# Research on the Recognition Algorithm concerning Geometric Boundary regarding Heat Conduction Based on BEM and CGM

Authors: Shoubin Wang; Huangchao Jia; Xiaogang Sun; Li Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An inverse algorithm on boundary element method and conjugate gradient method is proposed to solve the problem of thermal conduction inverse of geometric shape. The direct problem is solved with the boundary element method, while the solution to the inverse problem is obtained through optimizing the objective function in the conjugate gradient method. Taking into account the identification of different material specimens when the unknown boundary is sinusoidal, step function, or circular shape, the influence of initial value, temperature error, thermal conductivity, and thermal intensity on the precision of inversion solution is discussed. The experimental results show that the method can recognize various irregular boundaries and is insensitive to initial values, measurement errors, and heat intensity. The thermal conductivity has a certain effect on this method. The inversion accuracy is higher on the condition that the thermal conductivity is smaller.


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## Body

## 1. Introduction

The direct problem [1] of heat conduction is to determine the temperature field in domains, when the boundary condition, thermal performance, heat source, and geometric boundary shape regarding heat conduction are known. Conversely, the inverse problem [2] of heat conduction is to determine the boundary condition [3, 4], heat source [5], thermal performance [6], etc. through measuring the temperature in the domain or on the surface. When the geometric boundary shape of the heat conduction problem is changing or unknown, it needs to use the boundary recognition method to predict the unknown shape, which is a geometric inverse problem of heat conduction.As a branch of heat transfer inverse problem, thermal geometry inverse problem has a wide range of application prospects in many fields, such as industrial equipment detection, nondestructive testing [7], optimization of geometric shape [8], and biological focus [9]. When the inversion regarding the geometric boundary of the heat conduction system is conducted, the two-dimension steady-state space should be dispersed firstly; the discrete methods mainly include finite difference method [10] (FDM), finite element method [11] (FEM), and boundary element method [12] (BEM), and they can be used to solve the direct problem of heat transfer theory; and the inversion of the geometric boundary can be realized through various optimization technologies based on solving the direct problem. The inversion method based on optimized technology can be divided into the optimized algorithm based on gradient and the optimized algorithm based on nongradient; the optimized algorithm based on gradient mainly includes conjugate gradient method (CGM), Levenberg-Marquardt [13] (L-MM), and steepest descent method (SDM), while the optimized algorithm based on nongradient mainly includes genetic algorithm [14] (GA), neural network algorithm [15] (NNM), and particle swarm algorithm [16] (PSO).Fan Chunli et al. [17] used FEM to invert internal defects and discussed the influence of initial values and measurement errors on the inversion results. Chen Yi [18] et al. adopted the finite element method to monitor the erosion of the blast furnace bottom. These methods are all calculation methods based on grids. Because the shape of the solution domain varies, it needs to repeatedly divide the complex grids or units regarding the solution domain. Thus the solution process becomes extremely complex, and the calculation quantity is very large. This problem can be well avoided through BEM, the control equation of the described problem is transformed to the boundary integral equation only related to the boundary. The dimension of the problem is reduced by 1, and it does not need to scatter the interior domain. It only needs to scatter the boundary, and the shape of the unknown boundary can be easily modified. The complex reset calculation of grids is successfully avoided, and it also overcomes the difficulty caused by the distortion of grids or units in the geometric inversion process. From the aspect of geometric inversion problem, the boundary element method has obvious advantages compared with other value methods.There are many optimization algorithms in solving the inverse problem of heat conduction geometry. Fan Chunli et al. [19, 20] indicated that L-M method was very effective when there were few recognition parameters. Huang et al. [21] compared the L-M method with the CGM used for the geometric inversion problem of heat conduction, and the research indicated that CGM was more effective than the L-M method. This advantage was more obvious when there were many recognition parameters. It was mainly embodied in the following: (1) the calculation time was very short; (2) no exact initial value was required; (3) less measurement points were required. The optimized algorithm of CGM was adopted in the paper based on above literature study.In actual engineering application, the shape of the unknown boundary is mostly irregular. In order to improve the application range of boundary recognition, it is very important to study the boundary recognition of irregular shape. The BEM and CGM were combined in the paper, and they were used to solve the recognition problem regarding irregular geometric boundary shapes. The recognition of test pieces concerning different materials in the shapes such as sine wave, step function, and round on the unknown boundary was considered, and the influence of factors such as initial value, temperature measurement error, thermal conductivity, and heating intensity on the precision of inverse solution was discussed.


## 2. Direct Problem of Heat Conduction Based on BEM

### 2.1. Overview and Features of BEM

Boundary Element Method (BEM) is a calculation method developed on the basis of classical integral equation, which absorbs the discretization technology of finite element method. The basic principle of BEM is to solve the calculus equation in the method regarding the integral equation. After the first international conference of BEM in 1978, it was greatly applied and developed, and it was widely applied to many fields [22–24] such as elastic mechanics, fluid mechanics, rock mechanics, and heat transfer theory.The foundation of BEM is to establish the integral equation, and the solution process can be divided into the following two steps.The first step is to demarcate the problem and apply Green formula to transform the differential equation in the solution domain into the integral equation on the boundary by fundamental solution.In this way, the dimensionality of the equation solution problem is reduced by one dimension; for example, the three-dimensional space problem is reduced to a two-dimensional plane problem, and the two-dimensional plane problem is reduced to a one-dimensional problem. So the input data quantity and the unknown quantity of the algebraic equation set are greatly reduced, and this is significant feature of BEM compared with the FEM.The second step is the discretization of the boundary, which can use the discretization technique of finite element method. Moreover, since the discretization is only conducted on the boundary and error only exists on the boundary, while the unknown quantity in the domain can be calculated through the analytical equation, so the calculation precision can be higher. In addition, because of the singularity regarding the basic solution, the BEM has high precision in solving the singular problem. In conclusion, the BEM has outstanding advantages, such as reducing dimensionality, less input data, high calculation precision, especially suitable for large area, and singular problems.The solution to the direct problem regarding the heat conduction is the basis and premise to realize the inverse problem of heat transfer theory. When the geometric inverse problem of heat transfer theory is solved, the solution to the direct problem of the heat conduction is very complex because the geometric boundary shape is uncertain. At present in the geometric inverse problem of heat transfer theory, the boundary element method (BEM) and the finite element method (FEM) are two main methods to solve the direct problems. BEM is a developed calculation method based on the classic integral equation, and it absorbs the discrete technology of the value methods such as FEM; the basic thought is to solve the calculus equation with the method of the integral equation. Compared with FEM, the solution to the direct problem regarding the heat conduction with BEM has the following features [25]:( 1 ) BEM only needs to discretize the system on the boundary, which avoids the disadvantage that FEM needs to remesh the whole solution area every iteration and obviously reduces the computational dimension and workload.( 2 ) Because the basic solution adopted by BEM is applicable to infinite or semi-infinite domain, it is not necessary to determine the outer boundary when solving the problem in infinite or semi-infinite domain; thus the problem is greatly simplified.( 3 ) The solution error of the BEM is only generated on the boundary. The value of function and coefficient in the domain can be obtained by analytic formula; thus the error source is few and calculation precision is high.( 4 ) BEM fundamental solution itself is derivable and can be used to solve singularity problems.In the process of solving inverse heat transfer geometric problems, the objective function needs to be optimized continuously. Because of the change of the shape of the solution area, the finite difference method and the finite element method need to redivide the solution area into complicated meshes or elements in each iteration, which increases the difficulty and computational complexity of solving the direct problem. It only needs to conduct discretization regarding the system on the boundary for BEM; thus it avoids the shortcoming of repeatedly dividing grids in the overall solution domain in all iterations for FEM; it also obviously reduces calculation dimensionality, solution difficulty, and calculation quantity of the direct problems for BEM.


