIFs of heat conduction problems in FGMs on the Dirichlet boundary conditions is equal to Equation (17).

2.4. SBM Discretization Algebraic Equations

When the positions of boundary points ${\left\{{\mathit{x}}_i\right\}}_{i=1}^{N_1+N_2}$ ($N_1+N_2=N$) and the respective physical value is already known, one can obtain the following linear algebraic system

$$\left(\begin{array}{c}{\left[B_D\right]}_{N_1\times N}\\ {\left[B_N\right]}_{N_2\times N}\end{array}\right){\left[\begin{array}{c}\left\{\begin{array}{c}{\alpha }_1\\ ⋮\\ {\alpha }_{N_1}\end{array}\right\}\\ \left\{\begin{array}{c}{\alpha }_{N_1+1}\\ ⋮\\ {\alpha }_{N_1+N_2}\end{array}\right\}\end{array}\right]}_{N\times 1}={\left[\begin{array}{c}\left\{\begin{array}{c}\mathsf{\Phi }({\mathit{x}}_1)\\ ⋮\\ \mathsf{\Phi }({\mathit{x}}_{N_1})\end{array}\right\}\\ \left\{\begin{array}{c}{\mathsf{\Phi }}_q({\mathit{x}}_{N_1+1})\\ ⋮\\ {\mathsf{\Phi }}_q({\mathit{x}}_{N_1+N_2})\end{array}\right\}\end{array}\right]}_{N\times 1},$$

(18)

where

$${\left[B_D\right]}_{N_1\times N}=\left[\begin{array}{cccccc}{\vartheta }_{11}^D& G^{*}({\mathit{x}}^1,{\mathit{s}}^2)& \cdots & G^{*}({\mathit{x}}^1,{\mathit{s}}^{N_1})& \cdots & G^{*}({\mathit{x}}^1,{\mathit{s}}^N)\\ G^{*}({\mathit{x}}^2,{\mathit{s}}^1)& {\vartheta }_{22}^D& \cdots & G^{*}({\mathit{x}}^2,{\mathit{s}}^{N_1})& \cdots & G^{*}({\mathit{x}}^2,{\mathit{s}}^N)\\ ⋮& ⋮& \ddots & ⋮& & ⋮\\ G^{*}({\mathit{x}}^{N_1},{\mathit{s}}^1)& G^{*}({\mathit{x}}^{N_1},{\mathit{s}}^2)& \cdots & {\vartheta }_{N_1{N}_1}^D& \cdots & G^{*}({\mathit{x}}^{N_{1‘}},{\mathit{s}}^N)\end{array}\right],$$

(19)

and

$${\left[B_N\right]}_{N_2\times N}=-\left[\begin{array}{cccccc}{K}_{ij}{G}^{*}({\mathit{x}}^{N_1+1},{\mathit{s}}^1){,}_j{n}_i^{{\mathit{x}}^{N_1+1}}& \dots & {\vartheta }_{(N_1+1)(N_1+1)}^N& K_{ij}{G}^{*}({\mathit{x}}^{N_1+1},{\mathit{s}}^{N_1+2}){,}_j{n}_i^{{\mathit{x}}^{N_1+1}}& \cdots & K_{ij}{G}^{*}({\mathit{x}}^{N_1+1},{\mathit{s}}^N){,}_j{n}_i^{{\mathit{x}}^{N_1+1}}\\ K_{ij}{G}^{*}({\mathit{x}}^{N_1+2},{\mathit{s}}^1){,}_j{n}_i^{{\mathit{x}}^{N_1+2}}& \dots & K_{ij}{G}^{*}({\mathit{x}}^{N_1+2},{\mathit{s}}^{N_1+1}){,}_j{n}_i^{{\mathit{x}}^{N_1+2}}& {\vartheta }_{(N_1+2)(N_1+2)}^N& \cdots & K_{ij}{G}^{*}({\mathit{x}}^{N_1+2},{\mathit{s}}^N){,}_j{n}_i^{{\mathit{x}}^{N_1+2}}\\ ⋮& & ⋮& ⋮& \ddots & ⋮\\ K_{ij}{G}^{*}{({\mathit{x}}^N,{\mathit{s}}^1)}_{{,}_j}{n}_i^{{\mathit{x}}^N}& \cdots & K_{ij}{G}^{*}({\mathit{x}}^N,{\mathit{s}}^{N_1+1}){,}_j{n}_i^{{\mathit{x}}^N}& K_{ij}{G}^{*}({\mathit{x}}^N,{\mathit{s}}^{N_1+2}){,}_j{n}_i^{{\mathit{x}}^N}& \cdots & {\vartheta }_{(N_1+N_2)(N_1+N_2)}^N\end{array}\right],$$

