传热学反问题模糊推理方法的继续研究
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摘要
传热学反问题（Inverse Heat Transfer Problems，IHTP）是根据传热系统的部分输出信息反向求解热物性参数、几何形状、边界条件等未知参数。IHTP广泛存在于航空航天工程、动力工程、机械工程、建筑工程、生物医疗工程等领域。对传热学反问题开展深入的研究具有极其重要的科学和工程价值。
IHTP为一类典型的不确定性推理问题。传统的反演方法可归类为确定性推理方法，利用传统反演方法研究传热学反演问题不可避免的存在不足。分散模糊推理（DFI）方法是建立在模糊集合理论基础上的一种典型的不确定性推理方法，该方法对输入信息具有明显的抗干扰能力，可以有效利用不精确、不确定及不完备的输入信息进行推理和决策，表现出较好的抗不适定性。本文在我们已有研究成果的基础上，对应用DFI方法求解传热学反问题进行了更加深入的研究，主要研究内容及成果包括以下五个方面：
①以圆柱表面热流分布反演问题为例，建立基于灵敏度加权的分散模糊推理（SDFI）方法。通过数值试验，讨论了待反演热流分布的初始猜测值、观测点数目、测量误差以及测量误差与观测点数目耦合等条件对反演结果的影响，并与共轭梯度法（CGM）和遗传算法（GA）进行了对比，在此基础之上，总结了DFI方法的有效性和优越性。
②研究了模糊论域的选取问题，针对依赖于专家经验的固定论域DFI方法及现有变论域DFI方法所具有的局限性，借助目标函数的收敛特性对模糊论域进行自适应调整，提出新的变论域分散模糊推理（VDFI）方法。以二维平板边界温度分布反演为例，讨论了不同论域对反演结果的影响，并与定论域DFI方法及现有变论域DFI方法进行了比较，讨论了测量误差及测点数目对文VDFI方法反演结果的影响，证明了VDFI方法的有效性和优越性。
③研究了SDFI方法的综合协调问题，以三维平板传热反问题为例，指出了SDFI的反演结果容易受边界条件影响的不足，针对此问题，根据传热系统的局部影响特性，提出了基于空间正态分布加权的分散模糊推理（SND-DFI）方法。利用SDN-DFI反演了三维平板表面对流换热系数，系统讨论了方差参数对SND-DFI反演结果的影响，给出了方差参数的选择范围，同时也讨论了测量误差对反演结果的影响，并与SDFI方法的反演结果进行了对比，证明了SND-DFI方法的有效性和优越性。
④根据传热系统的局部影响特性，提出了基于测量空间分解的分散模糊推理（MSD-DFI）方法。该方法是针对具有明显空间分布特性、待反演参数及测点数较多的传热学反问题提出的。对于此类传热系统，DFI的加权矩阵通常比较庞大且难以建立。MSD-DFI方法不需建立综合协调矩阵，其基本思想为：通过分析传热学正问题，对测量空间（即测量信息）在空间域上进行分解，为每一个待反演参数构造一个测量子空间；在每个测量子空间内进行模糊推理，获得推理输出；最后，利用本文提出的模糊解耦方案，对模糊推理输出进行解耦协调，实现对待反演参数的反演。应用MSD-DFI方法研究了加热炉内壁温度分布反演问题，讨论了初始猜测值、测量误差对反演结果的影响，并与SDFI方法进行对比，说明了MSD-DFI的有效性。
⑤将DFI方法应用于非稳态传热学反问题研究。针对顺序函数法（SFSM）求解传热学反问题时反演结果严重依赖未来时间信息、最优未来时间步难以准确获得、对测量误差敏感等问题，在应用DFI研究稳态传热反问题基础上，通过对测量信息在时间域上进行分散与综合协调，提出了求解非稳态传热反问题的DFI方法。应用DFI研究了一维平板表面热流反演问题，并与SFSM进行对比，讨论了未来时间步、最优未来时间步、测量误差以及测点位置对反演结果的影响。试验结果表明，DFI能更加有效地利用未来时间的测量信息反演平板表面热流，降低了对未来时间信息的敏感性，即使利用非最优未来时间的测量信息也能获得非常好的反演结果；DFI能够有效利用不精确、不确定的信息进行推理和决策的优势得以体现，降低了对测量误差的敏感程度，表现出更好的抗不适定性。
The inverse heat transfer problems (IHTP) involves the estimation of the unknownparameters, such as the thermophysical properties, the geometrical shape, the boundaryconditions, based on the output information of the heat transfer systems. The IHTP iswidespread in the fields of aerospace, power engineering, mechanical engineering,constructional engineering and biomedical Engineering. It is of important scientific andengineering significance to further research the IHTP.