### 2.2. Boundary Integral Equation

As shown in Figure1, the boundary integral equation is deducted in the weighted residual method; because many projects can be described in the forms of Laplace’s equation, the definite solution problem of the Laplace’s equation is considered, namely. Boundary Γ1 and Γ2 are shown in Figure 1.Figure 1
BoundaryΓ1 and Γ2.In the domainΩ,(1)∇2u=0On the boundaryΓ1,(2)u=u¯On the boundaryΓ2,(3)q=∂u∂n=q¯Its expression of the weighted residual is(4)∫Ω∇2uWdΩ=∫Γ2q-q¯WdΓ-∫Γ2u-u¯∂W∂ndΓTake the weighted functionW as the basic solution u∗, and then correspondingly(5)∫Ω∇2uu∗dΩ=∫Γ2q-q¯u∗dΓ-∫Γ1u-u¯q∗dΓIntegration by parts is conducted regarding the Laplace operator, and Green Equation is applied, then(6)∫Ω∂P∂x+∂Ω∂ydΩ=∫ΓPdy+QdxSo the left item of (5) is(7)∫Ω∇2uu∗dΩ=∫Ω∇u∗∇udΩ-∫Ω∇u∇u∗dΩ=∫Γ∇uu∗dΓ-∫Ω∇u∇u∗dΩ=∫Γ∂u∂nu∗dΓ-∫Ω∇u∇u∗dΩIntegration by part and Green Equation are applied to the second item on the right of (7),(8)∫Ω∇u∇u∗dΩ=∫Ω∇u∇u∗dΩ-∫Ωu∇2u∗dΩ=∫Γu·∂u∗∂ndΓ-∫Ωu·∇2u∗dΩSo, there is(9)∫Ω∇2uu∗dΩ=∫Γ∂u∂nu∗dΓ-∫Γu·∂u∗∂ndΓ+∫Ωu·∇2u∗dΩSubstitute it into (5); because Γ=Γ1+Γ2+Γ3 and q=∂u/∂n, q∗=∂u∗/∂n, so(10)∫Ω∇2u∗udΩ=∫Γ1u¯q∗dΓ+∫Γ2uq∗dΓ-∫Γ1qu∗dΓ-∫Γ2q¯u∗dΓThe first, second, third, and fourth item on the right can be, respectively, combined together, namely.(11)∫Ω∇2u∗udΩ=∫Γuq∗dΓ-∫Γqu∗dΓWeighted functionu∗ is the basic solution of the Laplace’s equation, and it satisfies(12)∇2u∗+δr-r1=0In the equation, the point “i” indicates the focus point of the unit point source, so(13)∇2u∗=-δr-r1Substitute it into (10) or (11), and there is(14)ui+∫Γ1qu∗dΓ+∫Γ2q¯u∗dΓ=∫Γ1u¯q∗dΓ+∫Γ2uq∗dΓ(15)ui+∫Γuq∗dΓ=∫Γqu∗dΓEquations (14) and (15) are effective regarding all points in the domain Ω, and they are called the integral function of internal node. It indicates that the function value ui regarding all points in domain Ω can be indicated with the integral of u and its normal derivative q on the boundary; if all values of u and q on the boundary are obtained, u and q regarding all internal points can be calculated through (15).It needs to transfer the pointi to the boundary so as to obtain the integral equation regarding all points on the boundary, but the singularity of the integral ∂u∗/∂n shall be considered; the boundary near the point i is substituted by the hemisphere in radius ε to avoid that point i becomes singularity point; then the hemisphere ε tends to 0, so point i becomes the boundary point. Boundary point surrounded by the hemisphere is shown in Figure 2.Figure 2
Boundary point surrounded by the hemisphere.Without loss of generality, suppose the boundary belongs toΓ2 type, the point i is an internal point, and then (15) can be(16)ui+∫Γ2-εuq∗dΓ+∫Γεuq∗dΓ=∫Γ2-εqu∗dΓ+∫Γεqu∗dΓ(17)∫Γεuq∗dΓ=∫Γεu∂1/4πr∂ndΓ=-∫Γεu14πε2dΓSuppose the boundary is smooth, it can be obtained through the mean value theorem of integrals that(18)∫Γεuq∗dΓ=-∫Γεu14πε2dΓ=uξ-14πε22πε2whereξ is a point in Γε.Supposeε→0 and u(ξ)→ui, then (19)limε→0⁡∫Γεuq∗dΓ=-12uiEquation (19) is also applicable to smooth two-dimension boundary; similarly(20)limε→0⁡∫Γεqu∗dΓ=limε→0⁡∫Γε14πεqdΓ=limε→0⁡14πεqξ2πε2=limε→0⁡12qξε=0(21)limε→0⁡∫Γ2-εuq∗dΓ=∫Γ2uq∗dΓ(22)limε→0⁡∫Γ2-εqu∗dΓ=∫Γ2qu∗dΓWhen pointi is in the Γ1, the same result will be obtained, so (23)12ui+∫Γuq∗dΓ=∫Γqu∗dΓThis is the boundary integral equation of Laplace’s equation.The integral equation (15) of the internal node point and the boundary integral equation (23) regarding the boundary node point are combined, then(24)Ciui+∫Γuq∗dΓ=∫Γqu∗dΓWheni∈Ω, Ci=1; when i∈Γ (smooth boundary), Ci=1/2.


