《厦门大学学报（自然科学版）》

随着结构形式的日益大型复杂化,精细有限元分析由于受到网格尺寸和时间积分步长的限制,通常难以同时保证效率与精度.为了提升结构动力分析效率,本文首先根据有限元形函数影响域对有限元动力计算前期数据进行压缩提炼,构造动力计算的训练集.其次,通过引入激活函数和B样条基函数对训练集数据进行非线性预处理,提升结构关键节点的动力响应计算精度.最后,将预处理后的训练集和贝叶斯回归方法相融合,提出了一种结构动力响应贝叶斯回归代理模型.文中通过典型算例验证了所提代理模型在保证计算精度的条件下,能够明显提升结构动力分析的计算效率.

Objective: The increasing complexity of structures requires refined finite element analyses for better accuracies. Unfortunately, structural refinements inevitably demands small time steps, thus rendering dynamic analyses time-consuming. In this study, we strive to introduce Bayesian regression surrogate model so that both numerical properties, namely efficiencies and accuracies, are optimized.
Methods: A Bayesian regression surrogate model is constructed based on a nonlinear data preprocessing. In particular, the dynamic analysis input dataset is formulated with displacements and stresses from the direct finite element analysis. This input dataset is then compressed by using the influence domains of finite element shape functions. Subsequently, for the purpose of enhancing the accuracy at key finite element nodes, a nonlinear data preprocessing approach is proposed to transform the dynamic analysis training dataset via introducing the activation and B-spline basis functions. The resulting training dataset is then merged with the Bayesian regression method to formulate a Bayesian regression surrogate model for structural dynamic analyses.
Results:A Bayesian regression surrogate model for structural dynamic responses is established. Numerical results are presented for the dynamic analysis of both one-dimensional and three-dimensional examples. They show that structural response predictions by the proposed Bayesian surrogate model agree satisfactorily with reference solutions obtained by direct finite element dynamic analysis, in which the Sigmoid activation function is used as the activation function and the cubic B-spline basis function is employed to enrich the dataset. It is also found that the integrity of the training data exerts a significant impact on the prediction accuracy of the proposed Bayesian regression surrogate model trained directly using the original data. On the other hand, the proposed nonlinear data preprocessing approach can effectively improve prediction results under the condition of incomplete training data. In addition, by comparing the efficiency of the proposed Bayesian regression surrogate model and the direct finite element analysis, it is revealed that the proposed Bayesian regression surrogate model based on nonlinear data preprocessing can significantly improve the efficiency of structural dynamic analyses with satisfactory response accuracies.
Conclusions:In the proposed surrogate model, the Sigmoid activation function is used to normalize and process the data in a nonlinear fashion, and then the nonnegative B-spline basis function is selected to expand the training data around key locations. Finally, a Bayesian regression surrogate model for structural dynamic dynamics is formulated by combining the nonlinear preprocessed dataset and Bayesian regression scheme. Numerical results reveal that the proposed Bayesian regression surrogate model with nonlinear data preprocessing can accurately assess the structural dynamic response, which agrees satisfactorily with solutions obtained by the costly direct finite element analysis. In particular, it is shown that the nonlinear preprocessing step can significantly improve the prediction results under incomplete training data. An optimal balance between the efficiency and accuracy was achieved in the proposed Bayesian regression surrogate model for finite element structural dynamic analyses.

引言
1 贝叶斯回归模型的基本理论
2 训练集数据预处理
3 结构动力响应分析的贝叶斯回归模型
4 数值算例
5 结论

图1 贝叶斯回归概率图模型示意图<br/>Fig.1 Illustration of Bayesian regression probabilistic graphical model

图1 贝叶斯回归概率图模型示意图
Fig.1 Illustration of Bayesian regression probabilistic graphical model

图2 有限元离散及节点形函数示意图<br/>Fig.2 Illustration of finite element discretization and nodal shape functions

图2 有限元离散及节点形函数示意图
Fig.2 Illustration of finite element discretization and nodal shape functions

