# Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm

Authors: Xinfu Pang; Yang Yu; Haibo Li; Yuan Wang; Jinhui Zhao
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main work of this paper focuses on identifying the heat flux in inverse problem of two-dimensional nonhomogeneous parabolic equation, which has wide application in the industrial field such as steel-making and continuous casting. Firstly, the existence of the weak solution of the inverse problem is discussed. With the help of forward solution and dual equation, this paper proves the Lipchitz continuity of the cost function and derives the Lipchitz constant. Furthermore, in order to accelerate the convergence rate and reduce the running time, this paper presents a sufficient descent Levenberg–Marquard algorithm with adaptive parameter (SD-LMAP) to solve this inverse problem. At last, compared with other methods, the results of simulation experiment show that this algorithm can obviously reduce the running time and iterative number.


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## Body

## 1. Introduction

The most steel production is completed by the continuous casting machine. To guarantee the quality of casting billet, the spray cooling control system [1, 2] based on the heat transfer model is considered. Hence, the accurate heat transfer model counts a great deal for the production of continuous casting. Boundary conditions are essential to the accurate heat transfer model. Therefore, it is crucial to correct boundary conditions along the caster machine during the casting operations. Heat flux at the metal/cooling interface is an important one among these boundary conditions [3, 4]. The identification of heat flux of the heat transfer model in continuous casting is a kind of inverse problems of the parabolic equation. The inverse problems of the parabolic equation are encountered in many applications, in particular, in heat transfer [5, 6], filtration [7], and material science [8, 9]. We only consider the inverse problem in continuous casting.Recently, the inverse problems of the parabolic equation have been considered by many authors. Wang et al. [10] used a regularized optimization method for identifying the space-dependent source in a parabolic equation. Ruan et al. [11] investigated an inverse source problem for the time-fractional diffusion equation, which is formulated into an optimization problem, and a semismooth Newton (SSN) algorithm is developed to solve it. Chen et al. [12] applied an inverse algorithm based on the conjugate gradient method and the discrepancy principle to solve the inverse hyperbolic heat conduction problem. Zhao and Banda [13] deal with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and initial value simultaneously for the one-dimensional heat equation. Beck and Woodbury [14] estimated the time and/or space dependence of the surface heat flux or temperature utilizing interior temperature measurements at discrete times and/or locations. Lee [15] demonstrated the performance and efficiency of the repulsive particle swarm optimization (RPSO) method for solving the inverse heat conduction problem. Cui et al. [16, 17] developed an improved Levenberg–Marquardt (L-M) algorithm for solving inverse problems. Oliveira et al [18] used the Levenberg–Marquardt technique to determine the interfacial heat transfer coefficient.Our previous works [19, 20] focused on the identification of the heat transfer coefficients of the heat transfer model by the L-M algorithm. Based on the above works, this paper considers the inverse problem of two-dimensional nonhomogeneous parabolic equation associated with the continuous casting. This problem is described as a PDE optimization problem. The adjoint approach is used to prove the Lipchitz continuous of the gradient of cost function in PDE optimization problem with the two-dimensional nonhomogeneous parabolic equation. Under the condition of Lipchitz continuous, this paper presents a sufficient descent Levenberg–Marquard with adaptive parameter (SD-LMAP) algorithm to solve this problem. And, the simulation experiment is used to illustrate the validity of this algorithm.The construction of paper is organized as follows. In Section2, the heat transfer model of continuous casting is given. And, we prove the Lipchitz continuous of the gradient of cost function in Section 3. In Section 4, the sufficient descent Levenberg–Marquard (SD-LMAP) algorithm with an adaptive parameter is presented. In Section 5, the simulation experiments are given to illustrate the validity of the proposed method.


## 2. Heat Transfer Model for Continuous Casting and the Description of the Problem

### 2.1. Heat Transfer Mathematical Model for Continuous Casting

The pouring process of continuous casting is given in Figure1, and the solidification of the slab can be described by the heat transfer model. The unsteady state heat transfer mathematical model based on some assumptions can be used to predict the temperature distribution during the process of casting solidification. So, we assume as follows: (1) in the process of the continuous casting, convective heat transfer is equal to the thermal conduction, (2) the mold is considered uniform and the cooling temperature equal to the water-cooling temperature uw, and (3) the meniscus surface is considered to be flat. Based on the above assumptions, the two-dimensional heat transfer model [1] is given by(1)ρc∂u∂t=k∂2u∂x2+∂2u∂y2+Qx,y,t,x,y,t∈Ω,where Ω:=x,y,t∈ℜ3:x∈0,l,y∈0,l,t∈0,T,T>0, l is the width of billet (the width is equal to the thickness for billet), T is the casting time s, ρ is density kg/m3, c is specific heat J/kg⋅K, k is thermal conductivity W/m⋅K, u is temperature K, t is time s, and Q is the term associated to the internal heat generation because of the phase change.Figure 1
The pouring process of continuous casting.The boundary condition at beginning (onΓ0) is(2)ubeginx,y,t|t=0=ucast,x∈0,l,y∈0,l,where ucast is the pouring temperature K, which is considered as a constant.On the boundaryΓ1x=1 and Γ2y=1, the spray cooling method is used to solidify the billet, and these boundary conditions can be written as(3)−k∂ux,y,t∂x|x=l=qy,t,y∈0,l,t∈0,T,(4)−k∂ux,y,t∂y|y=l=qx,t,x∈0,l,t∈0,T,where qx,t=γ¯∗ux,l,t−uw is heat flux on Γ2=x∈0,l,t∈0,T, which is crucial to the heat transfer model. γ¯ is heat transfer coefficient and uw is the temperature of cooling water. However, the interface heat flux of the billet is difficult to be measured, so this paper considers to identify this parameter. Similarly, qy,t=γ¯∗ul,y,t−uw is the heat flux on Γ1=y∈0,l,t∈0,T. The cooling intensity on the surface of width is the same as the thickness for billet.Because of the symmetrical heat flux at the center of the slab, we can consider the 1/4 of steel billet, and the boundary conditions onΓ3x=0 and Γ4y=0 are given in the following:(5)−k∂ux,y,t∂x|x=0=0,y∈0,l,t∈0,T,(6)−k∂ux,y,t∂y|y=0=0,x∈0,l,t∈0,T.


### 2.2. The Description of the Problem

In the productive process of continuous casting, the surface temperature of billet (x=l or y=l) can be measured [3]. We can use these data to identify the heat flux q⋅,t. We define a=k/ρc and h⋅,t=q⋅,t/ρc, and the inverse problem can be described by the following PDE optimization problem:(7)Jh=J1h+J2h,s.t.ut=auxx+uyy+Qx,y,t,x,y,t∈Ω,ux,y,t|t=0=u0,x∈0,l,y∈0,l,auyx,y,t|y=l=−hx,t,auyx,y,t|y=0=0,t∈0,T,x∈0,l,auxx,y,t|x=l=−hy,t,auxx,y,t|x=0=0,t∈0,T,y∈0,l,where J1h=∫0l∫0Tucl,y,t;h−ueδl,y,t2dydt, J2h=∫0l∫0Tucx,l,t;h−ueδx,l,t2dxdt, ucl,y,t;h and ucx,l,t;h are calculated values of the surface temperature on Γ1 and Γ2, respectively, ueδl,y,t and ueδx,l,t are measured values of the surface temperature on Γ1 and Γ2, respectively, δ is the error level, and u0 is the initial temperature, where we define u0=ubegin−ucast=0.Remark 1.
Because the cooling intensity on the surface of width is the same as thickness for billet, as long as the heat flux on width side is estimated, the heat flux on the other side can be known. In the process of measurement, we only need to measure the surface temperature on one side.