(20)

in which $N_1$ is the number of boundary points on ${\mathsf{\Gamma }}_D$; $N_2$ is the number of boundary points on ${\mathsf{\Gamma }}_N$; ${\alpha }_i$ is the unknown coefficients to be determined; and the right item in the linear algebraic system (18) is the known physical values on the boundary. The unknown coefficients ${\alpha }_i$ can be calculated easily with specified boundary values. Then, through the inverse transformation of Equation (5), the solution of original problems (2)–(4) is obtained.

3. The Machine Learning Scheme for Thermal Conductivity Identification

The artificial neural network is a simplification of the biological neural network of the human brain, which is made up of many artificial neurons. The artificial neurons are normally arranged linearly into groups, and the group is called a layer. The behaviors of synapses between biological neurons are simulated by variable connections between units on artificial neurons. The variable connection is determined by weights on the artificial neurons. The outputs of artificial neurons are computed by the weighted sums of the inputs and bias. Figure 2 shows the schematic diagram of an artificial neuron in the neural network. The output of neurons is stated as follows:

$$f(X)=f\left[\left({\displaystyle \sum _{i=1}^n{\omega }_{ij}{x}_i}\right)+b_j\right],$$

(21)

where

$$X=\left({\displaystyle \sum _{i=1}^n{\omega }_{ij}{x}_i}\right)+b_j,$$

and $f$ is usually a nonlinear activation function such as sigmoid function

$$f(t)=\frac{1}{1+e^{-t}},$$

(22)

and tansig function

$$f(t)=\frac{sinh(t)}{cosh(t)}.$$

(23)

A basic neural network structure with two hidden layers is shown in Figure 3. This is generally divided into input layer, hidden layers, and output layer, where each layer contains a number of neurons. The mastery of knowledge is realized through the study of samples, and the network recorded by weight and bias has learned the sample knowledge and mastered the complex nonlinear relations between the input and output without the physical relationship, which are difficult to express in an analytical form. In theory, an artificial neural network with one hidden layer can simulate any function [50] and the deep neural network possesses higher performance. The process of adjusting weight and bias according to the samples by certain rules is called training. The training process is repeated until the mean square error (MSE) reduces to an acceptable value. The MSE is defined as follows:

$$MSE=\frac{{\displaystyle \sum _{i=1}^m{(k_{acu,i}-k_{est,i})}^2}}{m},$$

(24)

where $m$ is the parameter number of thermal conductivity; $k_{acu,i}$ is the i-th actual parameter; $k_{est,i}$ is the i-th estimated parameter. In this paper, the LM algorithm was employed to minimize the $MSE$.

In the training process, it demands some data on the mapping between the thermal conductivity parameters and measurement temperatures on the measurement points. Then, the temperature serves as inputs, the thermal conductivity serves as training targets, and the training process is to make the network outputs as close to the targets as possible through some algorithms. The input data used for training are usually close to the actual value.

It should be noted that, in this study, the numerical simulations replaced the complex practical experiments. The SBM was used to simulate the temperature fields, which were varied with different thermal conductivities, after obtaining the data including the relationship between the temperature fields and thermal conductivities. Then, the measurement temperatures are taken as the input data and the thermal conductivity parameters as the output data, so the trained ANN is obtained, and if the actual measurement temperatures are input to the ANN, the ANN will feedback the predicted thermal conductivity parameters. Figure 4 illustrates the flowchart of the parameter identification problem.

4. Numerical Results and Discussion

In this paper, two-dimensional FGMs with prescribed boundary conditions was considered. First, the accuracy and efficiency of the SBM for the direct heat conduction in FGMs were investigated.

To examine the numerical accuracy of the SBM, the following L2 error norm can be defined as follows:

$$Lerr(u)=\frac{\sqrt{{{\displaystyle {\sum }_{k=1}^{TN}\left|u({\mathit{x}}_k)-\overline{u}({\mathit{x}}_k)\right|}}^2}}{\sqrt{{{\displaystyle {\sum }_{k=1}^{TN}\left|u({\mathit{x}}_k)\right|}}^2}},$$

(25)

where $u({\mathit{x}}_k)$ and $\overline{u}({\mathit{x}}_k)$ are the analytical and numerical solution at position ${\mathit{x}}_k$, respectively, and TN is the number of uniform test points on the whole domain.

