多宗量热传导反问题的数值求解
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摘要
热传导反问题是指通过研究对象内部或者边界的温度相关信息,确定边界/初始条件、导热系数、内热源强度等宗量的未知部分,是一个涉及到传热学、物理、数学、计算机、实验技术等学科的交叉领域,在航空航天、核能工程、冶金铸造、化工领域、机械制造、无损探伤、疾病诊断、土木工程、冷冻储藏等工程中有许多重要应用。
由于热传导反问题的不适定性和非线性,使得其求解远比正问题复杂和困难。尽管目前对热传导反问题进行了大量的研究工作,并取得很多成果,但在理论、计算和应用上都需要进一步深入研讨。
考虑到实际问题中对待求宗量(导热系数、源项、边界/初始条件)的单一和组合反演需求,本文开展多宗量热传导反问题研究,对定常和非定常热传导反问题,分别建立了便于敏度分析的多宗量数值反演模型,提出了相应的求解策略和计算方法,并采用不同的敏度类算法进行求解,着重讨论了多宗量识别时一些因素对反演结果的影响,给出了令人满意的数值验证结果。目前多宗量热传导反问题的研究相对薄弱,相关文献很少。本文的工作,不仅为多宗量热传导反问题的理论研究,提供了有价值的参考,也为实际工程中多宗量热传导反问题的求解提供了可能的途径。
本文研究内容主要包括以下几个方面:
1 建立了定常多宗量热传导反问题以及湿热耦合反问题的数值求解模型,对导热系数/湿传输系数,以及边界条件进行单一和组合识别;对于非线性问题,提出了两级敏度分析的求解模式。
2 建立了一阶非定常多宗量热传导反问题的数值求解模型,对导热系数与边界条件进行单一和组合识别,时域上分别采用有限差分和精细算法进行正演数值分析。此外,对与时间相关的热源项也进行了单一和组合识别。
3 建立了二阶非定常多宗量热传导正/反问题的数值求解模型,对导热系数与边界条件进行单一和组合识别,时域上采用时域精细算法进行正演数值分析。
4 在以上所建模型中,考虑了非均质和参数空间分布的影响。在求解过程中,尝试采用不同的目标函数,如L_2范数、L_∞范数和基于D-函数的正则泛函;以及不同的敏度类算法,如高斯牛顿方法、同伦方法、同伦正则化方法、共轭梯度方法、BFGS方法等,取得了满意的数值结果。在多宗量情形下,探讨了所提算法的计算精度/效率和抗不适定性,以及误差水平、测点位置、时间步长和变量初值等因素对反演结果的影响,并与单宗量识别进行了比较,为多宗量热传导反问题的进一步研究,提供了有价值的参考。
The inverse heat conduction problem (IHCP) is usually defined as the estimations of boundary/initial conditions, thermal parameters and heat source by utilizing the known temperature measurements inside the body or on the surface. The study on IHCP is an interdisciplinary field related to the heat transfer, physics, mathematics, computing, and experiment technique, etc., and has significant applications in many engineering aspects, such as aerospace, nuclear engineering, metal casting, chemical, machine making, metal-lurgy, medical diagnostics, civil engineering and food science,etc.Due to the ill-posedness and nonlinearity, solving IHCP is usually much more difficult than solving direct heat conduction problem (DHCP). Although large amounts of achievement has been made in this area, further investigation and effort are greatly demanded.With the consideration of practical requirements of both single and combined identification of thermal parameters, heat source, and boundary/initial conditions, inverse heat conduction problems with multi-variables are investigated in this dissertation, a couple of numerical models, facilitating to the sensitivity analysis, are developed to solve inverse steady/transient heat conduction problems with multi-variables. By utilizing different sensitivity based algorithms, a number of numerical tests, considering the effects of some factors on the results, are carried out to verify the proposed models with satisfactory results. Since there seems to be quite few work directly relevant to inverse heat conduction problems with multi-variables by authors best knowledge, the work presented in this dissertation is not only theoretically valuable, but also practically applicable.The major work of this dissertation includes1 Numerical models solving inverse steady heat conduction and coupled heat-mass transfer problems with multi-variables are presented, and single/combined identifications are implemented for the thermal/moist parameters and boundary conditions etc. A solving strategy for nonlinear inverse steady heat conduction via 2-Clevel sensitivity analysis is presented.
2 A numerical model solving inverse one- order transient heat conduction problems with multi-variables is proposed, and single/combined identifications are carried out for the thermal parameters and boundary conditions etc. Finite difference technique and a precise algorithm in the time domain are employed in the time-dependent analysis. In addition, single/combined identifications are also carried out for the time-dependent heat source.3 Numerical models solving direct/inverse two-order transient heat conduction problems with multi-variables are presented, and single/combined identifications are conducted for the thermal parameters and boundary conditions etc. A precise algorithm in the time domain is employed in the time-dependent analysis.4 With the consideration of material inhomogeneity and spacial distribution of parameters, all the numerical models above are numerically verified with satisfactory results.A variety of object functions, such as L_2 ,L_x and D-function based regularizing functional,and a variety of sensitivity based algorithms, such as Gauss-Newton method, homotopy method, homotopy regularization method, conjugate gradient method. and Brayden-Fletcher-Goldfarb-Shanno (BFGS) method etc. are utilized to solve inverse problems. The computing accuracy/efficiency, and the ability of anti-noisy data as well are discussed. With the comparison with the case of single variable identification, a number of factors including noisy data, location of sample points, size of time step, and initial guess of unknowns are investigated with regard to their effects on the solutions, providing valuable reference for the further investigation on IHCP with multi-variables.