## 2.1. Heat Transfer Mathematical Model for Continuous Casting

The pouring process of continuous casting is given in Figure1, and the solidification of the slab can be described by the heat transfer model. The unsteady state heat transfer mathematical model based on some assumptions can be used to predict the temperature distribution during the process of casting solidification. So, we assume as follows: (1) in the process of the continuous casting, convective heat transfer is equal to the thermal conduction, (2) the mold is considered uniform and the cooling temperature equal to the water-cooling temperature uw, and (3) the meniscus surface is considered to be flat. Based on the above assumptions, the two-dimensional heat transfer model [1] is given by(1)ρc∂u∂t=k∂2u∂x2+∂2u∂y2+Qx,y,t,x,y,t∈Ω,where Ω:=x,y,t∈ℜ3:x∈0,l,y∈0,l,t∈0,T,T>0, l is the width of billet (the width is equal to the thickness for billet), T is the casting time s, ρ is density kg/m3, c is specific heat J/kg⋅K, k is thermal conductivity W/m⋅K, u is temperature K, t is time s, and Q is the term associated to the internal heat generation because of the phase change.Figure 1
The pouring process of continuous casting.The boundary condition at beginning (onΓ0) is(2)ubeginx,y,t|t=0=ucast,x∈0,l,y∈0,l,where ucast is the pouring temperature K, which is considered as a constant.On the boundaryΓ1x=1 and Γ2y=1, the spray cooling method is used to solidify the billet, and these boundary conditions can be written as(3)−k∂ux,y,t∂x|x=l=qy,t,y∈0,l,t∈0,T,(4)−k∂ux,y,t∂y|y=l=qx,t,x∈0,l,t∈0,T,where qx,t=γ¯∗ux,l,t−uw is heat flux on Γ2=x∈0,l,t∈0,T, which is crucial to the heat transfer model. γ¯ is heat transfer coefficient and uw is the temperature of cooling water. However, the interface heat flux of the billet is difficult to be measured, so this paper considers to identify this parameter. Similarly, qy,t=γ¯∗ul,y,t−uw is the heat flux on Γ1=y∈0,l,t∈0,T. The cooling intensity on the surface of width is the same as the thickness for billet.Because of the symmetrical heat flux at the center of the slab, we can consider the 1/4 of steel billet, and the boundary conditions onΓ3x=0 and Γ4y=0 are given in the following:(5)−k∂ux,y,t∂x|x=0=0,y∈0,l,t∈0,T,(6)−k∂ux,y,t∂y|y=0=0,x∈0,l,t∈0,T.


## 2.2. The Description of the Problem

In the productive process of continuous casting, the surface temperature of billet (x=l or y=l) can be measured [3]. We can use these data to identify the heat flux q⋅,t. We define a=k/ρc and h⋅,t=q⋅,t/ρc, and the inverse problem can be described by the following PDE optimization problem:(7)Jh=J1h+J2h,s.t.ut=auxx+uyy+Qx,y,t,x,y,t∈Ω,ux,y,t|t=0=u0,x∈0,l,y∈0,l,auyx,y,t|y=l=−hx,t,auyx,y,t|y=0=0,t∈0,T,x∈0,l,auxx,y,t|x=l=−hy,t,auxx,y,t|x=0=0,t∈0,T,y∈0,l,where J1h=∫0l∫0Tucl,y,t;h−ueδl,y,t2dydt, J2h=∫0l∫0Tucx,l,t;h−ueδx,l,t2dxdt, ucl,y,t;h and ucx,l,t;h are calculated values of the surface temperature on Γ1 and Γ2, respectively, ueδl,y,t and ueδx,l,t are measured values of the surface temperature on Γ1 and Γ2, respectively, δ is the error level, and u0 is the initial temperature, where we define u0=ubegin−ucast=0.Remark 1.
Because the cooling intensity on the surface of width is the same as thickness for billet, as long as the heat flux on width side is estimated, the heat flux on the other side can be known. In the process of measurement, we only need to measure the surface temperature on one side.


## 3. The Quasi-Solution of the Inverse Problem and Lipchitz Continuity of the Gradient of the Cost Function

### 3.1. The Quasi-Solution Approach Based on Weak Solution

We denote the set of admissible unknown heat fluxh⋅,t by HΩ, which satisfies the following equation:(8)h⋅,t∈HΩ,∀t∈0,T,x∈0,l,y∈0,l.The weak solution of the forward problem (7) will be defined to be the function u∈V˙1,0Ω, which satisfies the following integral identity:(9)∭Ω−uvt+avxux+avyuydxdydt=∭ΩQx,y,tvdxdydt,∀v∈HΩ,where vx,y,T=0 for x∈0,l and y∈0,l and vl,y,t=0 and vx,l,t=0 for t∈0,T. Obviously, under the above conditions with regards to the given data, the existence and uniqueness of this solution in u∈V˙1,0Ω can be found in [5, 21].Let us denote the solution of problem (7) by u∗x,y,t;h, corresponding to a given h⋅,t∈HΩ,∀t∈0,T,x∈0,l,y∈0,l. If this function satisfies the measured data ueδx,l,t or ueδl,y,t, then it should satisfy the nonlinear functional equation:(10)u∗x,y,t;h|x=l=ueδl,y,t,t∈0,T,y∈0,l,or(11)u∗x,y,t;h|y=l=ueδx,l,t,t∈0,T,x∈0,l.In practice, the measured dataueδx,0,t is usually given by the data with error level δ. The exact fulfillment of condition (10) may be not possible. According to this reason, a quasi-solution of the inverse problem (7) is defined as a solution of the following minimization problem:(12)Jh∗=infJh,where Jh is the cost functional, which is defined by equation (7).


