Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm

Abstract

The total heat exchange factor is one of the most important thermal physical parameters in the heat transfer model for a reheating furnace machine. In this paper, a novel general strategy, which is combined with the first-optimize-then-discretize (FOTD) approach and an improved hybrid conjugate gradient (IHCG) algorithm, is proposed to identify the total heat exchange factor by solving a nonlinear inverse heat conduction problem (IHCP). Firstly, a nonlinear IHCP with the Dirichlet-type boundary condition $T_m\left(t\right)=T(0,t)$ is built to determine the unknown total heat exchange factor $w\left(t\right)$. Secondly, the analysis of the Fréchet gradient of the cost functional is given and the gradient is proved as Lipschitz continuous by the FOTD approach. Thirdly, based on the gradient information by FOTD, a new IHCG algorithm, whose global convergence is proved by us, is proposed for fast solving of the optimization problem. Finally, simulation experiments are given to verify the effectiveness of the proposed strategy. Compared with the first-discretize-then-optimize (FDTO) approach, the FOTD approach can reduce running time and iteration number. Compared with other CG algorithms, the proposed IHCG algorithm has better convergence performance. The experimental data by the thermocouples experiments from a reheating furnace are also given to identify the total heat exchange factor.

1. Introduction

The method of total heat exchange factor is often used to build the temperature prediction model for the reheating furnace, and has been widely used in the online control due to its simplicity and lower computer cost [1]. The total heat exchange factor is one of the most important thermal physical parameters. It has significant effects on the accuracy of temperature fields in the calculation process of the prediction model [2]. However, the determination of the total heat exchange factor is very complicated due to the complex environment, in which many various factors (such as: structural parameters, operating parameters, thermal parameters, etc.) should be considered. Generally, the inverse heat conduction problem (IHCP), in which some characteristics are unknown, is often used to determine the unknown parameters. For instance, the unknown temperature-dependent parameters (such as: local convective heat transfer coefficients [3], surface heat flux [4,5], thermal conductivity [6], surface heat sources [7], etc.) can be evaluated in the solution of IHCP. Thus, a new IHCP is built to determine the total heat exchange factor, and it should be solved both quickly and accurately in this paper.

Methods for solving the IHCP include the conjugate gradient (CG) method [6], the maximum entropy (ME) method [8], the neural network (NN) method [9] and so on. These algorithms can be generally classified into two categories: the stochastic intelligent algorithms and the gradient-based algorithms [10]. The disadvantage of the stochastic intelligent methods is the low computation efficiency. Especially, a large number of iterations are required and only few parameters can be reconstructed [11]. The advantages of the gradient-based methods are the fast convergence speed and the high accuracy. However, it is not easy to obtain the gradient information for real applications. Since we want to solve the IHCP quickly and accurately, the gradient-based method is chosen to construct an effective gradient-based algorithm in this paper.

Current gradient-based numerical methods are divided into two groups: the first-discretize-then-optimize (FDTO) approach with the sensitivity matrix and the first-optimize-then-discretize (FOTD) approach with the adjoint equation [12]. In the past few decades, researchers pay more attention to the FDTO approach with the sensitivity matrix, and proposed a variety of numerical algorithms. For instance, Tseng et al. [13] developed a direct sensitivity coefficient (DSC) method to solve multidimensional IHCPs. Anderson et al. [14] determined the effective heat transfer coefficients by using an inverse method, in which the sensitivity coefficient is solved in an iterative manner. Han et al. [15] used the finite element method (FEM) to solve the sensitivity matrix, and the sensitivity analysis of the transient IHCP is carried out in this paper. Cui et al. [16] presented a complex variable differentiation method (CVDM) for accurately calculating sensitivity coefficients for a Levenberg–Marquardt (LM) algorithm to solve IHCPs. Zhang et al. [17] calculated the sensitivity matrix coefficients by the newly developing FEM and CVDM though ABAQUS. In general, sensitivity coefficients are difficult to accurately calculate. Besides, if the IHCP has some other properties (such as: multi-dimensions, transient states or nonlinearities), the calculation process will be a more difficult and challenging work.

Recently, some researchers have proposed a series of FOTD approaches and successfully introduced them into IHCPs. For instance, Azar et al. [18] proposed an iterative regularization method, in which the adjoint equation is given to determine the gradient of the cost function, to identify the unknown parameter by solving the IHCP. Xiong et al. [5] proposed a sequential CG method to reconstruct the undetermined surface heat flux for nonlinear IHCP. The adjoint partial differential equation (PDE) is introduced to describe the gradient of the objective function. On the bases of our earlier studies [19,20], an improved hybrid conjugate gradient (IHCG) algorithm based on the FOTD approach is proposed to determine the unknown total heat exchange factor $w\left(t\right)$ by solving the IHCP with the Dirichlet-type boundary condition $T_m=T(0,t)$.

This paper is composed as follows. In Section 2, mathematical models of reheating furnace and a nonlinear inverse heat conduction problem are built. Based on the FOTD approach, the explicit formula for the Fréchet gradient of the cost function is obtained, and the Lipschitz continuity of the gradient is proved in Section 3. In Section 4, an improved hybrid conjugate gradient (IHCG) algorithm is given, and a detailed mathematics proof of global convergence is published. In Section 5, simulation experiment and analysis are given to verify the effectiveness of the FOTD approach and the proposed IHCG algorithm. The experimental data by the thermocouples experiments are also given to identify the total heat exchange factor for the furnace. Finally, the conclusion is summarized in Section 6.

2. Problem Formulation

2.1. Mathematical Model for the Reheating Furnace

The heat transferred to the slab is extremely complicated. It is mainly in the form of heat convection, heat conduction and thermal radiation, as shown in Figure 1. To solve the model in practical engineering applications, some assumptions are given to build a low-dimensional mathematical model: 1. The heat conduction inside the slab is only transferred along the slab’s thickness direction. Because the temperature difference along the length and width direction is much smaller than the slab’s thickness direction [19]. 2. Thermal radiation is the only considered mode of heat exchange between the slabs and their environment. Other heat transfers such as conduction, skid marks, oxide scale layer, etc., are omitted [21]. 3. A half part of the slab is considered for the mathematical model. As the slab is symmetrically heated approximately in the furnace [4]. Finally, the 1-dimensional transient nonlinear mathematical model with the temperature-dependent material parameters is considered and shown as Equation (1). In this analysis, neither source term nor phase change inside the solid are considered.

$$\begin{array}{c}\rho c\left(T\right)\frac{\partial T}{\partial t}\left(y,t\right)-\frac{\partial }{\partial y}\left(\lambda \left(T\right)\frac{\partial T}{\partial y}\left(y,t\right)\right)=0,\left(y,t\right)\in {\Omega }_{yt}:=\left[0,\frac{l}{2}\right]\times \left[t_0,t_1\right]\hfill \\ T\left(y,t_0\right)=T_0\left(y\right),\hfill \\ -\lambda \left(T\right)\frac{\partial T}{\partial y}\left(0,t\right)=0,\hfill \\ \lambda \left(T\right)\frac{\partial T}{\partial y}\left(\frac{l}{2},t\right)=q\left(t\right)=\sigma w\left(t\right)\left[u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right].\hfill \end{array}$$

(1)

Here, $T(y,t)$ is the temperature field in the slab along the thickness dimension y with $y\in \left[0,\frac{l}{2}\right]$. The symbol l is the thickness of the slab $\left(m\right)$, and $t_0$ and $t_1$ are the charging and discharging time of the slab. $T_0\left(y\right)$ is the slab’s charging temperature (K), and $\sigma$ is the Stefan–Boltzmann constant, $5.67\times {10}^{-8}$ (W/m${}^2·$ K${}^4$). The symbol $q\left(t\right)$ defines the heat flux density, which is determined by the total heat exchange factor $w\left(t\right)$, upper furnace temperatures $u\left(t\right)$, and slab’s upper surface temperature $T(\frac{l}{2},t)$. Furthermore, $\rho$ is density (kg/m${}^3$), $c\left(T\right)$ and $\lambda \left(T\right)$ are the specific heat (J/(kg·K)) and the thermal conductivity (W/(m·K)), respectively. According to the paper [22], $c\left(T\right)$ and $\lambda \left(T\right)$ are temperature-dependent and shown in Figure 2 for the 20MnSi steel slab.

