非光滑分布参数系统参数辨识及其应用
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摘要
本文以夏季北极海冰、天然河冰以及油浸风冷三相变压器的热传导问题为背景,依据原型实测温度数据,分别建立了各温度场相应的热传导方程和初边值条件及其分布参数系统参数辨识模型,在实现参数辨识的基础上,对给定的温度场进行了数值模拟。对上述参数辨识问题的研究,是国家自然科学基金资助项目(10471014,40233032)的一部分,这项研究不仅可推动一类非线性抛物型(高阶导数项系数依赖于状态变量)分布参数系统参数辨识问题的研究,还可推动对天然冰导温系数等热力学参数的优化辨识研究。另外,本文依据实验室冰样拖曳运动试验数据,研究了冰样与水流之间摩拖曳系数和形拖曳系数的优化辨识方法。本文研究的内容与取得的主要结果概括如下:
1、针对夏季北极雪/冰层内的热量传递问题,建立了描述雪/冰相变过程中的焓度、比焓和导焓系数的概念、焓度热传导方程与定解条件以及焓度与温度之间的转换关系,构造了以焓扩散系数为参量的分布参数系统参数辨识模型及优化算法。焓度热传导方程是一类高阶导数项系数依赖于状态变量的非光滑非线性抛物方程,证明了该方程初边值问题弱解的存在唯一性和解对控制参数的连续依赖性;应用Polak提出的不可微函数的优化理论和方法,分析得到了系统及其解的一些性质,证明了该参数辨识问题最优解的存在性及一阶最优性条件。根据中国第二次北极科学考察现场采集的雪/冰温度数据进行了数值模拟,数值模拟结果与实测温度吻合良好。
2、构造了以天然河冰导温系数为参量的非光滑分布参数系统参数辨识模型与优化算法,针对该参数辨识问题给出了其最优解的存在性及一阶最优性条件。利用天然河冰垂直剖面温度实测数据,辨识了天然冰(-0.19～-14.02℃)导温系数随温度的变化关系。特别对高温冰层进行了多组次有效辨识,获取了-0.19～-2.12℃天然冰的导温系数,反映出当冰温低于-0.76℃时,天然冰导温系数与温度的关系符合前人报道的结果;当温度处于相变区间(0～-0.76℃)时,导温系数随温度呈现强非线性关系,并给出了该非线性关系的数学表达式。
3、丛龙飞等在国内首次全面考虑大型油浸风冷变压器的流场、绕组和铁芯诸介质的温升计算问题,根据变压器温度场计算区域分片光滑的特点,建立了一个直角坐标系和两个柱坐标系下的分片光滑的温度场方程,并给出边界条件和各界面之间的穿透性条件,构造了以变压器油流速分布函数为参量的分布参数系统参数辨识模型,构成了多区域耦合的高度非光滑分布参数系统参数辨识问题。本文证明了该非光滑抛物系统弱解的存在唯一性、解对控制参数的连续依赖性;参数辨识问题最优解的存在性与最优性条件。
With the background of the temperature fields of Arctic sea ice in summer, natural river ice and the oil-immersed wind-cooled three-phrase transformer, on the basis of the observed data, this dissertation establishes heat conduction equations with initial boundary conditions and distributed parameter identification models respectively. Then, the numerical simulation is performed. The study of the parameter identification problems, which is supported by the National Natural Science Foundation of China (Grant No. 10471014 and 40233032), can promote not only the study of parameter identification problems for a class of nonlinear parabolic (coefficients of higher derivative terms depend on state variants) distributed parameter systems, but also the study of optimization and parameter identification for the thermodynamics parameters of natural ice (i.e. sea ice and river ice). In addition, according to the data of current-ice drag experiments, the dissertation studies the optimization identification methods of the skin drag coefficients and the form drag coefficients between the ice samples and water current.The main contributions are as follows:1. For the heat transfer problems of the Arctic snow/ice layer in summer, the dissertation presents the definition of degree of enthalpy, specific enthalpy and enthalpy conduction, ceofficient, a heat conduction equation of enthalpy and the initial boundary conditions, and the transformation relationship of the enthalpy and temperature. The parameter identification model of distributed parameter systems of diffusion coefficients of enthalpy and optimistic algorithm are established. The heat conduction equation of degree of enthalpy is a non-smooth nonlinear parabolic equation which coefficients of higher derivative terms depend on degree of enthalpy. For the above parameter identification problems of distributed parameter systems, the dissertation proves the existence and uniqueness of weak solutions of the non-smooth parabolic equation and the continued dependence of solutions on control parameters. Some important properties of the system and its solutions are considered by using the optimization theory and methods of non-differentiable function of Polak, and the existence of optimal solutions and first-order optimality conditions are given. Based on the snow/ice temperature data presented by the Second China Arctic Research Expedition, numerical simulation results tally with the observed temperature.