### 3.2. Lipchitz Continuity of the Gradient of the Cost Function

We assume thath,h+Δh∈H, and the variation of the cost function ΔJ can be given in the following:(13)ΔJh=ΔJ1h+ΔJ2h=2∫0T∫0lucx,l,t;h−ueδΔux,l,t;Δhdxdt+∫0T∫0lΔux,l,t;Δh2dxdt+2∫0T∫0lucl,y,t;h−ueδΔul,y,t;Δhdydt+∫0T∫0lΔul,y,t;Δh2dydt,where Δux,y,t;h=ux,y,t;h+Δh−ux,y,t;h∈HΩ. Obviously, Δu=Δux,y,t;Δh is the solution of the following parabolic problem:(14)Δut=aΔuxx+Δuyy,x,y,t∈Ω,Δux,y,0=0,x∈0,l,y∈0,l,aΔuyx,y,t|y=l=−Δhx,t,aΔuyx,y,t|y=0=0,t∈0,T,x∈0,l,aΔuxx,y,t|x=l=−Δhy,t,aΔuxx,y,t|x=0=0,t∈0,T,y∈0,l.Lemma 1.
Ifu=ux,y,t;h∈HΩ is the corresponding solution of the direct problem (7) and φx,y,t;h∈HΩ is the solution of the adjoint problem,(15)φt=−aφxx+φyy,x,y,t∈Ω,φx,y,T=0,x∈0,l,y∈0,l,−aφyx,y,t|y=0=0,−aφyx,y,t|y=l=2ucx,l,t;h−ueδx,l,t,t∈0,T,x∈0,l,−aφxx,y,t|x=0=0,−aφxx,y,t|x=l=2ucl,y,t;h−ueδl,y,t,t∈0,T,y∈0,l.For allΔh, we have the following identity:(16)2∫0T∫0lucx,l,t;h−ueδx,l,tΔux,l,tdxdt=∫0T∫0lφx,l,tΔhx,tdxdt,2∫0T∫0lucl,y,t;h−ueδl,y,tΔul,y,tdydt=∫0T∫0lφl,y,tΔhy,tdydt.Proof.
Multiplying both sides of parabolic (14) by φx,y,t and integrating on Ω, based on the initial and boundary conditions, we have the following (17)∭ΩΔut−aΔuxx+Δuyyφx,y,tdxdydt=−∫0T∫0l∫0lφtx,y,tΔudxdydt−∫0T∫0l∫0laφxxx,y,tΔudxdydt−∫0T∫0l∫0laφyyx,y,tΔudxdydt−∫0T∫0laφx,y,tΔux|x=0x=ldydt+∫0T∫0laφxx,y,tΔu|x=0x=ldydt−∫0T∫0laφx,y,tΔuy|y=0y=ldydt+∫0T∫0laφyx,y,tΔu|y=0y=ldydt.
According to equation (15), we have(18)∭ΩΔut−aΔuxx+Δuyyφx,y,tdxdydt=−∫0T∫0laφx,y,tΔux|x=0x=ldydt+∫0T∫0laφxx,y,tΔu|x=0x=ldydt−∫0T∫0laφx,y,tΔuy|y=0y=ldydt+∫0T∫0laφyx,y,tΔu|y=0y=ldydt=0.
Using the conditions of equations (14) and (15), we have(19)∫0T∫0l2ucx,l,t;h−ueδx,l,tdxdt+∫0T∫0l2ucl,y,t;h−ueδl,y,tdydt=∫0T∫0lφl,y,tΔhy,tdydt+∫0T∫0lφx,l,tΔhx,tdxdt.
So, Lemma1 can be proved.Lemma 2.
Leth∈HΩ; then, for the solution Δu:=Δux,y,t;Δh∈V˙1,0Ω of the parabolic problem (14), there exists constants ε10<ε1<2/l and ε10<ε2<2/l, which satisfies c¯1=l/ε12−ε1l and c¯2=l/ε22−ε2l, and the following equation can be obtained for a given Δh⋅,t:(20)Δux,l,t2≤c¯1Δhx,t2,Δul,y,t2≤c¯2Δhy,t2,where Δux,l,t2=∫0T∫0lΔux,l,t2dxdt, Δul,y,t2=∫0T∫0lΔul,y,t2dydt, Δhx,t2=∫0T∫0lΔhx,t2dxdt, and Δhy,t2=∫0T∫0lΔhy,t2dydt.Proof.
If we multiply both sides of partial differential equation (14) by Δux,y,t and also simultaneously in time and space integration, we can obtain the following equation:(21)∭ΩΔut−aΔuxx−aΔuyyΔudxdydt=0.
It is easy to know that the following equation holds:(22)ΔutΔu=12Δu2t,ΔuxxΔu=ΔuxΔux−Δux2.
Hence, we have the following equations:(23)∭ΩΔutΔudxdydt=12∭Δu2tdxdydt=12∫0l∫0lΔux,y,T2dxdy,a∭ΩΔuxxΔudxdydt=a∭ΩΔuxΔuxdxdydt−a∭ΩΔux2dxdydt=a∫0l∫0TΔuxΔu|x=0x=ldydt−a∭ΩΔux2dxdydt=a∫0T∫0lΔhy,tΔul,y,tdydt−a∭ΩΔux2dxdydt.
Similarly, we also have(24)a∭ΩΔuyyΔudxdydt=a∭ΩΔuyΔuydxdydt−a∭ΩΔuy2dxdydt=a∫0l∫0TΔuyΔu|y=0y=ldxdt−a∭ΩΔuy2dxdydt=a∫0T∫0lΔhx,tΔux,l,tdydt−a∭ΩΔuy2dxdydt.
According to the boundary condition of equation (14), we have(25)∭ΩΔut−aΔuxx−aΔuyyΔudxdydt=∭ΩΔutΔudxdydt−a∭ΩΔuxxΔudxdydt−a∭ΩΔuyyΔudxdydt=12∫0l∫0lΔux,y,T2dxdy+a∭ΩΔux2+Δuy2dxdydt−a∫0T∫0lΔhx,tΔux,l,tdxdt−a∫0T∫0lΔhy,tΔul,y,tdxdt=0.
So, we have(26)12∫0l∫0lΔux,y,T2dxdy+a∭ΩΔux2+Δuy2dxdydt=a∫0T∫0lΔhx,tΔux,l,tdxdt+a∫0T∫0lΔhy,tΔul,y,tdxdt.
According to theε-inequality,(27)αβ≤εα22+β22ε,∀α,β∈R,∀ε>0.
The first item on the right side of equation (26) can be written as(28)∫0T∫0lΔhx,tΔux,l,tdxdt≤12ε1∫0T∫0lΔhx,t2dxdt+ε12∫0T∫0lΔux,l,t2dxdt.
Similarly, the second item on the right side of equation (26) can be shown in the following:(29)∫0T∫0lΔhy,tΔul,y,tdydt≤12ε2∫0T∫0lΔhy,t2dydt+ε22∫0T∫0lΔul,y,t2dxdt.
Furthermore, we estimate∫∫∫ΩΔux2+Δuy2dxdydt by applying Cauchy inequality:(30)Δul,y,t−Δu0,y,t2=∫0lΔuζζ,y,tdζ2≤l∫0lΔux2dx,x∈0,l,(31)Δux,l,t−Δux,0,t2=∫0lΔuζx,ζ,tdζ2≤l∫0lΔuy2dy,y∈0,l.
Now, we integrate both sides of inequality (30) on Ω, and we can obtain(32)∭ΩΔul,y,t2dxdydt≤l2∭ΩΔux2dxdydt.
Similarly, we have(33)∭ΩΔux,l,t2dxdydt≤l2∭ΩΔuy2dxdt.
And, we use these inequalities to estimate the left-hand side of inequality (26), and we can obtain the following inequality:(34)12∫0l∫0lΔux,y,T2dxdy+al∫0l∫0TΔul,y,t2dxdydt+al∫0l∫0TΔux,l,t2dxdydt≤a2ε1∫0T∫0lΔhx,t2dxdt+aε12∫0T∫0lΔux,l,t2dxdt+a2ε2∫0T∫0lΔhy,t2dydt+aε22∫0T∫0lΔul,y,t2dydt.
So, we have(35)al−ε22a∫0l∫0TΔul,y,t2dxdydt+al−ε12a∫0l∫0TΔux,l,t2dxdydt≤a2ε1∫0T∫0lΔhx,t2dxdt+a2ε2∫0T∫0lΔhy,t2dydt.
This case requiresa/l−ε1/2a>0 and a/l−ε2/2a>0. So, we obtain the following conditions:(36)0<ε1<2l,0<ε2<2l.
So, we have the following inequalities:(37)al−ε12a∫0T∫0lΔux,l,t2dxdt≤a2ε1∫0T∫0lΔhx,t2dxdt,al−ε22a∫0T∫0lΔul,y,t2dydt≤a2ε2∫0T∫0lΔhy,t2dydt.
So, we have(38)∫0T∫0lΔux,l,t2dxdt≤c¯1∫0T∫0lΔhx,t2dxdt,∫0T∫0lΔul,y,t2dydt≤c¯2∫0T∫0lΔhy,t2dydt,where c¯1=l/ε12−ε1l and c¯2=l/ε22−ε2l.
Lemma2 is been proved.