As verified in [23,24], a time-invariant transformation function, $\tilde{T}\left(T\right)={\tilde{T}}_0+\frac{1}{{\tilde{c}}_0}\underset{T_0}{\overset{T}{\int }}c\left(\tau \right)d\tau$, is chosen to isolate the nonlinear material characteristics in Equation (1). Then, after utilization of the transformation law into PDEs (1), the new PDE equations are obtained:

$$\begin{array}{c}\frac{\partial \tilde{T}}{\partial t}\left(y,t\right)-\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(y,t\right)\right)=0,\hfill \\ \tilde{T}\left(y,t_0\right)-T_0\left(y\right)=0,\hfill \\ k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(0,t\right)=0.\hfill \\ k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(\frac{l}{2},t\right)=q\left(t\right)={\sigma }_{1}w\left(t\right)\left[u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right].\hfill \end{array}$$

(2)

Here, $k\left(\tilde{T}\right)=\frac{\tilde{\lambda }\left(\tilde{T}\right)}{\rho c_0}$, $\tilde{\lambda }\left(\tilde{T}\right)=\frac{\lambda \left(\tilde{T}\right){\tilde{c}}_0}{c\left(\tilde{T}\right)}$, ${\sigma }_1=\frac{\sigma }{\rho c_0}$; $q\left(t\right)$ still defines the heat flux density, which does not depend on y.

2.2. Inverse Heat Conduction Problem

To obtain more accurate measurement data, the slab is equipped with a set of thermocouples to go through the reheating process [20]. The center temperatures in the slab are seldom affected by the environment relative to the surface temperatures. Thus, the measured output data can be used as the Dirichlet-type boundary condition $T_m\left(t\right)=T(0,t)$ in this paper. Then, the IHCP can be described by the following PDEs-constrained optimization problem:

$$\begin{array}{c}minJ={\int }_{t_0}^{t_1}{\left(\tilde{T}\left(0,t;w\right)-T_m\left(t\right)\right)}^{2}dt\hfill \\ st.\begin{array}{c}\frac{\partial \tilde{T}}{\partial t}\left(y,t\right)-\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(y,t\right)\right)=0,\hfill \\ \tilde{T}\left(y,t_0\right)-T_0\left(y\right)=0,\hfill \\ k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(0,t\right)=0,\hfill \\ k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(\frac{l}{2},t\right)=q\left(t\right)={\sigma }_{1}w\left(t\right)\left[u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right].\hfill \end{array}\hfill \end{array}$$

(3)

3. The Fréchet Gradient Based on the FOTD Approach

In this section, the FOTD approach based on the adjoint equation is introduced to obtain the Fréchet gradient of the cost functional. Afterward, the Lipschitz continuous is proved.

3.1. The Adjoint PDEs Based on Weak Solutions

To derive the adjoint PDEs, the Lagrange function-based approach [12] is applied. $p(y,t)$ is introduced to transfer all constraints into cost functional and formulate the strong version of the Lagrange function, which is shown as:

$$\begin{array}{c}L\left(\tilde{T},p,w\right)=\underset{t_0}{\overset{t_1}{\int }}{\left(\tilde{T}\left(0,t;w\right)-T_m\left(t\right)\right)}^{2}dt+\underset{0}{\overset{\frac{l}{2}}{\int }}\underset{t_0}{\overset{t_1}{\int }}\left(\frac{\partial \tilde{T}}{\partial t}\left(y,t\right)-\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(y,t\right)\right)\right)p\left(y,t\right)dtdy\hfill \\ +\underset{t_0}{\overset{t_1}{\int }}\left(-k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(0,t\right)\right)p\left(0,t\right)dt+\underset{t_0}{\overset{t_1}{\int }}\left(k\left(\tilde{T}\right)\frac{\partial \tilde{T}}{\partial y}\left(\frac{l}{2},t\right)-q\left(t\right)\right)p\left(\frac{l}{2},t\right)dt\hfill \end{array}$$

(4)

From the book [12], the adjoint equation is obtained by solving $\Delta L=0$ for $\tilde{T}$. It is shown as:

$$\begin{array}{c}\frac{\partial p}{\partial t}\left(y,t\right)+\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(y,t\right)\right)=0,\hfill \\ p\left(y,t_1\right)=0,\hfill \\ k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(0,t\right)=2\left(\tilde{T}\left(0,t\right)-T_m\left(t\right)\right),\hfill \\ k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(\frac{l}{2},t\right)=-4{\sigma }_{1}w\left(t\right)T^3\left(\frac{l}{2},t\right)p\left(\frac{l}{2},t\right).\hfill \end{array}$$

(5)

Denote by $\tilde{T}\left(y,t;w\right),\tilde{T}\left(y,t;w+\Delta w\right)$ are the solutions of problem (2). Then, $\Delta \tilde{T}\left(y,t;w\right):=\tilde{T}\left(y,t;w+\Delta w\right)-\tilde{T}\left(y,t;w\right)$ will be the weak solution of the following parabolic problem:

$$\begin{array}{c}\frac{\partial \Delta \tilde{T}}{\partial t}\left(y,t\right)-\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)=0,\hfill \\ \Delta \tilde{T}\left(y,t_0\right)=0,\hfill \\ k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(0,t\right)=0,\hfill \\ k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(\frac{l}{2},t\right)={\sigma }_1\left(u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right)\Delta w\left(t\right)-4{\sigma }_{1}w\left(t\right)T^3\left(\frac{l}{2},t\right)\Delta \tilde{T}\left(\frac{l}{2},t\right).\hfill \end{array}$$

(6)

3.2. Fréchet Gradient of the Cost Functional

The first variation $\Delta J\left(w\right)$ of the cost functional (3) is obtained as:

$$\begin{array}{c}\Delta J\left(w\right)=2{\int }_{t_0}^{t_1}\left(\tilde{T}\left(0,t;w\right)-T_m\left(t\right)\right)\Delta \tilde{T}\left(0,t;w\right)dt+{\int }_{t_0}^{t_1}\Delta {\tilde{T}}^2\left(0,t;w\right)dt\hfill \end{array}$$

(7)

By the FOTD approach, an explicit formula for the gradient of the cost functional will be defined by the solution $p(y,t)$ of the adjoint Equation (5). A simple derivation for the first term of the right-hand of Equation (7) is given in Lemma 1.

Lemma 1.

Let $w\left(t\right)$, $w\left(t\right)+\Delta w\left(t\right)$ be given functions. If $\tilde{T}(y,t;w)$ is the corresponding solution of the direct problem (2), and $p(y,t;w)$ is the solution of the adjoint Equation (5), for all $\Delta w\left(t\right)$, the following identity holds:

$$\begin{array}{c}2{\int }_{t_0}^{t_1}\left(\tilde{T}\left(0,t;w\right)-T_m\left(t\right)\right)\Delta \tilde{T}\left(0,t;w\right)dt=-{\sigma }_1{\int }_{t_0}^{t_1}\left(u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right)p\left(\frac{l}{2},t\right)\Delta w\left(t\right)dt\hfill \end{array}$$

(8)

Proof. 

