Abstract
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 of measuring error and the number of measuring points zero error which impact the measurement result are discussed and compared with L-MM method in inverse problems. Instance calculation and analysis prove that the method 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.
1. 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 [1–12]. 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.
2. 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 iswhere , , and are the boundary of the domain , , , is the internal heat source, is the heat conduction coefficient between solid and fluid, , temp, and is the normal vector outside the boundary. , 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 , 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, . Then the reduced result on the left side of (7) is
Because , is decomposed:
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 aswhere is the new circular border and is the boundary outside the new circular boundary in . Considering the smooth boundary, the integral mean value theorem is introduced. where is the point on . Let ; thenSynthesizing the above limit results, we haveConsidering (15) and (19) simultaneously, boundary integral equation of any point in the domain or on the boundary can be found out:
2.2. Calculation of the Unknown Value Based on Boundary Element Method
Discreteness of boundary integral equations: because the quantities and on the boundary are known, the only thing to do is divide the boundary discretely. Boundary is divided into boundary units , where there are units on the boundary , units on the boundary , and units on the boundary . .
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 aswhere is the interpolation node number in a boundary unit and is the interpolation function. When is in , (19) can be written asThis paper uses linear interpolation. Because , 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), whereLet and we haveSimilarly, when the point locates in , (24) becomesLetand the boundary integral equation is
Rearranging (30), we havewhere considering , (31) can be expressed as a matrix form . The boundary conditions and are written as , , where , are known temperature and heat flux. Equation (31) becomesPut the known quantities on the left.Appling the radial integration method, domain integration caused by heat source is converted to boundary integration [19–21].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 .
3. The Steady Inverse Problem
3.1. The Objective Function of Inverse Problem
For aforementioned heat conduction system, the heat transfer coefficient between solid and fluid 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 is the measurement of temperature. The minimum of the objective function is the parameter vector of inverse problem.
3.2. 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 th iteration calculates according to the following formula.
The iterative equation solving the inverse problem by conjugate gradient method includeswhere is step length, obtained by searching some one-dimensional 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-step-gradient paradigm:If the iteration step is and searching step length is , (37) giveswhere .
3.3. Complex Variable Derivation Method [22–24]
For any real function , a very small imaginary part is added to the real variable . It is expressed in complex function and its Taylor series isAs is very small,where is sensitivity coefficient and its matrix form isThe complex variable derivation method is used to calculate the matrix of sensitivity coefficient.
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.
3.4. Solving Process of the Inverse Problem
Initialization: , , 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
4. 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 m, and the thickness of casting concrete slab between two wooden templates is m, and coefficient of thermal conductivity is w/(mK), the air temperature is K, and D1 and D3 are the convective heat conduction boundary conditions. The heat transfer coefficients are W/(m2 K) and W/(m2 K). The thermal conductivity coefficient is W/(mK) and the power density of inner heat source is W/m3 (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.
4.1. The Influence of Measuring Points’ Number on Inversion
When the measurement error is zero, K, W/(m2 K), and W/(m2 K). The thermal conductivity coefficient is W/(mK) and the power density of inner heat source is W/m3. 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.
4.2. The Influence of the Measuring Error on Inversion
, K, W/(m2 K), and W/(m2 K). The thermal conductivity coefficient is W/(mK), and the power density of inner heat source is W/m3. The influence of different measuring error on inversion is shown as Figure 3. As the measuring standard deviation , the inversion results reveal that the method can obtain better inversion results for a relatively small measurement standard deviation.
4.3. 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: , , and , , and , (other conditions are the same as the aforementioned).
The inversions under the conditions , , and , 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 , 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.
5. 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was financially supported by the National Key Foundation for Exploring Scientific Instrument of China (2013YQ470767), Tianjin Science and Technology Committee for Science and Technology Development Strategy Research Project (15ZLZLZF00350), Tianjin Science and Technology Commissioner Project (16JCTPJC53000), Science and Technology Plan Project of Tianjin Science and Technology Committee (16ZLZXZF00270), and the 13th Five-Year Plan (2016–2020) of Science Education Project in Tianjin City (HE1017).