基于自适应嵌套抽样和贝叶斯理论的桥梁有限元模型修正

徐希堃; 洪彧; 许靖业; 周志达; 蒲黔辉; 文旭光

doi:10.3969/j.issn.0258-2724.20230358

基于自适应嵌套抽样和贝叶斯理论的桥梁有限元模型修正

doi: 10.3969/j.issn.0258-2724.20230358

1.
西南交通大学土木工程学院，四川成都 610031
2.
南宁学院广西中国-东盟综合交通国际联合重点实验室，，广西南宁 530000

基金项目: 广西科技计划（AA21077011）；中央高校基本科研业务费（2682022CX003）

详细信息

作者简介:
徐希堃（1993—），男，博士研究生，研究方向为桥梁智能检测与安全评定，E-mail：xuxikun@my.swjtu.edu.cn

通讯作者:
洪彧（1989—），女，副教授，博士，研究方向为桥梁健康监测和结构动力学，E-mail：hongyu@swjtu.edu.cn

中图分类号: U441
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- 被引次数: 0
出版历程
- 收稿日期: 2023-07-18
- 修回日期: 2023-11-06
- 网络出版日期: 2024-10-29

Finite Element Model Updating for Bridges Based on Adaptive Nested Sampling and Bayesian Theory

1.
School of Civil Engineering, Southwest Jiaotong University Chengdu, 610031, China
2.
Guangxi Key Laboratory of International Join for China-ASEAN Comprehensive Transportation, Nanning University Nanning, 530000, China

摘要

摘要:
在基于有限元模型的桥梁健康监测中，贝叶斯模型修正技术通常被用于量化有限元模型中重要参数的不确定性，以解决模型修正中由于测量误差、建模误差、计算误差等造成的非唯一解问题. 为解决由于大量调用有限元模拟运算，导致修正效率低下的问题，基于自适应嵌套抽样（ANS）算法，提出一种贝叶斯模型修正方法. 该方法利用模态参数构建概率目标函数，并采用 ANS 算法对其进行逼近，ANS 保留了嵌套抽样（NS）的性质，通过逐层缩小抽样范围，使得样本最终逼近最优参数；通过逐层近似，将高维积分问题转化为简单的一维积分问题，简化了证据值和后验概率密度值的计算过程；在此基础上，ANS 算法在迭代过程中通过自适应地调整样本数量，减少对有限元模型的调用；最后，对一座人行桁架桥进行了贝叶斯有限元模型修正试验. 结果表明：在相同算法参数设置下，ANS 算法相比传统 NS 算法降低了约 84% 的有限元模拟调用次数，节省了约 86% 计算时间，并能获得同等精度的不确定性修正结果.
- 有限元模型 /
- 贝叶斯模型修正 /
- 不确定性量化 /
- 嵌套抽样算法 /
- 自适应算法
Abstract:
In the context of bridge health monitoring based on finite element models, Bayesian model updating techniques are commonly used to quantify the uncertainties of important parameters in the finite element model. This is done to address the issue of non-uniqueness in model updating caused by measurement errors, modeling errors, computational errors, etc. To resolve the problem of low efficiency in model updating due to the large number of finite element simulations required, a Bayesian model updating method based on Adaptive Nested Sampling (ANS) algorithm is proposed. The method uses the modal parameters to construct the probability objective function and uses the ANS algorithm to approximate it. ANS retains the nature of nested sampling (NS), which makes the samples ultimately approximate the optimal parameters by narrowing the sampling range layer by layer, and simplifies the computation process of the evidence value and the a posteriori probability density value by transforming the high-dimensional integration problem into a simple one-dimensional integration problem through layer-by-layer approximation. On this basis, the ANS algorithm can also reduce the call of the finite element model by adaptively adjusting the number of samples during the iteration process. Finally, a pedestrian truss bridge was used as a case study for Bayesian finite element model updating experiments. The results demonstrate that, under the same algorithm parameter settings, the ANS algorithm reduces the number of finite element simulation calls by approximately 84% compared to the traditional NS algorithm. This leads to approximately 86% computational time savings while obtaining uncertainty updating results with equal accuracy.
- Finite element model /
- Bayesian model updating /
- parameter uncertainty quantification /
- nested sampling algorithm /
- adaptive algorithm

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图 1 嵌套抽样原理示意

Figure 1. Schematic diagram of the nested sampling

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图 2 超立方收缩法原理示意

Figure 2. Schematic diagram of hypercube shrinkage method

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图 3 自适应嵌套抽样方法计算流程

Figure 3. Calculation flow diagram of adaptive nested sampling method （ANS）

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图 4 人行桥

Figure 4. Steel truss footbridge

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图 5 修正参数各分量的收敛过程

Figure 5. Convergence processes of each component of the updating parameters

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图 6 修正参数各分量的收敛过程

Figure 6. Convergence processes of each component of the updating parameters

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图 7 修正参数范围收缩率

Figure 7. Range shrinkage rate of the updating parameters

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表 1 模态参数识别结果

Table 1. Modal parameter identification results

序号	模态阶数	频率	振型	方向
1	测试结果	4.140 Hz		横向弯曲
1	初始ANSYS模型模拟结果	4.484 Hz		横向弯曲
2	测试结果	4.620 Hz		竖向弯曲
2	初始ANSYS模型模拟结果	4.552 Hz		竖向弯曲
3	测试结果	6.895 Hz		横向剪切
3	初始ANSYS模型模拟结果	7.093 Hz		横向剪切
4	测试结果	8.598 Hz		纵向扭转
4	初始ANSYS模型模拟结果	9.307 Hz		纵向扭转
5	测试结果	10.448 Hz		竖向弯曲
5	初始ANSYS模型模拟结果	10.709 Hz		竖向弯曲

