样本分簇增强的快速近似建模方法及应用

doi:10.7527/S1000-6893.2025.31755

Abstract

Abstract:

To improve the analysis and optimization efficiency of stiffened composite cylindrical shells， a novel fast modeling method for the Augmented Radial Basis Function （ARBF） approximate model is proposed by fusing sample clustering and improved K-fold cross-validation. The K-means clustering algorithm is used to determine the optimal sample clustering， and the reference shape parameters of the samples are determined based on the local density of the samples. The scaling coefficient is introduced into each cluster to adjust the shape parameters adaptively， which effectively balance the optimization efficiency and accuracy of the shape parameters. The ARBF auxiliary approximation model is established by sample subset， and then the evaluation criteria of ARBF auxiliary approximation model generalization performance is established based on bias-variance decomposition， addressing the problem of insufficient utilization of traditional K-fold cross-validation sample information. Based on the block matrix inversion technique， a fast inversion method of the high-order coefficient matrix of the ARBF auxiliary approximation model is derived， and the scaling coefficient optimization method based on the improved K-fold cross-validation is proposed， which greatly reduces the computational complexity of determining the optimal shape parameters and improves the efficiency and accuracy of ARBF approximation modeling. Numerical and engineering examples validate that the optimal sample clustering and fast cross-validation have significant contributions to the approximate modeling efficiency and accuracy， and reduce the sensitivity of modeling efficiency to sample size and problem dimension. Moreover， using the same number of training samples， the modeling accuracy of the proposed method is significantly superior to that of other typical methods. The results verifies the effectiveness， advancemen and practical engineering value of the proposed method.

Key words: stiffened composite cylindrical shell, augmented radial basis function approximation model, sample clustering, K-fold cross-validation, block matrix inversion

CLC Number:

V214

Yue WANG, Zhixiang WANG, Daokui LI, Yongjun LEI. A fast approximate modeling method and application for sample cluster enhancement[J]. Acta Aeronautica et Astronautica Sinica, 2025, 46(22): 231755.

Figures/Tables 16

Table 1

Test function

编号	函数名	数学表达式	变量取值范围	维数	样本点个数
1	10-Ellipsoid	$f (x) = ∑ i = 1 m i x i 2$	$[- 5,5]$	$m = 10$	200
2	12-Griewank	$f (x) = ∑ i = 1 m x i 2 4 000 + ∏ i = 1 m c o s x i i + 1$	$[- 6,6]$	$m = 12$	240
3	20-Ellipsoid	$f (x) = ∑ i = 1 20 i x i 2$	$[- 5,5]$	$m = 20$	400
4	30-Dixo-Price	$f (x) = (x 1 - 1) 2 + ∑ i = 2 30 i (2 x i 2 - x i - 1) 2$	$[- 10,10]$	$m = 30$	600

Table 1

Table 2

Accuracy evaluation criteria for approximate models

精度准则	表达式	表征的近似能力
复相关系数	$R 2 = 1 - ∑ i = 1 N (y i - y ̂ i) 2 ∑ i = 1 N y i - 1 N ∑ j = 1 N y j 2$	全局近似能力
均方根误差	$E R M S E = 1 N ∑ i = 1 N (y i - y ̂ i) 2$	全局近似能力
相对最大绝对误差	$E R M A E = m a x y i - y ̂ i 1 N ∑ i = 1 N (y i - y ̂ i) 2$	局部近似能力

Table 2

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig.6

Fig.7

Fig.8

Table 3

Fig.9

Table 4

Design variable value range

变量	范围	初始取值	变量	范围	初始取值
$N R$ /根	$[20,100]$	62	$N V$ /根	$[20,100]$	12
h/mm	$[5,8]$	6.818	t/mm	$[4,6]$	4.162
$θ 1$ /（°）	$[- 90,90]$	$0$	$θ 2$ /（°）	$[- 90,90]$	$45$
$θ 3$ /（°）	$[- 90,90]$	$90$	$θ 4$ /（°）	$[- 90,90]$	$- 45$
$θ 5$ /（°）	$[- 90,90]$	$- 45$	$θ 6$ /（°）	$[- 90,90]$	$90$
$θ 7$ /（°）	$[- 90,90]$	$45$	$θ 8$ /（°）	$[- 90,90]$	$0$

Table 4

Table 5

Material properties of T700/BMI carbon fiber reinforced composites

参数	数值	参数	数值
纵向弹性模量/MPa	145 000	横向弹性模量/MPa	10 100
面内剪切模量/MPa	5 200	面内剪切强度/MPa	46
纵向抗拉强度/MPa	1 730	纵向抗压强度/MPa	916
横向抗拉强度/MPa	24	横向抗压强度/MPa	169
密度 $ρ$ /（kg·m^-3）	1.6×10³	泊松比 $ν 12$	0.28

Table 5

Fig.10

Fig.11

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函数类型	样本数量	本文方法计算耗时占比/%
10维Ellipsoid函数	10m	46.0
	15m	42.0
	20m	41.4
12维Griewank函数	10m	42.0
	15m	40.0
	20m	36.6
20维Ellipsoid函数	10m	37.9
	15m	36.0
	20m	33.2
30维Dixo-Price函数	10m	36.6
	15m	34.0
	20m	28.0

A fast approximate modeling method and application for sample cluster enhancement

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