### 2.3. Discretion of the Boundary Integral Equation

Formula (24) is discretized and the boundary is divided into N units, the constant element, linear element, or secondary element can be adopted to conduct uniform interpolation regarding u and q, and then the matrix is formed according to different interpolations regarding all boundary elements:(25)∑j=1NHijTj=∑j=1NGijqjNamely,(26)HT=GqH i i = C i + H ⌢ i i (when i=j) and Hij=H⌢ij (when i≠j); the unknown quantity is transferred to the right, and the known quantity is transferred to the left, then(27)AX=FThe unknown quantity inu and q can be obtained by solving above equation set.


### 2.4. Calculation Example regarding Direct Problems concerning Heat Conduction Based on BEM

The two-dimension steady-state model [24] of heat conduction without thermal internal source for internal surface flaw detection concerning industrial equipment is shown in Figure 3. The domain Ω is a two-dimension space composed of x = 0.0, x = 10.0, y = 0, and y = f(x). It is adiabatic on both sides of x = 0.0 and x = 10.0, and a known constant heat flux q0 flows at y = 0 through cooling; thus a constant known temperature T0 is maintained for y = f(x).Figure 3
Physical model of internal surface flaw detection for thermal equipment.The temperature distributionYi at y = 0 can be measured through the infrared scanning thermal-detector, and it aims to detect the fault in internal surface regarding industrial thermal equipment through back-stepping the shape f(x) of the internal surface flaw; the mathematical model is as follows:The heat conduction equation in domainΩ is(28a)∂2T∂x2+∂2T∂y2=0Whenx=0 and x=10, the heat conduction equation is(28b)∂T∂x=0Wheny=0, the heat conduction equation is(28c)-∂T∂y=q0Wheny=f(x), the heat conduction equation is(28d)T=T0(28e)Txi,0=Yii=1,2,···,mAccording to the boundary integral equation in Section2.2 and its discretion knowledge in Section 2.3, the unknown temperature and heat flux on the boundary can be obtained. Boundary temperature and heat flux obtained from boundary element are shown in Figure 4. Temperature of internal points through BEM is shown in Figure 5.Figure 4
Boundary temperature and heat flux obtained from boundary element.Figure 5
Temperature of internal points through BEM.


## 2.1. Overview and Features of BEM

Boundary Element Method (BEM) is a calculation method developed on the basis of classical integral equation, which absorbs the discretization technology of finite element method. The basic principle of BEM is to solve the calculus equation in the method regarding the integral equation. After the first international conference of BEM in 1978, it was greatly applied and developed, and it was widely applied to many fields [22–24] such as elastic mechanics, fluid mechanics, rock mechanics, and heat transfer theory.The foundation of BEM is to establish the integral equation, and the solution process can be divided into the following two steps.The first step is to demarcate the problem and apply Green formula to transform the differential equation in the solution domain into the integral equation on the boundary by fundamental solution.In this way, the dimensionality of the equation solution problem is reduced by one dimension; for example, the three-dimensional space problem is reduced to a two-dimensional plane problem, and the two-dimensional plane problem is reduced to a one-dimensional problem. So the input data quantity and the unknown quantity of the algebraic equation set are greatly reduced, and this is significant feature of BEM compared with the FEM.The second step is the discretization of the boundary, which can use the discretization technique of finite element method. Moreover, since the discretization is only conducted on the boundary and error only exists on the boundary, while the unknown quantity in the domain can be calculated through the analytical equation, so the calculation precision can be higher. In addition, because of the singularity regarding the basic solution, the BEM has high precision in solving the singular problem. In conclusion, the BEM has outstanding advantages, such as reducing dimensionality, less input data, high calculation precision, especially suitable for large area, and singular problems.The solution to the direct problem regarding the heat conduction is the basis and premise to realize the inverse problem of heat transfer theory. When the geometric inverse problem of heat transfer theory is solved, the solution to the direct problem of the heat conduction is very complex because the geometric boundary shape is uncertain. At present in the geometric inverse problem of heat transfer theory, the boundary element method (BEM) and the finite element method (FEM) are two main methods to solve the direct problems. BEM is a developed calculation method based on the classic integral equation, and it absorbs the discrete technology of the value methods such as FEM; the basic thought is to solve the calculus equation with the method of the integral equation. Compared with FEM, the solution to the direct problem regarding the heat conduction with BEM has the following features [25]:( 1 ) BEM only needs to discretize the system on the boundary, which avoids the disadvantage that FEM needs to remesh the whole solution area every iteration and obviously reduces the computational dimension and workload.( 2 ) Because the basic solution adopted by BEM is applicable to infinite or semi-infinite domain, it is not necessary to determine the outer boundary when solving the problem in infinite or semi-infinite domain; thus the problem is greatly simplified.( 3 ) The solution error of the BEM is only generated on the boundary. The value of function and coefficient in the domain can be obtained by analytic formula; thus the error source is few and calculation precision is high.( 4 ) BEM fundamental solution itself is derivable and can be used to solve singularity problems.In the process of solving inverse heat transfer geometric problems, the objective function needs to be optimized continuously. Because of the change of the shape of the solution area, the finite difference method and the finite element method need to redivide the solution area into complicated meshes or elements in each iteration, which increases the difficulty and computational complexity of solving the direct problem. It only needs to conduct discretization regarding the system on the boundary for BEM; thus it avoids the shortcoming of repeatedly dividing grids in the overall solution domain in all iterations for FEM; it also obviously reduces calculation dimensionality, solution difficulty, and calculation quantity of the direct problems for BEM.