### 3.3. Lipschitz Continuity of the Gradient of Cost Function

In this section, Lipschitz continuity of the gradient of cost functionJh is shown, and we also give the Lipschitz constant according to the constant of Lemma 1 and Lemma 2.Theorem 1.
If the functionφx,y,t;h is the solution of the adjoint problem (15), the function Δφx,y,t=φx,y,t;h+Δh−φx,y,t;h is the solution of the following problem:(39)Δφt=−aΔφxx+Δφyy,x,y,t∈Ω,Δφx,t,T=0,x∈0,l,y∈0,l,−aΔφyx,y,t|y=0=0,−aΔφyx,y,t|y=l=2Δux,l,t;Δh,t∈0,T,x∈0,l,−aΔφxx,y,t|x=0=0,−aΔφxx,y,t|x=l=2Δul,y,t;Δh,t∈0,T,y∈0,l.Then, we have the following estimate:(40)J′h+Δh−J′h,Δh≤C¯1∫0T∫0lΔhx,t2dxdt+C¯2∫0T∫0lΔhy,t2dydt,C¯1>0,C¯2>0.Proof.
Considering Lemma1 and Lemma 2, we multiply both sides of equation (39) by Δux,y,t and also simultaneously in time and space integration, and we have(41)J1′h+Δh−J1′h,Δh=∫0T∫0lΔφl,y,tΔhy,tdydt=2∫0T∫0lΔul,y,t2dydt≤2c¯2∫0T∫0lΔhy,t2dydt.
Similarly,(42)J2′h+Δh−J2′h,Δh=∫0T∫0lΔφx,l,tΔhx,tdxdt=2∫0T∫0lΔux,l,t2dxdt≤2c¯1∫0T∫0lΔhx,t2dxdt.
So, we have(43)J′h+Δh−J′h,Δh≤C¯1∫0T∫0lΔhx,t2dxdt+C¯2∫0T∫0lΔhy,t2dydt,where C¯1=2c¯1 and C¯2=2c¯2.
The theorem is been proved.


## 3.1. The Quasi-Solution Approach Based on Weak Solution

We denote the set of admissible unknown heat fluxh⋅,t by HΩ, which satisfies the following equation:(8)h⋅,t∈HΩ,∀t∈0,T,x∈0,l,y∈0,l.The weak solution of the forward problem (7) will be defined to be the function u∈V˙1,0Ω, which satisfies the following integral identity:(9)∭Ω−uvt+avxux+avyuydxdydt=∭ΩQx,y,tvdxdydt,∀v∈HΩ,where vx,y,T=0 for x∈0,l and y∈0,l and vl,y,t=0 and vx,l,t=0 for t∈0,T. Obviously, under the above conditions with regards to the given data, the existence and uniqueness of this solution in u∈V˙1,0Ω can be found in [5, 21].Let us denote the solution of problem (7) by u∗x,y,t;h, corresponding to a given h⋅,t∈HΩ,∀t∈0,T,x∈0,l,y∈0,l. If this function satisfies the measured data ueδx,l,t or ueδl,y,t, then it should satisfy the nonlinear functional equation:(10)u∗x,y,t;h|x=l=ueδl,y,t,t∈0,T,y∈0,l,or(11)u∗x,y,t;h|y=l=ueδx,l,t,t∈0,T,x∈0,l.In practice, the measured dataueδx,0,t is usually given by the data with error level δ. The exact fulfillment of condition (10) may be not possible. According to this reason, a quasi-solution of the inverse problem (7) is defined as a solution of the following minimization problem:(12)Jh∗=infJh,where Jh is the cost functional, which is defined by equation (7).


## 3.2. Lipchitz Continuity of the Gradient of the Cost Function

We assume thath,h+Δh∈H, and the variation of the cost function ΔJ can be given in the following:(13)ΔJh=ΔJ1h+ΔJ2h=2∫0T∫0lucx,l,t;h−ueδΔux,l,t;Δhdxdt+∫0T∫0lΔux,l,t;Δh2dxdt+2∫0T∫0lucl,y,t;h−ueδΔul,y,t;Δhdydt+∫0T∫0lΔul,y,t;Δh2dydt,where Δux,y,t;h=ux,y,t;h+Δh−ux,y,t;h∈HΩ. Obviously, Δu=Δux,y,t;Δh is the solution of the following parabolic problem:(14)Δut=aΔuxx+Δuyy,x,y,t∈Ω,Δux,y,0=0,x∈0,l,y∈0,l,aΔuyx,y,t|y=l=−Δhx,t,aΔuyx,y,t|y=0=0,t∈0,T,x∈0,l,aΔuxx,y,t|x=l=−Δhy,t,aΔuxx,y,t|x=0=0,t∈0,T,y∈0,l.Lemma 1.
Ifu=ux,y,t;h∈HΩ is the corresponding solution of the direct problem (7) and φx,y,t;h∈HΩ is the solution of the adjoint problem,(15)φt=−aφxx+φyy,x,y,t∈Ω,φx,y,T=0,x∈0,l,y∈0,l,−aφyx,y,t|y=0=0,−aφyx,y,t|y=l=2ucx,l,t;h−ueδx,l,t,t∈0,T,x∈0,l,−aφxx,y,t|x=0=0,−aφxx,y,t|x=l=2ucl,y,t;h−ueδl,y,t,t∈0,T,y∈0,l.For allΔh, we have the following identity:(16)2∫0T∫0lucx,l,t;h−ueδx,l,tΔux,l,tdxdt=∫0T∫0lφx,l,tΔhx,tdxdt,2∫0T∫0lucl,y,t;h−ueδl,y,tΔul,y,tdydt=∫0T∫0lφl,y,tΔhy,tdydt.Proof.
Multiplying both sides of parabolic (14) by φx,y,t and integrating on Ω, based on the initial and boundary conditions, we have the following (17)∭ΩΔut−aΔuxx+Δuyyφx,y,tdxdydt=−∫0T∫0l∫0lφtx,y,tΔudxdydt−∫0T∫0l∫0laφxxx,y,tΔudxdydt−∫0T∫0l∫0laφyyx,y,tΔudxdydt−∫0T∫0laφx,y,tΔux|x=0x=ldydt+∫0T∫0laφxx,y,tΔu|x=0x=ldydt−∫0T∫0laφx,y,tΔuy|y=0y=ldydt+∫0T∫0laφyx,y,tΔu|y=0y=ldydt.
According to equation (15), we have(18)∭ΩΔut−aΔuxx+Δuyyφx,y,tdxdydt=−∫0T∫0laφx,y,tΔux|x=0x=ldydt+∫0T∫0laφxx,y,tΔu|x=0x=ldydt−∫0T∫0laφx,y,tΔuy|y=0y=ldydt+∫0T∫0laφyx,y,tΔu|y=0y=ldydt=0.
Using the conditions of equations (14) and (15), we have(19)∫0T∫0l2ucx,l,t;h−ueδx,l,tdxdt+∫0T∫0l2ucl,y,t;h−ueδl,y,tdydt=∫0T∫0lφl,y,tΔhy,tdydt+∫0T∫0lφx,l,tΔhx,tdxdt.
So, Lemma1 can be proved.Lemma 2.
Leth∈HΩ; then, for the solution Δu:=Δux,y,t;Δh∈V˙1,0Ω of the parabolic problem (14), there exists constants ε10<ε1<2/l and ε10<ε2<2/l, which satisfies c¯1=l/ε12−ε1l and c¯2=l/ε22−ε2l, and the following equation can be obtained for a given Δh⋅,t:(20)Δux,l,t2≤c¯1Δhx,t2,Δul,y,t2≤c¯2Δhy,t2,where Δux,l,t2=∫0T∫0lΔux,l,t2dxdt, Δul,y,t2=∫0T∫0lΔul,y,t2dydt, Δhx,t2=∫0T∫0lΔhx,t2dxdt, and Δhy,t2=∫0T∫0lΔhy,t2dydt.Proof.
If we multiply both sides of partial differential equation (14) by Δux,y,t and also simultaneously in time and space integration, we can obtain the following equation:(21)∭ΩΔut−aΔuxx−aΔuyyΔudxdydt=0.
It is easy to know that the following equation holds:(22)ΔutΔu=12Δu2t,ΔuxxΔu=ΔuxΔux−Δux2.
Hence, we have the following equations:(23)∭ΩΔutΔudxdydt=12∭Δu2tdxdydt=12∫0l∫0lΔux,y,T2dxdy,a∭ΩΔuxxΔudxdydt=a∭ΩΔuxΔuxdxdydt−a∭ΩΔux2dxdydt=a∫0l∫0TΔuxΔu|x=0x=ldydt−a∭ΩΔux2dxdydt=a∫0T∫0lΔhy,tΔul,y,tdydt−a∭ΩΔux2dxdydt.
Similarly, we also have(24)a∭ΩΔuyyΔudxdydt=a∭ΩΔuyΔuydxdydt−a∭ΩΔuy2dxdydt=a∫0l∫0TΔuyΔu|y=0y=ldxdt−a∭ΩΔuy2dxdydt=a∫0T∫0lΔhx,tΔux,l,tdydt−a∭ΩΔuy2dxdydt.
According to the boundary condition of equation (14), we have(25)∭ΩΔut−aΔuxx−aΔuyyΔudxdydt=∭ΩΔutΔudxdydt−a∭ΩΔuxxΔudxdydt−a∭ΩΔuyyΔudxdydt=12∫0l∫0lΔux,y,T2dxdy+a∭ΩΔux2+Δuy2dxdydt−a∫0T∫0lΔhx,tΔux,l,tdxdt−a∫0T∫0lΔhy,tΔul,y,tdxdt=0.
So, we have(26)12∫0l∫0lΔux,y,T2dxdy+a∭ΩΔux2+Δuy2dxdydt=a∫0T∫0lΔhx,tΔux,l,tdxdt+a∫0T∫0lΔhy,tΔul,y,tdxdt.
According to theε-inequality,(27)αβ≤εα22+β22ε,∀α,β∈R,∀ε>0.
The first item on the right side of equation (26) can be written as(28)∫0T∫0lΔhx,tΔux,l,tdxdt≤12ε1∫0T∫0lΔhx,t2dxdt+ε12∫0T∫0lΔux,l,t2dxdt.
Similarly, the second item on the right side of equation (26) can be shown in the following:(29)∫0T∫0lΔhy,tΔul,y,tdydt≤12ε2∫0T∫0lΔhy,t2dydt+ε22∫0T∫0lΔul,y,t2dxdt.
Furthermore, we estimate∫∫∫ΩΔux2+Δuy2dxdydt by applying Cauchy inequality:(30)Δul,y,t−Δu0,y,t2=∫0lΔuζζ,y,tdζ2≤l∫0lΔux2dx,x∈0,l,(31)Δux,l,t−Δux,0,t2=∫0lΔuζx,ζ,tdζ2≤l∫0lΔuy2dy,y∈0,l.
Now, we integrate both sides of inequality (30) on Ω, and we can obtain(32)∭ΩΔul,y,t2dxdydt≤l2∭ΩΔux2dxdydt.
Similarly, we have(33)∭ΩΔux,l,t2dxdydt≤l2∭ΩΔuy2dxdt.
And, we use these inequalities to estimate the left-hand side of inequality (26), and we can obtain the following inequality:(34)12∫0l∫0lΔux,y,T2dxdy+al∫0l∫0TΔul,y,t2dxdydt+al∫0l∫0TΔux,l,t2dxdydt≤a2ε1∫0T∫0lΔhx,t2dxdt+aε12∫0T∫0lΔux,l,t2dxdt+a2ε2∫0T∫0lΔhy,t2dydt+aε22∫0T∫0lΔul,y,t2dydt.
So, we have(35)al−ε22a∫0l∫0TΔul,y,t2dxdydt+al−ε12a∫0l∫0TΔux,l,t2dxdydt≤a2ε1∫0T∫0lΔhx,t2dxdt+a2ε2∫0T∫0lΔhy,t2dydt.
This case requiresa/l−ε1/2a>0 and a/l−ε2/2a>0. So, we obtain the following conditions:(36)0<ε1<2l,0<ε2<2l.
So, we have the following inequalities:(37)al−ε12a∫0T∫0lΔux,l,t2dxdt≤a2ε1∫0T∫0lΔhx,t2dxdt,al−ε22a∫0T∫0lΔul,y,t2dydt≤a2ε2∫0T∫0lΔhy,t2dydt.
So, we have(38)∫0T∫0lΔux,l,t2dxdt≤c¯1∫0T∫0lΔhx,t2dxdt,∫0T∫0lΔul,y,t2dydt≤c¯2∫0T∫0lΔhy,t2dydt,where c¯1=l/ε12−ε1l and c¯2=l/ε22−ε2l.
Lemma2 is been proved.