2. A parameter identification model of non-smooth distributed parameter systems about coefficients of thermal diffusivity of natural river ice and optimistic algorithm are established, the existence of optimal solution and first-order optimality conditions are presented for the parameter identification problems. On the basis of the vertical temperature profile data of natural river ice, the relationship between the thermal diffusivity of natural river ice (_0.19~-14.02°C) and the ice temperature is identified. Especially a lot of effective identifications are performed for the ice layer of higher temperature, thermal diffusivity of the natural ice between -0.19 and -2.12°C is obtained, which shows that when the ice temperature lowers than -0.76°C, the relationship between thermal diffusivity of river ice and temperature corresponds with the previous;when the temperature is in phase-change range(0—0.76'>C),the relationship of thermal diffusivity and ice temperature is nonlinear, and for this relationship, the nonlinear mathematical equations are given.3. Cong Longfei et al. have first considered computation problems of temperature rise of flow field, coil and iron core in the oil-immersed wind-cooled three-phrase transformer. Based on the piece-smooth characteristic of the computation range of transformer temperature field, [60] established piece-smooth temperature equations of a rectangular coordinate system and two cylindrical coordinate systems, boundary conditions and penetrability conditions among the interfaces, and formulated the parameter identification model of distributed parameter systems of distributed function of flow velocity of transformer oil, which constituted a multi-domain coupled parameter identification problem of non-smooth distributed parameter system. The dissertation proves the existence and uniqueness of the weak solutions of non-smooth parabolic system, the continuous dependence of solutions on the control variables, the existence of optimal solution of optimal control problem and optimality conditions.4. Drag movement experiments are performed to obtain movement data of ice samples by using a wave-current tank in laboratory for freshwater ice samples. Current-ice drag coefficient is divided into skin drag coefficient and form drag coefficient. Two roughness indexes - profile bearing length rate and deviation of the profile are introduced. By analyzing the relationships of coefficients and indexes, it is found that skin drag coefficient is direct proportion to profile bearing length rate of ice sample bottom, and form drag coefficient increases nonlinearly with the change of sum of average length/draft ratio and root-mean-square deviation of the bottom profile. These relationships uniform the skin drag coefficients of ice floe and iceberg.
1、针对夏季北极雪/冰层内的热量传递问题,建立了描述雪/冰相变过程中的焓度、比焓和导焓系数的概念、焓度热传导方程与定解条件以及焓度与温度之间的转换关系,构造了以焓扩散系数为参量的分布参数系统参数辨识模型及优化算法。焓度热传导方程是一类高阶导数项系数依赖于状态变量的非光滑非线性抛物方程,证明了该方程初边值问题弱解的存在唯一性和解对控制参数的连续依赖性;应用Polak提出的不可微函数的优化理论和方法,分析得到了系统及其解的一些性质,证明了该参数辨识问题最优解的存在性及一阶最优性条件。根据中国第二次北极科学考察现场采集的雪/冰温度数据进行了数值模拟,数值模拟结果与实测温度吻合良好。
2、构造了以天然河冰导温系数为参量的非光滑分布参数系统参数辨识模型与优化算法,针对该参数辨识问题给出了其最优解的存在性及一阶最优性条件。利用天然河冰垂直剖面温度实测数据,辨识了天然冰(-0.19～-14.02℃)导温系数随温度的变化关系。特别对高温冰层进行了多组次有效辨识,获取了-0.19～-2.12℃天然冰的导温系数,反映出当冰温低于-0.76℃时,天然冰导温系数与温度的关系符合前人报道的结果;当温度处于相变区间(0～-0.76℃)时,导温系数随温度呈现强非线性关系,并给出了该非线性关系的数学表达式。
3、丛龙飞等在国内首次全面考虑大型油浸风冷变压器的流场、绕组和铁芯诸介质的温升计算问题,根据变压器温度场计算区域分片光滑的特点,建立了一个直角坐标系和两个柱坐标系下的分片光滑的温度场方程,并给出边界条件和各界面之间的穿透性条件,构造了以变压器油流速分布函数为参量的分布参数系统参数辨识模型,构成了多区域耦合的高度非光滑分布参数系统参数辨识问题。本文证明了该非光滑抛物系统弱解的存在唯一性、解对控制参数的连续依赖性;参数辨识问题最优解的存在性与最优性条件。
With the background of the temperature fields of Arctic sea ice in summer, natural river ice and the oil-immersed wind-cooled three-phrase transformer, on the basis of the observed data, this dissertation establishes heat conduction equations with initial boundary conditions and distributed parameter identification models respectively. Then, the numerical simulation is performed. The study of the parameter identification problems, which is supported by the National Natural Science Foundation of China (Grant No. 10471014 and 40233032), can promote not only the study of parameter identification problems for a class of nonlinear parabolic (coefficients of higher derivative terms depend on state variants) distributed parameter systems, but also the study of optimization and parameter identification for the thermodynamics parameters of natural ice (i.e. sea ice and river ice). In addition, according to the data of current-ice drag experiments, the dissertation studies the optimization identification methods of the skin drag coefficients and the form drag coefficients between the ice samples and water current.The main contributions are as follows:1. For the heat transfer problems of the Arctic snow/ice layer in summer, the dissertation presents the definition of degree of enthalpy, specific enthalpy and enthalpy conduction, ceofficient, a heat conduction equation of enthalpy and the initial boundary conditions, and the transformation relationship of the enthalpy and temperature. The parameter identification model of distributed parameter systems of diffusion coefficients of enthalpy and optimistic algorithm are established. The heat conduction equation of degree of enthalpy is a non-smooth nonlinear parabolic equation which coefficients of higher derivative terms depend on degree of enthalpy. For the above parameter identification problems of distributed parameter systems, the dissertation proves the existence and uniqueness of weak solutions of the non-smooth parabolic equation and the continued dependence of solutions on control parameters. Some important properties of the system and its solutions are considered by using the optimization theory and methods of non-differentiable function of Polak, and the existence of optimal solutions and first-order optimality conditions are given. Based on the snow/ice temperature data presented by the Second China Arctic Research Expedition, numerical simulation results tally with the observed temperature.
