Motivated by the idea of turbomachinery active subspace performance maps, this paper studies dimension reduction in turbomachinery 3D CFD simulations. First, we show that these subspaces exist across different blades—under the same parametrisation—largely independent of their Mach number or Reynolds number. This is demonstrated via a numerical study on three different blades. Then, in an attempt to reduce the computational cost of identifying a suitable dimension reducing subspace, we examine statistical sufficient dimension reduction methods, including sliced inverse regression, sliced average variance estimation, principal Hessian directions and contour regression. Unsatisfied by these results, we evaluate a new idea based on polynomial variable projection—a non-linear least-squares problem. Our results using polynomial variable projection clearly demonstrate that one can accurately identify dimension reducing subspaces for turbomachinery functionals at a fraction of the cost associated with prior methods. We apply these subspaces to the problem of comparing design configurations across different flight points on a working line of a fan blade. We demonstrate how designs that offer a healthy compromise between performance at cruise and sea-level conditions can be easily found by visually inspecting their subspaces.

中文翻译：

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支持多点风扇设计，降维

受涡轮机械主动子空间性能图思想的启发，本文研究了涡轮机械 3D CFD 模拟中的降维。首先，我们展示了这些子空间存在于不同的刀片上——在相同的参数化下——很大程度上独立于它们的马赫数或雷诺数。这通过对三种不同叶片的数值研究得到证明。然后，为了降低识别合适降维子空间的计算成本，我们研究了统计上足够的降维方法，包括切片逆回归、切片平均方差估计、主要 Hessian 方向和轮廓回归。对这些结果不满意，我们评估了一个基于多项式变量投影的新想法——一个非线性最小二乘问题。我们使用多项式变量投影的结果清楚地表明，可以以与现有方法相关的成本的一小部分准确识别涡轮机泛函的降维子空间。我们将这些子空间应用于比较风扇叶片工作线上不同飞行点的设计配置的问题。我们展示了如何通过目视检查其子空间来轻松找到在巡航性能和海平面条件之间提供健康折衷的设计。