污染物对流-扩散逆过程源项反演的伴随方法
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摘要
针对突发污染事故中确定污染源时反向时间积分求解对流-扩散方程易导致计算发散的问题,通过建立目标泛函,利用伴随方法同时求解伴随方程和原方程,逐次迭代得到目标泛函的极值,以实现污染源的反演。为验证该方法的有效性,分别利用污染物空间分布和时间序列对污染源进行反演,其中污染源包括单污染源和双污染源。结果显示,计算值与解析解之平均相对误差小于15%,满足工程应用的精度要求。
Determination of pollution source in sudden pollution accidents can be solved through convection-diffusion equation by reverse time integral. However, such method may result in divergence of numerical calculations. In this study, an objective function is established to solve the adjoint equation and the original equation by using the method of adjoint. Then, the extremums of the objective function are obtained by the successive iteration calculation method. In the end, the pollution source inversion can be realized. In order to verify the effectiveness of the method, pollution source was inverted by using the spatial and temporal distribution of the pollutant concentration, in which both single and double pollution source were included. The results show that the average relative errors between calculated values and analytical solutions are less than 15%, which can be accepted in engineering applications.
Determination of pollution source in sudden pollution accidents can be solved through convection-diffusion equation by reverse time integral. However, such method may result in divergence of numerical calculations. In this study, an objective function is established to solve the adjoint equation and the original equation by using the method of adjoint. Then, the extremums of the objective function are obtained by the successive iteration calculation method. In the end, the pollution source inversion can be realized. In order to verify the effectiveness of the method, pollution source was inverted by using the spatial and temporal distribution of the pollutant concentration, in which both single and double pollution source were included. The results show that the average relative errors between calculated values and analytical solutions are less than 15%, which can be accepted in engineering applications.
引文
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[17]吴自库,范海梅,陈秀荣.对流-扩散过程逆过程反问题的伴随同化研究[J].水动力学研究与进展:A辑,2008,23(2):5-9.(WU Ziku,FAN Meimei,CHENXiurong.The numerical study of the inverse problem in reverse process of convection-diffusion equation with adjoint assimilation method[J].Chinese Journal of Hydrodynamics,2008,23(2):5-9.(in Chinese))
[18]韩龙喜,易路,刘军英,等.连云港近岸海域污染物输移规律[J].河海大学学报(自然科学版),2011,39(3):248-253.(HAN Longxi,XI Lu,LIU Junying,et al.Transport laws for pollutants in coastal area of Lianyungang City[J].Journal of Hohai University(Natural Sciences),2011,39(3):248-253.(in Chinese))
[2]邱颖,顾斌杰,马强,等.永定河大宁水库水环境突发事件预防与应急处置方案[J].北京水务,2015(4):13-16.(QIU Ying,GU Binjie,MA Qiang,et al.Plan of emergency prevention and response for water environment of Daning Reservoir in Yongding River[J].Beijing Water,2015(4):13-16.(in Chinese))
[3]崔冬,何小燕,杨海燕.对流-扩散方程数值耗散的定量研究方法[J].水利水电科技进展,2014,34(5):8-11.(CUI Dong,HE Xiaoyan,YANG Haiyan.Quantitative research method of numerical dissipation levels of convective-diffusive equation[J].Advances in Science and Technology of Water Resources,2014,34(5):8-11.(in Chinese))
[4]LING L,YAMAMOTO M,HON Y C,et al.Identification of source locations in two-dimensional heat equations[J].Inverse Problems,2006,22(4):1289-1305.
[5]王泽文,邱淑芳.一类流域点污染源识别的稳定性与数值模拟[J].水动力学研究与进展:A辑,2008,23(4):364-371.(WANG Zewen,QIU Shufang.Stability and numerical simulation of pollution point source identification in a watershed[J].Chinese Journal of Hydrodynamics,2008,23(4):364-371.(in Chinese))
[6]BRUCKNER G,YAMAMOTO M.Determination of point wave sources by pointwise observations:stability and reconstruction[J].Inverse Problems,2000,16(3):723-748.
[7]朱嵩,刘国华,王立忠,等.水动力-水质耦合模型污染源识别的贝叶斯方法[J].四川大学学报(工程科学版),2009,41(5):30-35.(ZHU Hao,LIU Guohua,WANG Lizhong,et al.A Bayesian approach for the identification of pollution source in water quality model coupled with hydrodynamics[J].Journal of Sichuan University(Engineer Science edition),2009,41(5):30-35.(in Chinese))
[8]闵涛,周孝德,张世梅,等.对流-扩散方程源项识别反问题的遗传算法[J].水动力学研究与进展:A辑,2004,19(4):520-524.(MIN Tao,ZHOU Xiaode,ZHANG Shimei,et al.Genetic algorithm to an inverse problem of source term identification for convectiondiffusion equation[J].Chinese Journal of Hydrodynamics,2004,19(4):520-524.(in Chinese))
[9]辛小康,韩小波,李建,等.基于遗传算法的水污染事故污染源识别模型[J].水电能源科学,2014(7):52-55.(XIN Xiaokang,HAN Xiaobo,LI Jian,et al.Research on test of servomotor time constant of water turbine governor[J].Water Resources and Power,2014(7):52-55.(in Chinese))
[10]孙文心,江文胜,李磊.近海环境流体动力学数值模型[M].北京:科学出版社,2004.
[11]BENNETT A F,MCINTOSH P C.Open ocean modeling as an inverse problem:tidal theory[J].Journal of Physical Oceanography,1982,12(10):1004-1008.
[12]TZIPERMAN E,THACKER W C,LONG R B,et al.Oceanic data analysis using a general circulation model.PartⅠ:simulations[J].Journal of Physical Oceanography,1992,22(12):3349-3354.
[13]马继瑞,韩桂军.变分伴随数据同化在海表面温度预报中的应用研究[J].海洋学报,2002,24(5):1-7.(MA Jirui,HAN Guijun.A study on the application of variational adjoint data assimilation for numerical prediction of sea surface temperature[J].Acta Oceanologica Sinica,2002,24(5):1-7.(in Chinese))
[14]谷艺,吴自库,吕咸青.利用伴随同化方法估计渤海潮汐模式的底摩擦系数[J].海洋湖沼通报,2005(3):1-7.(GU Yi,WU Ziku,LXianqing.Estimation of bottom friction coefficient for a 2D nonlinear tidal model of the Bohai Sea by the adjoint method[J].Transactions of Oceanology and Limnology,2005(3):1-7.(in Chinese))
[15]AKCELIK V,BIROS G,GHATTAS O,et al.Avariational finite element method for source inversion for convective-diffusive transport[J].Finite Elements in Analysis&Design,2003,39(8):683-705.
[16]刘继军.不适定问题的正则化方法及应用[M].北京:科学出版社,2005.
[17]吴自库,范海梅,陈秀荣.对流-扩散过程逆过程反问题的伴随同化研究[J].水动力学研究与进展:A辑,2008,23(2):5-9.(WU Ziku,FAN Meimei,CHENXiurong.The numerical study of the inverse problem in reverse process of convection-diffusion equation with adjoint assimilation method[J].Chinese Journal of Hydrodynamics,2008,23(2):5-9.(in Chinese))
[18]韩龙喜,易路,刘军英,等.连云港近岸海域污染物输移规律[J].河海大学学报(自然科学版),2011,39(3):248-253.(HAN Longxi,XI Lu,LIU Junying,et al.Transport laws for pollutants in coastal area of Lianyungang City[J].Journal of Hohai University(Natural Sciences),2011,39(3):248-253.(in Chinese))