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摘要:
反距离权重插值方法在航空航天中有着广泛的应用，但其存在仅考虑距离关系而忽视方位关系的缺点，顾及方向遮蔽性的调和反距离权重插值方法弥补了这种不足，提高了插值精度，但仅适用于平面插值。借鉴该方法的基本假设，根据归一化后样本点的不同空间分布，以平面均匀角和球面均匀角为基准，制定统一的均匀性量化标准，提出一种更具普适性的三维空间插值方法。在搜索插值点的临近样本点时，提出一种最近邻搜索算法,极大提高了插值计算效率。通过测试函数计算发现，与反距离权重插值方法相比，所提插值方法误差显著降低。将所提插值方法应用于某型民用飞机短舱的气动载荷插值，结果表明，所提插值方法兼具高效和高精度的优点。
Abstract:The inverse distance weight interpolation method has a wide range of applications in aerospace, but it only considers the distance relationship and ignores the azimuth relationship. This shortcoming is addressed by the adjusted inverse distance weight interpolation method with position shading, although it is only appropriate for plane interpolation. Based on the basic assumption of this method, according to the different spatial distribution of normalized sample points, this paper formulates a unified uniformity quantization standard based on the plane uniform angle and spherical uniform angle, and proposes a three-dimensional spatial interpolation method. This study proposes a new technique that significantly increases the effectiveness of interpolation by searching sample points close to the interpolation points. Through the calculation of test functions, it is found that compared with the inverse distance weight interpolation method, the error of the proposed method issignificantly reduced. The proposed method is applied to the aerodynamic loads interpolation of a civil aircraft nacelle, and the results show that the proposed method has the advantages of high efficiency and high accuracy.
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图 8 插值平均误差随搜索点数的变化
Figure 8. Variation of interpolation average error with number of search points
图 9 插值平均误差随权指数的变化
Figure 9. Variation of interpolation average error with weight index
图 12 气动结点载荷和插值后有限元结点载荷
Figure 12. Aerodynamic nodal load and interpolated finite element nodal load
表 1 测试函数
Table 1. Test functions
函数编号 函数表达式 1 ${x^2} + {y^2} + {{\textit{z}}^2}$ 2 ${x^3} + {y^3} + {{\textit{z}}^3} + x{y^2} + x{{\textit{z}}^2} + y{{\textit{z}}^2} + xy{\textit{z}}$ 3 $3{\left( {1 - x} \right)^2}{ {\text{e} }^{ - {x^2} - { {\left( {y + 1} \right)}^2} } } - 10\left( {x - {x^3} - {y^5} } \right){ {\text{e} }^{ - {x^2} } } - { {\text{e} }^{ - { {\left( {x + 1} \right)}^2} - {y^{2} } } /3} + { {\text{e} }^{\textit{z}}}$ 
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表 2 不同插值方法的计算耗时对比
Table 2. Comparison of computation time of different interpolation methods
s 样本点数 IDW Lu’s AIDW AIDW-DP 20×20×20 2.73 2.71 0.11 40×40×40 21.94 21.90 0.13 60×60×60 74.38 74.48 0.14
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表 3 插值误差统计
Table 3. Statistics of interpolation error
插值方法 误差均值/${10^{ - 5}}$ 中误差/${10^{ - 4}}$ 最大误差/${10^{ - 3}}$ IDW 8.77 2.72 8.74 AIDW-DP 6.40 2.38 8.41
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