Then, the numerical feasibility, accuracy, and robustness of the proposed numerical inversion scheme for thermal conductivity in FGMs were investigated, and the effects of the number of measurement points, measurement point positions, and measurement errors were studied.

It must be pointed out that the measurement temperatures in practical problems always contain measurement errors, which are also often referred to as noise. To investigate the numerical stability of the proposed scheme, in this study, the artificial noisy data were generated by

$$mea{s}_{noise}(\mathit{x})=meas(\mathit{x})\ast \left(1+\tau \ast randn(N,1)\right),$$

(26)

where $meas(\mathit{x})$ represents the temperatures derived from the SBM solution; $randn(N,1)$ is a group of random numbers generated from the standard normal distribution; and $\tau$ denotes the noise level.

To examine the performance of the proposed inversion scheme, the relative error (RE) of the retrieved parameter by ANN is defined as follows:

$$RE=\left|\frac{p_{exa}-p_{ret}}{p_{exa}}\right|,$$

(27)

where $p_{exa}$ and $p_{ret}$ denote the effective and retrieved values of the thermal conductivity parameter for the FGMs, respectively.

If not specified, in the training process of an ANN, 90% random data are set as the training set, and the remaining 10% data of ANN are set as the test set. The LM algorithm is used as the optimizer to minimize the loss function, the learn rate was set to 1E-3, the training epochs were set to 1000, the activation function of hidden layer was the tansig function, and the activation function of the output layer was the purelin function. The model complexity of the ANN is described by the parameter number, which is the total number of weights and biases in the ANN.

The inversion scheme is based on the database generated by analyzing the direct heat conduction via the SBM, therefore, the accuracy of the direct problem determines the feasibility of the inversion scheme. The summarized Example 1 will show the accuracy and efficiency of the SBM for solving heat conduction problems in FGMs.

Example 1.

The accuracy and efficiency of SBM for direct heat conduction.

First, the accuracy and efficiency of the SBM to solve direct heat conduction problems in FGMs were investigated. In the computational square domain ${\mathsf{\Omega }}_s=(0,1)\times (0,1)$, the effective thermal conductivity parameters of FGMs are set as

$$K=\left[\begin{array}{cc}1& 0\\ 0& 4\end{array}\right], {\beta }_1=0.1, {\beta }_2=0.2,$$

and the corresponding analytical solution is

$$u\left(\mathit{x}\right)=\exp \left(\frac{\lambda \left(T_1+T_2\right)}{\tau }-{{\displaystyle \sum }}_{i=1}^2{\beta }_i{x}_i\right),$$

(28)

where

$$\begin{gathered} \tau =\sqrt{K_{11}{\left(\frac{\sqrt{{\Delta }_K}-K_{12}}{K_{11}}\right)}^2+2K_{12}\left(\frac{\sqrt{{\Delta }_K}-K_{12}}{K_{11}}\right)+K_{22}}, \\ {T}_1=\frac{x_1\sqrt{\Delta K}}{K_{11}},\hspace{1em}{T}_2=-\frac{x_1{K}_{12}}{K_{11}}+x_2. \end{gathered}$$

It should be pointed out that the corresponding Dirichlet and Neumann boundary conditions are imposed based on the analytical solution; and ${\mathsf{\Gamma }}_D:x_1=1, x_1=0$ and ${\mathsf{\Gamma }}_N:x_2=1, x_2=0$ are the boundaries subjected to Dirichlet and Neumann boundary conditions, respectively (Figure 5). A hundred test points were uniformly distributed in the computational domain.

Figure 6 illustrates the effect of the number of boundary collocation points. It can be seen from Figure 6a that with the increase in boundary collocation points, the $Lerr$ decreased rapidly, the temperature obtained by the SBM converged rapidly as the number of points increased, but the corresponding CPU time (s) increased. Figure 6b presents the effect of the number of boundary collocation points on the condition number of the SBM interpolation matrix, where the condition number is defined as the ratio between the largest and smallest singular values of the corresponding interpolation matrix. Figure 7 depicts the analytical and numerical solutions under a square domain with 400 boundary collocation points. Figure 8 shows the corresponding absolute and relative errors. The errors displayed in Figure 8 reflect the effectiveness of this method in solving direct heat conduction problems in FGMs.