From the boundary conditions of Equation (5), the left-hand side of Equation (8) can be rewritten as:

$$\begin{array}{c}2{\int }_{t_0}^{t_1}\left(\tilde{T}\left(0,t;w\right)-T_m\left(t\right)\right)\Delta \tilde{T}\left(0,t;w\right)dt={\int }_{t_0}^{t_1}\left(k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(0,t\right)\right)\Delta \tilde{T}\left(0,t;w\right)dt\hfill \\ =-{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(y,t\right)\Delta \tilde{T}\left(y,t\right)\right)dtdy+{\int }_{t_0}^{t_1}\left(k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(\frac{l}{2},t\right)\right)\Delta \tilde{T}\left(\frac{l}{2},t\right)dt\hfill \\ =-{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(y,t\right)\right)\Delta \tilde{T}\left(y,t\right)dtdy-{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(y,t\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)dtdy\hfill \\ +{\int }_{t_0}^{t_1}\left(k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(\frac{l}{2},t\right)\right)\Delta \tilde{T}\left(\frac{l}{2},t\right)dt\hfill \\ =-{\int }_{t_0}^{t_1}\left(k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(\frac{l}{2},t\right)\right)p\left(\frac{l}{2},t\right)dt+{\int }_{t_0}^{t_1}\left(k\left(\tilde{T}\right)\frac{\partial p}{\partial y}\left(\frac{l}{2},t\right)\right)\Delta \tilde{T}\left(\frac{l}{2},t\right)dt\hfill \end{array}$$

(9)

Introducing the boundary conditions of Equations (5) and (6) into Equation (9), we can obtain:

$$\begin{array}{c}2{\int }_{t_0}^{t_1}\left(\tilde{T}\left(0,t;w\right)-T_m\left(t\right)\right)\Delta \tilde{T}\left(0,t;w\right)dt=-{\sigma }_1{\int }_{t_0}^{t_1}\left(u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right)p\left(\frac{l}{2},t\right)\Delta w\left(t\right)dt\hfill \end{array}$$

(10)

Lemma 1 is proved. □

Then, Lemma 2 is given and proved for the second term of the right-hand of Equation (7).

Lemma 2.

Let $\Delta \tilde{T}=\Delta \tilde{T}\left(y,t;w\right)\in H^{1,0}\left({\mathsf{\Omega }}_{yt}\right)$ be the solution of the parabolic problem (6) for a given $w\in W$. Then, the following equation with a constant $L_1>0$ is obtained:

$${\int }_{t_0}^{t_1}\Delta {\tilde{T}}^2\left(0,t\right)dt\le L_1{\parallel \Delta w\parallel }_W^2$$

(11)

where $L_1=\left[\frac{{\sigma }_1}{k_{*}{\gamma }_1}+\frac{2}{{\gamma }_1\left(8w_{*}{T}_{*}-{\gamma }_1{r}^{*}\right)}\right]$, $0<{\gamma }_1\le \frac{8w_{*}{T}_{*}}{r^{*}}$, $w_{*}=min\left(w\left(t\right)\right)$, $T_{*}=min\left(T^3\left(\frac{l}{2},t\right)\right)$, $k_{*}=min\left(k\left(\tilde{T}\right)\right)$, $r^{*}=max{\left(u^4-T^4\left(\frac{l}{2},t\right)\right)}^2$ and ${\parallel \Delta w\parallel }_W^2:={\int }_{t_0}^{t_1}\Delta w^2\left(t\right)dt$.

Proof. 

Multiply both sides of $\frac{\partial \Delta \tilde{T}}{\partial t}\left(y,t\right)=\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)$ by $\Delta \tilde{T}(y,t)$, and integrate over ${\mathsf{\Omega }}_{yt}$, we can obtain the following equation:

$$\begin{array}{c}\frac{1}{2}{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}\frac{\partial }{\partial t}{\left(\Delta \tilde{T}\left(y,t\right)\right)}^{2}dtdy={\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}\frac{\partial }{\partial y}\left(k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)\Delta \tilde{T}\left(y,t\right)dtdy\hfill \end{array}$$

(12)

By $\Delta \tilde{T}\left(y,t_0\right)=0$, the left-hand of Equation (12) can be transferred as:

$$\frac{1}{2}{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}\frac{\partial }{\partial t}{\left(\Delta \tilde{T}\left(y,t\right)\right)}^{2}dtdy=\frac{1}{2}{\int }_0^{\frac{l}{2}}\Delta {\tilde{T}}^2\left(y,t_1\right)dy$$

(13)

The right-hand of Equation (12) can be converted to the following equation:

$$\begin{array}{c}{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}\frac{\partial }{\partial y}\left(k\left(T\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)\Delta \tilde{T}\left(y,t\right)dtdy\hfill \\ ={\int }_{t_0}^{t_1}\left(k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(\frac{l}{2},t\right)\Delta \tilde{T}\left(\frac{l}{2},t\right)\right)dt-{\int }_{t_0}^{t_1}\left(k\left(\tilde{T}\right)\frac{\partial \Delta \tilde{T}}{\partial y}\left(0,t\right)\Delta \tilde{T}\left(0,t\right)\right)dt-{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}k\left(\tilde{T}\right){\left(\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)}^{2}dtdy\hfill \\ ={\int }_{t_0}^{t_1}{\sigma }_1\left(u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right)\Delta w\left(t\right)\Delta \tilde{T}\left(\frac{l}{2},t\right)dt-{\int }_{t_0}^{t_1}4{\sigma }_{1}w\left(t\right)T^3\left(\frac{l}{2},t\right)\Delta {\tilde{T}}^2\left(\frac{l}{2},t\right)dt-{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}k\left(\tilde{T}\right){\left(\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)}^{2}dtdy\hfill \end{array}$$

(14)

Integrating Equations (13) and (14), we can obtain:

$$\begin{array}{c}\frac{1}{2}{\int }_0^{\frac{l}{2}}\Delta {\tilde{T}}^2\left(y,t_1\right)dy+{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}k\left(\tilde{T}\right){\left(\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)}^{2}dtdy+{\int }_{t_0}^{t_1}4{\sigma }_{1}w\left(t\right)T^3\left(\frac{l}{2},t\right)\Delta {\tilde{T}}^2\left(\frac{l}{2},t\right)dt\hfill \\ ={\sigma }_1{\int }_{t_0}^{t_1}\left(u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right)\Delta w\left(t\right)\Delta \tilde{T}\left(\frac{l}{2},t\right)dt\hfill \end{array}$$

(15)

Then, the $\epsilon$-inequality: $\alpha \beta ⩽\epsilon \frac{{\alpha }^2}{2}+\frac{{\beta }^2}{2\epsilon },\forall \alpha ,\beta \in R,\forall \epsilon >0$ is introduced for the right-hand side of Equation (15), which is converted to the following inequality:

$$\begin{array}{c}{\sigma }_1{\int }_{t_0}^{t_1}\left(u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right)\Delta w\left(t\right)\Delta \tilde{T}\left(\frac{l}{2},t\right)dt\le \frac{{\gamma }_1{\sigma }_1}{2}{\int }_{t_0}^{t_1}{\left(u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right)}^2\Delta {\tilde{T}}^2\left(\frac{l}{2},t\right)dt+\frac{{\sigma }_1}{2{\gamma }_1}{\int }_{t_0}^{t_1}\Delta w^2\left(t\right)dt\hfill \end{array}$$

(16)