2. A parameter identification model of non-smooth distributed parameter systems about coefficients of thermal diffusivity of natural river ice and optimistic algorithm are established, the existence of optimal solution and first-order optimality conditions are presented for the parameter identification problems. On the basis of the vertical temperature profile data of natural river ice, the relationship between the thermal diffusivity of natural river ice (_0.19~-14.02°C) and the ice temperature is identified. Especially a lot of effective identifications are performed for the ice layer of higher temperature, thermal diffusivity of the natural ice between -0.19 and -2.12°C is obtained, which shows that when the ice temperature lowers than -0.76°C, the relationship between thermal diffusivity of river ice and temperature corresponds with the previous;when the temperature is in phase-change range(0—0.76'>C),the relationship of thermal diffusivity and ice temperature is nonlinear, and for this relationship, the nonlinear mathematical equations are given.3. Cong Longfei et al. have first considered computation problems of temperature rise of flow field, coil and iron core in the oil-immersed wind-cooled three-phrase transformer. Based on the piece-smooth characteristic of the computation range of transformer temperature field, [60] established piece-smooth temperature equations of a rectangular coordinate system and two cylindrical coordinate systems, boundary conditions and penetrability conditions among the interfaces, and formulated the parameter identification model of distributed parameter systems of distributed function of flow velocity of transformer oil, which constituted a multi-domain coupled parameter identification problem of non-smooth distributed parameter system. The dissertation proves the existence and uniqueness of the weak solutions of non-smooth parabolic system, the continuous dependence of solutions on the control variables, the existence of optimal solution of optimal control problem and optimality conditions.4. Drag movement experiments are performed to obtain movement data of ice samples by using a wave-current tank in laboratory for freshwater ice samples. Current-ice drag coefficient is divided into skin drag coefficient and form drag coefficient. Two roughness indexes - profile bearing length rate and deviation of the profile are introduced. By analyzing the relationships of coefficients and indexes, it is found that skin drag coefficient is direct proportion to profile bearing length rate of ice sample bottom, and form drag coefficient increases nonlinearly with the change of sum of average length/draft ratio and root-mean-square deviation of the bottom profile. These relationships uniform the skin drag coefficients of ice floe and iceberg.
引文
[1] Omatu S, Seifeld J H. Distfibuted parameter system theory and application. Oxford Science Publication, 1989.
[2] Lions J L, Magenes E. Non-homogenous boundary value problems and applications(Ⅰ-Ⅲ). New York: Springer-Verlag, 1968.
[3] Lions J L. Optimal control of system governed by partial differential equations. New York: Springer-Verlag, 1971.
[4] Ahmed N U,Teo K L. Optimal control of distributed parameter systems. Amsterdam: North-Holland, 1981.
[5] Ladyzenskaya O A, Solonnikov. Linear and quasilinear equations of parabolic type. American mathematical society providence, Rhode Island, 1968.
[6] Ahmed N U. Optimization and identification of system governed by evolution equations on Banach space. Longman Scientific and Technical, Essex, England, 1989.
[7] Ahmed N U, Xiang X. Existence of solution for a class of nonlinear evolution equations with nonmonotone perturbations. Nonlinear analysis, 1994, 22(1):81-89.
[8] Ahmed N U, Xiang X. Nonlinear uncertain systems and necessary conditions of optimality. SIAM Journal on Control and Optimization, 1997, 35(5): 1755-1772.
[9] Xiang X. Optimal control for a class of strongly nonlinear evolution equations with constraints. Nonlinear Analysis, 2001, 47(1):57-66.
[10] Ahmed N U. Measure solutions for semilinear and quasilinear evolution equations and their optimal control. Nonlinear Analysis, 2000, 40(1-8):51-72.
[11] Ahmed N U. A general result on measure solutions for semilinear evolution equations. Nonlinear Analysis, 2000, 42(8):1335-1349.
[12] Xiang X, Ahmed N U. Necessary conditions of optimality for differential inclusions on banach space. Nonlinear Analysis, 1997, 30(8):5437-5445.
[13] Ahmed N U. Measure solutions for Impulsive system in Banach space and their control. Dynamics of Continuous, Discrete and Impulsive Systems, 1999, 6(4):519-535.
[14] Fattorini H O. The maximum principle for nonlinear non-convex systems in infinite dimensional space in distributed parabolic systems. Berlin: Springer-verlag, 1985.
[15] Fattorini H O, Frankowska H. Analysis and optimization of systems. In: Necessary conditions for infinite dimensional control problems. London: Springer-Verlag, 1988:381-392.
[16] Fattorini H O. Optimal control problems for distributed parameter systems in Banach spaces. Applied Mathematics and Optimization, 1993, 28(3):225-257.
[17] Fattorini H O, Murphy T. Optimal control problems for nonlinear parabolic boundary control systems: The Dirichlet boundary condition. Differential and Integral Equations, 1994, 7:1367-1388.
[18] Fattorini H O, Murphy T. Optimal problems for nonlinear parabolic boundary control systems. SIAM Journal on Control and Optimization, 1994, 32 (6): 1577-1596.
[19] Fattorini H O. Infinite dimensional optimization and control theory. Cambridge University Press, 1999.
[20] Raymond J P, Zidani H. Hamiltonian Pontryagin's principle for control problems governed by semilinear parabolic equations. Applied Mathematics and Optimization, 1999, 39(2): 143-177.
[21] Li X, Yong J. Optimal Control Theory for Infinite Dimensional Systems. New York: Springer-Verlag, 1994.
[22] 陈任昭,高夯.时变人口系统的李雅普诺夫稳定性.中国科学(A辑),1990,2:144-152.
[23] 陈任昭,李健全.一类时变人口系统正则解的惟一性.东北师大学报,1996,1:1-4.
[24] 陈任昭,张丹松.具有空间扩散且与年龄相关的时变种群系统的最优边界控制.系统工程理论与实践,2000,11:35-45.
[25] 王康宁.分布参数控制系统.北京:科学出版社,1986.
[26] 王康宁.最优控制的数学理论.北京:国防工业出版社,1995.
[27] 喻文焕.一类两相储层特性参数的辨识.数学物理学报,1985,5:367-377.
[28] Yu W H, Seinfeld J H. Identification of distributed parameter with pointwise constraints on the parameters. Journal of Mathematical Analysis and Applications, 1988, 136(2):497-520.
[29] Yu W H. Necessary condition for optimality in the identification of elliptic system with pointwise parameter constraints. Journal of Optimization Theory and Applications, 1996, 88(3):725-742.
