Complexity

Volume 2018, Article ID 8783946, 9 pages

https://doi.org/10.1155/2018/8783946

## Inversion of Thermal Conductivity in Two-Dimensional Unsteady-State Heat Transfer System Based on Boundary Element Method and Decentralized Fuzzy Inference

^{1}School of Control and Mechanical Engineering, Tianjin Chengjian University, Tianjin 300384, China^{2}School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Shoubin Wang; moc.621@008nibsw

Received 2 January 2018; Revised 9 April 2018; Accepted 18 April 2018; Published 22 May 2018

Academic Editor: Lucia Valentina Gambuzza

Copyright © 2018 Shoubin Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on the boundary element method and the decentralized fuzzy inference algorithm, the thermal conductivity in the two-dimensional unsteady-state heat transfer system changing with the temperature is deduced. The more accurate inversion results are obtained by introducing the variable universe method. The concrete method is as follows: using experimental means to obtain the instantaneous temperature in the material or on the boundary, to determine the thermal conductivity of the material by solving the inversion problem. The boundary element method is used to calculate the regional boundary and internal temperature in the direct problem. With the inversion problem, the decentralized fuzzy inference algorithm is used to compensate for the initial guess of the thermal conductivity by using the difference between the temperature measurement and the temperature calculation. In the inversion problem, the influence of the initial guess of different thermal conductivities, different numbers of measuring points, and the existence of measurement errors on the results is discussed. The example calculation and analysis prove that, with different initial guesses, existence of measurement errors, and the number of boundary measurements decrease, the methods adopted in this paper still maintain good validity and accuracy.

#### 1. Introduction

Inversion heat transfer problems refer to the fact that some of the output information of the heat transfer system is obtained through experimental methods to invert some structural features or input information in the system. For example, the inversion of information such as the shape of the temperature boundary layer, the thermal conductivity of the material, and the heat flux density are all typical inversion heat conduction problems. Inversion heat transfer problems have been widely used in many fields such as nondestructive testing, geometrical shape optimization, aerospace engineering, power engineering, mechanical engineering, constructional engineering, bioengineering, metallurgical engineering, material processing, biological medicine, and food engineering, all of which achieve great success [1–13]. For inversion heat conduction problems, a lot of researches have been done by scholars at home and abroad.

The boundary element method and the complex variable derivation method are applied by Yu to invert the thermal conductivity of heterogeneous materials, which can effectively identify the thermal conductivity of single or multiple parameters [14]. When the heat conduction boundary value of the stability boundary is inverted by Yaparova, Laplace and Fourier transforms [15] are applied. The boundary element method is used to analyze two-dimensional transient conduction problems by Zhou et al. and the conjugate gradient method is introduced to solve the thermal conduction coefficient, which verify the effectiveness and stability of this method [16]. Mera et al. use iterative BEM to generate a stable numerical solution, which increases the number of boundary elements and reduces the amount of noise added in the input data [17]. Chen and Tanaka use a coupling application of the dual reciprocity boundary element method and dynamic programming filter to some inversion heat conduction problems [18]. A new and simple boundary element method is presented by Gao and Wang; this method is called interface integral boundary element method for solving heat conduction problems consisting of multiple media [19]. Wang et al. apply a nonsingular indirect boundary element method for the solution of three-dimensional inversion heat conduction problems. The exact geometrical representation of computational domain is adopted by parametric equations to eliminate the errors in traditional approaches of polynomial shape functions [20].

The differential transformation is studied and a stable differential calculus method is proposed to solve the inversion heat conduction problem [21] by Baranov et al. A new method to invert the thermal conductivity of material with temperature is proposed Miao et al., by which the temperature of measurement point is obtained by finite element method, and the residual between calculated value and measured value of temperature at the measurement point is minimized to get numerical solution, proving the effectiveness and accuracy of the algorithm [22].

