CN111797535A - Structure reliability analysis self-adaptive point adding method for multiple agent models - Google Patents

Structure reliability analysis self-adaptive point adding method for multiple agent models Download PDF

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CN111797535A
CN111797535A CN202010665412.2A CN202010665412A CN111797535A CN 111797535 A CN111797535 A CN 111797535A CN 202010665412 A CN202010665412 A CN 202010665412A CN 111797535 A CN111797535 A CN 111797535A
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points
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李国发
陈泽权
何佳龙
钟瑞龄
陈传海
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Abstract

本发明公开了一种面向结构可靠性分析的序列加点方法，它包括：指定待分析结构的功能函数，确定变量及其公布信息；利用蒙特卡洛抽样，获取候选样本点集

Figure 223654DEST_PATH_IMAGE002
；利用拉丁超立方抽样获取初始点并组成初始样本集
Figure DEST_PATH_IMAGE003
；获取
Figure 553004DEST_PATH_IMAGE004
对应的功能函数的函数值并构建代理模型；对代理模型进行蒙特卡洛数值仿真，获取当前迭代步的失效率；获取最终的代理模型和失效概率。本发明拥有不俗的通用性和适用性，能适用于非kriging模型，大大扩展了适用于结构可靠性分析的代理模型方法，对可靠性分析领域有重要意义。

Figure 202010665412

The invention discloses a sequence point adding method oriented to structural reliability analysis, which includes: specifying the function function of the structure to be analyzed, determining variables and their published information; using Monte Carlo sampling to obtain a set of candidate sample points

Figure 223654DEST_PATH_IMAGE002
;Using Latin hypercube sampling to obtain initial points and form an initial sample set
Figure DEST_PATH_IMAGE003
;Obtain
Figure 553004DEST_PATH_IMAGE004
Calculate the function value of the corresponding function function and build a surrogate model; perform Monte Carlo numerical simulation on the surrogate model to obtain the failure rate of the current iteration step; obtain the final surrogate model and failure probability. The invention has good generality and applicability, can be applied to non-kriging models, greatly expands the surrogate model method suitable for structural reliability analysis, and is of great significance to the field of reliability analysis.

Figure 202010665412

Description

一种面向多种代理模型的结构可靠性分析自适应加点方法An Adaptive Point-Adding Method for Structural Reliability Analysis for Multiple Surrogate Models

技术领域technical field

本发明属于结构可靠性分析领域，尤其是涉及采用自适应加点方法的代理模型对结构可靠性进行分析的方面，具体的，是一种面向多种代理模型的结构可靠性分析自适应加点方法。The invention belongs to the field of structural reliability analysis, in particular to the aspect of analyzing structural reliability by adopting a surrogate model of an adaptive point addition method.

背景技术Background technique

结构可靠性设计和分析方法是考虑结构在寿命内的可靠性为量化指数，通过建立以上述载荷、边界条件等为影响参数的关于结构的功能函数，利用建立的功能函数精确量化计算结构的可靠性指数。与传统的确定性分析方法相比，充分考虑结构在寿命内的各种影响参数，有效避免传统方法的安全性欠缺或冗余的问题。且能通过灵敏度分析，有效分析影响结构可靠性的关键参数，通过改进这些参数能有效提高结构的可靠性。The structural reliability design and analysis method is to consider the reliability of the structure during its life as a quantitative index. By establishing a functional function about the structure with the above loads, boundary conditions, etc. as the influencing parameters, the established functional function is used to accurately quantify and calculate the reliability of the structure. Sex Index. Compared with the traditional deterministic analysis method, the various influence parameters of the structure in the life of the structure are fully considered, and the problem of lack of safety or redundancy of the traditional method is effectively avoided. And through the sensitivity analysis, the key parameters affecting the reliability of the structure can be effectively analyzed, and the reliability of the structure can be effectively improved by improving these parameters.

目前结构可靠性在工程应用上主要以一阶可靠性方法与二阶可靠性方法为主。一阶可靠性方法和二阶可靠性方法计算简便，但无论是一阶还是二阶可靠性方法在结构的功能函数强非线性的情况下精度很差，且在概率转换过程中也会产生误差。随着结构可靠性分析领域的问题往强非线性，高维度和高精度的方向发展，传统的一阶可靠性方法与二阶可靠性方法很大程度上已经无法满足实际工程需求。At present, structural reliability is mainly based on first-order reliability method and second-order reliability method in engineering application. The first-order reliability method and the second-order reliability method are easy to calculate, but both the first-order and second-order reliability methods have poor accuracy when the functional function of the structure is strongly nonlinear, and errors will also occur in the process of probability conversion. . With the development of problems in the field of structural reliability analysis towards strong nonlinearity, high dimension and high precision, the traditional first-order reliability methods and second-order reliability methods have largely been unable to meet the actual engineering needs.

目前，基于代理模型的结构可靠性方法正逐步应用于实际工程中，代理模型方法其本质是通过真实功能函数的若干个采样样本，通过插值或者拟合等方法，获取真实的结构功能函数在一定的范围内的替代模型，从而实现通过对代理模型的分析获取真实功能函数的相应结果。代理模型可以很好处理结构功能函数为隐式情况，且在功能函数为高非线性的情况下也有很高的计算精度。常用的代理模型有多项式响应面、多项式混沌展开、神经网络、支持向量机以及Kriging模型等等。At present, the structural reliability method based on surrogate model is gradually being applied in practical engineering. The essence of the surrogate model method is to obtain the real structural function function at a certain value through interpolation or fitting through several sampling samples of the real function function. The surrogate model within the scope of the surrogate model, so as to obtain the corresponding results of the real functional function through the analysis of the surrogate model. The surrogate model can handle the implicit structure-function function well, and also has high computational accuracy when the function function is highly nonlinear. Commonly used surrogate models include polynomial response surfaces, polynomial chaos expansion, neural networks, support vector machines, and Kriging models.

