二维可压缩流场下气动外形优化的Newton-Krylov方法

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"《最优气动布局设计:牛顿-克利夫算法》是一篇关于复杂气动外形优化设计的研究论文。作者Marjan Nemec针对非线性效应强烈的气流特性,如冲击波、边界层和分离,提出了一个基于梯度数值优化的牛顿-克利夫算法。研究的核心是将二维可压缩纳维-斯托克斯方程与一阶湍流模型结合,采用多块结构网格进行离散化处理。在计算目标函数梯度时,采用了离散雅可比方法,并利用预条件广义最小残差(GMRES)求解雅可比方程。论文创新地引入了一种新颖的预条件器,结合对离散纳维-斯托克斯方程和湍流模型的完整导数计算,极大地提高了梯度评估的精度和效率。通过这种方法,可以在达到收敛所需的五分之一到一半时间内得到梯度。 在实际应用方面,该算法展示了在提高升力、在多个跨音速工作点下实现升阻最小化、以及根据竞争目标计算帕累托前沿等设计问题上的性能。通过惩罚形式强加优化约束,将有约束问题转化为无约束问题,再利用拟牛顿方法求解。结果表明,新算法显著降低了梯度值,说明找到了局部最优解,从而证明了其在气动外形优化领域的高效性。 这项研究提出的方法对于当前的气动形状优化来说具有很高的速度优势,为实际气动设计提供了一种有效且实用的工具。它能够在保证设计精度的同时,显著缩短计算时间,使得复杂气动外形的设计过程更加便捷和高效。"

The starting configuration of this puzzle is a row of cells, with disks located on cells through . The goal is to move the disks to the end of the row using a constrained set of actions. At each step, a disk can only be moved to an adjacent empty cell, or to an empty cell two spaces away if another disk is located on the intervening square. Given these restrictions, it can be seen that in many cases, no movements will be possible for the majority of the disks. For example, from the starting position, the only two options are to move the last disk from cell to cell , or to move the second-to-last disk from cell to cell . 1. [15 points] Write a function solve_identical_disks(length, n) that returns an optimal solution to the above problem as a list of moves, where length is the number of cells in the row and n is the number of disks. Each move in the solution should be a twoelement tuple of the form (from, to) indicating a disk movement from the cell from to the cell to. As suggested by its name, this function should treat all disks as being identical. Your solver for this problem should be implemented using a breadth-first graph search. The exact solution produced is not important, as long as it is of minimal length. Unlike in the previous two sections, no requirement is made with regards to the manner in which puzzle configurations are represented. Before you begin, think carefully about which data structures might be best suited for the problem, as this choice may affect the efficiency of your search

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