Combining Equations (15) and (16), we can obtain:

$$\begin{array}{c}\frac{1}{2}{\int }_0^{\frac{l}{2}}\Delta {\tilde{T}}^2\left(y,t_1\right)dy+{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}k\left(\tilde{T}\right){\left(\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)}^{2}dtdy+{\int }_{t_0}^{t_1}\left[4{\sigma }_{1}w\left(t\right)T^3\left(\frac{l}{2},t\right)-\frac{{\gamma }_1{\sigma }_1{r}^{*}}{2}\right]\Delta {\tilde{T}}^2\left(\frac{l}{2},t\right)dt\le \frac{{\sigma }_1}{2{\gamma }_1}{\int }_{t_0}^{t_1}\Delta w^2\left(t\right)dt\hfill \end{array}$$

(17)

Equation (17) requires $0<{\gamma }_1\le \frac{8w_{*}{T}_{*}}{r^{*}}$. Then, with $k_{*}=min\left(k\left(\tilde{T}\right)\right)$, we can obtain two inequalities:

$$\begin{array}{c}{\int }_0^{\frac{l}{2}}{\int }_{t_0}^{t_1}{\left(\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right)}^{2}dtdy\le \frac{{\sigma }_1}{2k_{*}{\gamma }_1}{\int }_{t_0}^{t_1}\Delta w^2\left(t\right)dt,\hfill \\ {\int }_{t_0}^{t_1}\Delta {\tilde{T}}^2\left(\frac{l}{2},t\right)dt\le \frac{1}{{\gamma }_1\left[8w_{*}{T}_{*}-{\gamma }_1{r}^{*}\right]}{\int }_{t_0}^{t_1}\Delta w^2\left(t\right)dt.\hfill \end{array}$$

(18)

Finally, with ${\left|\Delta \tilde{T}\left(0,t\right)\right|}^2={\left|{\int }_0^{\frac{l}{2}}\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)dy-\Delta \tilde{T}\left(\frac{l}{2},t\right)\right|}^2\le 2{\int }_0^{\frac{l}{2}}{\left|\frac{\partial \Delta \tilde{T}}{\partial y}\left(y,t\right)\right|}^{2}dy$$+2{\left|\Delta \tilde{T}\left(\frac{l}{2},t\right)\right|}^2$, we obtain the following estimate:

$$\begin{array}{c}{\int }_{t_0}^{t_1}\Delta {\tilde{T}}^2\left(0,t\right)dt\le \left[\frac{{\sigma }_1}{k_{*}{\gamma }_1}+\frac{2}{{\gamma }_1\left(8w_{*}{T}_{*}-{\gamma }_1{r}^{*}\right)}\right]{\int }_{t_0}^{t_1}\Delta w^2\left(t\right)dt=L_1{\parallel \Delta w\parallel }_W^2\hfill \end{array}$$

(19)

Therefore, Lemma 2 is proved. Thus, the second term in the right-hand of Equation (7) is obviously bounded by the term $o\left({∥\Delta w∥}_W^2\right)$. By definition of the Fréchet -differential, Equations (8) and (11), we can conclude the following Theorem 1.

Theorem 1.

Let Lemmas 1 and 2 hold. Then the cost functional is the Fréchet -differential $J\left(w\right)\in {\mathrm{C}}^1\left(W\right)$. Moreover, the Fréchet derivative of the cost functional $J\left(w\right)$ at $w\in W$ can be defined by the solution $p\left(y,t\right)$ of the adjoint Equation (5) as follows:

$$J^{\prime }\left(w\right)=-{\sigma }_1\left[u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right]p\left(\frac{l}{2},t\right).$$

(20)

3.3. Lipschitz Continuous

Using lemmas in the previous section, we can show that the gradient $J^{\prime }\left(w\right)$ of the cost functional is Lipschitz continuous.

Lemma 3.

Let the condition of Theorem 1 hold. Then, there is a constant $L_2$ to make the following inequality hold:

$${∥J^{\prime }\left(w+\Delta w\right)-J^{\prime }\left(w\right)∥}_W^2⩽L_2{∥\Delta w∥}_W^2,$$

(21)

where, $L_2=\frac{{\sigma }_1^2{r}^{*}}{{\gamma }_2(2T_{*}{w}_{*}-{\gamma }_2{T}_{*}{p}_{*})}+\frac{16{\sigma }_1^2{\left(T^{*}\right)}^2{p}^{*}}{{\gamma }_1\left[8w_{*}{T}_{*}-{\gamma }_1{r}^{*}\right]}$, $0<{\gamma }_2\le \frac{2w_{*}}{T_{*}{p}_{*}}$, $T^{*}=max\left(T^3\left(\frac{l}{2},t\right)\right)$, $p^{*}=max\left(p^2\left(\frac{l}{2},t\right)\right)$, $p_{*}=min\left(p^2\left(\frac{l}{2},t\right)\right)$, and,

$$\begin{array}{c}{\parallel J^{\prime }\left(w+\Delta w\right)-J^{\prime }\left(w\right)\parallel }_W^2:={\sigma }_1^2{\int }_{t_0}^{t_1}{\left[u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right]}^2\Delta p^2\left(\frac{l}{2},t\right)dt+16{\sigma }_1^2{\int }_{t_0}^{t_1}\left[T^6\left(\frac{l}{2},t\right)p^2\left(\frac{l}{2},t\right)\right]\Delta {\tilde{T}}^2\left(\frac{l}{2},t\right)dt\hfill \end{array}$$

(22)

The process of proof is shown in Appendix A.

4. The Improved Hybrid Conjugate Gradient Algorithm

In Section 3, the gradient $J^{\prime }\left(w\right)$ can be calculated by the formula (20) based on the FOTD approach. This suggests an idea of constructing a new gradient-based optimization algorithm based on the FOTD approach.

There are many innovative CG algorithms with different values of ${\beta }_k$, such as the LS algorithm [25], HS algorithm [26], MCD algorithm [27], the variant Flétcher–Reeves (VFR) algorithm [28], the modified Polak–Ribière–Polyak (MPRP) algorithm [29], the spectral conjugate gradient (SCG) algorithm [30], the Nonlinear three-term conjugate gradient (NTTCG) algorithm [31]. In the past few years, some hybrid algorithms [32,33,34,35] have been developed based on the above algorithms. Inspired by these CG algorithms, this paper presents an improved hybrid conjugate gradient (IHCG) algorithm, which is given as follows:

$$d_k=\left\{\begin{array}{c}\begin{array}{cc}-g_k,& k=0\\ -{\eta }_k{g}_k+{\beta }_k{d}_{k-1},& k>0\end{array}\end{array}\right.$$

(23)

where,

$${\beta }_k=\frac{{\parallel g_k\parallel }^2-max\left\{0,\frac{\parallel g_k\parallel }{\parallel g_{k-1}\parallel }{g}_k^T{g}_{k-1}\right\}}{max\left\{-g_k^T{d}_k,d_{k-1}^T(g_k-g_{k-1})\right\}},{\eta }_k=1+\frac{{\parallel g_k-g_{k-1}\parallel }^2}{{\parallel g_k\parallel }^2}+\frac{{\parallel g_k\parallel }^2}{{\parallel g_{k-1}\parallel }^2}$$

(24)

Here, $g_k$ represents the gradient $J_w^{\prime }$ by Equation (20). The flowchart of IHCG algorithm is shown in Algorithm 1.

