Blade element momentum methods are widely used for initial aerodynamic analysis of propellers and wind turbines. A wide variety of correction methods exist, but common to all variations, a pair of residuals are converged to ensure compatibility between the two theories. This paper shows how to rearrange the sequence of calculations reducing to a single residual. This yields the significant advantage that convergence can be guaranteed and to machine precision. Both of these considerations are particularly important for gradient-based optimization where a wide variety of atypical inputs may be explored, and where tight convergence is necessary for accurate derivative computation. On a moderate-sized example optimization problem we show over an order of magnitude increase in optimization speed, with no changes to the physics. This is done by using the single residual form, providing numerically exact gradients using algorithmic differentiation with an adjoint, and by leveraging sparsity in the Jacobian using graph coloring techniques. Finally, we demonstrate a revised formulation for cases when no inflow exists in one of the directions (e.g., a hovering rotor or a parked rotor). These new residuals allow for robust convergence in optimization applications, avoiding the occasional numerical difficulties that exist with the standard formulation.

叶片要素动量法广泛用于螺旋桨和风力涡轮机的初始空气动力学分析。存在各种各样的校正方法，但是对于所有变体来说，一对残差会聚以确保两种理论之间的兼容性。本文展示了如何重新排列计算序列以减少到单个残差。这产生了显着的优点，即可以保证收敛并保证机器精度。这两个考虑因素对于基于梯度的优化特别重要，在梯度优化中，可以探索各种非典型输入，并且对于精确的导数计算，必须进行严格的收敛。在一个中等大小的示例优化问题上，我们显示了优化速度提高了一个数量级，而物理特性没有变化。这是通过使用单个残差形式，使用伴随算法的微分来提供数值精确的梯度以及使用图着色技术利用雅可比矩阵中的稀疏性来完成的。最后，我们针对其中一个方向（例如，悬停的转子或停放的转子）不存在流入的情况展示了一种修正的公式。这些新残差可以在优化应用程序中实现稳健的收敛，从而避免了标准配方偶尔出现的数值困难。悬停的转子或停放的转子）。这些新残差可以在优化应用程序中实现稳健的收敛，从而避免了标准配方偶尔出现的数值困难。悬停的转子或停放的转子）。这些新残差可在优化应用程序中实现稳健收敛，从而避免了标准公式编制中偶尔出现的数值困难。