## 3.3. Lipschitz Continuity of the Gradient of Cost Function

In this section, Lipschitz continuity of the gradient of cost functionJh is shown, and we also give the Lipschitz constant according to the constant of Lemma 1 and Lemma 2.Theorem 1.
If the functionφx,y,t;h is the solution of the adjoint problem (15), the function Δφx,y,t=φx,y,t;h+Δh−φx,y,t;h is the solution of the following problem:(39)Δφt=−aΔφxx+Δφyy,x,y,t∈Ω,Δφx,t,T=0,x∈0,l,y∈0,l,−aΔφyx,y,t|y=0=0,−aΔφyx,y,t|y=l=2Δux,l,t;Δh,t∈0,T,x∈0,l,−aΔφxx,y,t|x=0=0,−aΔφxx,y,t|x=l=2Δul,y,t;Δh,t∈0,T,y∈0,l.Then, we have the following estimate:(40)J′h+Δh−J′h,Δh≤C¯1∫0T∫0lΔhx,t2dxdt+C¯2∫0T∫0lΔhy,t2dydt,C¯1>0,C¯2>0.Proof.
Considering Lemma1 and Lemma 2, we multiply both sides of equation (39) by Δux,y,t and also simultaneously in time and space integration, and we have(41)J1′h+Δh−J1′h,Δh=∫0T∫0lΔφl,y,tΔhy,tdydt=2∫0T∫0lΔul,y,t2dydt≤2c¯2∫0T∫0lΔhy,t2dydt.
Similarly,(42)J2′h+Δh−J2′h,Δh=∫0T∫0lΔφx,l,tΔhx,tdxdt=2∫0T∫0lΔux,l,t2dxdt≤2c¯1∫0T∫0lΔhx,t2dxdt.
So, we have(43)J′h+Δh−J′h,Δh≤C¯1∫0T∫0lΔhx,t2dxdt+C¯2∫0T∫0lΔhy,t2dydt,where C¯1=2c¯1 and C¯2=2c¯2.
The theorem is been proved.