在构建代理模型之前，需要通过某种实验设计方法获取一系列的样本点或者训练点，主要的实验设计方法在目前可以分为两种，一次采样和序列采样。一次采样即是通过一次选取若干样本点，一般情况下为均匀分布，作为代理模型的训练样本点。序列采样则是通过多次按照一定的规律添加样本点，使得代理模型逐渐提高精度，直至满足设计要求为止。Before constructing a surrogate model, it is necessary to obtain a series of sample points or training points through a certain experimental design method. At present, the main experimental design methods can be divided into two types, one-time sampling and sequential sampling. One-time sampling is to select several sample points at a time, generally uniformly distributed, as the training sample points of the surrogate model. Sequence sampling is to add sample points according to certain rules for many times, so that the surrogate model gradually improves the accuracy until it meets the design requirements.

自适应加点作为序列采样中的一种，其主要通过代理模型以及已有样本点的情况来指导下一次采样或者加点的方法，从而实现代理模型的自适应加点。As a kind of sequence sampling, adaptive point addition mainly guides the next sampling or point addition method through the surrogate model and the situation of existing sample points, so as to realize the adaptive point addition of the surrogate model.

目前，主要的采用自适应加点方法的代理模型，很大程度上都局限于Kriging模型，其他代理模型在处理结构可靠性分析问题上，由于无法与Kriging模型一样，给出预测点的均方差误差，没有办法实现自适应加点。At present, the main surrogate models that use the adaptive point method are largely limited to the Kriging model. Other surrogate models cannot give the mean square error of the prediction points in the same way as the Kriging model when dealing with structural reliability analysis. , there is no way to achieve adaptive point addition.

因此提出一种适用性强，不单单局限于kriging模型的自适应序列加点方法对于结构可靠性分析领域，尤其是采用代理模型方法进行结构可靠性分析的方面具有重要意义。Therefore, to propose an adaptive sequence point addition method with strong applicability, which is not limited to kriging model only, is of great significance to the field of structural reliability analysis, especially the use of surrogate model method for structural reliability analysis.

发明内容SUMMARY OF THE INVENTION

本发明的目的是为了解决上述问题，而提供一种适用性强，能适用于多种代理模型的自适应加点方法，用于结构可靠性分析，实现了非kriging代理模型也可适用自适应加点方法，对非kriging代理模型也可适用自适应加点方法，实现了对结构可靠性分析方法的扩展。The purpose of the present invention is to solve the above problems, and to provide a self-adaptive point addition method with strong applicability and applicable to a variety of surrogate models for structural reliability analysis. The adaptive point method can also be applied to the non-kriging surrogate model, which realizes the extension of the structural reliability analysis method.

一种面向结构可靠性分析的序列加点方法，其包括以下步骤：A method for adding points to sequences for structural reliability analysis, comprising the following steps:

1)指定待分析结构的功能函数

Figure 203834DEST_PATH_IMAGE001
，确定功能函数的变量
Figure 745673DEST_PATH_IMAGE002
及其概率分布信息；1) Specify the functional function of the structure to be analyzed
Figure 203834DEST_PATH_IMAGE001
, which determines the variables of the function function
Figure 745673DEST_PATH_IMAGE002
and its probability distribution information;

2)根据步骤1）所确定的变量的概率密度分布函数

Figure 448050DEST_PATH_IMAGE003
，抽取
Figure 75341DEST_PATH_IMAGE004
个候选样本点，组成候选样本点集
Figure 583682DEST_PATH_IMAGE005
；2) According to the probability density distribution function of the variable determined in step 1)
Figure 448050DEST_PATH_IMAGE003
, extract
Figure 75341DEST_PATH_IMAGE004
candidate sample points to form a candidate sample point set
Figure 583682DEST_PATH_IMAGE005
;

3)根据步骤1）所确定的变量，采用拉丁超立方法在各变量的取值范围

Figure 232226DEST_PATH_IMAGE006
内抽取
Figure 105504DEST_PATH_IMAGE007
个初始随机样本点，组成初始的训练集
Figure 220090DEST_PATH_IMAGE008
，并令
Figure 532123DEST_PATH_IMAGE009
用于记录迭代次数。3) According to the variables determined in step 1), use the Latin hyper-dimension method in the value range of each variable
Figure 232226DEST_PATH_IMAGE006
internal extraction
Figure 105504DEST_PATH_IMAGE007
initial random sample points to form the initial training set
Figure 220090DEST_PATH_IMAGE008
, and let
Figure 532123DEST_PATH_IMAGE009
Used to record the number of iterations.

