Solution to Two-Dimensional Steady Inverse Heat Transfer Problems with Interior Heat Source Based on the Conjugate Gradient Method

The compound variable inverse problem which comprises boundary temperature distribution and surface convective heat conduction coefficient of two-dimensional steady heat transfer system with inner heat source is studied in this paper applying the conjugate gradient method.The introduction of complex variable to solve the gradient matrix of the objective function obtains more precise inversion results.This paper applies boundary element method to solve the temperature calculation of discrete points in forward problems.The factors ofmeasuring error and the number ofmeasuring points zero error which impact themeasurement result are discussed and comparedwith L-MMmethod in inverse problems. Instance calculation and analysis prove that themethod applied in this paper still has good effectiveness and accuracy even if measurement error exists and the boundary measurement points’ number is reduced. The comparison indicates that the influence of error on the inversion solution can be minimized effectively using this method.


Introduction
Inverse heat transfer problem (Inverse Heat Transfer Problems, IHTP) means inversion unknown characteristic parameters of heat conduction objects using the internal or surface local heat measurement [1,2], such as thermal physical parameters, thermal boundary conditions, geometric boundary shape, and heat conduction coefficient.Inverse heat conduction problem has a broad application prospect in nondestructive testing, geometry optimization, aerospace engineering, power engineering, mechanical engineering, construction engineering, biological engineering, metallurgical engineering, materials processing, biomedical and food engineering, and other fields [1][2][3][4][5][6][7][8][9][10][11][12].Focusing on this problem, domestic and foreign scholars have done a lot of research.Zhu et al. studied the disadvantages of optimization algorithms and inherent space distribution characteristics of the measurement in inverse heat conduction problems.They proposed a decentralized fuzzy reasoning mechanism and established a decentralized fuzzy inference system of two-dimensional steady inverse heat conduction problems [3,4].Cui et al. gave an improved conjugate gradient method.They introduced the complex variable derivation into the traditional conjugate gradient method, calculated the sensitivity coefficients accurately, and identified the boundary conditions [5].Yu combined the boundary element method with complex variables derivation to inverse the inhomogeneous material coefficients of thermal conductivity [6].Wang et al. applied particle swarm optimization algorithm and the least square method for solving the inverse problems.That improved the precision and shorted the solving time at the same time [13].Yaparova studied the heat conduction boundary value inverse problem by solving the stability boundary based on Laplace and Fourier transformation [14].Tian combined SPSO algorithm with the conjugate gradient method and the fast convergence of the traditional regularized gradient algorithm and global convergence of the intelligent optimization algorithm were highlighted [15].Using the boundary element method, Zhou et al. analyzed two-dimensional transient conduction problems, introduced the conjugate gradient method to find the heat conduction coefficient, and verified the validity and stability of the method [16].A stable differential method was proposed to solve the inverse heat conduction problems by Baranov et al. on the basis of the differential transform [17].In this paper, the two-dimensional steady forward problems with internal heat source are solved using the boundary element method.The existing surface convective thermal dissipation is analyzed, and the conjugate gradient method is applied to solve the inverse problems.And the complex variable derivation method is introduced to increase the sensitivity coefficient and the accuracy in the inversion.