In this example, the efficiency and accuracy of the SBM for the direct heat conduction problem in FGMs were verified. In the following numerical examples, the feasibility, accuracy, and robustness of the proposed inversion scheme are discussed.

Example 2.

The effect of the number of measurement points.

In this example, the effect of number of the measurement points in the inversion scheme was investigated. Figure 9 shows the I-shaped domain with different measurement points. The number of boundary collocation points was 608, and the effective thermal conductivity parameters were set as

$$K=\left[\begin{array}{cc}{K}_{11}& 0\\ 0& K_{22}\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right], \beta =\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]=\left[\begin{array}{c}0.15\\ 0.2\end{array}\right].$$

It should be stated that the effective thermal conductivity parameters are unknown in the inversion scheme, and the given effective thermal conductivity is to verify the feasibility and accuracy of the proposed inversion scheme. In the following studies, for convenience, the blue lines represent ${\mathsf{\Gamma }}_b$ and the red lines represent ${\mathsf{\Gamma }}_r$, as shown in Figure 9. The boundary condition on ${\mathsf{\Gamma }}_r$ was $u(\mathit{x})=0$, and on ${\mathsf{\Gamma }}_b$, it was $u(\mathit{x})=10$. The black points in the FGMs are thermal sensors; the actual temperatures on these points can be measured by these sensors.

The database on the mapping between the varied thermal conductivity parameters and corresponding measurement temperatures was generated by simulating the thermal behaviors with different thermal conductivities via the SBM with the above-mentioned boundary conditions. The parameters were set as $K_{11}=3\pm 0.1j,K_{22}=4\pm 0.1j,{\beta }_1=0.1j,{\beta }_2=0.1j,j=1,2,\cdots ,10$. Then, the parameter identification problem was solved by the ANN. In the training process, among these 40,000 groups of parameter settings, 90% data were randomly chosen as the training set, and the remaining 10% data were set as the testing set.

The parameter identification problem was to retrieve the unknown tensor K and gradient coefficient vector $\beta$, so the three-layer network was used.

Table 1. The structures of ANN and the number of parameters in Example 2.

	Input Layer	Hidden Layers	Output Layer	Parameter Numbers
Neuron number	5/10/15	[31 15]	4	730/885/1040
Activation function	/	tansig	purelin

represents the structures of ANN and the parameter numbers, the activation function of the hidden layer is the tansig function, and the activation function of the output layer is the purelin function, and the parameter numbers were 730/885/1040 for 5/10/15 measurement points. In

Table 2. The retrieved results with different measurement points in Example 2.

	Mn	Retrieved Values				Exact Values
		K11	K22	β1	β2	K11	K22	β1	β2
SBM-NN	5	3.049	4.016	0.143	0.217	3	4	0.15	0.2
	10	3.030	4.012	0.147	0.198
	15	2.973	4.014	0.154	0.201
SBM-LM	5	3.043	4.116	0.147	0.213
	10	3.020	4.068	0.151	0.199
	15	2.995	4.003	0.151	0.201

and

Table 3. The relative errors with different measurement points in Example 2.

	Mn	Relative Errors (%)
		K11	K22	β1	β2
SBM-ANN	5	1.63	0.40	4.67	8.50
	10	1.00	0.30	2.00	1.00
	15	0.90	0.35	2.67	0.50
SBM-LM	5	1.43	2.90	2.00	6.50
	10	0.67	1.70	0.67	0.50
	15	0.17	0.08	0.67	0.50

, Mn donates the number of measurement points. Table 2 represents the retrieved values of the thermal conductivity of the ANN and Levenberg–Marquardt (LM) algorithm; Table 3 presents the corresponding relative errors. From the tables, it can be seen that with the increase in measurement points, the predicted accuracy of ANN was enhanced, the retrieved results of 10 measurement points were more accurate than five measurement points, but with the number of measurement points increasing to 15, the accuracy did not significantly increase. Additionally, it was found that the LM algorithm retrieved a similar result compared with the ANN, and the appropriate initial values in the LM algorithm were set as $K_{11}=5,K_{22}=5,{\beta }_1=0.5,{\beta }_2=0.5$. In practice, however, the selection of a suitable initial value is not easy in the LM algorithm, and the convergence of the LM algorithm takes some time. In contrast, in the ANN, once the network is trained well, it obtains the retrieved results immediately.