其中

Figure 782976DEST_PATH_IMAGE010
表示标准正态分布的累积分布函数，而
Figure 827155DEST_PATH_IMAGE011
表示为变量的累积分布函数的逆函数；in
Figure 782976DEST_PATH_IMAGE010
represents the cumulative distribution function of the standard normal distribution, and
Figure 827155DEST_PATH_IMAGE011
is expressed as the inverse function of the cumulative distribution function of the variable;

4)根据训练集

Figure 179770DEST_PATH_IMAGE012
，获取
Figure 29915DEST_PATH_IMAGE013
对应的结构的功能函数的函数值，并构建代理模型；4) According to the training set
Figure 179770DEST_PATH_IMAGE012
,Obtain
Figure 29915DEST_PATH_IMAGE013
The function value of the function function of the corresponding structure, and build the proxy model;

5)利用步骤4所建立的代理模型，结合步骤2）的候选样本点集

Figure 400853DEST_PATH_IMAGE014
进行数值仿真，获取当前迭代步的代理模型预估的结构的失效概率
Figure 615934DEST_PATH_IMAGE015
；5) Using the surrogate model established in step 4, combined with the candidate sample point set of step 2)
Figure 400853DEST_PATH_IMAGE014
Carry out numerical simulation to obtain the failure probability of the structure estimated by the surrogate model of the current iteration step
Figure 615934DEST_PATH_IMAGE015
;

6)判断失效概率

Figure 705113DEST_PATH_IMAGE016
与上一次迭代的结果
Figure 342636DEST_PATH_IMAGE017
的相对误差是否小于
Figure 568081DEST_PATH_IMAGE018
，而
Figure 219642DEST_PATH_IMAGE019
为一足够小的常数，收敛条件为：6) Judging the probability of failure
Figure 705113DEST_PATH_IMAGE016
with the result of the previous iteration
Figure 342636DEST_PATH_IMAGE017
Is the relative error less than
Figure 568081DEST_PATH_IMAGE018
,and
Figure 219642DEST_PATH_IMAGE019
is a sufficiently small constant, the convergence condition is:

Figure 530538DEST_PATH_IMAGE020
Figure 530538DEST_PATH_IMAGE020

如果满足收敛要求，则获取最终的代理模型和最终预估的失效概率；If the convergence requirements are met, the final surrogate model and the final estimated failure probability are obtained;

如果不满足收敛要求或者

Figure 722485DEST_PATH_IMAGE021
，则进行下一步，进行自适应序列加点；If convergence requirements are not met or
Figure 722485DEST_PATH_IMAGE021
, then proceed to the next step to add points to the adaptive sequence;

7)利用学习函数

Figure 271278DEST_PATH_IMAGE022
对候选样本点集
Figure 375631DEST_PATH_IMAGE023
进行数值仿真，并获取最新的样本点
Figure 439402DEST_PATH_IMAGE024
，并更新训练集
Figure 169461DEST_PATH_IMAGE025
；其中学习函数
Figure 572760DEST_PATH_IMAGE026
的具体表达如下：7) Use the learning function
Figure 271278DEST_PATH_IMAGE022
For the candidate sample point set
Figure 375631DEST_PATH_IMAGE023
Run numerical simulations and get the latest sample points
Figure 439402DEST_PATH_IMAGE024
, and update the training set
Figure 169461DEST_PATH_IMAGE025
; where the learning function
Figure 572760DEST_PATH_IMAGE026
The specific expression is as follows:

Figure 97283DEST_PATH_IMAGE027
Figure 97283DEST_PATH_IMAGE027

其中，

Figure 900547DEST_PATH_IMAGE028
表示点
Figure 168717DEST_PATH_IMAGE029
对于训练集
Figure 426523DEST_PATH_IMAGE030
中各点欧式距离的平均值，
Figure 387526DEST_PATH_IMAGE031
表示点
Figure 425889DEST_PATH_IMAGE032
对于训练集
Figure 435433DEST_PATH_IMAGE033
中各点欧式距离的最小值；in,
Figure 900547DEST_PATH_IMAGE028
Representation point
Figure 168717DEST_PATH_IMAGE029
for the training set
Figure 426523DEST_PATH_IMAGE030
The mean value of the Euclidean distance of each point in the
Figure 387526DEST_PATH_IMAGE031
Representation point
Figure 425889DEST_PATH_IMAGE032
for the training set
Figure 435433DEST_PATH_IMAGE033
The minimum value of the Euclidean distance of each point;

8)将

Figure 360795DEST_PATH_IMAGE034
并入训练集
Figure 227120DEST_PATH_IMAGE035
，更新
Figure 752779DEST_PATH_IMAGE035
，令
Figure 362752DEST_PATH_IMAGE036
，回到步骤4）；8) will
Figure 360795DEST_PATH_IMAGE034
merge into the training set
Figure 227120DEST_PATH_IMAGE035
,renew
Figure 752779DEST_PATH_IMAGE035
,make
Figure 362752DEST_PATH_IMAGE036
, go back to step 4);

所述的步骤7）引入

Figure 595150DEST_PATH_IMAGE037
为了确保后续自适应的样本点用于改善当前代理模型的极限状态函数，而
Figure 412803DEST_PATH_IMAGE038
则是作为全局考虑因子，增加样本点最不密集处的权重，另一方面，
Figure 894599DEST_PATH_IMAGE039
则是用于避免样本点过分密集；而
Figure 511526DEST_PATH_IMAGE040
能够保证整个迭代过程中，所预估的失效概率能依
Figure 660747DEST_PATH_IMAGE041
收敛，确保迭代过程的鲁棒性。The step 7) introduced
Figure 595150DEST_PATH_IMAGE037
In order to ensure that the sample points of subsequent adaptation are used to improve the limit state function of the current surrogate model, while
Figure 412803DEST_PATH_IMAGE038
It is used as a global consideration factor to increase the weight of the least dense sample points. On the other hand,
Figure 894599DEST_PATH_IMAGE039
It is used to avoid excessively dense sample points; and
Figure 511526DEST_PATH_IMAGE040
It can ensure that the estimated failure probability can be
Figure 660747DEST_PATH_IMAGE041
Convergence to ensure robustness of the iterative process.