Algorithm 1: IHCG algorithm based on the FOTD approach.
Begin
1:	Set parameters $\delta =1e^{-2}\in (0,0.5),\sigma =0.2\in (\delta ,0.5),eps=1e^{-4},k_{max}=2000,k=0$. Choose an initial point $w_0$.
2:	Solve the direct problem (2) to determine the temperature distribution $T_k(y,t)$, and estimate the cost function $J\left(w_k\right)$;
3:	Solve the adjoint problem (5) to determine the gradient of the cost function $g_k$. If $\left|g_k\right|⩽eps$ or $k=k_{max}$, stop the algorithm.
4:	Estimate the descent depth $d_k$ based on $g_k$ and Equation (23).
5:	Determine a step length ${\alpha }_k$ by the standard Wolfe line search method. $$\begin{array}{c}J\left(w_k+{\alpha }_k{d}_k\right)\le J\left(w_k\right)+\delta {\alpha }_k{g}_k^T{d}_k,g{\left(w_k+{\alpha }_k{d}_k\right)}^T{d}_k>\sigma g_k^T{d}_k.\end{array}$$ (25)
6:	Update $w_{k+1}=w_k+{\alpha }_k{d}_k$, $k:=k+1$ and go to Step 2.
end

4.1. Property of the Proposed Algorithm

In this section, the global convergence of Algorithm 1 is proved under the Wolfe line search condition (25). Sufficient descent property is very important for the global convergence of the iterative method [36]. The following lemmas are given to state that the search directions in Algorithm 1 are all of descent.

Lemma 4.

If sequences $d_k$ and $g_k$ are generated by Equations (23) and (20), for all $k\ge 0$ the following equation holds:

$$g_k^T{d}_k\le 0.$$

(26)

Proof. 

Obviously, $g_0^T{d}_0=-{\parallel g_0\parallel }^2<0$ when $k=0$. When $k>0$, we divide the proof into the two situations by ${\beta }_k$.

Situation 1: If $k>0$ and ${\beta }_k=0$, we can obtain $g_k^T{d}_k=-{\eta }_k{\parallel g_k\parallel }^2<0$.

Situation 2: If $k>0$ and ${\beta }_k\ne 0$, the proof is divided by the following four cases:

Case 1: If $g_k^T{g}_{k-1}\le 0$ and $g_k^T{d}_{k-1}\ge 0$, we can obtain that ${\beta }_k=\frac{{\parallel g_k\parallel }^2}{d_{k-1}^T\left(g_k-g_{k-1}\right)}$. Then, the following inequality is derived:

$$\begin{array}{c}{g}_k^T{d}_k=-(1+\frac{{\parallel g_k-g_{k-1}\parallel }^2}{{\parallel g_k\parallel }^2}+\frac{{\parallel g_k\parallel }^2}{{\parallel g_{k-1}\parallel }^2}){\Vert g_k\Vert }^2+g_k^T\left({\beta }_k{d}_{k-1}\right)\hfill \\ =-\Vert g_k{\Vert }^2-{\parallel g_k-g_{k-1}\parallel }^2-\frac{{\parallel g_k\parallel }^4}{{\parallel g_{k-1}\parallel }^2}+\frac{{\parallel g_k\parallel }^2}{d_{k-1}^T\left(g_k-g_{k-1}\right)}{g}_k^T{d}_{k-1}\hfill \\ \le -\Vert g_k{\Vert }^2+\frac{{\parallel g_k\parallel }^2}{d_{k-1}^T\left(g_k-g_{k-1}\right)}{g}_k^T{d}_{k-1}\le \frac{{\parallel g_k\parallel }^2}{d_{k-1}^T\left(g_k-g_{k-1}\right)}{g}_{k-1}^T{d}_{k-1}\hfill \end{array}$$

(27)

From $d_{k-1}^T\left(g_k-g_{k-1}\right)>0$ by Equation (25) and induction assumption of Equation (27), we can notably conclude that $g_k^T{d}_k<0$.

Case 2: If $g_k^T{g}_{k-1}\le 0$ and $g_k^T{d}_{k-1}<0$, we can obtain that ${\beta }_k=\frac{{\parallel g_k\parallel }^2}{-g_{k-1}^T{d}_{k-1}}$. The following inequality is derived:

$$\begin{array}{c}{g}_k^T{d}_k=-(1+\frac{{\parallel g_k-g_{k-1}\parallel }^2}{{\parallel g_k\parallel }^2}+\frac{{\parallel g_k\parallel }^2}{{\parallel g_{k-1}\parallel }^2}){\Vert g_k\Vert }^2+g_k^T\left({\beta }_k{d}_{k-1}\right)\hfill \\ =-\Vert g_k{\Vert }^2-{\parallel g_k-g_{k-1}\parallel }^2-\frac{{\parallel g_k\parallel }^4}{{\parallel g_{k-1}\parallel }^2}+\frac{{\parallel g_k\parallel }^2}{-g_{k-1}^T{d}_{k-1}}{g}_k^T{d}_{k-1}\hfill \\ \le -\Vert g_k{\Vert }^2-\frac{{\parallel g_k\parallel }^2}{g_{k-1}^T{d}_{k-1}}{g}_k^T{d}_{k-1}\le -\frac{g_{k-1}^T{d}_{k-1}+g_k^T{d}_{k-1}}{g_{k-1}^T{d}_{k-1}}{\parallel g_k\parallel }^2<0\hfill \end{array}$$

(28)

Case 3: If $g_k^T{g}_{k-1}>0$ and $g_k^T{d}_{k-1}\ge 0$, we can obtain that ${\beta }_k=\frac{{\parallel g_k\parallel }^2-\frac{\parallel g_k\parallel }{\parallel g_{k-1}\parallel }{g}_k^T{g}_{k-1}}{d_{k-1}^T(g_k-g_{k-1})}$. Notice, that we have $0<cos{\theta }_k<1$, where ${\theta }_k$ is the angle between $g_k$ and $g_{k-1}$. The following inequality is derived:

$$\begin{array}{c}{g}_k^T{d}_k=-(1+\frac{{\parallel g_k-g_{k-1}\parallel }^2}{{\parallel g_k\parallel }^2}+\frac{{\parallel g_k\parallel }^2}{{\parallel g_{k-1}\parallel }^2}){\Vert g_k\Vert }^2+g_k^T\left({\beta }_k{d}_{k-1}\right)\hfill \\ \le -\Vert g_k{\Vert }^2+\frac{{\parallel g_k\parallel }^2-\frac{\parallel g_k\parallel }{\parallel g_{k-1}\parallel }{g}_k^T{g}_{k-1}}{d_{k-1}^T\left(g_k-g_{k-1}\right)}{g}_k^T{d}_{k-1}={\parallel g_k\parallel }^2\frac{g_{k-1}^T{d}_{k-1}-g_k^T{d}_{k-1}cos{\theta }_k}{d_{k-1}^T\left(g_k-g_{k-1}\right)}\hfill \\ \le {\parallel g_k\parallel }^2\frac{g_{k-1}^T{d}_{k-1}-g_{k-1}^T{d}_{k-1}cos{\theta }_k}{d_{k-1}^T\left(g_k-g_{k-1}\right)}=\frac{{\parallel g_k\parallel }^2\left(1-cos{\theta }_k\right)}{d_{k-1}^T\left(g_k-g_{k-1}\right)}{g}_{k-1}^T{d}_{k-1}<0\hfill \end{array}$$

(29)