Example 3.

The effect of the measurement points positions.

In this example, the effect of the measurement points’ positions was investigated. In Example 1, the measurement points were set in the inner of the FGMs, but sometimes in practical engineering, it is not easy to place those thermal sensors in the FGMs. To solve the problem, nondestructive testing schemes were proposed. The square domain was adopted, and 10 thermal sensors were placed on the Neumann boundary, as shown in Figure 10. The number of boundary collocation points was 400, and the effective thermal conductivity parameters of FGMs were set as

$$K=\left[\begin{array}{cc}{K}_{11}& 0\\ 0& K_{22}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 4\end{array}\right], \beta =\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]=\left[\begin{array}{c}0.1\\ 0.2\end{array}\right].$$

The boundary condition on ${\mathsf{\Gamma }}_b$ was $u(\mathit{x})=15$, and on ${\mathsf{\Gamma }}_r$, it was $q(\mathit{x})=100$. In the generation of the ANN database, the parameters were set as $K_{11}=1\pm 0.1j,K_{22}=4\pm 0.1j,{\beta }_1=0.1j,{\beta }_2=0.1j,j=1,2,\cdots ,10 (K_{11}\ne 0).$ In the training process, among these 38,000 groups of parameter settings, 90% data were randomly chosen as the training set, and the remaining 10% data were set as the testing set. Then, the parameter identification problem was solved by the ANN.

Table 4. The structure of ANN and the parameter numbers in Example 3.

	Input Layer	Hidden Layers	Output Layer	Parameter Numbers
Neuron number	10	[31 15]	4	885
Activation function	/	tansig	purelin

gives the structure of the ANN and the parameter numbers.

Table 5. The retrieved results with different measurement point positions in Example 3.

	(a)	(b)	(c)	Exact
K11	1.021	0.963	1.034	1
K22	3.984	3.891	3.886	4
β1	0.099	0.096	0.102	0.1
β2	0.198	0.206	0.203	0.2

shows the retrieved results with different measurement point positions and

Table 6. The relative error (%) with different measurement point positions in Example 3.

	(a)	(b)	(c)
K11	2.10	3.70	3.40
K22	0.40	2.73	2.85
β1	1.00	4.00	2.00
β2	1.00	3.00	1.50

presents the corresponding relative errors with different measurement errors. From the tables, it can be concluded that the parameter identification based on nondestructive testing schemes is feasible. However, the scheme with built-in measurement points (a) had a higher performance than the nondestructive testing schemes (b) and (c), and it was also found that bilaterally placed measurement points had a higher performance than the unilaterally placed measurement points.

Example 4.

The effect of measurement errors.

An armor-shaped domain with 10 measurement points was considered in this example, as shown in Figure 11. The boundary conditions were ${\mathsf{\Gamma }}_r:u(\mathit{x})=20, {\mathsf{\Gamma }}_b:u(\mathit{x})=0.$ The number of boundary collocation points was 400, and the effective thermal conductivity parameters were set as

$$K=\left[\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right]=\left[\begin{array}{cc}3& 1\\ 1& 5\end{array}\right], \beta =\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]=\left[\begin{array}{c}0.2\\ 0.4\end{array}\right].$$

It should be pointed out that the parameters $K_{11}, K_{12}$, and $K_{22}$ were identified in this example, so the parameters ${\beta }_1$ and ${\beta }_2$ were known. In the generation of the database of ANN, the parameters were set as $K_{11}=3\pm 0.1j,K_{22}=5\pm 0.1j,$ $K_{12}=K_{21}=1\pm 0.1, j=1,2,\cdots ,10$. The parameters $K_{11}, K_{12}$, and $K_{22}$ consisted of 20 groups that were then combined, so the thermal conductivities of 8000 groups were established. The measurement errors were taken as 0%, 1%, and 3%.

In the inverse process, the four-layer network was employed because of the measurement noise.

Table 7. The structure of the ANN and the parameter numbers in Example 4.

	Input Layer	Hidden Layers	Output Layer	Parameter Numbers
Neuron number	10	[31 15 10]	3	1014
Activation function	/	tansig	purelin

represents the structure of ANN and the parameter numbers. The retrieved results with different measurement errors are exhibited in

Table 8. The retrieved results with different measurement errors in Example 4.

