Finite Element Model Updating for Bridges Based on Adaptive Nested Sampling and Bayesian Theory
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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.
Abstract:In 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 models, so as 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 an adaptive nested sampling (ANS) algorithm was proposed. The method used the modal parameters to construct the probability objective function and adopted the ANS algorithm to approximate it. ANS retained the nature of nested sampling (NS), which made the samples ultimately approximate the optimal parameters by narrowing the sampling range layer by layer, and it simplified 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 could 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.
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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 updated parameters
图 6 修正参数各分量的收敛过程
Figure 6. Convergence processes of each component of updated parameters
表 1 模态参数识别结果
Table 1. Modal parameter identification results
序号 方向 模态阶数 频率/Hz 振型 1 横向弯曲 测试结果 4.140 
初始 ANSYS
模型模拟结果4.484 
2 竖向弯曲 测试结果 4.620 
初始 ANSYS
模型模拟结果4.552 
3 横向剪切 测试结果 6.895 
初始 ANSYS
模型模拟结果7.093 
4 纵向扭转 测试结果 8.598 
初始 ANSYS
模型模拟结果9.307 
5 竖向弯曲 测试结果 10.448 
初始 ANSYS
模型模拟结果10.709 
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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 updated parameters
参数 初始值 下限 上限 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 0.88 3.25 × 1010 1.63 ky1/（N·m−1） 1.50 × 107 1.00 4.95 × 106 0.33 2.00 × 107 1.33 kz1/（N·m−1） 1.00 × 108 1.00 5.00 × 107 0.50 1.50 × 108 1.50 ky2/（N·m−1） 1.50 × 107 1.00 4.95 × 106 0.33 2.00 × 107 1.33 kz2/（N·m−1） 1.00 × 108 1.00 5.00 × 107 0.50 1.50 × 108 1.50 ρ2/（kg·m−3） 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 ANS method
参数 最大后验概率参数 90% 置信下限 90% 置信上限 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 ky1/（N·m−1） 0.84 × 107 0.56 −44 0.79 × 107 0.52 1.00 × 107 0.67 kz1/（N·m−1） 0.61 × 108 0.61 −39 0.51 × 108 0.51 1.04 × 108 1.04 ky2/（N·m−1） 1.32 × 107 0.88 −12 1.17 × 107 0.78 1.44 × 107 0.96 kz2/（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 NS method
参数 最大后验概率参数 90% 置信下限 90% 置信上限 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 ky1/（N·m−1） 0.83 × 107 0.55 −45 0.81 × 107 0.54 0.97 × 107 0.65 kz1/（N·m−1） 0.62 × 108 0.62 −38 0.53 × 108 0.53 1.03 × 108 1.03 ky2/（N·m−1） 1.32 × 107 0.88 −12 1.22 × 107 0.81 1.53 × 107 1.02 kz2/（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 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 4.140 4.484 8.30 4.162 0.53 4.159 0.47 2 4.620 4.552 −1.47 4.627 0.15 4.617 −0.06 3 6.895 7.093 2.87 6.930 0.51 6.938 0.63 4 8.598 9.307 8.24 8.580 −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＝（修正值−测试值）/测试值.
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