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表 2 修正参数初始值及取值范围

Table 2. Initial values and range of the updating parameters

参数	单位	初始值		下限		上限
参数	单位	数值	θ值	数值	θ值	数值	θ值
E₁	Pa	2.00 × 10¹¹	1.00	1.80 × 10¹¹	0.90	2.30 × 10¹¹	1.15
E₂	Pa	2.00 × 10¹¹	1.00	1.80 × 10¹¹	0.90	2.30 × 10¹¹	1.15
E₃	Pa	2.00 × 10¹⁰	1.00	1.75 × 10¹⁰	0.875	3.25 × 10¹⁰	1.625
k_y1	N/m	1.50 × 10⁷	1.00	4.95 × 10⁶	0.33	2.00 × 10⁷	1.33
k_z1	N/m	1.00 × 10⁸	1.00	5.00 × 10⁷	0.50	1.50 × 10⁸	1.50
k_y2	N/m	1.50 × 10⁷	1.00	4.95 × 10⁶	0.33	2.00 × 10⁷	1.33
k_z2	N/m	1.00 × 10⁸	1.00	5.00 × 10⁷	0.50	1.50 × 10⁸	1.50
ρ₂	kg/m³	2.48 × 10³	1.00	2.26 × 10³	0.91	2.68 × 10³	1.08

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表 3 ANS修正结果

Table 3. Updated results of the ANS method

参数	最大后验概率参数			90% 置信下限		90% 置信上限
参数	数值	θ值	变化率/%	数值	θ值	数值	θ值
E₁/Pa	2.19 × 10¹¹	1.10	+ 10	2.04 × 10¹¹	1.02	2.26 × 10¹¹	1.13
E₂/Pa	1.82 × 10¹¹	0.91	−9	1.81 × 10¹¹	0.91	1.89 × 10¹¹	0.95
E₃/Pa	3.06 × 10¹⁰	1.53	+ 53	1.94 × 10¹⁰	0.97	3.21 × 10¹⁰	1.60
k_y1/（N·m⁻¹）	0.84 × 10⁷	0.56	−44	0.79 × 10⁷	0.52	1.00 × 10⁷	0.67
k_z1/（N·m⁻¹）	0.61 × 10⁸	0.61	−39	0.51 × 10⁸	0.51	1.04 × 10⁸	1.04
k_y2/（N·m⁻¹）	1.32 × 10⁷	0.88	−12	1.17 × 10⁷	0.78	1.44 × 10⁷	0.96
k_z2/（N·m⁻¹）	1.02 × 10⁸	1.02	+ 2	0.59 × 10⁸	0.59	1.36 × 10⁸	1.36
ρ₂/（kg·m⁻³）	2.46 × 10³	0.99	−1	2.38 × 10³	0.96	2.56 × 10³	1.03

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表 4 NS修正结果

Table 4. Updated results of the NS method

参数	最大后验概率参数			90% 置信下限		90% 置信上限
参数	数值	θ值	变化率/%	数值	θ值	数值	θ值
E₁/Pa	2.14 × 10¹¹	1.07	+ 7	2.06 × 10¹¹	1.03	2.14 × 10¹¹	1.14
E₂/Pa	1.82 × 10¹¹	0.91	−9	1.82 × 10¹¹	0.91	1.97 × 10¹¹	0.99
E₃/Pa	3.20 × 10¹⁰	1.60	+ 60	2.19 × 10¹⁰	1.10	3.20 × 10¹⁰	1.61
k_y1/（N·m⁻¹）	0.83 × 10⁷	0.55	−45	0.81 × 10⁷	0.54	0.97 × 10⁷	0.65
k_z1/（N·m⁻¹）	0.62 × 10⁸	0.62	−38	0.53 × 10⁸	0.53	1.03 × 10⁸	1.03
k_y2/（N·m⁻¹）	1.32 × 10⁷	0.88	−12	1.22 × 10⁷	0.81	1.53 × 10⁷	1.02
k_z2/（N·m⁻¹）	1.06 × 10⁸	1.06	+ 6	0.61 × 10⁸	0.61	1.38 × 10⁸	1.38
ρ₂/（kg·m⁻³）	2.42 × 10³	0.98	−2	2.38 × 10³	0.96	2.60 × 10³	1.05

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表 5 修正后模态频率对比

Table 5. Comparison of the updated modal frequencies

模态频率/Hz	测试值/Hz	初始值/Hz	初始误差/%	NS修正值/Hz	NS误差/%	ANS修正值/Hz	ANS误差/%
1	4.14	4.484	8.30	4.162	0.53	4.159	0.47
2	4.62	4.552	−1.47	4.627	0.15	4.617	−0.06
3	6.895	7.093	2.87	6.93	0.51	6.938	0.63
4	8.598	9.307	8.24	8.58	−0.21	8.552	−0.54
5	10.448	10.709	2.50	10.492	0.42	10.463	0.15
注：相对误差＝（x－测试值）/测试值

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