基于贝叶斯方法的有限元模型修正研究
发布时间：2018-07-26 21:07
【摘要】：有限元模型是进行结构静动力响应分析、结构健康监测和优化设计的基础，但由于几何尺寸误差、材料参数误差和网格离散误差，需要对有限元模型进行修正才能使其与实际结构的响应相符。由于结构材料参数存在随机性、空间位置存在不确定性、施工质量存在变易性，以及实测数据、有限元模型本身存在不确定性，导致有限元模型修正的结果充满着不确定性，而现有基于确定性的模型修正方法无法考虑这些不确定性因素的影响。本文依据贝叶斯统计理论和马尔科夫蒙特卡洛模拟（MCMC），建立了基于设计参数的贝叶斯有限元模型修正方法，并提出基于相关向量机的回归分析法以克服所提方法计算效率低下的问题，从而实现大型土木工程结构有限元模型的修正工作。 论文的主要研究内容和主要结论如下： ①针对贝叶斯统计理论中先验分布选择易受人为因素影响的问题，基于最大信息熵原理推导了先验分布的选取原则：当待修正设计参数均值和标准差均已知时，最大信息熵的先验分布为高斯分布；当仅已知待修正设计参数的变化区间时，最大信息熵分布与贝叶斯假定都为均匀分布。然后采用数值模拟讨论了不同先验选取对模型修正结果的影响，结果表明随测试次数的增加，均匀先验分布是高斯先验分布的极限状态。 ②基于贝叶斯统计理论，推导了待修正设计参数的后验概率密度函数，并采用Metropolis-Hastings算法进行后验概率密度函数的数值计算，从而建立了基于贝叶斯的不确定性有限元模型修正方法。通过简支梁数值算例的结果表明：标准MH算法模拟后验样本，能够解决待修正设计参数个数较少的贝叶斯模型修正，但存在采样易停滞的缺陷。 ③将DRAM算法引入到贝叶斯有限元模型修正中，通过自适应算法（AM）实现自主调整采样步距；通过延缓拒绝（DR）算法提高新样本接受概率，从而有效克服了标准MH算法对高维待修正参数收敛较慢或无法收敛的问题。五层剪切型框架结构数值算例结果表明：DRAM算法能够成功计算多待修正设计参数后验概率密度函数。通过对悬臂梁模型试验修正的结果表明：DRAM算法的贝叶斯模型修正，使模态频率误差由10%降低至1%以内，同时模态相关系数MAC由最小的0.9提高至1.0附近，达到了较好的修正效果；且在相同的条件下，DRAM算法修正后的模态频率最大误差为0.8%，较一阶优化算法的2.56%更小，实现了更高精度的修正效果。 ④针对模型修正反问题普遍存在的计算效率低下问题，将有限元模型的响应计算由有限元软件实现转化为通过RVM数学回归实现，提出了基于相关向量机（RVM）的贝叶斯快速计算方法，并对RVM回归精度影响因素进行分析，模拟计算的结果表明，，修正速度提高60倍左右，实现了快速贝叶斯模型修正。 ⑤通过脉冲荷载单点激励四层两跨实验钢框架模型结构，并采用频域分解法识别了该框架结构的频率和振型，然后根据测得的模态信息，对初始有限元模型进行了贝叶斯修正，修正结果表明：贝叶斯法模型修正，使误差项由初始模型的24%降低至2%左右，表明贝叶斯模型修正能应用于实际结构。
[Abstract]:Finite element model is the basis of structural static and dynamic response analysis, structural health monitoring and optimization design, but due to geometric error, material parameter error and grid discrete error, it is necessary to modify the finite element model in order to match the response of the actual structure. In the uncertainty, the quality of the construction is variable and the measured data, the finite element model itself has uncertainty. The result of the finite element model correction is full of uncertainty, and the existing deterministic model correction method can not consider the influence of these uncertainties. This paper is based on Bayesian statistics theory and Marco. J Montecarlo simulation (MCMC), a Bayesian finite element model correction method based on the design parameters is established, and the regression analysis method based on the correlation vector machine is proposed to overcome the problem of inefficient calculation, thus the modification of the finite element model of large civil engineering structure is realized.
The main research contents and conclusions are as follows:
(1) in view of the problem that the prior distribution selection is susceptible to human factors in Bayesian statistical theory, the principle of selecting prior distribution based on the maximum information entropy principle is derived. When the mean and standard difference of the design parameters are known to be corrected, the prior distribution of the maximum information entropy is Gauss distribution; when only the change area of the design parameters to be corrected is known to be corrected Both the maximum information entropy distribution and the Bayesian hypothesis are uniformly distributed. Then the influence of the different prior selection on the model correction results is discussed by numerical simulation. The results show that the uniform prior distribution is the limit state of Gauss's prior distribution with the increase of the test times.
Secondly, based on the Bayesian statistics theory, the posterior probability density function of the modified design parameters is derived, and the Metropolis-Hastings algorithm is used to calculate the posterior probability density function, and the Bayesian based finite element model correction method is established. The results of a simple supported beam numerical example show that the standard MH is calculated. The posterior sample can be used to solve the Bayesian model updating with less number of design parameters to be corrected, but there is a defect that the sampling is easy to stagnate.
