跨音速翼型反设计的一种大范围收敛方法

求解跨音速翼型的反设计问题时，传统的梯度型方法一般均为局部收敛. 为增大求解的收敛范围，依据同伦方法的思想，通过构造不动点同伦，将原问题的求解 转化为其同伦函数的求解，并依据拟Sigmoid函数调整同伦参数以提高计算效率，进而构造 出一种具有较高计算效率的大范围收敛反设计方法. 数值算例以RAE2822翼型的表面压力分 布为拟合目标，分别采用B样条方法, PARSEC方法及正交形函数方法等3种不同的 参数化方法，并分别以NACA0012, OAF139及VR15翼型为初始翼型进行迭代计 算. 计算结果证明，该方法适用于多种参数化方法，且具有较好的计算效率，从多 个不同的初始翼型出发，经较少次数迭代后， 均能与目标翼型很好地拟合，是一种高效的大范围收敛方法.

A global convergence method on inverse design of transonic airfoil

Inverse design problem can be represented by a nonlinear optimization problem or a complex nonlinear equation. Some traditional algorithms, such as nonlinear least square method, can be used to solve inverse design problems. But unfortunately, they are all local convergence and an unsuitable initial value of iteration may lead to a divergent computational procedure. In this paper, a global convergence method was developed to overcome the shortcoming. Based on the idea of a homotopy method, the original formulation for inverse design was replaced by a homotopy function, and then an iterative local linearization method was used to solve the equation. Moreover, a quasi-sigmoid method was utilized to adjust the homotopy parameter during the iteration, which can assure a stable and efficient iteration procedure. As numerical examples, the surface pressure distribution of RAE2822 airfoil was taken as the target pressure distribution, three parameterization methods, which include B-Spline, PARSEC and Orthogonal Shape Function methods were individual used to parameterize the airfoil shape, and three different airfoil shapes, including NACA0012, OAF139 and VR15 airfoils, were used as the initial shapes, all inverse design results demonstrate the feature of global convergence and efficiency of the presented new method.