IHTP is a typical uncertain reasoning subject, the traditional optimizationalgorithms are can be classed as certain methods. The deficiencies are unavoidablewhen these traditional optimization algorithms are applied to research the IHTP. Thedecentralized fuzzy inference (DFI) method is a typical uncertain reasoning methodbased on the fuzzy set theory. The DFI method owns strong capacity of resistingdisturbance to input information. It can effectively use the imprecise and incompleteinformation to perform the reasoning process and possesses better anti-ill-posedcharacter. In this paper, the DFI method for IHTP is further researched on the basis ofour prior research, and the main works of this paper are as follows:
①Take the problem of determining the heat flux on the surface of a cylinder forexample. The sensitivity weighting decentralized fuzzy inference (SDFI) method isapplied to solve this IHTP. Numerical experiments are conducted, the influence of theinitial guesses of the unknown heat flux, the measurement errors, and the couplingeffects of measurement errors and measurement points number on the inversion resultsare discussed. Comparisons with the conjugate gradient method (CGM) and the geneticalgorithm (GA) are also conducted. Finally, the validity and superiority of SDFI aresummarized.
②The determining of the universes of discourse of DFI is researched. For thelimitations of the DFI method based on the fixed universes of discourse scheme relyingon the expertise and the existing DFI method based on the variable universes ofdiscourse scheme, the converging character of the objective function is considered to setup the self-adaptive adjustment strategy for the universes of discourse, and a new DFImethod based variable universes of discourse, the VDFI method, is proposed. The VDFIis performed to image the temperature boundary of a two-dimensional flat. The effect ofdifferent universes of discourse on the inverse results are researched, comparisons with the DFI method based on the fixed and the existing variable universes of discourse areconducted. The results show the practicality of our VDFI method. The influence of themeasurement errors and measurement points number on the inversion results are studied,and the validity and superiority of our VDFI method are proved.
③The synthetic issue of SDFI is studied, taking a three-dimensional IHTP forexample, the shortcoming that the inverse results are sensitive to the boundaryconditions of SDFI method is point out. For this problem, we propose the spatialnormal distribution weighting DFI (SND-DFI) method. The SND-DFI method isapplied to estimate the convective heat transfer coefficient of a three-dimensional flat.The effects of variance parameters on the SND-DFI method are detailed and the suitableranges of variance parameters are given. The influence of the measurement errors on theinversion results are studied, comparisons with the SDFI method are researched. Theresults show the validity and superiority of our SND-DFI method.
④The DFI method based on the measured space decomposition (MSD-DFImethod) is presented relying on the local influence characteristic of heat transfer system.The MSD-DFI is proposed for the IHTP that the local effect characteristic is apparent,and the number of unknown parameters and measurement points are large. For this kindof heat transfer systems, the synthesizing matrix is large and often difficult to set up.The MSD-DFI method has no need of the synthesizing matrix. The basic ideas ofMSD-DFI method are: Firstly, the measured space (namely, the measured information)decomposition is conducted according to the local influence by analyzing the direct heattransfer problem, and the measured subspaces corresponding to every estimatedparameter is built; Then, the fuzzy inference are performed for each measuredsubspaces to obtain the fuzzy inference outputs; Finally, the fuzzy decoupling algorithmproposed in this paper are applied to the fuzzy inference outputs, and the inverseprocess is accomplished. The MSD-DFI method is applied to estimate the temperaturedistribution of furnace inner surface. The influences of the initial guesses of thetemperature distribution and the measurement errors on the inversion results areresearched. Comparisons with the SDFI method are researched. The results show thevalidity of MSD-DFI method.
⑤The DFI method is applied to the unsteady IHTP. For the problems that theinverse results are depending on the number of future time steps, and the optimalnumber of future time steps is difficult to obtain, and the inverse results are sensitive tothe measurement errors when the Sequential Function Specification Method (SFSM) is used for solving the unsteady IHTP, the DFI method for the unsteady IHTP bydecomposing and synthesizing the measured information in the temporal domain basedon our previous study on the steady IHTP. The heat flux of one-dimensional flat isdetermined by the DFI method, the influence of the number of future time steps, theoptimal number of future time steps, measurement errors and measurement position onthe estimated results are discussed. Comparisons with the SFSM are discussed. Theresults show that the DFI can more effective use the measured infoemation in the futuretime to estimate the heat flux availabl, even without the optimal number of future timesteps, and significantly reduces the dependence of the estimated results on the numberof future time steps, and also weakens the effects of measurement errors. The DFIpossesses higher accuracy than the SFSM. The advantages that DFI can effectivelyutilize imprecise, uncertain and incomplete information to infer and make strategicdecisions are reflected, DFI possesses better anti-ill-posed character.
