论文标题
平板平行于其表面浸入液体中的平板运动
Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid
论文作者
论文摘要
我们确定在工作预算中平行于其表面平行的薄层平板所需的全球最低时间$ t $。假定流动的雷诺数很大，因此板上的阻力是由薄粘性边界层中的皮肤摩擦引起的。最小值是使用最陡的下降来确定的，其中使用伴随公式来计算梯度。由于管理流体力学的方程是非线性的，因此可能存在多个局部最小值。利用目标函数的二次性质和约束微分方程，我们得出并应用“光谱条件”，以表明融合的局部最佳选择为全局。条件指出，如果使用收敛的伴随字段构建的状态变量中的拉格朗日人的黑森州，则最佳是全局。板的全球最佳运动学从休息开始，速度$ \ propto t^{1/4} $，并随时间$ t $的函数而与speed $ \ propto（t-t）^{1/4} $休息。对于距离的时间比板的长得多，工作最小的运动学包括最佳启动，恒定巡航和最佳停止。光谱条件也可以用于优化受到其他二次偏微分方程约束的优化问题。
We determine the globally minimum time $T$ needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined using a steepest descent, where an adjoint formulation is used to compute the gradients. Because the equations governing fluid mechanics for this problem are nonlinear, multiple local minima could exist. Exploiting the quadratic nature of the objective function and the constraining differential equations, we derive and apply a "spectral condition" to show that the converged local optimum to be a global one. The condition states that the optimum is global if the Hessian of the Lagrangian in the state variables constructed using the converged adjoint field is positive semi-definite at every instance. The globally optimum kinematics of the plate starts from rest with speed $\propto t^{1/4}$ and comes to rest with speed $\propto (T-t)^{1/4}$ as a function of time $t$. For distances much longer than the plate, the work-minimizing kinematics consists of an optimum startup, a constant-speed cruising, and an optimum stopping. The spectral condition can also be used for optimization problems constrained by other quadratic partial differential equations.