Case 4: If $g_k^T{g}_{k-1}>0$ and $g_k^T{d}_{k-1}<0$, we can obtain ${\beta }_k=\frac{{\parallel g_k\parallel }^2-\frac{\parallel g_k\parallel }{\parallel g_{k-1}\parallel }{g}_k^T{g}_{k-1}}{-g_{k-1}^T{d}_{k-1}}$ and $g_{k_1}^T{d}_{k-1}<0$. The following inequality is derived:

$$\begin{array}{c}{g}_k^T{d}_k=-(1+\frac{{\parallel g_k-g_{k-1}\parallel }^2}{{\parallel g_k\parallel }^2}+\frac{{\parallel g_k\parallel }^2}{{\parallel g_{k-1}\parallel }^2}){\Vert g_k\Vert }^2+g_k^T\left({\beta }_k{d}_{k-1}\right)\hfill \\ \le -\Vert g_k{\Vert }^2-\frac{{\parallel g_k\parallel }^2-\frac{\parallel g_k\parallel }{\parallel g_{k-1}\parallel }{g}_k^T{g}_{k-1}}{g_{k-1}^T{d}_{k-1}}{g}_k^T{d}_{k-1}=-{\Vert g_k\Vert }^2-\frac{{\parallel g_k\parallel }^2\left(1-cos{\theta }_k\right)}{g_{k-1}^T{d}_{k-1}}{g}_k^T{d}_{k-1}\hfill \\ =-\frac{g_{k-1}^T{d}_{k-1}+g_k^T{d}_{k-1}\left(1-cos{\theta }_k\right)}{g_{k-1}^T{d}_{k-1}}{\parallel g_k\parallel }^2<0\hfill \end{array}$$

(30)

Finally, for all $k\ge 0$, $g_k^T{d}_k\le 0$ always holds. □

Meanwhile, we can easily obtain the following important property.

Lemma 5.

For any $k\ge 0$, the relation $0\le {\beta }_k\le \frac{g_k^T{d}_k}{g_{k-1}^T{d}_{k-1}}$ always holds.

Proof. 

From the definition of ${\beta }_k$ in Equation (23), one knows ${\beta }_k\ge 0$. Next, we divide the proof into the two situations by ${\beta }_k$.

Situation 1: If ${\beta }_k=0$, by Lemma 4, we can obtain $\frac{g_k^T{d}_k}{g_{k-1}^T{d}_{k-1}}>0={\beta }_k$.

Situation 2: If ${\beta }_k>0$, the proof is divided by the following four cases:

Case 1: If $g_k^T{g}_{k-1}\le 0$ and $g_k^T{d}_{k-1}\ge 0$, from Lemma 4.1 $g_k^T{d}_k\le 0$ and Equation (27), we have:

$${\beta }_k=\frac{{\parallel g_k\parallel }^2}{d_{k-1}^T\left(g_k-g_{k-1}\right)}\le \frac{g_k^T{d}_k}{g_{k-1}^T{d}_{k-1}}$$

(31)

Case 2: If $g_k^T{g}_{k-1}\le 0$ and $g_k^T{d}_{k-1}<0$, from Lemma 4.1 and Equation (28), we have: $g_k^T{d}_k\le \frac{g_{k-1}^T{d}_{k-1}+g_k^T{d}_{k-1}}{-g_{k-1}^T{d}_{k-1}}{\parallel g_k\parallel }^2\le \left(g_{k-1}^T{d}_{k-1}+g_k^T{d}_{k-1}\right){\beta }_k$, then the following inequality is obtained.

$${\beta }_k\le \left(1+\frac{g_k^T{d}_{k-1}}{g_{k-1}^T{d}_{k-1}}\right){\beta }_k\le \frac{g_k^T{d}_k}{g_{k-1}^T{d}_{k-1}}$$

(32)

Case 3: If $g_k^T{g}_{k-1}>0$ and $g_k^T{d}_{k-1}\ge 0$, from Lemma 4.1 and Equation (29), we have:

$${\beta }_k=\frac{{\parallel g_k\parallel }^2\left(1-cos{\theta }_k\right)}{d_{k-1}^T\left(g_k-g_{k-1}\right)}\le \frac{g_k^T{d}_k}{g_{k-1}^T{d}_{k-1}}$$

(33)

Case 4: If $g_k^T{g}_{k-1}>0$ and $g_k^T{d}_{k-1}<0$, from Lemma 4.1 and Equation (30), we have $g_k^T{d}_k\le \left(g_{k-1}^T{d}_{k-1}+g_k^T{d}_{k-1}\left(1-cos{\theta }_k\right)\right){\beta }_k$, and the following inequality is obtained.

$${\beta }_k\le \left(1+\frac{g_k^T{d}_{k-1}\left(1-cos{\theta }_k\right)}{g_{k-1}^T{d}_{k-1}}\right){\beta }_k\le \frac{g_k^T{d}_k}{g_{k-1}^T{d}_{k-1}}$$

(34)

Finally, Lemma 5 is proved. □

4.2. Global Convergence

As demonstrated in [34], the following assumptions are needed to prove the global convergence:

1. The objective function $J\left(w\right)$ is bounded from below on the level set $\Lambda$, where $\Lambda =\left\{w\in W^n|J\left(w\right)\le J\left(w_0\right)\right\}$, and $w_0$ is the initial point.

2. The gradient of the cost functional $J^{\prime }\left(w\right)$ is Lipchitz continuous and a constant exists, which has been given by Lemma 3.

Lemma 6.

Suppose that Assumptions 1 and 2 hold. Considering the iteration $g=J^{\prime }\left(w\right)$, the direction and step length satisfy the Wolfe line search condition (25), we can obtain:

$$\sum _{k=0}^{\infty }\frac{{\Vert g_k\Vert }^4}{{\Vert d_k\Vert }^2}<\infty$$

(35)

Proof. 

If sequences $\left\{d_k\right\}$ and $\left\{w_k\right\}$ are calculated by Equations (23) and (25), the following equation holds:

$$\underset{k\to \infty }{lim\; inf}\parallel g_k\parallel =0$$

(36)

By contradiction, if Equation (36) does not hold, there exists a positive constant ${\lambda }_1>0$ satisfying ${\parallel g_k\parallel }^2\ge {\lambda }_1$ for all $k>0$. From $d_k=-{\eta }_k{g}_k+{\beta }_k{d}_{k-1}$ and Lemma 5, we can obtain:

$${\parallel d_k\parallel }^2\le -{\eta }_k^2{\parallel g_k\parallel }^2-2{\eta }_k{g}_k^T{d}_k+{\left(\frac{g_k^T{d}_k}{g_{k-1}^T{d}_{k-1}}\right)}^2{\parallel d_{k-1}\parallel }^2$$

(37)

Divided both sides of Equation (37) by ${\left(g_k^T{d}_k\right)}^2$, the following equation is obtained:

$$\begin{array}{c}\frac{{\parallel d_k\parallel }^2}{{\left(g_k^T{d}_k\right)}^2}\le -{\eta }_k^2\frac{{\parallel g_k\parallel }^2}{{\left(g_k^T{d}_k\right)}^2}-\frac{2{\eta }_k}{g_k^T{d}_k}+\frac{{\parallel d_{k-1}\parallel }^2}{{\left(g_{k-1}^T{d}_{k-1}\right)}^2}\le \frac{1}{{\parallel g_k\parallel }^2}+\frac{{\parallel d_{k-1}\parallel }^2}{{\left(g_{k-1}^T{d}_{k-1}\right)}^2}\hfill \end{array}$$

(38)

By a recurrence of Equation (38) and ${\parallel g_k\parallel }^2\ge {\lambda }_1$, we can have:

$$\begin{array}{c}\frac{{\parallel d_k\parallel }^2}{{\left(g_k^T{d}_k\right)}^2}\le \frac{1}{{\parallel g_k\parallel }^2}+\frac{{\parallel d_{k-1}\parallel }^2}{{\left(g_{k-1}^T{d}_{k-1}\right)}^2}\le \sum _{i=1}^k\frac{1}{{\parallel g_i\parallel }^2}+\frac{{\parallel d_0\parallel }^2}{{\left(g_0^T{d}_0\right)}^2}=\sum _{i=0}^k\frac{1}{{\parallel g_i\parallel }^2}\le \frac{k}{{\lambda }_1}\hfill \end{array}$$