The thermal conductivity of material with temperature changes is piecewise discrete by temperature range by Tang et al., and the genetic algorithms and the adjoint equation are used to carry out the inversion [23] of the thermal conductivity of the full temperature range. Based on the semi-infinite one-dimensional thermal model, the thermal conductivity inversion algorithm is studied by Lei; by changing the mathematical model, different intensity of noise is simulated, and the impact of noise on the accuracy of inversion is observed and studied; the method to improve the accuracy is proposed [24].

The decentralized fuzzy inference algorithm is successfully applied to the unsteady-state heat transfer system by Ran, which shows good anti-ill-posedness and obvious advantages and effectiveness [25]. In this paper, the boundary element method is used to solve the boundary and internal temperature in the two-dimensional unsteady-state heat transfer system, and the decentralized fuzzy inference algorithm is used to compensate for the initial guess of the thermal conductivity in order to minimize the residual between the calculated and the measured values of the temperature, and the true thermal conductivity is obtained.

#### 2. Direct Heat Conduction Problem

##### 2.1. The Boundary Integral Equation

The mathematical model of the two-dimensional unsteady-state heat transfer problem [26]:

In the mathematic model formula, there are , and the boundary of domain , which meets . And is the thermal conductivity coefficient , is the density of the object, and is the specific heat capacity of the object. And is the heat transfer coefficient of the object changing with temperature, is the temperature, and is the outer normal vector of the boundary. And is the ambient temperature, and is the heat flux density. The letter with “−” denotes the known quantity.

Weight function is introduced into the expression of weighted residual of governing equation [27].

The left side of the equation is decomposed to get:

In Green’s theorem for the Laplacian, of which is the boundary curve of plane closed Region and is the arc differential.

According to Laplace Green function, the following is obtained:

Equation (4) is taken into (3) and (2) becomes

It is further simplified, and is transformed to get

One Integration by parts () is carried out in the equation for to get

Equation (6) is taken into (5) to get

Because of and , , is discomposed and taken into (8) to get

The corresponding basic solution to this formula iswhere is the dimensionality of space, for two-dimensional problem, .

Differential derivation of (10) is done to get

In the formula, is the vertical distance from the source point “” to the boundary element line.

The basic solution has the following characteristics:

Equations (12) and (11) are taken into (10) and a good merger of similar items is done to get

##### 2.2. The Boundary Element Equation

The change of the function , over time is small enough to be negligible compared to that of , , which can be reasonably approximated as a constant over small time intervals, and (13) can be segmented into time integration [28].

And the interval integral for is

In the formula, is the exponential integral function, which can be calculated by the series, which is

In the formula, is Euler function, , for ; generally the first five approximations are taken.

According to the above formula, (17) can be written as

For the spatial domain division, the boundary is divided into units and the domain is divided into units. Equation (17) becomes

Linear element interpolation is adopted and the interpolation function of linear element is . Therefore, the boundary curve is approximated by a straight line. The values of and in the unit are approximated by the linearity with two endpoint values.

Equation (18) is done as

In the formula, . The formula is written in matrix form:

and at the boundary node can be obtained by (21). Take and the inner point temperature is obtained by (13) and (18).

#### 3. Decentralized Fuzzy Inversion

##### 3.1. Inversion of Thermal Conductivity

The thermal conductivity inverted in this paper varies with the temperature of the material, the function of which is known. By the known measured temperature at a particular measurement point, the inversion algorithm is used to determine the constant coefficients of the function.

The difference between the temperature measurement and the temperature calculation is taken as the objective function, which is minimized as

is the inversion parameter in the objective function; and , respectively, represent the temperature measurement and the temperature calculation at the measuring point at time ; is the number of temperature measuring points; is the number of future time steps; and the minimum value of the objective function is calculated as the parameter vector in the inversion problem.

##### 3.2. Decentralized Fuzzy Inference Method

The difference between the temperature measurement and the temperature calculation is used to correspondingly compensate for the initial guess of the thermal conductivity. Therefore, a multiple-input multiple-output fuzzy inference system is established. Each independent measurement point is a single fuzzy inference unit (FIU). Fuzzy inference unit is shown in Figure 1. The independent fuzzy inference controller is divided into four parts: fuzzy interface, knowledge base, inference engine, and fuzzy decision interface (defuzzy).