## 4. Sufficient Descent Levenberg–Marquard Algorithm for Solving the Inverse Heat Transfer Problem

### 4.1. Sufficient Descent Levenberg–Marquard with Adaptive Parameter (SD-LMAP) Algorithm

Based on the above sections, some gradient methods, such as conjugate gradient method [5], can be used to solve the optimization problem. In practice, the inverse heat transfer problem is ill-posed [22, 23], which means that small errors in the measured values of surface temperature can lead to large disturbance in the inverse results. In present, many research works [16, 17, 24, 25] apply the Levenberg–Marquard (L-M) algorithm to solve the inverse heat transfer problem and analyze convergence of a regularizing L-M scheme for nonlinear ill-posed problems. Therefore, this paper considers the L-M algorithm to solve the inverse heat transfer problem.Because the width and thickness of billet have the same heat flux, the surface temperature on one of it is required to be measured. In order to identify the heat fluxh⋅,t, we assume that the surface temperature on the Γ1 can be measured. We have the following cost function:(44)J1h=12∫0l∫0Tucl,y,t;h−ueδl,y,t2dydt.The iterative process of L-M algorithm is written as(45)hk+1=hk+αksk,k=0,1,2,…,where αk>0 is an iteration parameter; in this paper, the exact linear search is used:(46)Jhk−αksk=minα>0Jhk−αksk,where h0 is a given initial iteration and sk is the search direction determined by L-M algorithm, which is given in the following:(47)sk=∇ucx,y,t;hkT∇ucx,y,t;hk+μ¯I−1dk,where dk=∇J1hk is the gradient, which decides the iterative direction of L-M algorithm, μ¯ is the L-M parameter, the adaptive parameter method [19] μk=Jhk2δ is used, δ is the error level, and I is the identity matrix.Sufficient descent property plays a key role in the global convergence of the iterative method [26]. The sufficient descent condition is defined as(48)gkTdk≤−cgk2,where dk is the iterative direction at step k, gk is the gradient of cost function at step k, and c is a positive constant.In order to accelerate the convergence rate and reduce the running time, this paper presents a sufficient descent Levenberg–Marquard with adaptive parameter (SD-LMAP) algorithm. The iterative formula is given in the following:(49)hk+1=hk+αk∇ucx,y,t;hkT∇ucx,y,t;hk+μ¯I−1dk,k=0,1,2,…,where(50)dk=−∇J1hk,k=0,−ηk∇J1hk+βk∇J1hk−1−θkyk−1,k>0,with ηk=γ+yk−12/∇J1hk2, βk=∇J1hkTyk−1/∇J1hk−12, θk=∇J1hk2/∇J1hk−12, and yk=∇J1hk−∇J1hk−1, γ≥1.Theorem 2.
If sequenceshk and s¯k are generated by equations (48) and (49), for all k≥0, the following equation holds:(51)∇J1hkTdk≤−∇J1hk2.Proof.
Evidently, it is true for inequality (50) when k=0. If k>0, multiplying equation (49) by ∇J1hkT, we can obtain the following equation:(52)∇J1hkTdk=−ηk∇J1hk2+∇J1hkTβk∇J1hk−1−θkyk−1=−γ+yk−12∇J1hk2∇J1hk2+∇J1hkTyk−1∇J1hk−12∇J1hkT∇J1hk−1−∇J1hk2∇J1hk−12∇J1hkTyk−1=−γ+yk−12∇J1hk2∇J1hk2+∇J1hkTyk−1∇J1hk−12∇J1hkT∇J1hk−1−∇J1hk=−γ+yk−12∇J1hk2∇J1hk2−∇J1hkTyk−12∇J1hk−12≤−γ+yk−12∇J1hk2∇J1hk2≤−∇J1hk2.
Theorem2 is been proved.Remark 2.
The SD-LMAP algorithm is proposed in the framework of LM algorithm because [25, 27] prove the regularizing of LM algorithm. Therefore, the SD-LMAP algorithm also has regularizing. Furthermore, most of the references [16, 28] used the LM and the modified LM algorithms to solve the inverse problems in many fields and overcame the ill-posed problem. So, the SD-LMAP algorithm can be used to solve the ill-posed problem.Remark 3.
For the SD-LMAP algorithm, the item∇ucx,y,t;hkT∇ucx,y,t;hk+μ¯I has little effect on the iterative direction, but it should be nonsingular. Obviously, this item is not nonsingular when L-M parameter μ¯ is positive. The direction dk of L-M algorithm adopts the gradient scheme, but the sufficient descent direction is used in this paper, and the sufficient descent property of SD-LMAP algorithm is analyzed. The SD-LMAP algorithm can guarantee descent at each step and global convergence.


### 4.2. Model Discretization and Solving Process of the Inverse Heat Transfer Problem

#### 4.2.1. Numerical Solution of the Direct Problem

An implementation of the iterative method (SD-LM) requires solving the direct problem effectively in each step. For the numerical solution of direct problem (1)–(6), the discrete model with its boundary conditions is shown in the following based on the finite difference method:(53)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn.Boundary condition onΓ1x=l is shown in the following:(54)ui,jn+1=2Aui−1,jn−ui,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn−2ρcΔxqyi,tn.Boundary condition onΓ2y=l is shown in the following:(55)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+2Bui,j+1n−ui,jn+Qxi,yi,tn−2ρcΔyqxi,tn.Boundary condition onΓ3x=0 is shown in the following:(56)ui,jn+1=2Aui−1,jn−ui,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn.Boundary condition onΓ4y=0 is shown in the following(57)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+2Bui,j+1n−ui,jn+Qxi,yi,tn,where we define Δx=l/M1,i=1,2,…,M1, Δy=l/M2,j=1,2,…,M2, M1 is the number of grid at x direction, M2 is the number of grid at y direction, M1 is equal to M2 for the billet, A=kΔt/ρcΔx, B=kΔt/ρcΔy, Δt=t/N is time step, n=1,2,…,N, N is the number of time grid. ui,jn is abbreviation of uxi,yj,tn, and uxi,yj,tn is the approximate value of ux,y,t at the mesh point xi,yi,tn.


#### 4.2.2. Solving Process of the Inverse Heat Transfer Problem

We assume that the number of measured point isM on Γ1 at space. The measured time interval is ΔTm and the number of measured point is P=T/ΔTm at space. So, the discrete model of cost function is shown in the following:(58)J1h=∑m=1M∑p=1Pucl,ym,tp;h−ueδl,ym,tp2,where ueδl,ym,tp;h is the measured value at point l,ym,tp and ucl,ym,tp;h is the calculated value corresponding to the measured point, which is computed by solving the direct problem (1)–(6). The solving process of the inverse heat transfer problem (7) based on the SD-LM algorithm is given in the following.Step 1:
setk=1, given an initial value h0∈Rn, the maximum iterative number kmax, and stopping criteria εspStep 2:
compute the temperatureucl,ym,tp;hk according to equations (53)–(57)Step 3:
calculate the cost functionJ1hk; if J1hk<εsp or k=kmax, stop; else k=k+1, go to Step 4Step 4:
calculate the L-M parameterμk=Jhk2δ and the iteration parameter αk>0 by equation (46)Step 5:
update the heat fluxhk+1 according to equation (48), and go to Step 2