IHTP为一类典型的不确定性推理问题。传统的反演方法可归类为确定性推理方法，利用传统反演方法研究传热学反演问题不可避免的存在不足。分散模糊推理（DFI）方法是建立在模糊集合理论基础上的一种典型的不确定性推理方法，该方法对输入信息具有明显的抗干扰能力，可以有效利用不精确、不确定及不完备的输入信息进行推理和决策，表现出较好的抗不适定性。本文在我们已有研究成果的基础上，对应用DFI方法求解传热学反问题进行了更加深入的研究，主要研究内容及成果包括以下五个方面：
①以圆柱表面热流分布反演问题为例，建立基于灵敏度加权的分散模糊推理（SDFI）方法。通过数值试验，讨论了待反演热流分布的初始猜测值、观测点数目、测量误差以及测量误差与观测点数目耦合等条件对反演结果的影响，并与共轭梯度法（CGM）和遗传算法（GA）进行了对比，在此基础之上，总结了DFI方法的有效性和优越性。
②研究了模糊论域的选取问题，针对依赖于专家经验的固定论域DFI方法及现有变论域DFI方法所具有的局限性，借助目标函数的收敛特性对模糊论域进行自适应调整，提出新的变论域分散模糊推理（VDFI）方法。以二维平板边界温度分布反演为例，讨论了不同论域对反演结果的影响，并与定论域DFI方法及现有变论域DFI方法进行了比较，讨论了测量误差及测点数目对文VDFI方法反演结果的影响，证明了VDFI方法的有效性和优越性。
③研究了SDFI方法的综合协调问题，以三维平板传热反问题为例，指出了SDFI的反演结果容易受边界条件影响的不足，针对此问题，根据传热系统的局部影响特性，提出了基于空间正态分布加权的分散模糊推理（SND-DFI）方法。利用SDN-DFI反演了三维平板表面对流换热系数，系统讨论了方差参数对SND-DFI反演结果的影响，给出了方差参数的选择范围，同时也讨论了测量误差对反演结果的影响，并与SDFI方法的反演结果进行了对比，证明了SND-DFI方法的有效性和优越性。
④根据传热系统的局部影响特性，提出了基于测量空间分解的分散模糊推理（MSD-DFI）方法。该方法是针对具有明显空间分布特性、待反演参数及测点数较多的传热学反问题提出的。对于此类传热系统，DFI的加权矩阵通常比较庞大且难以建立。MSD-DFI方法不需建立综合协调矩阵，其基本思想为：通过分析传热学正问题，对测量空间（即测量信息）在空间域上进行分解，为每一个待反演参数构造一个测量子空间；在每个测量子空间内进行模糊推理，获得推理输出；最后，利用本文提出的模糊解耦方案，对模糊推理输出进行解耦协调，实现对待反演参数的反演。应用MSD-DFI方法研究了加热炉内壁温度分布反演问题，讨论了初始猜测值、测量误差对反演结果的影响，并与SDFI方法进行对比，说明了MSD-DFI的有效性。
⑤将DFI方法应用于非稳态传热学反问题研究。针对顺序函数法（SFSM）求解传热学反问题时反演结果严重依赖未来时间信息、最优未来时间步难以准确获得、对测量误差敏感等问题，在应用DFI研究稳态传热反问题基础上，通过对测量信息在时间域上进行分散与综合协调，提出了求解非稳态传热反问题的DFI方法。应用DFI研究了一维平板表面热流反演问题，并与SFSM进行对比，讨论了未来时间步、最优未来时间步、测量误差以及测点位置对反演结果的影响。试验结果表明，DFI能更加有效地利用未来时间的测量信息反演平板表面热流，降低了对未来时间信息的敏感性，即使利用非最优未来时间的测量信息也能获得非常好的反演结果；DFI能够有效利用不精确、不确定的信息进行推理和决策的优势得以体现，降低了对测量误差的敏感程度，表现出更好的抗不适定性。
The inverse heat transfer problems (IHTP) involves the estimation of the unknownparameters, such as the thermophysical properties, the geometrical shape, the boundaryconditions, based on the output information of the heat transfer systems. The IHTP iswidespread in the fields of aerospace, power engineering, mechanical engineering,constructional engineering and biomedical Engineering. It is of important scientific andengineering significance to further research the IHTP.