[30] 高夯.半线性抛物方程支配系统的最优性条件.数学学报,1999,42(4):705-714.
[31] Wang Q, Feng D, Cheng D. Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems. Computers and Mathematics with Applications, 2004, 48:1847-1861.
[32] 冯恩民,李春发.三维古地温度场方程的参数化辨识.高等学校应用数学学报,2001,16(1):75-80.
[33] 李春发,冯恩民.一类热传导方程非线性源项的识别问题.大连理工大学学报,2002,42(4):391-395.
[34] 李春发,冯恩民,刘金旺.一类弱耦合动力系统的参数识别问题.高等学校应用数学学报,2002.17(4):425-432.
[35] Campbell W J. The wind-driven circulation of ice and water in a polar ocean. Journal of Geophysical Research, 1965, 70:3279-3301.
[36] Maykut G A, Untersteiner N. Some results from a time dependent thermodynamic model of sea ice. Journal of Geophysical Research, 1971, 76(6): 1550-1575.
[37] Parmerter P R, Coon M D. Mechanical models of ridging in the Arctic sea ice cover. AIDJEX Bull, 1973, 19:59-112.
[38] Thorndike A S, Rothrock D A, Maykut G A. The thickness distritution of sea ice. Journal of Geophysical Research, 1975, 80(33):4501-4513.
[39] Hibler W D. A dynamic thermodynamic sea ice model. Journal of Physics Oceanography, 1979, 9(4): 815-846.
[40] Hibler W D. Modeling a variable thickness sea ice cover. Monthly Weather Review, 1980, 108(12): 1943-1973.
[41] Hibler W D, Walsh J E. On modeling seasonal and interannual fluctuations of Arctic sea ice. Journal of Physics Oceanography, 1982, 12(12): 1514-1523.
[42] Hibler W D, Bryan K. A diagnostic ice-ocean model. Journal of Physics Oceanography, 1987, 17(7): 987-1015.
[43] Holland D M, Mysak L A, Manak D K. Sensitivity study of a dynamic thermodynamic sea ice model. Journal of Geophysical Research, 1993, 98(C2):2561-2586.
[44] 大清 敏.海冰数值.氣象研究,1993,177:131-170.
[45] Chapman W L, Welch W J, Bowman K P. Arctic sea ice variability: Model sensitivities and a multidecadal simulation. Journal of Geophysical Research, 1994, 99(C1):919-935.
[46] Feistel R, Hagen E. A Gibbs thermodynamic potential of sea ice. Cold Regions Science and Technology, 1998, 28(2):83-142.
[47] Wang L R, Ikeda M. A Lagrangian description of sea ice dynamics using the finite element method. Ocean Modeling, 2004, 7(1-2):21-38.
[48] 王仁树.渤海海冰的数值试验.海洋学报,1984,6(4):572-580.
[49] 吴辉碇.海冰的动力-热力过程的数学处理.海洋与湖沼,1991,22(4):321-327.
[50] 白珊,吴辉碇.渤海的海冰数值预报.气象学报,1998,56(2):139-153.
[51] 吴辉碇,白珊,张占海.海冰动力学过程的数值模拟.海洋学报,1998,20(2):1-13.
[52] 刘钦政,白珊,黄嘉佑等.一种冰-海洋模式的热力耦合方案.海洋学报,2004,26(6):13-21.
[53] 苏洁,吴辉碇,刘钦政等.渤海冰-海洋耦合模式.海洋学报,2005,27(1):19-26.
[54] Oliver A J. Estimation of Transformer Wind in Temperature and Coolant Flows with a General Network Method. IEE. Pt. C., 1980, 127(6):54-59.
[55] Weedy B M. Some Aspects of Temperature Rises in a Hatural Coold Oil-filled Three-phase Transformer. Electric Machines and Power System, 1985, 10:395.
[56] Lindsay J F. Temperature Rise of an oil-filled Transformer with Varying Load. IEEE PAS-103, 1984(9):88.
[57] 王文,顾昌,陈汝庆.温度场的数值模拟法在干式电力变压器热设计中的应用.变压器,1997,34(9):18-21.
[58] 王晓远,李保林,刘丽霞.变压器三维温度场的分析与计算.天津大学学报,1996,29(3):320-323.
[59] 颜寒,郭永基,林兆庄.树脂绝缘干式变压器内部温度场分布仿真研究.清华大学学报(自然科学版),1999,39(7):1-4.
[60] 丛龙飞,冯恩民,郭振岩等.油浸风冷变压器温度场的数值模拟.变压器,2003,40(5):1-6.
[61] 欧阳涛,张智勇.变压器绕组温度场初探.电机电器技术,1996,3:9-11.
[62] 吕涛,石济民,林振宝.区域分解算法.北京:科学出版社,1999.
[63] Adams R.著,叶其孝等译.索伯列夫空间.北京:人民教育出版社,1981.
[64] 伍卓群,尹景学,王春明.椭圆与抛物型方程引论.北京:科学出版社,2003.
[65] 王耀东.偏微分方程的L~2理论.北京:北京大学出版社,1989.
[66] 董光昌.非线性二阶偏微分方程.北京:清华大学出版社,1988.
[67] Polak E. Optimization Algorithm and Consistent Approximations. New York: Springer-Verlag, 1997.
[68] 陈宝林.最优化理论与算法.北京:清华大学出版社,1989.
[69] Walsh J E, Hibler W D, Ross B. Numerical simulation of northern hemisphere sea ice variability. Journal of Geophysical Research, 1985, 90(C7):4847-4865.
[70] Semtner A J. A model for the thermodynamic growth of sea ice in numerical investigations of climate. Journal of Physics Oceanography, 1976, 6(3):379-389.
[71] Parkinson C L, Washington W M. A large-scale numerical model of sea ice. Journal of Geophysical Research, 1979, 84(C1):311-337.