## 4.1. Sufficient Descent Levenberg–Marquard with Adaptive Parameter (SD-LMAP) Algorithm

Based on the above sections, some gradient methods, such as conjugate gradient method [5], can be used to solve the optimization problem. In practice, the inverse heat transfer problem is ill-posed [22, 23], which means that small errors in the measured values of surface temperature can lead to large disturbance in the inverse results. In present, many research works [16, 17, 24, 25] apply the Levenberg–Marquard (L-M) algorithm to solve the inverse heat transfer problem and analyze convergence of a regularizing L-M scheme for nonlinear ill-posed problems. Therefore, this paper considers the L-M algorithm to solve the inverse heat transfer problem.Because the width and thickness of billet have the same heat flux, the surface temperature on one of it is required to be measured. In order to identify the heat fluxh⋅,t, we assume that the surface temperature on the Γ1 can be measured. We have the following cost function:(44)J1h=12∫0l∫0Tucl,y,t;h−ueδl,y,t2dydt.The iterative process of L-M algorithm is written as(45)hk+1=hk+αksk,k=0,1,2,…,where αk>0 is an iteration parameter; in this paper, the exact linear search is used:(46)Jhk−αksk=minα>0Jhk−αksk,where h0 is a given initial iteration and sk is the search direction determined by L-M algorithm, which is given in the following:(47)sk=∇ucx,y,t;hkT∇ucx,y,t;hk+μ¯I−1dk,where dk=∇J1hk is the gradient, which decides the iterative direction of L-M algorithm, μ¯ is the L-M parameter, the adaptive parameter method [19] μk=Jhk2δ is used, δ is the error level, and I is the identity matrix.Sufficient descent property plays a key role in the global convergence of the iterative method [26]. The sufficient descent condition is defined as(48)gkTdk≤−cgk2,where dk is the iterative direction at step k, gk is the gradient of cost function at step k, and c is a positive constant.In order to accelerate the convergence rate and reduce the running time, this paper presents a sufficient descent Levenberg–Marquard with adaptive parameter (SD-LMAP) algorithm. The iterative formula is given in the following:(49)hk+1=hk+αk∇ucx,y,t;hkT∇ucx,y,t;hk+μ¯I−1dk,k=0,1,2,…,where(50)dk=−∇J1hk,k=0,−ηk∇J1hk+βk∇J1hk−1−θkyk−1,k>0,with ηk=γ+yk−12/∇J1hk2, βk=∇J1hkTyk−1/∇J1hk−12, θk=∇J1hk2/∇J1hk−12, and yk=∇J1hk−∇J1hk−1, γ≥1.Theorem 2.
If sequenceshk and s¯k are generated by equations (48) and (49), for all k≥0, the following equation holds:(51)∇J1hkTdk≤−∇J1hk2.Proof.
Evidently, it is true for inequality (50) when k=0. If k>0, multiplying equation (49) by ∇J1hkT, we can obtain the following equation:(52)∇J1hkTdk=−ηk∇J1hk2+∇J1hkTβk∇J1hk−1−θkyk−1=−γ+yk−12∇J1hk2∇J1hk2+∇J1hkTyk−1∇J1hk−12∇J1hkT∇J1hk−1−∇J1hk2∇J1hk−12∇J1hkTyk−1=−γ+yk−12∇J1hk2∇J1hk2+∇J1hkTyk−1∇J1hk−12∇J1hkT∇J1hk−1−∇J1hk=−γ+yk−12∇J1hk2∇J1hk2−∇J1hkTyk−12∇J1hk−12≤−γ+yk−12∇J1hk2∇J1hk2≤−∇J1hk2.
Theorem2 is been proved.Remark 2.
The SD-LMAP algorithm is proposed in the framework of LM algorithm because [25, 27] prove the regularizing of LM algorithm. Therefore, the SD-LMAP algorithm also has regularizing. Furthermore, most of the references [16, 28] used the LM and the modified LM algorithms to solve the inverse problems in many fields and overcame the ill-posed problem. So, the SD-LMAP algorithm can be used to solve the ill-posed problem.Remark 3.
For the SD-LMAP algorithm, the item∇ucx,y,t;hkT∇ucx,y,t;hk+μ¯I has little effect on the iterative direction, but it should be nonsingular. Obviously, this item is not nonsingular when L-M parameter μ¯ is positive. The direction dk of L-M algorithm adopts the gradient scheme, but the sufficient descent direction is used in this paper, and the sufficient descent property of SD-LMAP algorithm is analyzed. The SD-LMAP algorithm can guarantee descent at each step and global convergence.


## 4.2. Model Discretization and Solving Process of the Inverse Heat Transfer Problem

### 4.2.1. Numerical Solution of the Direct Problem

An implementation of the iterative method (SD-LM) requires solving the direct problem effectively in each step. For the numerical solution of direct problem (1)–(6), the discrete model with its boundary conditions is shown in the following based on the finite difference method:(53)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn.Boundary condition onΓ1x=l is shown in the following:(54)ui,jn+1=2Aui−1,jn−ui,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn−2ρcΔxqyi,tn.Boundary condition onΓ2y=l is shown in the following:(55)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+2Bui,j+1n−ui,jn+Qxi,yi,tn−2ρcΔyqxi,tn.Boundary condition onΓ3x=0 is shown in the following:(56)ui,jn+1=2Aui−1,jn−ui,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn.Boundary condition onΓ4y=0 is shown in the following(57)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+2Bui,j+1n−ui,jn+Qxi,yi,tn,where we define Δx=l/M1,i=1,2,…,M1, Δy=l/M2,j=1,2,…,M2, M1 is the number of grid at x direction, M2 is the number of grid at y direction, M1 is equal to M2 for the billet, A=kΔt/ρcΔx, B=kΔt/ρcΔy, Δt=t/N is time step, n=1,2,…,N, N is the number of time grid. ui,jn is abbreviation of uxi,yj,tn, and uxi,yj,tn is the approximate value of ux,y,t at the mesh point xi,yi,tn.


### 4.2.2. Solving Process of the Inverse Heat Transfer Problem

We assume that the number of measured point isM on Γ1 at space. The measured time interval is ΔTm and the number of measured point is P=T/ΔTm at space. So, the discrete model of cost function is shown in the following:(58)J1h=∑m=1M∑p=1Pucl,ym,tp;h−ueδl,ym,tp2,where ueδl,ym,tp;h is the measured value at point l,ym,tp and ucl,ym,tp;h is the calculated value corresponding to the measured point, which is computed by solving the direct problem (1)–(6). The solving process of the inverse heat transfer problem (7) based on the SD-LM algorithm is given in the following.Step 1:
setk=1, given an initial value h0∈Rn, the maximum iterative number kmax, and stopping criteria εspStep 2:
compute the temperatureucl,ym,tp;hk according to equations (53)–(57)Step 3:
calculate the cost functionJ1hk; if J1hk<εsp or k=kmax, stop; else k=k+1, go to Step 4Step 4:
calculate the L-M parameterμk=Jhk2δ and the iteration parameter αk>0 by equation (46)Step 5:
update the heat fluxhk+1 according to equation (48), and go to Step 2


## 4.2.1. Numerical Solution of the Direct Problem

An implementation of the iterative method (SD-LM) requires solving the direct problem effectively in each step. For the numerical solution of direct problem (1)–(6), the discrete model with its boundary conditions is shown in the following based on the finite difference method:(53)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn.Boundary condition onΓ1x=l is shown in the following:(54)ui,jn+1=2Aui−1,jn−ui,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn−2ρcΔxqyi,tn.Boundary condition onΓ2y=l is shown in the following:(55)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+2Bui,j+1n−ui,jn+Qxi,yi,tn−2ρcΔyqxi,tn.Boundary condition onΓ3x=0 is shown in the following:(56)ui,jn+1=2Aui−1,jn−ui,jn+Bui,j+1n−2ui,jn+ui,j−1n+Qxi,yi,tn.Boundary condition onΓ4y=0 is shown in the following(57)ui,jn+1=Aui+1,jn−2ui,jn+ui−1,jn+2Bui,j+1n−ui,jn+Qxi,yi,tn,where we define Δx=l/M1,i=1,2,…,M1, Δy=l/M2,j=1,2,…,M2, M1 is the number of grid at x direction, M2 is the number of grid at y direction, M1 is equal to M2 for the billet, A=kΔt/ρcΔx, B=kΔt/ρcΔy, Δt=t/N is time step, n=1,2,…,N, N is the number of time grid. ui,jn is abbreviation of uxi,yj,tn, and uxi,yj,tn is the approximate value of ux,y,t at the mesh point xi,yi,tn.