Thermal Steady State Forward Problems
2.1.The Boundary Integral Equation.The mathematical model of two-dimensional steady heat conduction problems with internal heat source is where Γ 1 , Γ 2 , and Γ 3 are the boundary of the domain Ω, Γ = Γ 1 +Γ 2 +Γ 3 ,  = −/,  is the internal heat source, ℎ is the heat conduction coefficient between solid and fluid,  = HTC,  = temp, and  is the normal vector outside the boundary.  = (ℎ/)  ,   is environment temperature, and  is heat flux.Weight function  * is introduced into weighted residual of control equation.
Green's function of Laplace equation is where  is the boundary curve of plane closed area  and   is arc differential [18].
To decompose the left side of the equation, it becomes According to Laplace Green's function, we can have Further reduction gives Substituting (5) for the same term in (3), (2) becomes Further reduction gives * is the basic solution of Laplace equation and it can make ∇ * +(−  ) = 0, where the point "" refers to the unit point concentration.Because  * / =  * and / = , (7) can be further reduced.
The left side of (7) becomes Calculating furthermore, it is shown as follows: For the selectivity of  function, ∫ +∞ −∞ ()( −  0 )  = ( 0 ).Then the reduced result on the left side of ( 7) is Substituting ( 10) into (7), Because Substituting ( 12) for the same term in (11), Further reduction gives The second and the third terms are combined, which makes The two-dimensional steady Moving the source point  to the boundary, the integral equation at any point on the boundary can be obtained.Thus, the basic solution  * and the singularity of the integral   * /  need to be considered.Assuming that a circular arc centering on the point  near the border and  is the small radius of the circle, (15) can be written as where Γ is the new circular border and Γ −  is the boundary outside the new circular boundary in Γ. Considering the smooth boundary, the integral mean value theorem is introduced.
where  is the point on Γ.Let  → 0 () →   ; then lim Synthesizing the above limit results, we have Considering ( 15) and ( 19) simultaneously, boundary integral equation of any point in the domain Ω or on the boundary can be found out: The unknown points in the boundary units are called "nodes."For the requirements of different difference, the expressions of  and  to be found out can be written as where  is the interpolation node number in a boundary unit and  is the interpolation function.When  is in Γ 1  Γ 2 , ( 19) can be written as This paper uses linear interpolation.Because  1 () = (1 − )/ 2,  2 () = (1 + )/2 is the interpolation function of linear unit, the straight line is used to approximate the boundary curve, and the two linearized terminal values are used to approximate the values of  and .So the node values can be converted into the following: Substituting (23a) and (23b) into ( 22), where Let   = ℎ −1 (2) + ℎ  (1) ,   =  −1 (2) +   (1) , and we have Similarly, when the point locates in Γ 3 , (24) becomes Let and the boundary integral equation is Rearranging (30), we have where considering   = ∫ Ω  *  Ω , (31) can be expressed as a matrix form  =  −   .The boundary conditions  and  are written as where  1 ,  2 are known temperature and heat flux.Equation (31) becomes Put the known quantities on the left.
Appling the radial integration method, domain integration caused by heat source is converted to boundary integration [19][20][21].
where is the distance between the source and the field point and  is radial integration which is expressed as Because  is a known function and the function form is simple,  * is the basic function, the radial integration can be found out through (36), and the boundary integration can be obtained from (35).
Thus, (34) can be written as  =  to solve the above equations and to find  and .

The Objective Function of Inverse Problem.
For aforementioned heat conduction system, the heat transfer coefficient between solid and fluid ℎ and the thermal conductivity  are known.The boundary temperature to be found out is unknown which can be determined based on the known measured boundary temperature and the known condition of forward problem.
The following objective function is defined as In the objective function,  is a parameter vector whose temperature needs to be inversed and  is the number of temperature measuring points on the boundary.
() is the calculation of measuring point  in the forward problem and  mea is the measurement of temperature.The minimum of the objective function () is the parameter vector  of inverse problem.

Conjugate Gradient Method of Inverse Problem.
Conjugate gradient method is a method which combines the conjugacy and the steepest descent method.It derives from perturbation principle and the inverse problem is converted into three questions, such as forward problem, sensitivity problem, and adjoint problem.In order to solve the effects of these three questions, this paper introduces the complex variable derivation method into the traditional conjugate gradient method, which makes the calculation of the sensitivity coefficient accurate.
During the calculation, unconstrained optimization algorithm is achieved by iteration.Considering that the th iteration point   has been available, the ( + 1)th iteration calculates according to the following formula.
The iterative equation solving the inverse problem by conjugate gradient method includes where   is step length, obtained by searching some onedimensional line;   is the searching direction in which   = ∇(  ) and   is a scalar.Different   corresponds to different nonlinear conjugate gradient methods.The searching step length   , the conjugate coefficient   , and gradient ∇(  ) are needed to be found out.
Conjugate coefficient   equals the ratio of the square of normal form between the current and the previous-stepgradient paradigm: If the iteration step is  + 1 and searching step length is   , (37 where ∇  = (  / 1 ,   / 2 , . . .,   / +1 ).[22][23][24].For any real function (), a very small imaginary part ℎ is added to the real variable .It is expressed in complex function ( + ℎ) and its Taylor series is

Complex Variable Derivation Method
As ℎ is very small, where   (  )/  is sensitivity coefficient and its matrix form is The complex variable derivation method is used to calculate the matrix of sensitivity coefficient.