[72] Melia D S. A global coupled sea ice-ocean model. Ocean Modelling, 2002, 4(2): 137-172.
[73] 刘钦政,黄嘉佑,白珊等.全球冰海洋耦合模式的海冰模拟.地学前缘,2000,7(增刊):219-230.
[74] 邹进,黄素逸.相变热传导的计算.能源技术,2000,1:11-14.
[75] Hammerschmidt U. Thermal transport properties of water and ice from one single experiment. International Journal of Thermophysics, 2002, 23(4):975-996.
[76] 李晓红.微生物发酵非线性系统的稳定性与最优控制:(博士学位论文).大连:大连理工大学,2005.
[77] 陆金甫.偏微分方程差分方法.北京:高等教育出版社,1988.
[78] 钱炜祺,蔡金狮.用灵敏度法辨识热传导系数及热流参数.空气动力学学报,1998,16(2):226-231.
[79] Launiainen J, Cheng B. Modeling of ice thermodynamics in natural water bodies. Cold Regions Science and Technology, 1998, 27(3): 153-178.
[80] White K D. Review of prediction methods for break ice jam. Canadian Journal of Civil Engineering, 2003, 30(1):1-12.
[81] 茅泽育,吴剑疆,张磊等.天然河道冰塞演变发展的数值模拟.水科学进展,2003,14(6):700-705.
[82] 吴剑疆,茅泽育,王爱民等.河道中水内冰演变的数值计算.清华大学学报,2003,43(5):702-705.
[83] 程斌.一维海冰热力模式的守恒型差分格式和数值模拟.海洋通报,1996,15(4):8-16.
[84] 王军.河冰水力学研究进展.水利水电技术,2004,35(5):111-113.
[85] 秦大河,Yong N W,Thaites R J et al.南极洲洛多姆冰帽表面活动层内的温度特征和热学性质.南极科学考察论文集(五):冰川学研究,北京:科学出版社,1988.
[86] Langlebea M P. On the thermal diffusivity of sea ice. IAHR Ice Symposium, 1986, 569-578.
[87] Ling F, Zhang T J. A numerical model for surface energy balance and thermal regime of the active layer and permafrost containing unfrozen water. Cold Regions Science and Technology, 2004, 38(1):1-15.
[88] 顾元宪,周业涛,陈飚松等.基于灵敏度的热传导辨识问题的求解方法.土木工程学报,2002,35(3):94-98.
[89] 王登刚,刘迎曦,李守臣等.识别材料导热系数和导温系数的温度场逆分析.上海理工大学学报,2001,23(3):233-237.
[90] 肖建民,金龙海,谢永刚等.寒区水库冰盖形成与消融机理分析.水利学报,2004,6:80-85.
[91] 周明,孙树栋.遗传算法原理及应用.北京:国防工业出版社,1999.
[92] Yen Y C. Review of thermal properties of snow, ice and sea ice. CRREL Report 81-10, USA Cold Regions Research and Engineering Laboratory, Hanover, NH, USA, 1981.
[93] Koubyshkin N V, Sazonov K E. Evaluation of loads due to partial freezing of seawater trapped in enclosed cavities. Proceedings of 17th International Symposium on Ice, Saint Petersburg, Russia, 2004, 1: 100-107.
[94] 傅晨钊,汲胜昌,王世山等.变压器绕组温度场的二维数值计算.高电压技术,2002,28(5):10-12.
[95] Jean-Michel Mufuta M B. Comparison of Experimental Values and Numerical Simulation on a Set-up Simulating the Cross-section of a Disc-type transformer. International Journal of Thermal Sciences, 1999, 38(5): 424-435.
[96] Yamaguchi M, Kumasaka T, inui Y et al. The Flow Rate in Selt-cooled Transformer, IEEE Transactions on Power Systems, 1981, 100 (3).
[97] Nakadate M, Toda K, Sato K et al. Gas Cooling Performance in Disc Winding of Large-capacity Gas-insulated Transformer. IEEE Transactions on Power Delivery, 1996, 11 (2): 903-908.
[98] Guest P S, Davidson K L. The Aerodynamic Roughness of Different Types of Sea Ice. Journal of Geophysical Research, 1991, 96(C3):4709-4721.
[99] Andreas E L. Roughness of Weddle sea ice and estimate of air-ice drag coefficient. Journal of Geophysical Research, 1993, 98(C7): 12439-12452.
[100] Martinson D G. Ice Drift and Momentum Exchange in Winter Antarctic Pack Ice. Journal of Geophysical Research, 1990, 94(C5):1741-1755.
[101] Wu Huiding, Bai Shan, Zhang Zhanhai et al. Numerical simulation for dynamical process of sea ice. Acta Oceanologica Sinica, 1997, 16(2):303-325.
[102] Hibler W D. Ice dynamics. In: The Geophysics of sea ice. New York: Plenum Press, 1986.
[103] 岳前进,张希,季顺迎.辽东湾海冰漂移的动力要素分析.海洋环境科学,2001,20(4):34-39.
[104] 苏洁,吴辉碇,Margaret A K等.渤海海冰和海洋的动力相互作用模拟试验.冰川冻土,2003,25(增刊2):292-298.
[105] 季顺迎,王瑞学,毕祥军等.海冰拖曳系数的确定方法研究.冰川冻土,2003,25(增刊2):299-303.
[106] Prinsenberg S, Peterson I K. Variations in air-ice drag coefficient due to ice surface roughness. International Journal of offshore and polar Engineering, 2002, 12(2):1-5.
[107] Steiner N, Harder M, Lemke P. Sea-ice roughness and drag coefficients in a dynamicthermodynamic sea-ice model for the Arctic. Tellus, 1999, 51A(5):964-978.