(39)

It concludes that ${\displaystyle \sum _{k=0}^{\infty }}\frac{\Vert g_k{\Vert }^4}{\Vert d_k{\Vert }^2}=\infty$, which contradicts with Equation (35), so Lemma 6 is proved. □

5. Simulation and Analysis

In this section, some simulations are made to verify the proposed IHCG algorithm based on the FOTD approach. Firstly, the comparison between the FDTO approach and FOTD approach is given. Secondly, the comparison between IHCG and other popular variant CG algorithms (LS [25], HS [26], MCD [27], VFR [28], MPRP [29], SCG [30], NTTCG [31]) is given. Finally, the application of the reheating furnace with actual measured data is considered to identify the total heat exchange factor. All simulations are programmed by MATLAB R2018b and implemented on a computer with Intel i5-11400F GPU, 2.60 GHz, 16GB RAM.

5.1. FOTD Approach vs. FDTO Approach

Here, the FDTO approach is introduced as the comparative object. In the FDTO approach, the explicit finite difference method [37] is used to discretize the direct PDEs (2). By the FDTO approach with the sensitivity method [12], the gradient $g_k$ is obtained in terms of the Jacobian matrix: $g_k=J^{\prime }\left(\mathit{w}\right)=\left[\begin{array}{cccc}\frac{\partial J}{\partial w_1}& \frac{\partial J}{\partial w_2}& \cdots & \frac{\partial J}{\partial w_n}\end{array}\right]$$=\left[\begin{array}{cccc}\frac{\partial J}{\partial T}\frac{\partial T}{\partial w_1}& \frac{\partial J}{\partial T}\frac{\partial T}{\partial w_2}& \cdots & \frac{\partial J}{\partial T}\frac{\partial T}{\partial w_n}\end{array}\right]$, where, $\mathit{w}$ is the discrete form of $w\left(t\right)$: $\mathit{w}=\left[\begin{array}{cccc}{w}_1& w_2& \cdots & w_n\end{array}\right]$, and n is the total number of discrete points. Generally, $n+1$ times solution of nonlinear PDEs (2) are required to calculate the Jacobian matrix.

In this case, the total heat exchange factor of PDEs (2) is defined as the functional form: $w\left(t\right)=0.7+0.2\times \sin (2.17\times \pi \times t/t_1)$. Furthermore, other parameter values of the PDE Equation (2) are shown as: $l=0.2$ (m), $T_0\left(y\right)=298$ (K), $\rho =7850$ (kg/m${}^3$), $t_0=0,t_1=180$ (min). Then, the accurate measured values $T_m\left(t\right)=T(0,t)$ of the center temperature are obtained by solving the direct PDEs (2) with $w\left(t\right)$. The initial value of $w_0\left(t\right)$ is given as $w_0\left(t\right)=0.6+0.4\times \sin (2.17\times \pi \times t/t_1)$. The details of $w\left(t\right),w_0\left(t\right)$ in forms of the sine function are illustrated in Figure 3. The same numerical algorithm (LS [25]) is applied for both the FDTO and FOTD approaches. The numerical values of the computational results are given in Figure 4 and

Table 1. Comparison of simulation performance between FOTD and FDTO approaches.

	Simulation Time (s)	Iteration Steps	Final Cost Functional (J)
FOTD	300.92	713	0.6761
FDTO	42035.76	317	0.3710

.

Combing Figure 4 and Table 1, it is clear that the FDTO approach yields less iteration steps than the FOTD approach; meanwhile, the value of the final cost functional J is a little smaller than the FOTD approach. However, the simulation time of the FDTO approach is 42035.76 s, which is 140 times larger than the 300.92 s by the FOTD approach because it wastes exceedingly large computation time to obtain $n+1$ times the solution of nonlinear PDEs for the Jacobian matrix of the FDTO approach.

In general, the FOTD approach takes less time to obtain the almost same value of the final cost functional than the FDTO approach.

5.2. Comparisons between IHCG and Other CG Algorithms

The proposed IHCG algorithm is compared with other popular variant CG algorithms (LS [25], HS [26], MCD [27], VFR [28], MPRP [29], SCG [30], NTTCG [31]). Here, the comparisons are shown in the following two cases.

Case 1: In this case, the physical parameters of Section 5.1 are used. The numerical performances of the computational results are given in Figure 5, Figure 6 and Figure 7 and

Table 2. Comparison of simulation performance by different CG algorithms.

Algorithms	VFR	SCG	HS	LS	MCD	MPRP	NTTCG	IHCG
Iteration steps	708	896	971	713	783	1044	791	334
Simulation time (s)	306.72	409	456.44	300.92	334.64	474.74	331.32	140.38

. Notice, the Y-coordinate direction is a logarithmic function of the cost function in Figure 6.

Figure 5 shows that the performance of our proposed IHCG is better than other CG algorithms in the early stage of the converged process, and Figure 6 shows that our proposed IHCG needs less iteration steps (only 334) to satisfy the convergence criteria, which is much lower than all the other CG algorithms. Combining Table 2, the running time by our proposed IHCG is 140.38 s, which is lower than all the other CG algorithms.

The symbol $|w^c\left(t\right)-w\left(t\right)|$ in Figure 7 means the absolute prediction errors by different CG algorithms. In Figure 7, the smallest two prediction errors are obtained by by the VFR algorithm and our IHCG algorithm. However, the iteration steps and simulation time by the VFR algorithm are much bigger than that by the IHCG algorithm as shown in Table 2. Notice, there is a phenomenon in Figure 7 that the prediction errors near the final time $t=t_1$ are relatively higher than other periods. This phenomenon may be caused by the FOTD approach, in which the Fréchet gradient $J^{\prime }\left(w\right)=-{\sigma }_1\left[u^4\left(t\right)-T^4\left(\frac{l}{2},t\right)\right]p\left(\frac{l}{2},t\right)$ at the final time $t=t_1$ is strongly effected by $p(y,t_1)=0$ from the weak solution of the adjoint PDE (2).

Compared with other CG algorithms, the proposed IHCG algorithm can solve the IHCP quickly and accurately.

Case 2: In this case, the physical parameters of Section 5.1 are also used. Furthermore, some errors are added for the measured values $T_m\left(t\right)=T(0,t)$. It assumes that the temperature values at 10 points are known with random errors:

$$T_m^e\left(t\right)=T_m\left(t\right)+\delta \times rand\left(100\right)$$

(40)

Considering the measured values with errors, the IHCG and other CG algorithms are used to identify the given $w\left(t\right)$, and the simulation results are given in

Table 3. Comparison under different error levels by different CG algorithms.