Thirdly, the DRAM algorithm is introduced into the Bayesian finite element model correction, the self-tuning sampling step is realized by the adaptive algorithm (AM), and the new sample acceptance probability is improved by the delay rejection (DR) algorithm. Thus the standard MH algorithm is effectively overcome the problem of slow convergence or inability to converge to the high dimensional modified parameters. The number of five layers of shear frame structure can be effectively overcome. The results of a numerical example show that the DRAM algorithm can successfully calculate the posterior probability density function of the modified design parameters. Through the correction of the cantilever beam model test, it is shown that the Bayesian model correction of the DRAM algorithm reduces the modal frequency error from 10% to less than 1%, and the modal relation number MAC is increased from the minimum of 0.9 to 1. In the same condition, the maximum error of the modal frequency modified by DRAM algorithm is 0.8%, which is smaller than the 2.56% of the first order optimization algorithm, and the correction effect of higher precision is achieved.
(4) aiming at the problem of inefficient calculation in the inverse problem of model correction, the response calculation of the finite element model is converted from the finite element software to the realization of RVM mathematical regression. A Bayesian fast calculation method based on the correlation vector machine (RVM) is proposed, and the influencing factors of the RVM regression precision are analyzed and the results of the simulation calculation are simulated. It shows that the correction speed is increased by about 60 times, and a fast Bayesian model updating is achieved.
Through the impulse load single point excitation four layers and two span experimental steel frame model structure, and the frequency domain decomposition method is used to identify the frequency and vibration mode of the frame structure. Then, according to the measured modal information, the initial finite element model is modified by Bias. The correction results show that the Bayesian Yesfa model is modified to make the error term from the initial model. 24% reduced to about 2%, indicating that Bayesian model updating can be applied to practical structures.
【学位授予单位】：重庆大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TU31
本文编号：2147290
[Abstract]:Finite element model is the basis of structural static and dynamic response analysis, structural health monitoring and optimization design, but due to geometric error, material parameter error and grid discrete error, it is necessary to modify the finite element model in order to match the response of the actual structure. In the uncertainty, the quality of the construction is variable and the measured data, the finite element model itself has uncertainty. The result of the finite element model correction is full of uncertainty, and the existing deterministic model correction method can not consider the influence of these uncertainties. This paper is based on Bayesian statistics theory and Marco. J Montecarlo simulation (MCMC), a Bayesian finite element model correction method based on the design parameters is established, and the regression analysis method based on the correlation vector machine is proposed to overcome the problem of inefficient calculation, thus the modification of the finite element model of large civil engineering structure is realized.
The main research contents and conclusions are as follows:
(1) in view of the problem that the prior distribution selection is susceptible to human factors in Bayesian statistical theory, the principle of selecting prior distribution based on the maximum information entropy principle is derived. When the mean and standard difference of the design parameters are known to be corrected, the prior distribution of the maximum information entropy is Gauss distribution; when only the change area of the design parameters to be corrected is known to be corrected Both the maximum information entropy distribution and the Bayesian hypothesis are uniformly distributed. Then the influence of the different prior selection on the model correction results is discussed by numerical simulation. The results show that the uniform prior distribution is the limit state of Gauss's prior distribution with the increase of the test times.
Secondly, based on the Bayesian statistics theory, the posterior probability density function of the modified design parameters is derived, and the Metropolis-Hastings algorithm is used to calculate the posterior probability density function, and the Bayesian based finite element model correction method is established. The results of a simple supported beam numerical example show that the standard MH is calculated. The posterior sample can be used to solve the Bayesian model updating with less number of design parameters to be corrected, but there is a defect that the sampling is easy to stagnate.
Thirdly, the DRAM algorithm is introduced into the Bayesian finite element model correction, the self-tuning sampling step is realized by the adaptive algorithm (AM), and the new sample acceptance probability is improved by the delay rejection (DR) algorithm. Thus the standard MH algorithm is effectively overcome the problem of slow convergence or inability to converge to the high dimensional modified parameters. The number of five layers of shear frame structure can be effectively overcome. The results of a numerical example show that the DRAM algorithm can successfully calculate the posterior probability density function of the modified design parameters. Through the correction of the cantilever beam model test, it is shown that the Bayesian model correction of the DRAM algorithm reduces the modal frequency error from 10% to less than 1%, and the modal relation number MAC is increased from the minimum of 0.9 to 1. In the same condition, the maximum error of the modal frequency modified by DRAM algorithm is 0.8%, which is smaller than the 2.56% of the first order optimization algorithm, and the correction effect of higher precision is achieved.
(4) aiming at the problem of inefficient calculation in the inverse problem of model correction, the response calculation of the finite element model is converted from the finite element software to the realization of RVM mathematical regression. A Bayesian fast calculation method based on the correlation vector machine (RVM) is proposed, and the influencing factors of the RVM regression precision are analyzed and the results of the simulation calculation are simulated. It shows that the correction speed is increased by about 60 times, and a fast Bayesian model updating is achieved.
Through the impulse load single point excitation four layers and two span experimental steel frame model structure, and the frequency domain decomposition method is used to identify the frequency and vibration mode of the frame structure. Then, according to the measured modal information, the initial finite element model is modified by Bias. The correction results show that the Bayesian Yesfa model is modified to make the error term from the initial model. 24% reduced to about 2%, indicating that Bayesian model updating can be applied to practical structures.
【学位授予单位】：重庆大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TU31
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