[108] 季顺迎,沈洪道,王志联等.基于Mohr_Coulomb准则的黏弹_塑性海冰动力学本构模型.海洋学报,2005,27(4):19-30.
[109] Shirasawa K, Ingram R G. Currents and turbulent fluxes under the first-year sea ice in Resolute Passage, Northwest Territories, Canada. Journal of Marine Systems, 1997, 11:21-32.
[110] Smith S D, Banke E G. The influence of winds, currents and towing forces on the drift of icebergs. Cold Regions Science and Technology, 1983, 6:241-255.
[111] Smith S D. Hindcasting iceberg drift using current profiles and winds. Cold Regions Science and Technology, 1993, 22(1):33-45.
[112] Li Zhijun, Lu Peng. Measurement of iceberg draft with marine radar. 6th International Symposium on Test and Measurement, Dalian, China, 2005:2972-2975.
[2] Lions J L, Magenes E. Non-homogenous boundary value problems and applications(Ⅰ-Ⅲ). New York: Springer-Verlag, 1968.
[3] Lions J L. Optimal control of system governed by partial differential equations. New York: Springer-Verlag, 1971.
[4] Ahmed N U,Teo K L. Optimal control of distributed parameter systems. Amsterdam: North-Holland, 1981.
[5] Ladyzenskaya O A, Solonnikov. Linear and quasilinear equations of parabolic type. American mathematical society providence, Rhode Island, 1968.
[6] Ahmed N U. Optimization and identification of system governed by evolution equations on Banach space. Longman Scientific and Technical, Essex, England, 1989.
[7] Ahmed N U, Xiang X. Existence of solution for a class of nonlinear evolution equations with nonmonotone perturbations. Nonlinear analysis, 1994, 22(1):81-89.
[8] Ahmed N U, Xiang X. Nonlinear uncertain systems and necessary conditions of optimality. SIAM Journal on Control and Optimization, 1997, 35(5): 1755-1772.
[9] Xiang X. Optimal control for a class of strongly nonlinear evolution equations with constraints. Nonlinear Analysis, 2001, 47(1):57-66.
[10] Ahmed N U. Measure solutions for semilinear and quasilinear evolution equations and their optimal control. Nonlinear Analysis, 2000, 40(1-8):51-72.
[11] Ahmed N U. A general result on measure solutions for semilinear evolution equations. Nonlinear Analysis, 2000, 42(8):1335-1349.
[12] Xiang X, Ahmed N U. Necessary conditions of optimality for differential inclusions on banach space. Nonlinear Analysis, 1997, 30(8):5437-5445.
[13] Ahmed N U. Measure solutions for Impulsive system in Banach space and their control. Dynamics of Continuous, Discrete and Impulsive Systems, 1999, 6(4):519-535.
[14] Fattorini H O. The maximum principle for nonlinear non-convex systems in infinite dimensional space in distributed parabolic systems. Berlin: Springer-verlag, 1985.
[15] Fattorini H O, Frankowska H. Analysis and optimization of systems. In: Necessary conditions for infinite dimensional control problems. London: Springer-Verlag, 1988:381-392.
[16] Fattorini H O. Optimal control problems for distributed parameter systems in Banach spaces. Applied Mathematics and Optimization, 1993, 28(3):225-257.
[17] Fattorini H O, Murphy T. Optimal control problems for nonlinear parabolic boundary control systems: The Dirichlet boundary condition. Differential and Integral Equations, 1994, 7:1367-1388.
[18] Fattorini H O, Murphy T. Optimal problems for nonlinear parabolic boundary control systems. SIAM Journal on Control and Optimization, 1994, 32 (6): 1577-1596.
[19] Fattorini H O. Infinite dimensional optimization and control theory. Cambridge University Press, 1999.
[20] Raymond J P, Zidani H. Hamiltonian Pontryagin's principle for control problems governed by semilinear parabolic equations. Applied Mathematics and Optimization, 1999, 39(2): 143-177.
[21] Li X, Yong J. Optimal Control Theory for Infinite Dimensional Systems. New York: Springer-Verlag, 1994.
[22] 陈任昭,高夯.时变人口系统的李雅普诺夫稳定性.中国科学(A辑),1990,2:144-152.
[23] 陈任昭,李健全.一类时变人口系统正则解的惟一性.东北师大学报,1996,1:1-4.
[24] 陈任昭,张丹松.具有空间扩散且与年龄相关的时变种群系统的最优边界控制.系统工程理论与实践,2000,11:35-45.
[25] 王康宁.分布参数控制系统.北京:科学出版社,1986.
[26] 王康宁.最优控制的数学理论.北京:国防工业出版社,1995.
[27] 喻文焕.一类两相储层特性参数的辨识.数学物理学报,1985,5:367-377.
[28] Yu W H, Seinfeld J H. Identification of distributed parameter with pointwise constraints on the parameters. Journal of Mathematical Analysis and Applications, 1988, 136(2):497-520.
[29] Yu W H. Necessary condition for optimality in the identification of elliptic system with pointwise parameter constraints. Journal of Optimization Theory and Applications, 1996, 88(3):725-742.
[30] 高夯.半线性抛物方程支配系统的最优性条件.数学学报,1999,42(4):705-714.
[31] Wang Q, Feng D, Cheng D. Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems. Computers and Mathematics with Applications, 2004, 48:1847-1861.
[32] 冯恩民,李春发.三维古地温度场方程的参数化辨识.高等学校应用数学学报,2001,16(1):75-80.
[33] 李春发,冯恩民.一类热传导方程非线性源项的识别问题.大连理工大学学报,2002,42(4):391-395.
[34] 李春发,冯恩民,刘金旺.一类弱耦合动力系统的参数识别问题.高等学校应用数学学报,2002.17(4):425-432.
[35] Campbell W J. The wind-driven circulation of ice and water in a polar ocean. Journal of Geophysical Research, 1965, 70:3279-3301.