由于热传导反问题的不适定性和非线性,使得其求解远比正问题复杂和困难。尽管目前对热传导反问题进行了大量的研究工作,并取得很多成果,但在理论、计算和应用上都需要进一步深入研讨。
考虑到实际问题中对待求宗量(导热系数、源项、边界/初始条件)的单一和组合反演需求,本文开展多宗量热传导反问题研究,对定常和非定常热传导反问题,分别建立了便于敏度分析的多宗量数值反演模型,提出了相应的求解策略和计算方法,并采用不同的敏度类算法进行求解,着重讨论了多宗量识别时一些因素对反演结果的影响,给出了令人满意的数值验证结果。目前多宗量热传导反问题的研究相对薄弱,相关文献很少。本文的工作,不仅为多宗量热传导反问题的理论研究,提供了有价值的参考,也为实际工程中多宗量热传导反问题的求解提供了可能的途径。
本文研究内容主要包括以下几个方面:
1 建立了定常多宗量热传导反问题以及湿热耦合反问题的数值求解模型,对导热系数/湿传输系数,以及边界条件进行单一和组合识别;对于非线性问题,提出了两级敏度分析的求解模式。
2 建立了一阶非定常多宗量热传导反问题的数值求解模型,对导热系数与边界条件进行单一和组合识别,时域上分别采用有限差分和精细算法进行正演数值分析。此外,对与时间相关的热源项也进行了单一和组合识别。
3 建立了二阶非定常多宗量热传导正/反问题的数值求解模型,对导热系数与边界条件进行单一和组合识别,时域上采用时域精细算法进行正演数值分析。
4 在以上所建模型中,考虑了非均质和参数空间分布的影响。在求解过程中,尝试采用不同的目标函数,如L_2范数、L_∞范数和基于D-函数的正则泛函;以及不同的敏度类算法,如高斯牛顿方法、同伦方法、同伦正则化方法、共轭梯度方法、BFGS方法等,取得了满意的数值结果。在多宗量情形下,探讨了所提算法的计算精度/效率和抗不适定性,以及误差水平、测点位置、时间步长和变量初值等因素对反演结果的影响,并与单宗量识别进行了比较,为多宗量热传导反问题的进一步研究,提供了有价值的参考。
The inverse heat conduction problem (IHCP) is usually defined as the estimations of boundary/initial conditions, thermal parameters and heat source by utilizing the known temperature measurements inside the body or on the surface. The study on IHCP is an interdisciplinary field related to the heat transfer, physics, mathematics, computing, and experiment technique, etc., and has significant applications in many engineering aspects, such as aerospace, nuclear engineering, metal casting, chemical, machine making, metal-lurgy, medical diagnostics, civil engineering and food science,etc.Due to the ill-posedness and nonlinearity, solving IHCP is usually much more difficult than solving direct heat conduction problem (DHCP). Although large amounts of achievement has been made in this area, further investigation and effort are greatly demanded.With the consideration of practical requirements of both single and combined identification of thermal parameters, heat source, and boundary/initial conditions, inverse heat conduction problems with multi-variables are investigated in this dissertation, a couple of numerical models, facilitating to the sensitivity analysis, are developed to solve inverse steady/transient heat conduction problems with multi-variables. By utilizing different sensitivity based algorithms, a number of numerical tests, considering the effects of some factors on the results, are carried out to verify the proposed models with satisfactory results. Since there seems to be quite few work directly relevant to inverse heat conduction problems with multi-variables by authors best knowledge, the work presented in this dissertation is not only theoretically valuable, but also practically applicable.The major work of this dissertation includes1 Numerical models solving inverse steady heat conduction and coupled heat-mass transfer problems with multi-variables are presented, and single/combined identifications are implemented for the thermal/moist parameters and boundary conditions etc. A solving strategy for nonlinear inverse steady heat conduction via 2-Clevel sensitivity analysis is presented.
2 A numerical model solving inverse one- order transient heat conduction problems with multi-variables is proposed, and single/combined identifications are carried out for the thermal parameters and boundary conditions etc. Finite difference technique and a precise algorithm in the time domain are employed in the time-dependent analysis. In addition, single/combined identifications are also carried out for the time-dependent heat source.3 Numerical models solving direct/inverse two-order transient heat conduction problems with multi-variables are presented, and single/combined identifications are conducted for the thermal parameters and boundary conditions etc. A precise algorithm in the time domain is employed in the time-dependent analysis.4 With the consideration of material inhomogeneity and spacial distribution of parameters, all the numerical models above are numerically verified with satisfactory results.A variety of object functions, such as L_2 ,L_x and D-function based regularizing functional,and a variety of sensitivity based algorithms, such as Gauss-Newton method, homotopy method, homotopy regularization method, conjugate gradient method. and Brayden-Fletcher-Goldfarb-Shanno (BFGS) method etc. are utilized to solve inverse problems. The computing accuracy/efficiency, and the ability of anti-noisy data as well are discussed. With the comparison with the case of single variable identification, a number of factors including noisy data, location of sample points, size of time step, and initial guess of unknowns are investigated with regard to their effects on the solutions, providing valuable reference for the further investigation on IHCP with multi-variables.
引文
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