## 2.2. Boundary Integral Equation

As shown in Figure1, the boundary integral equation is deducted in the weighted residual method; because many projects can be described in the forms of Laplace’s equation, the definite solution problem of the Laplace’s equation is considered, namely. Boundary Γ1 and Γ2 are shown in Figure 1.Figure 1
BoundaryΓ1 and Γ2.In the domainΩ,(1)∇2u=0On the boundaryΓ1,(2)u=u¯On the boundaryΓ2,(3)q=∂u∂n=q¯Its expression of the weighted residual is(4)∫Ω∇2uWdΩ=∫Γ2q-q¯WdΓ-∫Γ2u-u¯∂W∂ndΓTake the weighted functionW as the basic solution u∗, and then correspondingly(5)∫Ω∇2uu∗dΩ=∫Γ2q-q¯u∗dΓ-∫Γ1u-u¯q∗dΓIntegration by parts is conducted regarding the Laplace operator, and Green Equation is applied, then(6)∫Ω∂P∂x+∂Ω∂ydΩ=∫ΓPdy+QdxSo the left item of (5) is(7)∫Ω∇2uu∗dΩ=∫Ω∇u∗∇udΩ-∫Ω∇u∇u∗dΩ=∫Γ∇uu∗dΓ-∫Ω∇u∇u∗dΩ=∫Γ∂u∂nu∗dΓ-∫Ω∇u∇u∗dΩIntegration by part and Green Equation are applied to the second item on the right of (7),(8)∫Ω∇u∇u∗dΩ=∫Ω∇u∇u∗dΩ-∫Ωu∇2u∗dΩ=∫Γu·∂u∗∂ndΓ-∫Ωu·∇2u∗dΩSo, there is(9)∫Ω∇2uu∗dΩ=∫Γ∂u∂nu∗dΓ-∫Γu·∂u∗∂ndΓ+∫Ωu·∇2u∗dΩSubstitute it into (5); because Γ=Γ1+Γ2+Γ3 and q=∂u/∂n, q∗=∂u∗/∂n, so(10)∫Ω∇2u∗udΩ=∫Γ1u¯q∗dΓ+∫Γ2uq∗dΓ-∫Γ1qu∗dΓ-∫Γ2q¯u∗dΓThe first, second, third, and fourth item on the right can be, respectively, combined together, namely.(11)∫Ω∇2u∗udΩ=∫Γuq∗dΓ-∫Γqu∗dΓWeighted functionu∗ is the basic solution of the Laplace’s equation, and it satisfies(12)∇2u∗+δr-r1=0In the equation, the point “i” indicates the focus point of the unit point source, so(13)∇2u∗=-δr-r1Substitute it into (10) or (11), and there is(14)ui+∫Γ1qu∗dΓ+∫Γ2q¯u∗dΓ=∫Γ1u¯q∗dΓ+∫Γ2uq∗dΓ(15)ui+∫Γuq∗dΓ=∫Γqu∗dΓEquations (14) and (15) are effective regarding all points in the domain Ω, and they are called the integral function of internal node. It indicates that the function value ui regarding all points in domain Ω can be indicated with the integral of u and its normal derivative q on the boundary; if all values of u and q on the boundary are obtained, u and q regarding all internal points can be calculated through (15).It needs to transfer the pointi to the boundary so as to obtain the integral equation regarding all points on the boundary, but the singularity of the integral ∂u∗/∂n shall be considered; the boundary near the point i is substituted by the hemisphere in radius ε to avoid that point i becomes singularity point; then the hemisphere ε tends to 0, so point i becomes the boundary point. Boundary point surrounded by the hemisphere is shown in Figure 2.Figure 2
Boundary point surrounded by the hemisphere.Without loss of generality, suppose the boundary belongs toΓ2 type, the point i is an internal point, and then (15) can be(16)ui+∫Γ2-εuq∗dΓ+∫Γεuq∗dΓ=∫Γ2-εqu∗dΓ+∫Γεqu∗dΓ(17)∫Γεuq∗dΓ=∫Γεu∂1/4πr∂ndΓ=-∫Γεu14πε2dΓSuppose the boundary is smooth, it can be obtained through the mean value theorem of integrals that(18)∫Γεuq∗dΓ=-∫Γεu14πε2dΓ=uξ-14πε22πε2whereξ is a point in Γε.Supposeε→0 and u(ξ)→ui, then (19)limε→0⁡∫Γεuq∗dΓ=-12uiEquation (19) is also applicable to smooth two-dimension boundary; similarly(20)limε→0⁡∫Γεqu∗dΓ=limε→0⁡∫Γε14πεqdΓ=limε→0⁡14πεqξ2πε2=limε→0⁡12qξε=0(21)limε→0⁡∫Γ2-εuq∗dΓ=∫Γ2uq∗dΓ(22)limε→0⁡∫Γ2-εqu∗dΓ=∫Γ2qu∗dΓWhen pointi is in the Γ1, the same result will be obtained, so (23)12ui+∫Γuq∗dΓ=∫Γqu∗dΓThis is the boundary integral equation of Laplace’s equation.The integral equation (15) of the internal node point and the boundary integral equation (23) regarding the boundary node point are combined, then(24)Ciui+∫Γuq∗dΓ=∫Γqu∗dΓWheni∈Ω, Ci=1; when i∈Γ (smooth boundary), Ci=1/2.


## 2.3. Discretion of the Boundary Integral Equation

Formula (24) is discretized and the boundary is divided into N units, the constant element, linear element, or secondary element can be adopted to conduct uniform interpolation regarding u and q, and then the matrix is formed according to different interpolations regarding all boundary elements:(25)∑j=1NHijTj=∑j=1NGijqjNamely,(26)HT=GqH i i = C i + H ⌢ i i (when i=j) and Hij=H⌢ij (when i≠j); the unknown quantity is transferred to the right, and the known quantity is transferred to the left, then(27)AX=FThe unknown quantity inu and q can be obtained by solving above equation set.


## 2.4. Calculation Example regarding Direct Problems concerning Heat Conduction Based on BEM

The two-dimension steady-state model [24] of heat conduction without thermal internal source for internal surface flaw detection concerning industrial equipment is shown in Figure 3. The domain Ω is a two-dimension space composed of x = 0.0, x = 10.0, y = 0, and y = f(x). It is adiabatic on both sides of x = 0.0 and x = 10.0, and a known constant heat flux q0 flows at y = 0 through cooling; thus a constant known temperature T0 is maintained for y = f(x).Figure 3
Physical model of internal surface flaw detection for thermal equipment.The temperature distributionYi at y = 0 can be measured through the infrared scanning thermal-detector, and it aims to detect the fault in internal surface regarding industrial thermal equipment through back-stepping the shape f(x) of the internal surface flaw; the mathematical model is as follows:The heat conduction equation in domainΩ is(28a)∂2T∂x2+∂2T∂y2=0Whenx=0 and x=10, the heat conduction equation is(28b)∂T∂x=0Wheny=0, the heat conduction equation is(28c)-∂T∂y=q0Wheny=f(x), the heat conduction equation is(28d)T=T0(28e)Txi,0=Yii=1,2,···,mAccording to the boundary integral equation in Section2.2 and its discretion knowledge in Section 2.3, the unknown temperature and heat flux on the boundary can be obtained. Boundary temperature and heat flux obtained from boundary element are shown in Figure 4. Temperature of internal points through BEM is shown in Figure 5.Figure 4
Boundary temperature and heat flux obtained from boundary element.Figure 5
Temperature of internal points through BEM.