[36] Maykut G A, Untersteiner N. Some results from a time dependent thermodynamic model of sea ice. Journal of Geophysical Research, 1971, 76(6): 1550-1575.
[37] Parmerter P R, Coon M D. Mechanical models of ridging in the Arctic sea ice cover. AIDJEX Bull, 1973, 19:59-112.
[38] Thorndike A S, Rothrock D A, Maykut G A. The thickness distritution of sea ice. Journal of Geophysical Research, 1975, 80(33):4501-4513.
[39] Hibler W D. A dynamic thermodynamic sea ice model. Journal of Physics Oceanography, 1979, 9(4): 815-846.
[40] Hibler W D. Modeling a variable thickness sea ice cover. Monthly Weather Review, 1980, 108(12): 1943-1973.
[41] Hibler W D, Walsh J E. On modeling seasonal and interannual fluctuations of Arctic sea ice. Journal of Physics Oceanography, 1982, 12(12): 1514-1523.
[42] Hibler W D, Bryan K. A diagnostic ice-ocean model. Journal of Physics Oceanography, 1987, 17(7): 987-1015.
[43] Holland D M, Mysak L A, Manak D K. Sensitivity study of a dynamic thermodynamic sea ice model. Journal of Geophysical Research, 1993, 98(C2):2561-2586.
[44] 大清 敏.海冰数值.氣象研究,1993,177:131-170.
[45] Chapman W L, Welch W J, Bowman K P. Arctic sea ice variability: Model sensitivities and a multidecadal simulation. Journal of Geophysical Research, 1994, 99(C1):919-935.
[46] Feistel R, Hagen E. A Gibbs thermodynamic potential of sea ice. Cold Regions Science and Technology, 1998, 28(2):83-142.
[47] Wang L R, Ikeda M. A Lagrangian description of sea ice dynamics using the finite element method. Ocean Modeling, 2004, 7(1-2):21-38.
[48] 王仁树.渤海海冰的数值试验.海洋学报,1984,6(4):572-580.
[49] 吴辉碇.海冰的动力-热力过程的数学处理.海洋与湖沼,1991,22(4):321-327.
[50] 白珊,吴辉碇.渤海的海冰数值预报.气象学报,1998,56(2):139-153.
[51] 吴辉碇,白珊,张占海.海冰动力学过程的数值模拟.海洋学报,1998,20(2):1-13.
[52] 刘钦政,白珊,黄嘉佑等.一种冰-海洋模式的热力耦合方案.海洋学报,2004,26(6):13-21.
[53] 苏洁,吴辉碇,刘钦政等.渤海冰-海洋耦合模式.海洋学报,2005,27(1):19-26.
[54] Oliver A J. Estimation of Transformer Wind in Temperature and Coolant Flows with a General Network Method. IEE. Pt. C., 1980, 127(6):54-59.
[55] Weedy B M. Some Aspects of Temperature Rises in a Hatural Coold Oil-filled Three-phase Transformer. Electric Machines and Power System, 1985, 10:395.
[56] Lindsay J F. Temperature Rise of an oil-filled Transformer with Varying Load. IEEE PAS-103, 1984(9):88.
[57] 王文,顾昌,陈汝庆.温度场的数值模拟法在干式电力变压器热设计中的应用.变压器,1997,34(9):18-21.
[58] 王晓远,李保林,刘丽霞.变压器三维温度场的分析与计算.天津大学学报,1996,29(3):320-323.
[59] 颜寒,郭永基,林兆庄.树脂绝缘干式变压器内部温度场分布仿真研究.清华大学学报(自然科学版),1999,39(7):1-4.
[60] 丛龙飞,冯恩民,郭振岩等.油浸风冷变压器温度场的数值模拟.变压器,2003,40(5):1-6.
[61] 欧阳涛,张智勇.变压器绕组温度场初探.电机电器技术,1996,3:9-11.
[62] 吕涛,石济民,林振宝.区域分解算法.北京:科学出版社,1999.
[63] Adams R.著,叶其孝等译.索伯列夫空间.北京:人民教育出版社,1981.
[64] 伍卓群,尹景学,王春明.椭圆与抛物型方程引论.北京:科学出版社,2003.
[65] 王耀东.偏微分方程的L~2理论.北京:北京大学出版社,1989.
[66] 董光昌.非线性二阶偏微分方程.北京:清华大学出版社,1988.
[67] Polak E. Optimization Algorithm and Consistent Approximations. New York: Springer-Verlag, 1997.
[68] 陈宝林.最优化理论与算法.北京:清华大学出版社,1989.
[69] Walsh J E, Hibler W D, Ross B. Numerical simulation of northern hemisphere sea ice variability. Journal of Geophysical Research, 1985, 90(C7):4847-4865.
[70] Semtner A J. A model for the thermodynamic growth of sea ice in numerical investigations of climate. Journal of Physics Oceanography, 1976, 6(3):379-389.
[71] Parkinson C L, Washington W M. A large-scale numerical model of sea ice. Journal of Geophysical Research, 1979, 84(C1):311-337.
[72] Melia D S. A global coupled sea ice-ocean model. Ocean Modelling, 2002, 4(2): 137-172.
[73] 刘钦政,黄嘉佑,白珊等.全球冰海洋耦合模式的海冰模拟.地学前缘,2000,7(增刊):219-230.
[74] 邹进,黄素逸.相变热传导的计算.能源技术,2000,1:11-14.
[75] Hammerschmidt U. Thermal transport properties of water and ice from one single experiment. International Journal of Thermophysics, 2002, 23(4):975-996.
[76] 李晓红.微生物发酵非线性系统的稳定性与最优控制:(博士学位论文).大连:大连理工大学,2005.
[77] 陆金甫.偏微分方程差分方法.北京:高等教育出版社,1988.