Algorithms	$\mathit{\delta }=0.01$		$\mathit{\delta }=0.03$		$\mathit{\delta }=0.05$
	S	${\mathit{S}}_{max}$	S	${\mathit{S}}_{max}$	S	${\mathit{S}}_{max}$
VFR	2.73$\times {\mathbf{10}}^{-\mathbf{6}}$	1.48$\times {\mathbf{10}}^{-\mathbf{4}}$	6.52$\times {10}^{-6}$	3.52$\times {10}^{-4}$	4.15$\times {10}^{-4}$	1.13$\times {10}^{-2}$
SCG	3.18$\times {10}^{-6}$	1.72$\times {10}^{-4}$	5.74$\times {10}^{-6}$	3.10$\times {10}^{-4}$	3.89$\times {10}^{-4}$	8.05$\times {10}^{-3}$
HS	3.16$\times {10}^{-6}$	1.70$\times {10}^{-4}$	5.90$\times {10}^{-6}$	3.19$\times {10}^{-4}$	3.88$\times {10}^{-4}$	8.07$\times {10}^{-3}$
LS	3.10$\times {10}^{-6}$	1.67$\times {10}^{-4}$	5.73$\times {10}^{-6}$	3.09$\times {10}^{-4}$	3.89$\times {10}^{-4}$	8.07$\times {10}^{-3}$
MCD	3.12$\times {10}^{-6}$	1.68$\times {10}^{-4}$	5.72$\times {10}^{-6}$	3.09$\times {10}^{-4}$	3.88$\times {10}^{-4}$	7.99$\times {10}^{-3}$
MPRP	2.95$\times {10}^{-6}$	1.59$\times {10}^{-4}$	9.62$\times {10}^{-6}$	5.19$\times {10}^{-4}$	3.89$\times {10}^{-4}$	8.07$\times {10}^{-3}$
NTTCG	7.15$\times {10}^{-6}$	3.86$\times {10}^{-4}$	6.51$\times {10}^{-6}$	3.52$\times {10}^{-4}$	1.48$\times {10}^{-3}$	3.23$\times {10}^{-2}$
IHCG	2.94$\times {10}^{-6}$	1.58$\times {10}^{-4}$	5.71$\times {\mathbf{10}}^{-\mathbf{6}}$	3.08$\times {\mathbf{10}}^{-\mathbf{4}}$	3.85$\times {\mathbf{10}}^{-\mathbf{4}}$	7.98$\times {\mathbf{10}}^{-\mathbf{3}}$

The best values are shown in bold.

and Figure 8, Figure 9 and Figure 10. In Table 3, S denotes the standard deviation, and $S_{max}$ is the maximum standard deviation [35]. They are defined as:

$$\begin{array}{c}S=\frac{1}{N_n}\sqrt{{\sum }_{i=1}^{N_n}{\left(\frac{T_c\left(i\right)-T_m^e\left(i\right)}{T_m^e\left(i\right)}\right)}^2},S_{max}={\displaystyle \underset{i\in \left[1.N_n\right]}{max}}\left(\sqrt{{\left(\frac{T_c\left(i\right)-T_m^e\left(i\right)}{T_m^e\left(i\right)}\right)}^2}\right)\hfill \end{array}$$

(41)

where $T_c\left(i\right)$ is the calculated value by solving the IHCP (3), $T_m^e\left(i\right)$ is the true measured value with random errors, and $N_n$ is the total number of cells for the whole reheating time.

In Table 3, the best values are shown in bold. The smallest values of S and $S_{max}$ are obtained by the VFR algorithm when the random errors are given by $\delta =0.01$. Except for the VFR algorithm, the IHCG algorithm performs better than all the other CG algorithms. With the increase of the random errors $\delta =0.03,0.05$, the smallest values of S and $S_{max}$ are obtained by the IHCG algorithm, which means the proposed IHCG algorithm can overcome the bad affect caused by random errors for the inverse heat conduction problem.

In contrast to other CG algorithms, the IHCG algorithm needs the smallest iteration step to accelerate the convergence rate as shown in Figure 8, Figure 9 and Figure 10. This goes to prove that the IHCG algorithm can solve the proposed IHCP quickly.

Finally, the IHCG algorithm can effectively eliminate the ill-posedness of the inverse problem.

5.3. Application in Reheating Furnace

In this case, the measured values of the center temperature were obtained from a walking beam reheating furnace. The charging state and discharging state of the trial steel slab, whose thickness is $0.23$ m, is shown in Figure 11. The slab enters into the reheating furnace at 15:40, and is discharged from the furnace at 18:13. Thus, the total reheating time is 153 min. In the trial steel slab, the thermocouples (Type K, $\varphi =15$ mm) are set to record the center and surface temperatures of the slab. The center temperatures are shown in Figure 12 and defined as Dirichlet type boundary condition $T_m\left(t\right)$ to obtain and identify the total heat exchange factor. The identified total heat exchange factor will be introduced into the transient nonlinear heat conduction model in Section 2. The measured surface temperatures in Figure 12 will be compared with the calculated surface temperatures to verify the necessity and accuracy of identifying the total heat exchange factor.

First, the IHCG and other CG algorithms are used to identify the total heat exchange factor based on actual measured data. The initial guess $w_0$ is also defined as the functional form: $w\left(t\right)=0.7+0.2\times \sin (2.17\times \pi \times t/t_1)$. The convergence rate of these algorithms is shown in Figure 13, and the iterative number and running time are listed in

Table 4. Comparison of simulation performance by different CG algorithms with measured data.

Algorithms	Cost Function (J)	Iteration Steps	Simulation Time (s)
VFR	1308.07	935	508.15
SCG	30.31	948	495.18
HS	13.53	1477	823.55
LS	89.32	360	179.94
MCD	28.56	980	525.86
MPRP	31.6	808	408.31
NTTCG	7258.97	60	29.72
IHCG	7.58	781	407.42

. In Figure 13 and Table 4, the VFR and NTTCG algorithm cannot obtain enough small cost function J when the convergence process is stopped at $\left|g_k\right|⩽eps$. The IHCG algorithm performs best among these CG algorithms. It needs 407.42 s to obtain the value of cost function J is 7.58 when the convergence process is stopped at 781 iteration steps. The identified total heat exchange factor by IHCG algorithm based on actual measured data is shown in Figure 14.

Second, the PDE model with the identified total heat exchange factor in Figure 14 is solved to calculate the surface temperature. Here, a new PDE model, in which the total heat exchange factor is given as constant $w=0.7$, is introduced as the comparative object. The calculated surface temperatures by two PDE models are compared with the measured surface temperatures, which is shown in Figure 15. In Figure 15, $T_m^s$ is the measured surface temperatures, $T_c^1$ stands for the calculated surface temperatures with the identified total heat exchange factor, and $T_c^2$ represents the calculated surface temperatures with the constant $w=0.7$. The temperature differences $|T_c^{1,2}-T_m^s|$ with different total heat exchange factors are shown in Figure 16. It is clear the calculated surface temperatures with the identified total heat exchange factor are closer to the measured surface temperatures than that with the constant $w=0.7$.

6. Conclusions

In the present work, a new strategy combined the FOTD approach and a new IHCG algorithm is proposed to identify the total heat exchange factor for the reheating furnace by solving a typical inverse heat conduction problem. The main contributions of this paper are given as follows:

1. A nonlinear IHCP with the Dirichlet-type boundary condition $T_m\left(t\right)=T(0,t)$ is proposed to identify the unknown total heat exchange factor for the reheating furnace.

2. The FOTD approach is applied to obtain the Fréchet gradient $g_k$ of the cost function, which can be described by the weak solution $p(y,t)$ of the adjoint equations. Simulation results prove that the FOTD approach takes less time to obtain nearly the same value of the final cost functional than the FDTO approach.

3. Based on the Fréchet gradient $g_k$ of the cost function by the FOTD approach, a new IHCG algorithm, which combines the hybrid scalar parameter ${\beta }_k$ with spectrum ${\eta }_k$, is developed for the numerical solution of the proposed IHCP. The simulation results show that the IHCG algorithm is accurate and can effectively eliminate the ill-posedness of the inverse problem.

Moreover, the new strategy may provide a theoretical basis for identifying the unknown thermophysical properties for other applications.