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The Inverse Heat Conduction Problem (IHCP) refers to the inversion of the internal characteristics or thermal boundary conditions of a heat transfer system by using other known conditions of the system and according to some information that the system can observe. It has been extensively applied in the fields of engineering related to heat-transfer measurement, such as the aerospace, atomic energy technology, mechanical engineering, and metallurgy. The paper adopts Finite Difference Method (FDM) and Model Predictive Control Method (MPCM) to study the inverse problem in the third-type boundary heat-transfer coefficient involved in the two-dimensional unsteady heat conduction system. The residual principle is introduced to estimate the optimized regularization parameter in the model prediction control method, thereby obtaining a more precise inversion result. Finite difference method (FDM) is adopted for direct problem to calculate the temperature value in various time quanta of needed discrete point as well as the temperature field verification by time quantum, while inverse problem discusses the impact of different measurement errors and measurement point positions on the inverse result. As demonstrated by empirical analysis, the proposed method remains highly precise despite the presence of measurement errors or the close distance of measurement point position from the boundary angular point angle.

The Inverse Heat Conduction Problem is able to retrieve the unknown parameters such as boundary conditions, material thermophysical parameters [

Regarding the boundary heat transfer in the heat conduction system, in the paper, FDM is adopted to solve the direct problem of the two-dimensional unsteady-state heat conduction without internal heat source and model prediction control method is used to solve the inverse problem. Besides, residual principle is introduced to optimize the regularization parameter during the inversion process, thereby improving the efficiency of inversion in terms of speed and time.

The Inverse Heat Conduction Problem usually involves the multiple deduction of the forward problem, and its inversion accuracy is directly affected by the calculation accuracy of the forward problem. Positive problem refers to the solution of historical temperature field through given boundary conditions, initial temperature, and thermal conductivity differential equation. Common solutions are Lattice Boltzmann Method, Finite Volume Method, Adomain Decomposition Method, Boundary Element Method, and Finite Difference Method.

LBM (Lattice Boltzmann Method) [

The basic idea of FVM (Finite Volume Method) [

ADM (Adomain Decomposition Method) [

BEM (Boundary Element Method) [

In this paper, a square rectangular plate is selected as an experimental physical model, which is a very common physical model. The finite-difference method [

The mathematical model of two-dimensional unsteady-state heat conduction without internal heat source is expressed as follows.

where

The discrete rules of two-dimensional unsteady-state heat conduction problem without internal heat source in geometry and time domain are as follows.

Assuming that after the domain of uniform discretization, the step length of

n(

The explicit difference array of the two-dimensional unsteady-state heat conduction without internal heat source is expressed as follows.

Applying the first heat conduction equation in (

The partial differential in the two sides of (

The second-order partial differential in the left of the equal sign can be approximated by the central difference quotient.

Substituting (

Equation (

Assuming that

where

The stability condition of explicit finite difference equation of two-dimensional unsteady-state heat conduction without internal heat source is in interior node,

First-type boundary condition, i.e., the temperature, is given. In general, when the FDM is used for calculation, it shall be processed as in the moment of the initial, the boundary node temperature is

In the second- and third-type boundary condition, it is necessary to set virtual node outside the boundary to make the boundary node into interior node. The node numbering is shown in Figure

Second-type boundary condition, i.e., the heat flow boundary, is given,

(

Node 1 is changed into interior node and (

Substituting (

Third-type boundary condition, i.e., the heat transfer boundary, is given,

(

Node 1 is changed into interior node and (

Substituting (

where

Adiabatic boundary condition is

Boundary angular point is 0 node. Virtual nodes 1’ and 3’ are set in the symmetric position of node 1 and node 3, respectively, and the central different quotient is applied in the

Equation (

Node 1 is changed into interior node and (

By (

Figure

Figures

The length Lx and width Ly of the plate is

Predictive control is a model-based control algorithm, which focuses on the function of the model rather than the form of the model. Compared with other control methods, its characteristics are reflected in the use of rolling optimization and rolling implementation of the control mode to achieve the control effect, but also did not give up the traditional control feedback. Therefore, the predictive control algorithm is based on the future dynamic behavior of the process model prediction system under a certain control effect, uses the rolling optimization to obtain the control effect under the corresponding constraint conditions and performance requirements, and corrects the prediction of future dynamic behavior in the rolling optimization process by detecting real-time information.

The step response of heat-transfer coefficient in the

According to the principle of linear superposition [

Equation (

Equation (

Where,

The step response system function from the time moment

From (

The corresponding discrete value

Measurement value and predictive value of temperature can be seen in the time range from

where

After the derivation of

The optimal heat-transfer coefficient at moment

where

The residual principle [

To invert the boundary heat-transfer coefficient, it is necessary to firstly solve the direct problem using the predictive value of heat-transfer coefficient, to get the temperature calculation at

where

where the constant K is the number of iterations.

The residual of heat-transfer coefficient in the whole inversion time domain is defined as

In (

Since

In ideal condition,

Similarily,

From the residual principle, the regularization parameter is the optimal when both (

Numerical experiments are performed to validate whether the proposed method is effective, with the focus on analyzing the impact of different measurement errors and measurement point positions on the inversion result. Also, the inversion result obtained in the condition without measurement error is compared with the practical result, which verifies the precision of the proposed method.

The two-dimensional plate heat transfer model (Figure _{4}.

Heat conduction model.

Boundary node.

The model of two-dimensional unsteady-state heat conduction system without internal heat source.

The temperature field in the t=50s.

The temperature field in the t=100s.

The temperature field in the t=150s.

The temperature field in the t=200s.

The curve of the measuring point with time.

Given the measurement error _{4} boundary and the future time step

The heat transfer coefficient of the measuring point without error.

Figure

Given the future time step _{4} boundary, the inversion results when the measurement error is

The heat transfer coefficient of the measuring point with

The heat transfer coefficient of the measuring point with

The heat transfer coefficient of the measuring point with

According to Table

Relative average errors of inversion result under different measurement errors given

Measurement error | 0.001 | 0.005 | 0.01 |

| |||

Relative average error | 7.01% | 11.29% | 16.01% |

Given measurement error _{4} boundary and when the measurement point is _{4} boundary are shown in Figures

The heat transfer coefficient of the measuring point at the boundary

The heat transfer coefficient of the measuring point at the boundary

The heat transfer coefficient of the measuring point at the distance from the boundary

Analyze the contents of Table

Relative average errors of inversion result in different measurement point positions given

Measurement point | | | |

| |||

Relative average error | 7.65% | 7.06% | 21.05% |

The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed by the FDM and model prediction control method. By solving and analyzing the algorithm example, it demonstrates that the proposed methods have higher accuracy in the inversion process. Model predictive control method focuses on the model function rather than the structural form, so that we only need to know the step response or impulse response of the object; we can directly get the prediction model and skip the derivation process. It absorbs the idea of optimization control and replaces global optimization by rolling time-domain optimization combined with feedback correction, which avoids a lot of calculation required by global optimization and constantly corrects the influence caused by uncertain factors in the system. At the same time, by discussing the impacts of error free, measuring point positions, and measuring errors on the results, it demonstrates that the obtained inversion results, except the early oscillation, can better represent the stability of the exact solution.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Shoubin Wang and Rui Ni contributed to developing the ideas of this research. All of the authors were involved in preparing this manuscript.

This work was financially supported by the National Key Foundation for Exploring Scientific Instrument of China (2013YQ470767), Tianjin Municipal Education Commission Project for Scientific Research Items (2017KJ059), and Tianjin Science and Technology Commissioner Project (18JCTPJC62200, 18JCTPJC64100).