所述的步骤7）在候选样本点集

Figure 134454DEST_PATH_IMAGE042
中，使得学习函数
Figure 854279DEST_PATH_IMAGE043
最小的点将被自适应序列选择为新的样本点；The step 7) in the candidate sample point set
Figure 134454DEST_PATH_IMAGE042
, so that the learning function
Figure 854279DEST_PATH_IMAGE043
The smallest point will be selected as a new sample point by the adaptive sequence;

所述的步骤7）在候选样本点集

Figure 9317DEST_PATH_IMAGE044
中，具体的数学表达如下：The step 7) in the candidate sample point set
Figure 9317DEST_PATH_IMAGE044
The specific mathematical expression is as follows:

Figure 13045DEST_PATH_IMAGE045
。
Figure 13045DEST_PATH_IMAGE045
.

步骤2）所述的抽取

Figure 923232DEST_PATH_IMAGE046
个候选样本点，采用蒙特卡洛抽样方法抽取候选样本点。The extraction described in step 2)
Figure 923232DEST_PATH_IMAGE046
The candidate sample points are selected by Monte Carlo sampling method.

本发明提供了一种面向结构可靠性分析的序列加点方法，它包括：指定待分析结构的功能函数，确定变量及其公布信息；利用蒙特卡洛抽样，获取候选样本点集

Figure 582884DEST_PATH_IMAGE047
；利用拉丁超立方抽样获取初始点并组成初始样本集
Figure 338350DEST_PATH_IMAGE048
；获取
Figure 194922DEST_PATH_IMAGE049
对应的功能函数的函数值并构建代理模型；对代理模型进行蒙特卡洛数值仿真，获取当前迭代步的失效率；获取最终的代理模型和失效概率。本发明拥有不俗的通用性和适用性，能适用于非kriging模型，大大扩展了适用于结构可靠性分析的代理模型方法，对可靠性分析领域有重要意义。The invention provides a sequence point adding method for structural reliability analysis.
Figure 582884DEST_PATH_IMAGE047
;Using Latin hypercube sampling to obtain initial points and form an initial sample set
Figure 338350DEST_PATH_IMAGE048
;Obtain
Figure 194922DEST_PATH_IMAGE049
Calculate the function value of the corresponding function function and build a surrogate model; perform Monte Carlo numerical simulation on the surrogate model to obtain the failure rate of the current iteration step; obtain the final surrogate model and failure probability. The invention has good generality and applicability, can be applied to non-kriging models, greatly expands the surrogate model method suitable for structural reliability analysis, and is of great significance to the field of reliability analysis.

本发明具有以下优点The present invention has the following advantages

1)本发明所提的学习函数

Figure 276010DEST_PATH_IMAGE050
，通过引入变量的概率分布函数
Figure 219695DEST_PATH_IMAGE051
，确保所预估的失效概率能稳定收敛，确保迭代过程的鲁棒性；1) The learning function proposed by the present invention
Figure 276010DEST_PATH_IMAGE050
, by introducing the probability distribution function of the variable
Figure 219695DEST_PATH_IMAGE051
, to ensure that the estimated failure probability can be stably converged and the robustness of the iterative process is ensured;

2)本发明所提的学习函数

Figure 982115DEST_PATH_IMAGE052
，考虑了自适应加点与当前模型的样本点的信息，确保后续自适应序列添加的新样本点，能够充分从代理模型的全局性和局部性考虑，很大限度上改善当前代理模型的预估误差，保证整个自适应序列加点迭代进行结构可靠性分析过程的高效性与准确性；2) The learning function proposed by the present invention
Figure 982115DEST_PATH_IMAGE052
, considering the information of the adaptive points and the sample points of the current model, to ensure that the new sample points added by the subsequent adaptive sequence can fully consider the globality and locality of the surrogate model, and greatly improve the prediction of the current surrogate model error, to ensure the efficiency and accuracy of the structural reliability analysis process of the whole adaptive sequence plus point iteration;

3)本发明所提的自适应序列加点方法，拥有不俗的通用性和适用性，能适用于非kriging模型，大大扩展了适用于结构可靠性分析的代理模型方法，对可靠性分析领域有重要意义；3) The self-adaptive sequence point adding method proposed in the present invention has good generality and applicability, can be applied to non-kriging models, greatly expands the surrogate model method suitable for structural reliability analysis, and is useful in the field of reliability analysis. important meaning;

4)本发明所提的自适应序列加点方法，能够采用少量样本点的前提下，高效高精度地预估出结构的失效概率，且不同的代理模型方法均能获得不俗的精度和效率，表明本发明方法的通用与可靠。4) The adaptive sequence point adding method proposed by the present invention can estimate the failure probability of the structure with high efficiency and high precision under the premise of using a small number of sample points, and different surrogate model methods can obtain good accuracy and efficiency, It shows the generality and reliability of the method of the present invention.