## 3. CGM Algorithm regarding the Geometric Inverse Problem concerning Heat Transfer Theory

### 3.1. The Principle of the CGM

The solution to the inverse problems can be transformed to the following optimization control problem concerning functional variation mathematically:(29)Jfx=Ti-Yi2=∑i=1mTix,0-Yix,02Y i ( x , 0 ) is the actual measured temperature value regarding the measurement point on the model surface; Ti(x,0) is the calculated temperature value concerning the measurement point from above equation set based on the guessed boundary shape.The iterative solution method is used to search the standard problem regarding boundary shapef(x) involved with iteration stop, and it is related to the error of temperature measurement. If there is no measurement error, the stop standard is (30)Jfk+1x<εε is a minor number such as 0.01, and it should be determined based on the specific convergence. If there is a measurement error and the standard deviation of the temperature measurement is σ, the lowest standard can be determined based on the error principle(31)ε=3σThe conjugate gradient [26] transforms the inverse problem to three problems: direct problem, sensitivity problem, and adjoint problem. The direct problem is based on the assumed boundary shape equations (28a), (28b), (28c), (28d), and (28e) and other boundary conditions; then the equations of boundary conditions are solved through BEM, and the minimum of the objective function (29) is calculated. The cyclic way of the CGM is(32)fn+1x=fnx-βnpnxn=0,1,2···βn is the step size in search from the nth to the (n+1)th iteration; pn(x) is the search direction; namely,(33)pnx=J′nx+γnpn-1xJ′n(x) is the gradient direction, so the nth search direction pn(x) is the conjugate between the gradient direction J′n(x) and the (n-1)th search direction pn-1(x); γn is the conjugate coefficient, and it can be calculated as follows:(34)γn=∫x=0LJ′n2dx∫x=0LJ′n-12dxγ0=0.


### 3.2. Sensitivity Problem

To confirm the step sizeβn in search of (32), a so-called sensitivity problem, namely, the increment problem needs to be solved. The sensitivity problem refers to the variation ΔT(x) of surface temperature T, when the boundary shape f(x) has an increment Δf(x).(35)∂J∂fT=∂J∂f1,∂J∂f2,···,∂J∂fnThe specific way is in (29), and the original T is replaced by T+ΔT, while the original f(x) is replaced by f(x)+Δf(x); then the sensitivity problem is obtained by subtracting the original equations; namely,The heat conduction problem is as follows in the domainΩ:(36a)∂2ΔT∂x2+∂2ΔT∂y2=0The heat conduction problem is as follows whenx=0: (36b)∂ΔT∂x=0The heat conduction problem is as follows whenx=10:(36c)∂ΔT∂x=0The heat conduction problem is as follows wheny=0:(36d)∂ΔT∂y=0The heat conduction problem is as follows wheny=f(x):(36e)ΔT=Δf∂T∂yThe above equation set can be solved by discretization through BEM similarly, and the temperature incrementΔT can be obtained. J(fn+1) can be as follows regarding the (n+1)th iteration according to (29)(37)Jfn+1=∑i=1mTifn-βnpn-Yi2Conduct Taylor expansion regardingTi(fn-βnpn) to remove 2 linear items to obtain(38)Jfn+1=∑i=1mTifn-βnΔTipn-Yi2Derive and setβn at 0, and then the step size Γ in search can be obtained; namely,(39)βn=∑i=1mTi-Yi2ΔTi∑i=1mΔTi2


### 3.3. Adjoint Problem

The new boundary shape functionfn+1(x) can be calculated according to the boundary shape function fn(x) of last iteration in (32), and the search direction pn(x) can also be obtained; so the gradient direction J′n(x) should be solved according to (33), namely, the so-called functional derivation problem, and it is also called adjoint problem. To derive the equation in the adjoint direction, the basic control equation (28a) is multiplied by a Lagrange operator λ(x,y) (also called adjoint function); then integration is applied to the spatial domain, and the result is added to the right of (29); thus the expression of the functional Jf(x) can be obtained.(40)Jfx=∫x=0LT-Y2δx-xidx+∫x=0L∫y=0fxλ∂2T∂x2+∂2T∂y2dydxSubstitute T withT+ΔT and substitute f with f+Δf, and it is obtained through a series of transformation that(41)∫abT-Y2dx=∫ab∫0LT-Y2δx-xidxdyThen(42)ΔJ=ΔJq+Δq-Jq=∫x=0L2T-Yδx-xidx=∫x=0L∫y=0fxλ∂2ΔT∂x2+∂2ΔT∂y2idydxThe integration by parts is conducted regarding the second item on the right of (42) twice, and then the boundary condition of the sensitivity problem is used and then makes ΔJ tend to 0 to obtain the adjoint problem.In the domainΩ,(43a)∂2λ∂x2+∂2λ∂y2=0Whenx=0,(43b)∂λ∂x=0Whenx=10,(43c)∂λ∂y=0Wheny=0,(43d)∂λ∂y=-2T-Yδx-xiWheny=f(x),(43e)λ=0The corresponding value of the adjoint functionλ(x,y) is obtained through the BEM, and then the functional increment is(44)ΔJ=∫0L-∂λ∂y·∂T∂yy=fxΔfxdxSo there is the following equation according to the definition of Alifanov(45)ΔJ=∫0LJ′xΔfxdxThe derivative of the functionalJ is(46)J′x=-∂λ∂y·∂T∂yy=fxAbove three problems including direct problem, sensitivity problem and adjoint problem are solved, then the iterative calculation steps of CGM can be given.( 1 ) Select a guessed initial value f(x) of internal surface shape to solve direct problems (28a), (28b), (28c), (28d), and (28e) and calculate the temperature distribution T(x,t) in the domain.( 2 ) Judge whether it satisfies the standard equation J[fk+1(x)]<ε of convergence stop according to Ti and Yi; if it satisfies, stop the iteration or calculate based on the next step.( 3 ) Solve the adjoint problem equations (43a), (43b), (43c), (43d), and (43e) to obtain the adjoint function λ(x).(4) Calculate the gradientJ′(x).(5) Calculate the coefficientγk and the descent direction pk of the conjugate gradient.( 6 ) Suppose Δf(x)=-pk(x), solve the sensitivity problem equations (36a), (36b), (36c), (36d), and (36e) to obtain the temperature increment ΔT(x).(7) Calculate the step sizeβk in search based on (39).( 8 ) Suppose k=k+1 and obtain the new boundary shape fk(x); the new Ti returns to the 2nd step.


