Application of Digital Control Theory to Computational Fluid Dynamics

提出了用数字控制理论研究计算流体力学的新方法。众所周知，计算流体力学包括两个主要部分，即(1)计算网格的生成和(2)求解守恒律的数值格式。本文提出了通过数字控制的最优调节器生成自适应网格的新方法，并尝试用最优方案数值求解一个守恒律。线性最优调节器问题的性能指标与自适应网格生成的特性密切相关。考虑到这一点，在第一种方法中，将一维一维泊松方程和二维泊松方程转化为状态空间形式。NACA-0012翼型流场的数值算例表明，通过这三种适应，激波时网格线有很强的聚集性。一个守恒律的数值解具有来自数字控制理论的状态空间。利用数值误差的二次范数构造性能指标，通过求解最优调节器问题来获得最小化该误差的反馈增益。

New methods for computational fluid dynamics through digital control theory are presented. As well known, computational fluid namics consists of two main parts, i. e. (1) generation of computational grid, and (2) numerical scheme to solve the conservation law. In this investigation, new methods for adaptive grid generation through the optimal regulator of digital control are presented, and, moreover, an attempt to make optimal scheme to solve a conservation law numerically. The performance index of linear optimal regulator problem has a strong relation to the characteristics of adaptive grid generation. Paying attention to this point, one quasi-one, and two dimensional Poisson equations are transformed into the statespace form in the first method. The numerical examples of NACA-0012 airfoil flow field show strong clustering of grid lines at shock waves through these three kinds of adaptation. The numerical solution to a conservation law has state-space from of digital control theory. Constructing a performance index by a second norm for the numerical error, feedback gain minimizing this error should be obtained by solving an optimal regulator problem.