IHTP is a typical uncertain reasoning subject, the traditional optimizationalgorithms are can be classed as certain methods. The deficiencies are unavoidablewhen these traditional optimization algorithms are applied to research the IHTP. Thedecentralized fuzzy inference (DFI) method is a typical uncertain reasoning methodbased on the fuzzy set theory. The DFI method owns strong capacity of resistingdisturbance to input information. It can effectively use the imprecise and incompleteinformation to perform the reasoning process and possesses better anti-ill-posedcharacter. In this paper, the DFI method for IHTP is further researched on the basis ofour prior research, and the main works of this paper are as follows:
①Take the problem of determining the heat flux on the surface of a cylinder forexample. The sensitivity weighting decentralized fuzzy inference (SDFI) method isapplied to solve this IHTP. Numerical experiments are conducted, the influence of theinitial guesses of the unknown heat flux, the measurement errors, and the couplingeffects of measurement errors and measurement points number on the inversion resultsare discussed. Comparisons with the conjugate gradient method (CGM) and the geneticalgorithm (GA) are also conducted. Finally, the validity and superiority of SDFI aresummarized.
②The determining of the universes of discourse of DFI is researched. For thelimitations of the DFI method based on the fixed universes of discourse scheme relyingon the expertise and the existing DFI method based on the variable universes ofdiscourse scheme, the converging character of the objective function is considered to setup the self-adaptive adjustment strategy for the universes of discourse, and a new DFImethod based variable universes of discourse, the VDFI method, is proposed. The VDFIis performed to image the temperature boundary of a two-dimensional flat. The effect ofdifferent universes of discourse on the inverse results are researched, comparisons with the DFI method based on the fixed and the existing variable universes of discourse areconducted. The results show the practicality of our VDFI method. The influence of themeasurement errors and measurement points number on the inversion results are studied,and the validity and superiority of our VDFI method are proved.
③The synthetic issue of SDFI is studied, taking a three-dimensional IHTP forexample, the shortcoming that the inverse results are sensitive to the boundaryconditions of SDFI method is point out. For this problem, we propose the spatialnormal distribution weighting DFI (SND-DFI) method. The SND-DFI method isapplied to estimate the convective heat transfer coefficient of a three-dimensional flat.The effects of variance parameters on the SND-DFI method are detailed and the suitableranges of variance parameters are given. The influence of the measurement errors on theinversion results are studied, comparisons with the SDFI method are researched. Theresults show the validity and superiority of our SND-DFI method.
④The DFI method based on the measured space decomposition (MSD-DFImethod) is presented relying on the local influence characteristic of heat transfer system.The MSD-DFI is proposed for the IHTP that the local effect characteristic is apparent,and the number of unknown parameters and measurement points are large. For this kindof heat transfer systems, the synthesizing matrix is large and often difficult to set up.The MSD-DFI method has no need of the synthesizing matrix. The basic ideas ofMSD-DFI method are: Firstly, the measured space (namely, the measured information)decomposition is conducted according to the local influence by analyzing the direct heattransfer problem, and the measured subspaces corresponding to every estimatedparameter is built; Then, the fuzzy inference are performed for each measuredsubspaces to obtain the fuzzy inference outputs; Finally, the fuzzy decoupling algorithmproposed in this paper are applied to the fuzzy inference outputs, and the inverseprocess is accomplished. The MSD-DFI method is applied to estimate the temperaturedistribution of furnace inner surface. The influences of the initial guesses of thetemperature distribution and the measurement errors on the inversion results areresearched. Comparisons with the SDFI method are researched. The results show thevalidity of MSD-DFI method.
⑤The DFI method is applied to the unsteady IHTP. For the problems that theinverse results are depending on the number of future time steps, and the optimalnumber of future time steps is difficult to obtain, and the inverse results are sensitive tothe measurement errors when the Sequential Function Specification Method (SFSM) is used for solving the unsteady IHTP, the DFI method for the unsteady IHTP bydecomposing and synthesizing the measured information in the temporal domain basedon our previous study on the steady IHTP. The heat flux of one-dimensional flat isdetermined by the DFI method, the influence of the number of future time steps, theoptimal number of future time steps, measurement errors and measurement position onthe estimated results are discussed. Comparisons with the SFSM are discussed. Theresults show that the DFI can more effective use the measured infoemation in the futuretime to estimate the heat flux availabl, even without the optimal number of future timesteps, and significantly reduces the dependence of the estimated results on the numberof future time steps, and also weakens the effects of measurement errors. The DFIpossesses higher accuracy than the SFSM. The advantages that DFI can effectivelyutilize imprecise, uncertain and incomplete information to infer and make strategicdecisions are reflected, DFI possesses better anti-ill-posed character.
引文
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