基于Kriging模型和提升小波变换的随机模型修正

吴雨程; 殷红; 彭珍瑞

doi:10.21656/1000-0887.420128

基于Kriging模型和提升小波变换的随机模型修正

doi: 10.21656/1000-0887.420128

兰州交通大学机电工程学院，兰州 730070

基金项目: 国家自然科学基金 (51768035)

详细信息

作者简介:
吴雨程（1997—），男，硕士生（E-mail：821085844@qq.com）

殷红（1978—），女，教授，博士，硕士生导师（E-mail：yinhong@mail.lzjtu.cn）

彭珍瑞（1972—），男，教授，博士，博士生导师（通讯作者. E-mail：pzrui@163.com）

中图分类号: O327
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- 文章访问数: 697
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- 被引次数: 0
出版历程
- 收稿日期: 2021-05-10
- 修回日期: 2021-06-17
- 刊出日期: 2022-07-15

Stochastic Model Updating Based on Kriging Model and Lifting Wavelet Transform

School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China

摘要

摘要:
为提高随机模型修正效率，减小计算量，提出了一种基于Kriging模型和提升小波变换的随机模型修正方法。首先，对加速度频响函数进行提升小波变换，提取第5层近似系数代替原频响函数。其次，采用拉丁超立方抽样抽取待修正样本，将其作为Kriging模型的输入，对应的近似系数作为输出，构建Kriging模型。提出了一种引入莱维飞行(Lévy flight)的蝴蝶优化算法(LBOA)，并将其应用于Kriging模型相关参数的寻优中，提高Kriging模型的精度。最后，以最小化Wasserstein距离为目标，通过鲸鱼优化算法求解待修正参数的均值。测试函数结果表明，LBOA在寻优能力、收敛精度和稳定性等方面有了很大的提升。数值算例的修正误差均低于0.4%，验证了所提模型修正方法具有较高的修正精度和效率。
- 模型修正 /
- Kriging模型 /
- 提升小波变换 /
- Wasserstein距离 /
- 蝴蝶优化算法
Abstract:
In order to improve the efficiency of stochastic model updating and reduce the amount of calculation, a stochastic model updating method based on Kriging model and lifting wavelet transform was proposed. Firstly, the lifting wavelet transform was performed on the acceleration frequency response function, and the 5th-level approximate coefficients were extracted to replace the original frequency response function; secondly, the Latin hypercube sampling was applied to sample the parameters to be updated and the corresponding approximate coefficients as the outputs to build the Kriging model. A butterfly optimization algorithm with Lévy flight (LBOA) was proposed and used to improve the accuracy of Kriging model; finally, with the goal of minimizing the Wasserstein distance, the mean values of the parameters to be updated were solved with the whale optimization algorithm. The results of the test function show that, the LBOA greatly improves in terms of optimization, convergence accuracy and stability. The updating errors of the numerical examples are all less than 0.4%, and indicate the high accuracy and efficiency of the proposed model updating method.
- model updating /
- Kriging model /
- lifting wavelet transform /
- Wasserstein distance /
- butterfly optimization algorithm

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图 1 f₆收敛曲线

Figure 1. Convergence curves of $ {f_6} $

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图 2 提升小波变换过程

Figure 2. The process of the lifting wavelet transform

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图 3 近似系数提取流程

Figure 3. The flow chart of extracting approximate coefficients

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图 4 模型修正流程图

Figure 4. The flowchart of model updating

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图 5 二维桁架结构

Figure 5. The 2D truss structure

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图 6 参数对结构AFRF的灵敏度

Figure 6. Sensitivity of structure AFRF to parameters

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图 7 Kriging模型精度评估

Figure 7. Accuracy evaluation of the Kriging model

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图 8 修正前后加速度频响函数曲线

Figure 8. AFRF curves before and after updating

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图 9 三维桁架结构

Figure 9. The 3D truss structure

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图 10 第1、3、5层近似系数

Figure 10. Approximate coefficients for the 1st, 3rd and 5th levels

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图 11 Kriging模型参数寻优曲线

Figure 11. Optimization curves of the Kriging model parameter

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图 12 Kriging模型精度评估

Figure 12. Accuracy evaluation of the Kriging model

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图 13 修正前后加速度频响函数曲线：(a)实部曲线；(b)虚部曲线

Figure 13. AFRF curves before and after updating: (a) real part curves; (b) imaginary part curves

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表 1 BOA改进算法寻优结果

Table 1. Optimization results of the improved BOA algorithm

function	BOA			LBOA
function	mean value	standard deviation	successful rate $\delta$/%	mean value	standard deviation	successful rate $\delta$/%
$ {f_1} $	1.80E−14	1.25E−15	100	0	0	100
$ {f_2} $	9.43E−12	3.40E−12	100	0	0	100
$ {f_3} $	7.67E−4	3.48E−4	0	4.37E−5	4.93E−5	86
$ {f_4} $	1.81E−15	2.24E−15	100	0	0	100
$ {f_5} $	1.90E+1	5.81E+1	80	0	0	100
$ {f_6} $	1.19E−11	1.95E−12	100	8.88E−16	0	100

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表 2 寻优结果对比

Table 2. Comparison of optimization results

	LBOA	BOA
${\theta _k}$	4.064 5 × 10⁻³	3.320 5
fitting value e	7.396 2 × 10⁻¹⁴	6.941 9 × 10⁻⁸
running time t/s	27.2	27.7

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表 3 桁架结构修正前后参数均值及误差

Table 3. Parameter mean values and errors of the truss structure before and after updating

updated parameter	test value	finite element value	updated value	relative error δ/%
E_㉓/GPa	190	171	189.881	0.0626
E_⑱/GPa	190	209	190.009	0.0047
E_㉒/GPa	190	171	189.937	0.0331
E_⑫/GPa	190	209	189.971	0.0155

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表 4 不同响应指标下的结果对比

Table 4. Comparison of results under different response indicators

response indicator	relative error of E_㉓ δ₁/%	relative error of E_⑱ δ₂/%	relative error of E_㉒ δ₃/%	relative error of E_⑫ δ₄/%	time consumed t/s
AFRF	1.8092	1.8907	0.9921	0.7739	92
approximate coefficient	0.0626	0.0047	0.0331	0.0155	27

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表 5 桁架结构修正前后结构参数均值及误差

Table 5. Parameter mean values and errors of the truss structure before and after updating

updated parameter	test value	finite element value	updated value	relative error δ/%
E/GPa	190	209	189.607	0.207
$\rho /({\text{kg} } \cdot { {\text{m} }^{ { { - 3} } } })$	7800	7020	7773.562	0.339
A/mm²	85.5	95	85.662	0.189

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