Quadratic programming equivalent mapping method for external aerodynamic force in flight load analysis
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摘要:
传统的气动力映射方法中，守恒型方法能保证总积分载荷的守恒，但计算复杂；非守恒型方法实现简单，能保证气动压力的分布，但不能保证总积分载荷的守恒。利用等式约束保证积分载荷相等，并基于非守恒型加入线性基的径向基函数（RPIM）构造最小二乘保持原气动压力的分布特征，将气动力等效传递问题转化为一个带等式约束的二次规划问题。通过在非守恒方法的基础上构造出新的守恒型方法，兼顾了2种方法的优点。对某一大展弦比机翼上表面的三维气动力等效映射至二维平板面元气动力的结果表明，所提方法在牺牲了一定气动压力分布精度的基础上，严格保证了总积分载荷的相等，同时也继承了非守恒插值方法的高效性。所提方法可用于各类复杂外部气动力数据的快速等效映射，以及其他学科类似的数据映射问题。
Abstract:In conventional aerodynamic force mapping methods, conservative interpolation method can ensure the equivalence of the integral load, but the calculation is complex. While the non-conservative interpolation method is simple to implement with consistency of the distribution of aerodynamic pressure, but no conservation of the overall load. In this paper, equality constraints are used to ensure that the integral loads are equal, with the least squares based on point interpolation method using radical basis function (RPIM) to maintain the distribution characteristics of the original aerodynamic pressure, thus transforming the aerodynamic force mapping problem into a quadratic programming problem with equality constraints. This operation builds up a new conservative method on an original non-conservative method, which has preserved the advantages of each method. Mapping the aerodynamic force from a three-dimensional mesh to low-order panel mesh of a high-aspect-ratio wing is taken as an example. The results show that the method proposed in this paper can not only strictly guarantee the equality of integral loads, but also inherit the high efficiency of non-conservative interpolation method, despite the loss of some accuracy in the distribution of aerodynamic force. Besides, it is suitable for fast equivalent mapping of various complex external aerodynamic data, and similar data mapping problem in other disciplines.
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图 3 原气动网格、RPIM插值方法及二次规划方法得到的压力系数分布
Figure 3. Pressure coefficient distribution obtained by original aerodynamic mesh, RPIM interpolation method and quadratic programming method
图 4 20%、40%、60%和80%展向位置截面压力系数
Figure 4. Section pressure coefficient in 20%, 40%, 60% and 80% spanwise position
图 5 升力系数、扭矩系数和弯矩系数沿展向积分结果
Figure 5. Integral lift coefficient, torque coefficient and bending moment coefficient along spanwise direction
表 1 径向基函数
Table 1. Radial basis function
类型 R(x)定义式 高斯型(G) e－α||x||2 对数型(TPS) ||x||2ln||x|| 复合2次型 (c2+||x||2)1/2 逆复合2次型 1/(c2+||x||2)1/2 Wendland’s C0 (1－||x||)+2 Wendland’s C2 (1－||x||)+4(4||x||+1) Wendland’s C4 (1－||x||)+6(35||x||2+18||x||+3) 
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表 2 映射前后总积分载荷系数
Table 2. Total integral load coefficient before and after mapping
系数 原气动
网格RPIM插值方法
(误差)二次规划方法
(误差)Cz 0.149 08 0.147 0(1.4%) 0.149 08(0%) CMx 0.027 657 5 0.027 081(2.08%) 0.027 657 5(0%) CMy -0.060 124 -0.059 259(1.44%) -0.060 124(0%)
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