Optimal control problems in unsteady flow computation and their variational solutions
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摘要: 为综合利用各类可用的信息源(如水流运动规律,观测资料和统计信息),在反问题的框架下构建了非恒定水流计算的变分模型.针对模型预测的可能误差来源,给出了初始条件、边界条件、物理参数三类独立参量及综合控制问题的定义.在此基础上,阐述了变分法求解这类偏微分方程最优化控制问题的基本原理及模型求解步骤.同时,为便于应用参考,选取初始条件、边界条件和物理参数作为控制向量,以实际常用的水流运动微分方程为基础,导出了一、二维非恒定水流计算的通用伴随方程.并以珠江黄浦至大虎段二维潮流计算为例,展示了变分模型的应用.Abstract: Dynamical equations,observations and their statistics are all important information for presenting the states of unsteady flows.Under the framework of inverse problems,a variational model is developed to fuse all available information for improving the computation of unsteady flows.In the paper,the appropriate optimal control problems are firstly defined for controlling possible numerical errors from inaccurate initial conditions,boundary conditions and unknown physical parameters.Then,the general variational method and its numerical algorithm are suggested to solve these optimal control problems of unsteady flows.Also,we derive the adjoint equations of one-and two-dimensional shallow equations with the forms for practical purpose.It directly serves for real engineering applications.With the proposed variational model,the tidal flows in one reach of the Pearl River are investigated.Results show that the model can well reconstruct tidal flow states when data are unavailable at some boundaries,unlike the direct model.
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Key words:
- unsteady flows /
- inverse problem /
- variational method /
- optimal control /
- data assimilation /
- mathematical model
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