## 4.2.2. Solving Process of the Inverse Heat Transfer Problem

We assume that the number of measured point isM on Γ1 at space. The measured time interval is ΔTm and the number of measured point is P=T/ΔTm at space. So, the discrete model of cost function is shown in the following:(58)J1h=∑m=1M∑p=1Pucl,ym,tp;h−ueδl,ym,tp2,where ueδl,ym,tp;h is the measured value at point l,ym,tp and ucl,ym,tp;h is the calculated value corresponding to the measured point, which is computed by solving the direct problem (1)–(6). The solving process of the inverse heat transfer problem (7) based on the SD-LM algorithm is given in the following.Step 1:
setk=1, given an initial value h0∈Rn, the maximum iterative number kmax, and stopping criteria εspStep 2:
compute the temperatureucl,ym,tp;hk according to equations (53)–(57)Step 3:
calculate the cost functionJ1hk; if J1hk<εsp or k=kmax, stop; else k=k+1, go to Step 4Step 4:
calculate the L-M parameterμk=Jhk2δ and the iteration parameter αk>0 by equation (46)Step 5:
update the heat fluxhk+1 according to equation (48), and go to Step 2


## 5. Simulation

In this section, the estimation of heat flux in two-dimensional nonhomogeneous parabolic equation is discussed in two cases. During the process of slab production, the heat transfer coefficient is described by the piecewise linear; thus, the piecewise linear of the heat transfer coefficient is selected in the first case. Furthermore, in order to test the performance of algorithm (SD-LM with adaptive parameter), in the second case, the heat transfer coefficient changing with time is selected. In the next, these two cases and the analyses of simulation results are shown in the following.Case 1.
We consider the process described by means of (1)–(6) for the following values of parameters: l=3m, k=31W/m⋅K, c=660J/kg⋅K, ρ=7800kg/m3, u0=1000C0, uw=30C0, and Qx,y,t=80sinπxsinπysint. Because of hy,t=γ¯ul,y,t−uw, if the heat transfer coefficient γ¯ is known, the heat flux h⋅,t can be obtained. The coefficient γ¯ is sought in the form given below:(59)γ¯=γ¯1,0≤x≤0.3,γ¯2,0.3<x≤0.7,γ¯3,0.7<x≤1.2,γ¯4,1.2<x≤1.8,γ¯5,1.8<x≤2.4,γ¯6,2.4<x≤3.0,where the exact values of coefficient γ¯i are known and they are equal to 1500W/m2⋅K, 1000W/m2⋅K, 800W/m2⋅K, 1800W/m2⋅K, 1200W/m2⋅K, and 950W/m2⋅K, respectively.
In order to calculate this problem, the finite difference method is used to solve direct heat transfer equation. In executed experiment, we assumed that the values of temperature in six pointsNm=6 are known. The accurate measured values uel,ym,tp of surface temperature are calculated by the solving the direct heat transfer equation. In order to investigate the stability of numerical solution, the values of measurement temperature were perturbed as(60)ueδl,ym,tp=uel,ym,tp+δe∗r,n∗uel,ym,tp,where δe is the error level.
In this simulation experiment, we measure the temperature valuesuel,l/2,…,tp at the point l,l/2,tptp=1,…,P,P=6 with the time interval ΔTm=20s and choose the maximum iterative number Nmax=100. The gradient algorithm (GA), conjugate gradient algorithm (CGA), improved conjugate gradient algorithm (ICGA) [29], LM algorithm [19], sufficient descent Levenberg–Marquard (SD-LM) algorithm, and SD-LMAP are used to estimate the heat transfer coefficient γ¯i,i=1,2,…,6. The convergence of the cost function with the iterative number is given in Figure 2. It is can be seen from this figure that, comparing with other methods, the SD-LMAP algorithm obviously accelerates the convergence rate. Furthermore, we choose the stopping criteria εsp=1.0, and the results of simulation experiment are shown in Table 1. It is can be seen from this table that the iterative number and running time obviously reduced.
In order to investigate the performance of algorithm for overcoming the error, this simulation experiment used the error levelsδe=0.0, δe=0.01, δe=0.03, and δe=0.05 to generate the measuring error. The standard deviation S¯ is given in the following:(61)S¯=1n∑i=1nhei−hrihri2,where hei is the estimated value of boundary condition, hri is the true value of boundary condition, and n is the number of sampling point. The inverse results are shown in Table 2. From this table, we can see that the standard deviation obtained by the SD-LMAP algorithm is less than other methods, which can eliminate the ill-posedness of inverse problem.Figure 2
The convergence of the cost function with the number of iterations.
(a)(b)Table 1
The comparison of running time and iterative number.
AlgorithmError levelδe=0.0δe=0.01δe=0.03δe=0.05Iterative numberRunning timeIterative numberRunning timeIterative numberRunning timeIterative numberRunning timeGA59101.8100273.3100213.4100212.9CGA4663.35273.35587.967151.9ICGA [29]2452.73754.24164.125886.4LM [19]3158.952119.05487.975181.1SD-LM1625.61839.51520.32846.2SD-LMAP65.8188.3087.761010.05Table 2
The comparison of standard deviation.
AlgorithmError levelδe = 0.0δe = 0.01δe = 0.03δe = 0.05GAS¯=0.0123S¯=0.90S¯=0.31S¯=0.36CGAS¯=0.0074S¯=0.15S¯=0.33S¯=0.39ICGA [20]S¯=0.0012S¯=0.11S¯=0.31S¯=0.58LM [19]S¯=0.0018S¯=0.085S¯=0.38S¯=0.41SD-LMS¯=2.2×10−5S¯=0.057S¯=0.26S¯=0.32SD-LMAPS¯=1.7561×10−15S¯=0.0541S¯=0.1063S¯=0.2551Case 2.
In this case, the main parameters and physical parameters are given in Case1, and the heat flux are changing with time t, which is given in the following:(62)ht=1100−1300∗sint1000.
Two hundred points are selected at midsurface as the measured temperature values at positionsl,l/2,…,tp,p=1,2,…,50. The SD-LMAP algorithm with other methods is used to estimate the boundary condition ht. In this simulation, the maximum iterative number kmax is 200 and the inverse results are shown in Figure 3. Furthermore, we choose the maximum iterative number kmax is 1000 and the stop criteria εsp is 2, and the performance of SD-LMAP with other methods is listed in Table 3. From Figure 3 and Table 3, it can be seen that the iterative number and running time of SD-LMAP algorithm are obviously reduced.
Furthermore, the measured data with different error levels is used to test the performance of SD-LMAP algorithm, which is generated by equation (60). The SD-LMAP algorithm with other methods are used to estimate the heat flux ht. The results of simulation experiments are shown in Table 4, and it can be seen that the SD-LMAP algorithm eliminates the ill-posedness of this inverse heat transfer problem effectively.Figure 3
The convergence of the cost function with the number of iterations.
(a)(b)Table 3
The comparison of the running time and iterative number.
AlgorithmError levelδe=0.0δe=0.01δe=0.03δe=0.05Iterative numberRunning timeIterative numberRunning timeIterative numberRunning timeIterative numberRunning timeGA2864.4442×1043545.5008×10410001.6247×10510001.6358×105CGA1802.8520×1042233.3721×1048651.3441×10510001.6549×105ICGA [29]1081.9320×1041332.1007×1045017.7851×1049451.5479×105LM [19]1442.2639×1041792.8560×1045719.1108×10410001.5956×105SD-LM580.9565×104701.1026×1042093.2816×1044026.201×104SD-LMAP150.2283×104170.2622×104350.5758×104460.7241×104Table 4
The comparison of standard deviation.
AlgorithmError levelδe=0.0δe=0.01δe=0.03δe=0.05GAS¯=0.1289S¯=0.1420S¯=0.41S¯=0.45CGAS¯=0.0766S¯=0.0841S¯=0.343S¯=0.46ICGA [29]S¯=0.0322S¯=0.0415S¯=0.451S¯=0.51LM [19]S¯=0.065S¯=0.0761S¯=0.468S¯=0.47SD-LMS¯=0.01664S¯=0.0213S¯=0.036S¯=0.087SD-LMAPS¯=0.0071S¯=0.0086S¯=0.0107S¯=0.021


## 6. Conclusion

The two-dimensional nonhomogeneous inverse heat transfer problem is considered. This problem is transformed into the PDE optimization problem. The Lipchitz continuous of the gradient of cost function in the PDE optimization problem with the two-dimensional nonhomogeneous heat transfer equation is proved. Furthermore, a sufficient descent Levenberg–Marquard with adaptive parameter (SD-LMAP) algorithm is presented to solve this problem. At last, comparing with other methods, the results of the simulation experiment show that the SD-LMAP algorithm obviously reduces the iterative numbers and running time.



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