附图说明Description of drawings

图1是本发明提供的一种用于结构可靠性分析的序列加点方法流程图；Fig. 1 is a kind of sequence adding method flow chart for structural reliability analysis provided by the present invention;

图2是本发明实施例1中采用高斯过程回归神经网络作为代理模型方法所拟合的结构的极限状态函数的示意图；2 is a schematic diagram of a limit state function of a structure fitted by a Gaussian process regression neural network as a proxy model method in Embodiment 1 of the present invention;

图3是本发明实施例1中采用RBF神经网络作为代理模型方法所拟合的结构的极限状态函数的示意图；Fig. 3 is the schematic diagram of the limit state function of the structure that adopts RBF neural network as the surrogate model method fitting in the embodiment of the present invention 1;

图4是本发明实施例2中的无阻尼单自由度振荡系统的结构简图；4 is a schematic structural diagram of an undamped single-degree-of-freedom oscillation system in Embodiment 2 of the present invention;

图5 是本发明实施例2中采用高斯过程回归神经网络作为代理模型方法进行自适应序列加点预测失效概率收敛图；FIG. 5 is a convergence diagram of the failure probability prediction of adaptive sequence adding points using a Gaussian process regression neural network as a surrogate model method in Embodiment 2 of the present invention;

图6 是本发明实施例1中采用RBF神经网络作为代理模型方法进行自适应序列加点预测失效概率收敛图。FIG. 6 is a convergence diagram of the failure probability prediction of adaptive sequence adding points using an RBF neural network as a surrogate model method in Embodiment 1 of the present invention.

具体实施方式Detailed ways

为了使本发明的目的、技术方案及优点更加清楚明白，以下结合附图及实施例，对本发明进行进一步详细说明。应当理解，此处所描述的具体实施例仅仅用以解释本发明，并不用于限定本发明。In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, but not to limit the present invention.

一种面向结构可靠性分析的序列采样方法，其包括以下步骤：A sequence sampling method for structural reliability analysis, comprising the following steps:

1)指定待分析结构的功能函数

Figure 694856DEST_PATH_IMAGE053
，确定功能函数的变量
Figure 431999DEST_PATH_IMAGE054
及其概率分布信息；此处所决定的结构功能函数可以通过利用力学理论或者有限元仿真方法等途径获取，为现有技术，不在赘述；1) Specify the functional function of the structure to be analyzed
Figure 694856DEST_PATH_IMAGE053
, which determines the variables of the function function
Figure 431999DEST_PATH_IMAGE054
and its probability distribution information; the structure function function determined here can be obtained by using mechanical theory or finite element simulation method, which is the prior art, and will not be repeated here;

2)根据步骤1所确定的变量的概率密度分布函数

Figure 862980DEST_PATH_IMAGE055
，利用蒙特卡洛抽样方法，抽取
Figure 163512DEST_PATH_IMAGE056
个候选样本点，组成候选样本点集
Figure 996339DEST_PATH_IMAGE057
；2) According to the probability density distribution function of the variable determined in step 1
Figure 862980DEST_PATH_IMAGE055
, using Monte Carlo sampling, to extract
Figure 163512DEST_PATH_IMAGE056
candidate sample points to form a candidate sample point set
Figure 996339DEST_PATH_IMAGE057
;

3)根据步骤1所确定的变量，采用拉丁超立方法在各变量的取值范围

Figure 153650DEST_PATH_IMAGE058
内抽取
Figure 321195DEST_PATH_IMAGE059
个初始随机样本点，组成初始的训练集
Figure 425418DEST_PATH_IMAGE060
，并令
Figure 847172DEST_PATH_IMAGE061
用于记录迭代次数。3) According to the variables determined in step 1, the Latin hyper-dimension method is used in the value range of each variable
Figure 153650DEST_PATH_IMAGE058
internal extraction
Figure 321195DEST_PATH_IMAGE059
initial random sample points to form the initial training set
Figure 425418DEST_PATH_IMAGE060
, and let
Figure 847172DEST_PATH_IMAGE061
Used to record the number of iterations.

其中

Figure 440964DEST_PATH_IMAGE062
表示标准正态分布的累积分布函数，而
Figure 580958DEST_PATH_IMAGE063
表示为变量的累积分布函数的逆函数；in
Figure 440964DEST_PATH_IMAGE062
represents the cumulative distribution function of the standard normal distribution, and
Figure 580958DEST_PATH_IMAGE063
is expressed as the inverse function of the cumulative distribution function of the variable;

4）根据训练集

Figure 488872DEST_PATH_IMAGE064
，获取
Figure 47023DEST_PATH_IMAGE065
对应的结构的功能函数的函数值，并构建代理模型；4) According to the training set
Figure 488872DEST_PATH_IMAGE064
,Obtain
Figure 47023DEST_PATH_IMAGE065
The function value of the function function of the corresponding structure, and build the proxy model;

5）利用步骤4所建立的代理模型，结合步骤2的候选样本点集

Figure 546137DEST_PATH_IMAGE066
进行数值仿真，获取当前迭代步的代理模型预估的结构的失效概率
Figure 376690DEST_PATH_IMAGE067
；5) Using the surrogate model established in step 4, combined with the candidate sample point set in step 2
Figure 546137DEST_PATH_IMAGE066
Carry out numerical simulation to obtain the failure probability of the structure estimated by the surrogate model of the current iteration step
Figure 376690DEST_PATH_IMAGE067
;

6）判断失效概率

Figure 619453DEST_PATH_IMAGE068
与上一次迭代的结果
Figure 281378DEST_PATH_IMAGE069
的相对误差是否小于
Figure 203591DEST_PATH_IMAGE070
，而
Figure 521440DEST_PATH_IMAGE071
为一足够小的常数，收敛条件为：6) Determine the probability of failure
Figure 619453DEST_PATH_IMAGE068
with the result of the previous iteration
Figure 281378DEST_PATH_IMAGE069
Is the relative error less than
Figure 203591DEST_PATH_IMAGE070
,and
Figure 521440DEST_PATH_IMAGE071
is a sufficiently small constant, the convergence condition is:

Figure 567894DEST_PATH_IMAGE072
Figure 567894DEST_PATH_IMAGE072

如果满足收敛要求，则获取最终的代理模型和最终预估的失效概率；If the convergence requirements are met, the final surrogate model and the final estimated failure probability are obtained;

如果不满足收敛要求或者

Figure 84325DEST_PATH_IMAGE073
，则进行下一步，进行自适应序列加点；If convergence requirements are not met or
Figure 84325DEST_PATH_IMAGE073
, then proceed to the next step to add points to the adaptive sequence;

7）利用学习函数

Figure 128505DEST_PATH_IMAGE074
对候选样本点集
Figure 995967DEST_PATH_IMAGE075
进行数值仿真，并获取最新的样本点
Figure 331264DEST_PATH_IMAGE076
，并更新训练集
Figure 905465DEST_PATH_IMAGE077
。学习函数
Figure 917283DEST_PATH_IMAGE078
的具体表达如下：7) Use the learning function
Figure 128505DEST_PATH_IMAGE074
For the candidate sample point set
Figure 995967DEST_PATH_IMAGE075
Run numerical simulations and get the latest sample points
Figure 331264DEST_PATH_IMAGE076
, and update the training set
Figure 905465DEST_PATH_IMAGE077
. learning function
Figure 917283DEST_PATH_IMAGE078
The specific expression is as follows:

Figure 740883DEST_PATH_IMAGE079
Figure 740883DEST_PATH_IMAGE079

其中，

Figure 394718DEST_PATH_IMAGE080
表示点
Figure 869431DEST_PATH_IMAGE081
对于训练集
Figure 255413DEST_PATH_IMAGE082
中各点欧式距离的平均值，
Figure 831888DEST_PATH_IMAGE083
表示点
Figure 23834DEST_PATH_IMAGE081
对于训练集
Figure 103786DEST_PATH_IMAGE084
中各点欧式距离的最小值。in,
Figure 394718DEST_PATH_IMAGE080
Representation point
Figure 869431DEST_PATH_IMAGE081
for the training set
Figure 255413DEST_PATH_IMAGE082
The mean value of the Euclidean distance of each point in the
Figure 831888DEST_PATH_IMAGE083
Representation point
Figure 23834DEST_PATH_IMAGE081
for the training set
Figure 103786DEST_PATH_IMAGE084
The minimum value of the Euclidean distance for each point in .

更近一步，在候选样本点集

Figure 926248DEST_PATH_IMAGE085
中，使得学习函数
Figure 475173DEST_PATH_IMAGE086
最小的点将被自适应序列选择为新的样本点，具体的数学表达如下：Going one step further, in the candidate sample point set
Figure 926248DEST_PATH_IMAGE085
, so that the learning function
Figure 475173DEST_PATH_IMAGE086
The smallest point will be selected as a new sample point by the adaptive sequence, and the specific mathematical expression is as follows:

Figure 470810DEST_PATH_IMAGE087
Figure 470810DEST_PATH_IMAGE087

8）将

Figure 405268DEST_PATH_IMAGE088
并入训练集
Figure 398632DEST_PATH_IMAGE089
，更新
Figure 949699DEST_PATH_IMAGE090
，令
Figure 470067DEST_PATH_IMAGE091
，回到步骤4；8) will
Figure 405268DEST_PATH_IMAGE088
merge into the training set
Figure 398632DEST_PATH_IMAGE089
,renew
Figure 949699DEST_PATH_IMAGE090
,make
Figure 470067DEST_PATH_IMAGE091
, go back to step 4;

图1为本发明提供的一种用于结构可靠性分析的序列加点方法流程图，共包括八个步骤。本实施例1以一个二维应用实例对本发明进行进一步阐述。FIG. 1 is a flow chart of a sequence point adding method for structural reliability analysis provided by the present invention, which includes eight steps in total. The present embodiment 1 further illustrates the present invention with a two-dimensional application example.

实施例1Example 1

1）指定待分析结构的功能函数，确定功能函数的变量及其概率分布信息；二维应用实例的功能函数如下：1) Specify the function function of the structure to be analyzed, and determine the variables of the function function and their probability distribution information; the function function of the two-dimensional application example is as follows:

Figure 727873DEST_PATH_IMAGE092
Figure 727873DEST_PATH_IMAGE092

其中,

Figure 688876DEST_PATH_IMAGE093
表示实施例1的功能函数，而
Figure 461659DEST_PATH_IMAGE094
为功能函数的变量，
Figure 533521DEST_PATH_IMAGE095
服从均值1.5，标准差1的正态分布，而
Figure 911412DEST_PATH_IMAGE096
服从均值为2.5，标准差为1的正态分布；in,
Figure 688876DEST_PATH_IMAGE093
represents the functional function of Example 1, and
Figure 461659DEST_PATH_IMAGE094
is the variable of the function function,
Figure 533521DEST_PATH_IMAGE095
follows a normal distribution with a mean of 1.5 and a standard deviation of 1, while
Figure 911412DEST_PATH_IMAGE096
Obey a normal distribution with a mean of 2.5 and a standard deviation of 1;