[78] 钱炜祺,蔡金狮.用灵敏度法辨识热传导系数及热流参数.空气动力学学报,1998,16(2):226-231.
[79] Launiainen J, Cheng B. Modeling of ice thermodynamics in natural water bodies. Cold Regions Science and Technology, 1998, 27(3): 153-178.
[80] White K D. Review of prediction methods for break ice jam. Canadian Journal of Civil Engineering, 2003, 30(1):1-12.
[81] 茅泽育,吴剑疆,张磊等.天然河道冰塞演变发展的数值模拟.水科学进展,2003,14(6):700-705.
[82] 吴剑疆,茅泽育,王爱民等.河道中水内冰演变的数值计算.清华大学学报,2003,43(5):702-705.
[83] 程斌.一维海冰热力模式的守恒型差分格式和数值模拟.海洋通报,1996,15(4):8-16.
[84] 王军.河冰水力学研究进展.水利水电技术,2004,35(5):111-113.
[85] 秦大河,Yong N W,Thaites R J et al.南极洲洛多姆冰帽表面活动层内的温度特征和热学性质.南极科学考察论文集(五):冰川学研究,北京:科学出版社,1988.
[86] Langlebea M P. On the thermal diffusivity of sea ice. IAHR Ice Symposium, 1986, 569-578.
[87] Ling F, Zhang T J. A numerical model for surface energy balance and thermal regime of the active layer and permafrost containing unfrozen water. Cold Regions Science and Technology, 2004, 38(1):1-15.
[88] 顾元宪,周业涛,陈飚松等.基于灵敏度的热传导辨识问题的求解方法.土木工程学报,2002,35(3):94-98.
[89] 王登刚,刘迎曦,李守臣等.识别材料导热系数和导温系数的温度场逆分析.上海理工大学学报,2001,23(3):233-237.
[90] 肖建民,金龙海,谢永刚等.寒区水库冰盖形成与消融机理分析.水利学报,2004,6:80-85.
[91] 周明,孙树栋.遗传算法原理及应用.北京:国防工业出版社,1999.
[92] Yen Y C. Review of thermal properties of snow, ice and sea ice. CRREL Report 81-10, USA Cold Regions Research and Engineering Laboratory, Hanover, NH, USA, 1981.
[93] Koubyshkin N V, Sazonov K E. Evaluation of loads due to partial freezing of seawater trapped in enclosed cavities. Proceedings of 17th International Symposium on Ice, Saint Petersburg, Russia, 2004, 1: 100-107.
[94] 傅晨钊,汲胜昌,王世山等.变压器绕组温度场的二维数值计算.高电压技术,2002,28(5):10-12.
[95] Jean-Michel Mufuta M B. Comparison of Experimental Values and Numerical Simulation on a Set-up Simulating the Cross-section of a Disc-type transformer. International Journal of Thermal Sciences, 1999, 38(5): 424-435.
[96] Yamaguchi M, Kumasaka T, inui Y et al. The Flow Rate in Selt-cooled Transformer, IEEE Transactions on Power Systems, 1981, 100 (3).
[97] Nakadate M, Toda K, Sato K et al. Gas Cooling Performance in Disc Winding of Large-capacity Gas-insulated Transformer. IEEE Transactions on Power Delivery, 1996, 11 (2): 903-908.
[98] Guest P S, Davidson K L. The Aerodynamic Roughness of Different Types of Sea Ice. Journal of Geophysical Research, 1991, 96(C3):4709-4721.
[99] Andreas E L. Roughness of Weddle sea ice and estimate of air-ice drag coefficient. Journal of Geophysical Research, 1993, 98(C7): 12439-12452.
[100] Martinson D G. Ice Drift and Momentum Exchange in Winter Antarctic Pack Ice. Journal of Geophysical Research, 1990, 94(C5):1741-1755.
[101] Wu Huiding, Bai Shan, Zhang Zhanhai et al. Numerical simulation for dynamical process of sea ice. Acta Oceanologica Sinica, 1997, 16(2):303-325.
[102] Hibler W D. Ice dynamics. In: The Geophysics of sea ice. New York: Plenum Press, 1986.
[103] 岳前进,张希,季顺迎.辽东湾海冰漂移的动力要素分析.海洋环境科学,2001,20(4):34-39.
[104] 苏洁,吴辉碇,Margaret A K等.渤海海冰和海洋的动力相互作用模拟试验.冰川冻土,2003,25(增刊2):292-298.
[105] 季顺迎,王瑞学,毕祥军等.海冰拖曳系数的确定方法研究.冰川冻土,2003,25(增刊2):299-303.
[106] Prinsenberg S, Peterson I K. Variations in air-ice drag coefficient due to ice surface roughness. International Journal of offshore and polar Engineering, 2002, 12(2):1-5.
[107] Steiner N, Harder M, Lemke P. Sea-ice roughness and drag coefficients in a dynamicthermodynamic sea-ice model for the Arctic. Tellus, 1999, 51A(5):964-978.
[108] 季顺迎,沈洪道,王志联等.基于Mohr_Coulomb准则的黏弹_塑性海冰动力学本构模型.海洋学报,2005,27(4):19-30.
[109] Shirasawa K, Ingram R G. Currents and turbulent fluxes under the first-year sea ice in Resolute Passage, Northwest Territories, Canada. Journal of Marine Systems, 1997, 11:21-32.
[110] Smith S D, Banke E G. The influence of winds, currents and towing forces on the drift of icebergs. Cold Regions Science and Technology, 1983, 6:241-255.
[111] Smith S D. Hindcasting iceberg drift using current profiles and winds. Cold Regions Science and Technology, 1993, 22(1):33-45.
[112] Li Zhijun, Lu Peng. Measurement of iceberg draft with marine radar. 6th International Symposium on Test and Measurement, Dalian, China, 2005:2972-2975.