## 3.1. The Principle of the CGM

The solution to the inverse problems can be transformed to the following optimization control problem concerning functional variation mathematically:(29)Jfx=Ti-Yi2=∑i=1mTix,0-Yix,02Y i ( x , 0 ) is the actual measured temperature value regarding the measurement point on the model surface; Ti(x,0) is the calculated temperature value concerning the measurement point from above equation set based on the guessed boundary shape.The iterative solution method is used to search the standard problem regarding boundary shapef(x) involved with iteration stop, and it is related to the error of temperature measurement. If there is no measurement error, the stop standard is (30)Jfk+1x<εε is a minor number such as 0.01, and it should be determined based on the specific convergence. If there is a measurement error and the standard deviation of the temperature measurement is σ, the lowest standard can be determined based on the error principle(31)ε=3σThe conjugate gradient [26] transforms the inverse problem to three problems: direct problem, sensitivity problem, and adjoint problem. The direct problem is based on the assumed boundary shape equations (28a), (28b), (28c), (28d), and (28e) and other boundary conditions; then the equations of boundary conditions are solved through BEM, and the minimum of the objective function (29) is calculated. The cyclic way of the CGM is(32)fn+1x=fnx-βnpnxn=0,1,2···βn is the step size in search from the nth to the (n+1)th iteration; pn(x) is the search direction; namely,(33)pnx=J′nx+γnpn-1xJ′n(x) is the gradient direction, so the nth search direction pn(x) is the conjugate between the gradient direction J′n(x) and the (n-1)th search direction pn-1(x); γn is the conjugate coefficient, and it can be calculated as follows:(34)γn=∫x=0LJ′n2dx∫x=0LJ′n-12dxγ0=0.


## 3.2. Sensitivity Problem

To confirm the step sizeβn in search of (32), a so-called sensitivity problem, namely, the increment problem needs to be solved. The sensitivity problem refers to the variation ΔT(x) of surface temperature T, when the boundary shape f(x) has an increment Δf(x).(35)∂J∂fT=∂J∂f1,∂J∂f2,···,∂J∂fnThe specific way is in (29), and the original T is replaced by T+ΔT, while the original f(x) is replaced by f(x)+Δf(x); then the sensitivity problem is obtained by subtracting the original equations; namely,The heat conduction problem is as follows in the domainΩ:(36a)∂2ΔT∂x2+∂2ΔT∂y2=0The heat conduction problem is as follows whenx=0: (36b)∂ΔT∂x=0The heat conduction problem is as follows whenx=10:(36c)∂ΔT∂x=0The heat conduction problem is as follows wheny=0:(36d)∂ΔT∂y=0The heat conduction problem is as follows wheny=f(x):(36e)ΔT=Δf∂T∂yThe above equation set can be solved by discretization through BEM similarly, and the temperature incrementΔT can be obtained. J(fn+1) can be as follows regarding the (n+1)th iteration according to (29)(37)Jfn+1=∑i=1mTifn-βnpn-Yi2Conduct Taylor expansion regardingTi(fn-βnpn) to remove 2 linear items to obtain(38)Jfn+1=∑i=1mTifn-βnΔTipn-Yi2Derive and setβn at 0, and then the step size Γ in search can be obtained; namely,(39)βn=∑i=1mTi-Yi2ΔTi∑i=1mΔTi2


## 3.3. Adjoint Problem

The new boundary shape functionfn+1(x) can be calculated according to the boundary shape function fn(x) of last iteration in (32), and the search direction pn(x) can also be obtained; so the gradient direction J′n(x) should be solved according to (33), namely, the so-called functional derivation problem, and it is also called adjoint problem. To derive the equation in the adjoint direction, the basic control equation (28a) is multiplied by a Lagrange operator λ(x,y) (also called adjoint function); then integration is applied to the spatial domain, and the result is added to the right of (29); thus the expression of the functional Jf(x) can be obtained.(40)Jfx=∫x=0LT-Y2δx-xidx+∫x=0L∫y=0fxλ∂2T∂x2+∂2T∂y2dydxSubstitute T withT+ΔT and substitute f with f+Δf, and it is obtained through a series of transformation that(41)∫abT-Y2dx=∫ab∫0LT-Y2δx-xidxdyThen(42)ΔJ=ΔJq+Δq-Jq=∫x=0L2T-Yδx-xidx=∫x=0L∫y=0fxλ∂2ΔT∂x2+∂2ΔT∂y2idydxThe integration by parts is conducted regarding the second item on the right of (42) twice, and then the boundary condition of the sensitivity problem is used and then makes ΔJ tend to 0 to obtain the adjoint problem.In the domainΩ,(43a)∂2λ∂x2+∂2λ∂y2=0Whenx=0,(43b)∂λ∂x=0Whenx=10,(43c)∂λ∂y=0Wheny=0,(43d)∂λ∂y=-2T-Yδx-xiWheny=f(x),(43e)λ=0The corresponding value of the adjoint functionλ(x,y) is obtained through the BEM, and then the functional increment is(44)ΔJ=∫0L-∂λ∂y·∂T∂yy=fxΔfxdxSo there is the following equation according to the definition of Alifanov(45)ΔJ=∫0LJ′xΔfxdxThe derivative of the functionalJ is(46)J′x=-∂λ∂y·∂T∂yy=fxAbove three problems including direct problem, sensitivity problem and adjoint problem are solved, then the iterative calculation steps of CGM can be given.( 1 ) Select a guessed initial value f(x) of internal surface shape to solve direct problems (28a), (28b), (28c), (28d), and (28e) and calculate the temperature distribution T(x,t) in the domain.( 2 ) Judge whether it satisfies the standard equation J[fk+1(x)]<ε of convergence stop according to Ti and Yi; if it satisfies, stop the iteration or calculate based on the next step.( 3 ) Solve the adjoint problem equations (43a), (43b), (43c), (43d), and (43e) to obtain the adjoint function λ(x).(4) Calculate the gradientJ′(x).(5) Calculate the coefficientγk and the descent direction pk of the conjugate gradient.( 6 ) Suppose Δf(x)=-pk(x), solve the sensitivity problem equations (36a), (36b), (36c), (36d), and (36e) to obtain the temperature increment ΔT(x).(7) Calculate the step sizeβk in search based on (39).( 8 ) Suppose k=k+1 and obtain the new boundary shape fk(x); the new Ti returns to the 2nd step.


## 4. Simulation Result of the Internal Surface Shape Recognition

To verify the effectiveness of the internal boundary shape through CGM, the two boundaries in shapes of sine curve (example A) step function curve (example B) are, respectively, researched. The outside diameter of pipe in-wall is 0.18 m; surface coefficient regarding heat transfer of the internal surface is 1000 W/(m2·K); surface coefficient regarding heat transfer of the external surface is 10 W/(m2·K). Two kinds of materials are adopted for the pipe: one is chromium-nickel steel regarding thermal conductivity 15.2 W/(m2·K); the other is cement concerning thermal conductivity 0.5 W/(m2·K). The corresponding examples to the two materials are respectively marked as A(1), B(1) and A(2), B(2).