2）根据步骤1所确定的变量的概率密度分布函数

Figure 794049DEST_PATH_IMAGE097
，利用蒙特卡洛抽样方法，抽取
Figure 54129DEST_PATH_IMAGE098
个候选样本点，组成候选样本点集
Figure 867364DEST_PATH_IMAGE099
；本实施例1中，抽取
Figure 896500DEST_PATH_IMAGE100
个候选样本点。图2和图3均显示了所获得的蒙特卡洛抽样的候选样本点集
Figure 199305DEST_PATH_IMAGE101
；2) According to the probability density distribution function of the variable determined in step 1
Figure 794049DEST_PATH_IMAGE097
, using Monte Carlo sampling, to extract
Figure 54129DEST_PATH_IMAGE098
candidate sample points to form a candidate sample point set
Figure 867364DEST_PATH_IMAGE099
; In this embodiment 1, extract
Figure 896500DEST_PATH_IMAGE100
candidate sample points. Both Figures 2 and 3 show the obtained candidate sample point sets for Monte Carlo sampling
Figure 199305DEST_PATH_IMAGE101
;

3）根据步骤1所确定的变量，采用拉丁超立方抽样在各变量的取值范围

Figure 149944DEST_PATH_IMAGE102
内抽取
Figure 812875DEST_PATH_IMAGE103
个初始随机样本点，组成初始的训练集
Figure 962097DEST_PATH_IMAGE104
，并令
Figure 435804DEST_PATH_IMAGE105
用于记录迭代次数。其中
Figure 404897DEST_PATH_IMAGE106
表示标准正态分布的累积分布函数，而
Figure 107404DEST_PATH_IMAGE107
表示为变量的累积分布函数的逆函数；3) According to the variables determined in step 1, Latin hypercube sampling is used in the value range of each variable
Figure 149944DEST_PATH_IMAGE102
internal extraction
Figure 812875DEST_PATH_IMAGE103
initial random sample points to form the initial training set
Figure 962097DEST_PATH_IMAGE104
, and let
Figure 435804DEST_PATH_IMAGE105
Used to record the number of iterations. in
Figure 404897DEST_PATH_IMAGE106
represents the cumulative distribution function of the standard normal distribution, and
Figure 107404DEST_PATH_IMAGE107
is expressed as the inverse function of the cumulative distribution function of the variable;

本实施例中，抽取

Figure 314395DEST_PATH_IMAGE108
个初始随机样本点，图2和图3均显示了所获得的初始点；In this embodiment, the extraction
Figure 314395DEST_PATH_IMAGE108
initial random sample points, Figure 2 and Figure 3 both show the obtained initial points;

4）根据训练集

Figure 693424DEST_PATH_IMAGE109
，获取
Figure 415392DEST_PATH_IMAGE110
对应的结构的功能函数的函数值，并构建代理模型
Figure 374121DEST_PATH_IMAGE111
；在构建代理模型时，可以采用现有的代理模型方法进行构建，例如多项式响应面、多项式混沌展开、神经网络、支持向量机以及Kriging模型等等。代理模型方法等为现有的方法，本处不在赘述。在本实施例中分别采用高斯过程回归神经网络与RBF神经网络方法构建代理模型，以表示本发明适用性强，且可以适用于非kriging模型的代理模型方法；4) According to the training set
Figure 693424DEST_PATH_IMAGE109
,Obtain
Figure 415392DEST_PATH_IMAGE110
The function value of the function function of the corresponding structure, and build the surrogate model
Figure 374121DEST_PATH_IMAGE111
; When constructing a surrogate model, existing surrogate model methods can be used for construction, such as polynomial response surface, polynomial chaotic expansion, neural network, support vector machine and Kriging model, etc. The proxy model method and the like are existing methods, which will not be repeated here. In this embodiment, the Gaussian process regression neural network and the RBF neural network method are respectively used to construct the surrogate model to indicate that the present invention has strong applicability and can be applied to the surrogate model method of the non-kriging model;

5）利用步骤4所建立的代理模型，结合步骤2的候选样本点集

Figure 232355DEST_PATH_IMAGE112
进行蒙特卡洛数值仿真，获取当前迭代步的代理模型预估的结构的失效概率
Figure 565641DEST_PATH_IMAGE113
。利用蒙特卡洛方法进行结构可靠性分析为成熟方法，这里不再阐述；5) Using the surrogate model established in step 4, combined with the candidate sample point set in step 2
Figure 232355DEST_PATH_IMAGE112
Perform a Monte Carlo numerical simulation to obtain the failure probability of the structure estimated by the surrogate model of the current iteration step
Figure 565641DEST_PATH_IMAGE113
. The use of Monte Carlo method for structural reliability analysis is a mature method, and will not be described here;

6）判断失效概率

Figure 509326DEST_PATH_IMAGE113
与上一次迭代的结果
Figure 6167DEST_PATH_IMAGE114
的相对误差是否小于
Figure 984487DEST_PATH_IMAGE115
，即：6) Determine the probability of failure
Figure 509326DEST_PATH_IMAGE113
with the result of the previous iteration
Figure 6167DEST_PATH_IMAGE114
Is the relative error less than
Figure 984487DEST_PATH_IMAGE115
,which is:

Figure 970898DEST_PATH_IMAGE116
；
Figure 970898DEST_PATH_IMAGE116
;

如果满足收敛要求，则获取最终的代理模型和最终预估的失效概率；如果不满足收敛要求，则进行下一步，进行自适应序列加点；另外，当

Figure 605141DEST_PATH_IMAGE117
时，跳过这一步，进行下一步；If the convergence requirements are met, the final surrogate model and the final estimated failure probability are obtained; if the convergence requirements are not met, proceed to the next step and add points to the adaptive sequence; in addition, when
Figure 605141DEST_PATH_IMAGE117
, skip this step and go to the next step;

本实施例中，设置

Figure 453143DEST_PATH_IMAGE118
。In this embodiment, set
Figure 453143DEST_PATH_IMAGE118
.