图3 数据非线性预处理流程<br/>Fig.3 Flowchart for nonlinear data preprocessing operations

图3 数据非线性预处理流程
Fig.3 Flowchart for nonlinear data preprocessing operations

图4 结构动力响应的贝叶斯回归代理模型的计算框架与模块<br/>Fig.4 Computational framework and module of the Bayesian regression surrogate model for structural dynamic response

图4 结构动力响应的贝叶斯回归代理模型的计算框架与模块
Fig.4 Computational framework and module of the Bayesian regression surrogate model for structural dynamic response

图5 一维弹性杆问题<br/>Fig.5 1D elastic rod problem

图5 一维弹性杆问题
Fig.5 1D elastic rod problem

图6 基于数据非线性预处理的结构动力响应贝叶斯回归代理模型的详细数据传递过程<br/>Fig.6 Detailed data transfer procedure of the Bayesian regression surrogate model for structural dynamic response with nonlinear data preprocessing

图6 基于数据非线性预处理的结构动力响应贝叶斯回归代理模型的详细数据传递过程
Fig.6 Detailed data transfer procedure of the Bayesian regression surrogate model for structural dynamic response with nonlinear data preprocessing

图7 有无数据预处理的代理模型对f=1 000sin(t)/(t+1)的预测结果对比<br/>Fig.7 Comparison of predicted results by the surrogate model with and without data preprocessing under f=1 000sin(t)/(t+1)

图7 有无数据预处理的代理模型对f=1 000sin(t)/(t+1)的预测结果对比
Fig.7 Comparison of predicted results by the surrogate model with and without data preprocessing under f=1 000sin(t)/(t+1)

图8 贝叶斯回归代理模型对f=1 000sin(x+t)的预测结果对比<br/>Fig.8 Comparison of predicted results by the Bayesian regression surrogate model under f=1 000sin(x+t)

图8 贝叶斯回归代理模型对f=1 000sin(x+t)的预测结果对比
Fig.8 Comparison of predicted results by the Bayesian regression surrogate model under f=1 000sin(x+t)

图9 动力学响应预测的3D空间框架模型及解观测节点选择<br/>Fig.9 3D spatial frame structure and selection of observation nodes for dynamic response prediction

图9 动力学响应预测的3D空间框架模型及解观测节点选择
Fig.9 3D spatial frame structure and selection of observation nodes for dynamic response prediction

图10 空间框架及观测节点局部区域t=1 s时刻位移预测解(单位: m)<br/>Fig.10 Displacement prediction for the spatial frame structure and regions near observation nodes at t=1 s(unit: m)

图10 空间框架及观测节点局部区域t=1 s时刻位移预测解(单位: m)
Fig.10 Displacement prediction for the spatial frame structure and regions near observation nodes at t=1 s(unit: m)

图11 空间框架观测点动力响应预测结果对比<br/>Fig.11 Comparison of dynamic response prediction results at observation nodes for the spatial frame structure

图11 空间框架观测点动力响应预测结果对比
Fig.11 Comparison of dynamic response prediction results at observation nodes for the spatial frame structure

图12 空间框架观测点局部区域动力响应代理模型预测解与有限元数值解对比(单位: m)<br/>Fig.12 Comparison between the predicted solutions by surrogate model and the numerical solutions by finite element dynamic analysis near observation nodes for the spatial frame structure(unit: m)

图12 空间框架观测点局部区域动力响应代理模型预测解与有限元数值解对比(单位: m)
Fig.12 Comparison between the predicted solutions by surrogate model and the numerical solutions by finite element dynamic analysis near observation nodes for the spatial frame structure(unit: m)

表1 空间框架问题的有限元动力分析和代理模型预测计算效率对比<br/>Tab.1 Computational efficiency comparison between finite element dynamic analysis and surrogate model prediction for the spatial frame structure

表1 空间框架问题的有限元动力分析和代理模型预测计算效率对比
Tab.1 Computational efficiency comparison between finite element dynamic analysis and surrogate model prediction for the spatial frame structure

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引言

1 贝叶斯回归模型的基本理论

2 训练集数据预处理

3 结构动力响应分析的贝叶斯回归模型

4 数值算例

5 结论

学报简介

备注