### 4.1. Recognition Result of Different Initial Boundary Hypothesis

Figures6 and 7 are the identification results of the boundary shapes of the inner wall of examples A and B under different initial boundary assumptions. Influence of the initial boundary hypothesis on the recognition result of in-wall boundary A (ε=1.0) is shown in Figure 6. Influence of the initial boundary hypothesis on the recognition result of in-wall boundary B (ε=1.0) is shown in Figure 7.Figure 6
Influence of the initial boundary hypothesis on the recognition result of in-wall boundary A (ε=1.0).
(a)
Example A(1) (b)
Example A(2)Figure 7
Influence of the initial boundary hypothesis on the recognition result of in-wall boundary B (ε=1.0).
(a)
Example B(1) (b)
Example B(2)The average relative error of the recognition result for different initial values (the internal round boundary of different radiuses is the initial boundary hypothesis) is shown in Table1. The iterative convergent condition is ε=1.0.Table 1
Average relative error of recognition result under the condition of different initial values.
Initial number Initial value/mm Average relative error/% Initial value Average relative error/% A(1) A(2) B(1) B(2) 1 60 0.8 0.07 50 3.8 0.8 2 80 0.9 0.05 80 3.8 0.8It can be found from the recognition result of Figures6 and 7 and Table 1 that the boundary shape of pipe in-wall can be precisely recognized through CGM, and the initial boundary hypothesis has little influence on the recognition result. It can be found that the recognition precision is low if the thermal conductivity is large, when the recognition result regarding the thermal conductivity of different pipes is compared. It is mainly because the thermal resistance is small when the thermal conductivity of pipes is large. Therefore the temperature variation is small upon changing the same pipeline wall thickness.


### 4.2. Recognition Result of Different Temperature Measurement Errors

Influence of temperature measurement error on the recognition result regarding the in-wall boundary A is shown in Figure8. Influence of temperature measurement error on the recognition result concerning the in-wall boundary B is shown in Figure 9. The average relative error of recognition results concerning different temperature measurement errors is shown in Table 2.Table 2
Average relative error of recognition results concerning different temperature measurement errors.
σ/°C Average relative error/% A(1) A(2) B(1) B(2) 0.5 3.4 0.4 6.2 1.4 1.0 7.2 0.7 9.8 2.1Figure 8
Influence of temperature measurement error on the recognition result regarding the in-wall boundary A.
(a)
Example A(1) (b)
Example A(2)Figure 9
Influence of temperature measurement error on the recognition result concerning the in-wall boundary B.
(a)
Example B(1) (b)
Example B(2)It can be found that the temperature measurement error has little influence on the recognition result regarding pipeline of small thermal conductivity [example A(2) and example B (2)], when the recognition results of different thermal conductivity are compared on the basis of Figures 8 and 9 and Table 2. The error of the recognition result is large, when the thermal conductivity is high [example A (1) and example B (1)].The results show that CGM can identify the inner wall shape of pipeline successfully according to the temperature distribution of the testing surface. The initial boundary hypothesis has little influence on the recognition result. The measurement error has certain influence on the final recognition result, and the recognition precision is lower if the measurement error is larger. The adopted boundary condition in the detection process should be changed to obtain precise recognition result. The maximum temperature difference regarding the detection surface should be increased as much as possible, and then the influence of errors on the recognition precision can be reduced.


## 4.1. Recognition Result of Different Initial Boundary Hypothesis

Figures6 and 7 are the identification results of the boundary shapes of the inner wall of examples A and B under different initial boundary assumptions. Influence of the initial boundary hypothesis on the recognition result of in-wall boundary A (ε=1.0) is shown in Figure 6. Influence of the initial boundary hypothesis on the recognition result of in-wall boundary B (ε=1.0) is shown in Figure 7.Figure 6
Influence of the initial boundary hypothesis on the recognition result of in-wall boundary A (ε=1.0).
(a)
Example A(1) (b)
Example A(2)Figure 7
Influence of the initial boundary hypothesis on the recognition result of in-wall boundary B (ε=1.0).
(a)
Example B(1) (b)
Example B(2)The average relative error of the recognition result for different initial values (the internal round boundary of different radiuses is the initial boundary hypothesis) is shown in Table1. The iterative convergent condition is ε=1.0.Table 1
Average relative error of recognition result under the condition of different initial values.
Initial number Initial value/mm Average relative error/% Initial value Average relative error/% A(1) A(2) B(1) B(2) 1 60 0.8 0.07 50 3.8 0.8 2 80 0.9 0.05 80 3.8 0.8It can be found from the recognition result of Figures6 and 7 and Table 1 that the boundary shape of pipe in-wall can be precisely recognized through CGM, and the initial boundary hypothesis has little influence on the recognition result. It can be found that the recognition precision is low if the thermal conductivity is large, when the recognition result regarding the thermal conductivity of different pipes is compared. It is mainly because the thermal resistance is small when the thermal conductivity of pipes is large. Therefore the temperature variation is small upon changing the same pipeline wall thickness.


## 4.2. Recognition Result of Different Temperature Measurement Errors

Influence of temperature measurement error on the recognition result regarding the in-wall boundary A is shown in Figure8. Influence of temperature measurement error on the recognition result concerning the in-wall boundary B is shown in Figure 9. The average relative error of recognition results concerning different temperature measurement errors is shown in Table 2.Table 2
Average relative error of recognition results concerning different temperature measurement errors.
σ/°C Average relative error/% A(1) A(2) B(1) B(2) 0.5 3.4 0.4 6.2 1.4 1.0 7.2 0.7 9.8 2.1Figure 8
Influence of temperature measurement error on the recognition result regarding the in-wall boundary A.
(a)
Example A(1) (b)
Example A(2)Figure 9
Influence of temperature measurement error on the recognition result concerning the in-wall boundary B.
(a)
Example B(1) (b)
Example B(2)It can be found that the temperature measurement error has little influence on the recognition result regarding pipeline of small thermal conductivity [example A(2) and example B (2)], when the recognition results of different thermal conductivity are compared on the basis of Figures 8 and 9 and Table 2. The error of the recognition result is large, when the thermal conductivity is high [example A (1) and example B (1)].The results show that CGM can identify the inner wall shape of pipeline successfully according to the temperature distribution of the testing surface. The initial boundary hypothesis has little influence on the recognition result. The measurement error has certain influence on the final recognition result, and the recognition precision is lower if the measurement error is larger. The adopted boundary condition in the detection process should be changed to obtain precise recognition result. The maximum temperature difference regarding the detection surface should be increased as much as possible, and then the influence of errors on the recognition precision can be reduced.


## 5. Recognition of Thermal Internal Flaw Shape through CGM

The schematic diagram of the test piece with the heat-type defect studied in this section is shown in Figure10. The inside of the test piece is steady-state heat transfer, and the temperature at the junction of the defect and the test piece is the same, and the heat flow is equal.Figure 10
Sketch map of the test piece concerning thermal flaw.Heat convection is conducted between the outside surface of the test piece and air, and the surface coefficient of heat transfer is 10 W/(m2·K). Two kinds of materials are adopted in this section concerning the round test piece for comparison research: the heat transfer coefficient of test piece A is 52 W/(m2·K); the heat transfer coefficient of test piece B is 0.34 W/(m2·K). The external diameter of the test piece is 0.04 mm. The flaw emits heat evenly, and the thermal power is qv. The external environmental temperature Ta is 25°C.