		0%	1%	3%	Exact
	K11	3.023	3.123	2.754	3
SBM-ANN	K22	5.126	4.896	5.226	5
	K12	0.986	0.943	0.921	1
	K11	3.011	3.479	4.546	3
SBM-LM	K22	5.044	6.146	3.012	5
	K12	1.011	0.656	1.654	1

, and the corresponding relative errors are exhibited in

Table 9. The relative errors (%) with different measurement errors in Example 4.

		0%	1%	3%
	K11	0.77	4.10	8.20
SBM-ANN	K22	2.52	2.08	4.52
	K12	1.40	5.70	7.90
	K11	0.37	16.0	51.5
SBM-LM	K22	0.88	22.9	39.8
	K12	110	34.4	65.4

. It can be seen from the tables that the relative error of ANN decays with the decreasing noisy data, and the retrieved results achieved the best accuracy with noise-free measurement data. However, the LM algorithm cannot retrieve a satisfactory result when the measurement noise exists.

Example 5.

A complex engineering case.

A complex Gear-shaped pipeline with 10 built-in thermal sensors, as shown in Figure 12, was considered in this example, with the boundary collocation point number set to 604, and the effective thermal conductivity parameters of FGMs were set as

$$K=\left[\begin{array}{cc}{K}_{11}& 0\\ 0& K_{22}\end{array}\right]=\left[\begin{array}{cc}5& 0\\ 0& 4\end{array}\right], \beta =\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]=\left[\begin{array}{c}0.5\\ 0.3\end{array}\right].$$

The boundary condition on ${\mathsf{\Gamma }}_1$ was $q(\mathit{x})=-100$, and the boundary condition on ${\mathsf{\Gamma }}_2$ was $u(\mathit{x})=50$. In the training process, the parameters were set as $K_{11}=5\pm 0.1j,K_{22}=4\pm 0.1j,{\beta }_1=0.5j,{\beta }_2=0.3j,j=1,2,\cdots ,10$. So 40,000 groups thermal conductivities are established. In the example above, just the actual thermal conductivity has cared, but the retrieved results of other thermal conductivities are not discussed.

Table 10. The structure of ANN and the parameter numbers in Example 5.

	Input Layer	Hidden Layers	Output Layer	Parameter Numbers
Neuron number	10	[20 15 10]	3	728
Activation function	/	tansig	purelin

shows the structure of ANN and the parameter numbers. The retrieved results and corresponding relative errors are exhibited in

Table 11. Retrieved results and corresponding relative errors in Example 5.

	Retrieved Results	Relative Errors (%)	Exact Value
K11	5.220	4.40	5
K22	4.124	3.10	4
β1	0.503	0.60	0.5
β2	0.294	2.00	0.3

, where the table shows that the proposed scheme is feasible in complex engineering cases. Figure 13 shows the linear regressions of the test set, R donates the determination coefficient, where the closer the value of R is to one, the better the network’s performance. It must be stated that all values are normalized in the ANN. It can be seen that almost all the retrieved results of the test set matched the actual values pretty well, and the determination coefficients were very close to 1, so demonstrates that the network performs well and possesses good applicability for different thermal conductivities.

5. Conclusions

A machine learning strategy based on the artificial neural network and the semi-analytical SBM was presented for the thermal conductivity identification of functionally graded materials. In this study, only the temperature on the FGMs’ interior or boundary can be measured via the thermal sensors. Instead of extensive experiments, the SBM is introduced to construct the database of the relationship between the thermal conductivity and the temperature in the heat conduction of FGMs. Based on the aforementioned constructed database, the artificial neural network was implemented to identify the thermal conductivity of FGMs. Several benchmark examples were presented to verify the effects of measurement point numbers, measurement point positions, and measurement errors were studied. The following conclusions can be drawn:

• The machine learning strategy is feasible to identify the unknown thermal conductivities of FGMs.
• The number of measurement points has an effect on the retrieved results. The proposed machine learning strategy with 10 measurement points was better than the one with five measurement points, but with the further increase in measurement point number, the accuracy of the retrieved results will not be improved.
• Nondestructive testing schemes are feasible, but the accuracy only reduced a little.
• The measurement errors also affect the retrieved results. The results became worse with the decrease in measurement error, while the LM algorithm did not work with the measurement noise.
• The proposed inversion scheme had good applicability for different thermal conductivities.