Mathematical Problems in Engineering
The complex variable derivation method is introduced into the traditional conjugate gradient method, which makes the calculation of the sensitivity coefficient accurate and avoids the sensitivity problem and the adjoint problem.

Solving Process of the Inverse Problem.
Initialization:  = 0,  0 ,  0 = 0, and let  be a small positive number.
Solve the forward problem given by ( 1), calculate temperature   , and judge if the following condition is true: The equation means the condition required by the instruction DO WHILE has not been satisfied.
Calculating the gradient ∇ using (41) Calculating the searching-down direction   using (27) Calculating the searching step length   using (43) Calculating the new estimation using (34) Solving the forward problem given by ( 1) and obtaining   : End do

The Instance Calculation and Analysis
To testify the availability of the aforementioned method, the simulation experiment having two sets of data was designed.
The experiment discussed the effect of the testing points' number and the measuring error on the inversion results and compared CGM and L-MM.The schematic diagram is shown as Figure 1.
In the experiment, the concrete slab's thickness is  = 0.3 m, and the thickness of casting concrete slab between two wooden templates is  1 =  2 = 0.02 m, and coefficient of thermal conductivity is  1 =  2 = 0.16 w/(mK), the air temperature is   = 293 K, and D1 and D3 are the convective heat conduction boundary conditions.The heat transfer coefficients are ℎ  = 8 W/(m 2 K) and ℎ  = 20 W/(m 2 K).The thermal conductivity coefficient is  = 1.5 W/(mK) and the power density of inner heat source is  = 1000 W/m 3 (similar to even-distributed and constant in short time).The real temperature distribution of the boundary D4 to be solved is According to the real temperatures, the calculated temperatures   of testing points on the known boundary D2 are counted through the forward question.
where  is standard-normal-distributed random number and  is the standard deviation of measuring temperatures.

The Influence of Measuring Points' Number on Inversion.
When the measurement error is zero,   = 293 K, ℎ  = 8 W/(m 2 K), and ℎ  = 20 W/(m 2 K).The thermal conductivity coefficient is  = 1.5 W/(mK) and the power density of inner heat source is  = 1000 W/m 3 .The influence of measuring points' number on inversion is shown as Figure 2. When the measuring point  = 5, 9, or 12, the inversion results reveal that the more points are measured, the more accuracy of inversion can be achieved.

The Influence of the Measuring Error on Inversion
The thermal conductivity coefficient is  = 1.5 W/(mK), and the power density of inner heat source is  = 1000 W/m 3 .The influence of different measuring error on inversion is shown as Figure 3.As the measuring standard deviation  = 0, 0.2, 0.4, the inversion results reveal that the method can obtain better inversion results for a relatively small measurement standard deviation.

The Comparison between CGM and L-MM.
This paper applies CGM and L-MM to calculate the inversion and compare with each other under three conditions:  = 0,  = 9, and  = 0,  = 12, and  = 12,  = 0.2 (other conditions are the same as the aforementioned).
The inversions under the conditions  = 0,  = 9, and  = 0,  = 12 are shown as Figures 4 and 5. Figures 4 and 5 show that the inversions using CGM and L-MM are similar and both have high accuracy.Both of them are satisfied.
The inversions under the conditions  = 0.2,  = 9 are shown as Figure 6.Comparing with Figure 5, the inversions using L-MM cause obvious fluctuation when the measuring error increases, while the inversions using CGM are satisfied.That means CGM has the better stability.

Conclusion
This paper applies CGM based on complex variable derivation to study the multivariable inverse problem which combines boundary distribution with convection coefficient of two-dimensional steady system with inner heat source.The inversions applying CGM and L-MM are compared.The influence of measuring points' number and measuring error on inversion is tested.The simulation reveals that applying boundary element method and conjugate gradient method to solve the inverse heat conduction problem is successful and has good stability.