### 5.1. Influence of the Thermal Conductivity regarding the Test Piece and Flaw on the Recognition Result

The recognition result of the internal round flaws for two test pieces at different initial values is shown in Figure11. When the heat liberation rate qv of the unit area is 1×103 kW/m2 and the thermal conductivity of flaws is 0.34 W/(m·k).Figure 11
Recognition result of the internal round flaws for two test pieces at different initial values.
(a)
Test piece A (b)
Test piece BIt can be found from the figure that the shape of this internal flaw can be precisely recognized, and the average error is all 0.24%. The influence of initial values on the recognition result can be ignored.Recognition result concerning flaw boundary shapes regarding two test pieces at the same initial value is shown in Figure12. The iterative convergent condition is ε=2.0.Figure 12
Recognition result concerning flaw boundary shapes regarding two test pieces at the same initial value.It can be found from the figure that, for specimen B with small thermal conductivity, the accuracy of boundary shape recognition is higher, and the average relative error is 2.4%. While the recognition result precision of boundary shape is relatively low for the test piece A with large thermal conductivity, and the average relative error is 40%. The main reason is that the difference of thermal conductivity makes the temperature distribution of the same shape defect on the outer surface quite different.


### 5.2. Influence of Estimated Error regarding Thermal Intensity and Error concerning Temperature Measurement on the Recognition Result

The intensity of heat source is a known condition for the identification and calculation of defect boundary. It can be calculated according to the measurement results of surface temperature and the correlation of natural convection heat transfer of ambient temperature through the surface.Influence of the calculation error regarding thermal intensity on the recognition result flaws (ε=2.0) is shown in Figure 13. Influence of random temperature measurement error on the recognition result regarding flaws is shown in Figure 14. Influence of temperature measurement error on the average relative error regarding recognition algorithm is shown in Table 3.Table 3
Influence of temperature measurement error on the average relative error regarding recognition algorithm.
Random error σ/°C The average relative error of the results/%  1.0 5.0  2.0 9.5  3.0 12.6Figure 13
Influence of the calculation error regarding thermal intensity on the recognition result concerning flaws (ε=2.0).Figure 14
Influence of random temperature measurement error on the recognition result regarding flaws.Figure13 shows the influence of the calculation error concerning the heat source intensity on the recognition result, and the average error of the recognition result is within 2.0% when the estimated error of the heat source intensity is ±10%. Thus its influence can be ignored. It can be found from Figure 14 and Table 3 that when the measurement errors are 1.0, 2.0 and 3.0, the average errors are 5.0%, 9.5%, and 12.6%, respectively, indicating that the temperature measurement errors have little influence on the identification results.


## 5.1. Influence of the Thermal Conductivity regarding the Test Piece and Flaw on the Recognition Result

The recognition result of the internal round flaws for two test pieces at different initial values is shown in Figure11. When the heat liberation rate qv of the unit area is 1×103 kW/m2 and the thermal conductivity of flaws is 0.34 W/(m·k).Figure 11
Recognition result of the internal round flaws for two test pieces at different initial values.
(a)
Test piece A (b)
Test piece BIt can be found from the figure that the shape of this internal flaw can be precisely recognized, and the average error is all 0.24%. The influence of initial values on the recognition result can be ignored.Recognition result concerning flaw boundary shapes regarding two test pieces at the same initial value is shown in Figure12. The iterative convergent condition is ε=2.0.Figure 12
Recognition result concerning flaw boundary shapes regarding two test pieces at the same initial value.It can be found from the figure that, for specimen B with small thermal conductivity, the accuracy of boundary shape recognition is higher, and the average relative error is 2.4%. While the recognition result precision of boundary shape is relatively low for the test piece A with large thermal conductivity, and the average relative error is 40%. The main reason is that the difference of thermal conductivity makes the temperature distribution of the same shape defect on the outer surface quite different.


## 5.2. Influence of Estimated Error regarding Thermal Intensity and Error concerning Temperature Measurement on the Recognition Result

The intensity of heat source is a known condition for the identification and calculation of defect boundary. It can be calculated according to the measurement results of surface temperature and the correlation of natural convection heat transfer of ambient temperature through the surface.Influence of the calculation error regarding thermal intensity on the recognition result flaws (ε=2.0) is shown in Figure 13. Influence of random temperature measurement error on the recognition result regarding flaws is shown in Figure 14. Influence of temperature measurement error on the average relative error regarding recognition algorithm is shown in Table 3.Table 3
Influence of temperature measurement error on the average relative error regarding recognition algorithm.
Random error σ/°C The average relative error of the results/%  1.0 5.0  2.0 9.5  3.0 12.6Figure 13
Influence of the calculation error regarding thermal intensity on the recognition result concerning flaws (ε=2.0).Figure 14
Influence of random temperature measurement error on the recognition result regarding flaws.Figure13 shows the influence of the calculation error concerning the heat source intensity on the recognition result, and the average error of the recognition result is within 2.0% when the estimated error of the heat source intensity is ±10%. Thus its influence can be ignored. It can be found from Figure 14 and Table 3 that when the measurement errors are 1.0, 2.0 and 3.0, the average errors are 5.0%, 9.5%, and 12.6%, respectively, indicating that the temperature measurement errors have little influence on the identification results.


## 6. Conclusions

The direct problem is solved in BEM in this paper, the inverse algorithm based on BEM and CGM is adopted to solve the thermal inverse problems, such as the recognition of boundary shape regarding the pipe in-wall and thermal flaw in test piece. The recognition of test pieces concerning different materials in the shapes such as sinusoidal, step function, or circular shape at the unknown boundary is considered, and the influence of factors such as initial value, temperature measurement error, thermal conductivity, and heating intensity on the precision concerning the inverse solution is discussed. The conclusions are as follows:( 1 ) The CGM can accurately identify the boundary shape of the inner wall of the pipeline by using the temperature profiler on the outer surface of the pipeline. The initial value has little effect on the recognition results. The smaller the thermal conductivity of the pipeline is, the higher the accuracy of the identification results.( 2 ) For heating defects, the initial assumption of the defect boundary, the estimation error of the heat source strength, and the measurement error of the testing surface temperature have little influence on the identification results. It is proved that the CGM can accurately identify the boundary contour of the thermal defect.



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*Source: 3723949-2018-10-03.xml*