7）利用学习函数LF对候选样本点集

Figure 285969DEST_PATH_IMAGE119
进行数值仿真，并获取最新的样本点
Figure 443281DEST_PATH_IMAGE120
，并更新训练集
Figure 564821DEST_PATH_IMAGE121
；其中学习函数LF的具体表达如下：7) Use the learning function LF to analyze the candidate sample point set
Figure 285969DEST_PATH_IMAGE119
Run numerical simulations and get the latest sample points
Figure 443281DEST_PATH_IMAGE120
, and update the training set
Figure 564821DEST_PATH_IMAGE121
; The specific expression of the learning function LF is as follows:

Figure 200202DEST_PATH_IMAGE122
Figure 200202DEST_PATH_IMAGE122

其中，

Figure 402382DEST_PATH_IMAGE123
表示
Figure 730595DEST_PATH_IMAGE124
点对于训练集
Figure 73852DEST_PATH_IMAGE125
中各点欧式距离的平均值，
Figure 778502DEST_PATH_IMAGE126
表示点
Figure 320342DEST_PATH_IMAGE127
对于训练集
Figure 835768DEST_PATH_IMAGE128
中各点欧式距离的最小值。in,
Figure 402382DEST_PATH_IMAGE123
express
Figure 730595DEST_PATH_IMAGE124
points for the training set
Figure 73852DEST_PATH_IMAGE125
The mean value of the Euclidean distance of each point in the
Figure 778502DEST_PATH_IMAGE126
Representation point
Figure 320342DEST_PATH_IMAGE127
for the training set
Figure 835768DEST_PATH_IMAGE128
The minimum value of the Euclidean distance for each point in .

更近一步，在候选样本点集

Figure 666321DEST_PATH_IMAGE129
中，使得学习函数
Figure 643504DEST_PATH_IMAGE130
最小的点将被自适应序列选择为新的样本点，具体的数学表达如下：Going one step further, in the candidate sample point set
Figure 666321DEST_PATH_IMAGE129
, so that the learning function
Figure 643504DEST_PATH_IMAGE130
The smallest point will be selected as a new sample point by the adaptive sequence, and the specific mathematical expression is as follows:

Figure 305430DEST_PATH_IMAGE131
Figure 305430DEST_PATH_IMAGE131

8）将

Figure 241025DEST_PATH_IMAGE132
并入训练集
Figure 558874DEST_PATH_IMAGE133
，更新
Figure 580226DEST_PATH_IMAGE134
，令
Figure 362238DEST_PATH_IMAGE135
，回到步骤4；8) will
Figure 241025DEST_PATH_IMAGE132
merge into the training set
Figure 558874DEST_PATH_IMAGE133
,renew
Figure 580226DEST_PATH_IMAGE134
,make
Figure 362238DEST_PATH_IMAGE135
, go back to step 4;

图2以及图3给出了实施例1中最终的采样结果和代理模型与结构的真实极限状态函数的拟合情况。其中图2表示为采用高斯过程回归神经网络的结果，图3表示为采用RBF神经网络作为代理模型方法的结果。更为详细的结果表示在表1中。FIG. 2 and FIG. 3 show the final sampling results in Example 1 and the fitting situation of the surrogate model and the real limit state function of the structure. Among them, Figure 2 shows the result of using Gaussian process regression neural network, and Figure 3 shows the result of using RBF neural network as a surrogate model method. More detailed results are shown in Table 1.

表1实施例1中本发明方法与蒙特卡洛方法详细结果对比The detailed results comparison between the method of the present invention and the Monte Carlo method in the embodiment 1 of Table 1

Figure 203155DEST_PATH_IMAGE136
Figure 203155DEST_PATH_IMAGE136

根据图2、图3与表1可以得知，采用本发明方法结合不同的代理模型，在分析结构可靠性问题中，对比蒙特卡洛仿真，能高效且精确地估计结构的失效概率，满足工程实际需求。According to Fig. 2, Fig. 3 and Table 1, it can be known that the method of the present invention combined with different surrogate models can efficiently and accurately estimate the failure probability of the structure in the analysis of structural reliability problems by comparing with Monte Carlo simulation. Actual demand.

实施例2Example 2

为了进一步表明本发明所提方法的有效性，通过提出一个常见的工程系统作为实施例，对本发明所提方法进行详细说明。In order to further demonstrate the effectiveness of the method proposed in the present invention, the method proposed in the present invention is described in detail by taking a common engineering system as an example.

9）指定待分析结构的功能函数，确定功能函数的变量及其概率分布信息；本实施例2中为一个常见的无阻尼单自由度振荡系统，其结构简图为图4。振荡器的功能函数

Figure 8300DEST_PATH_IMAGE137
定义为9) Specify the function function of the structure to be analyzed, and determine the variable of the function function and its probability distribution information; this embodiment 2 is a common undamped single-degree-of-freedom oscillation system, and its structural diagram is shown in Figure 4. Oscillator function
Figure 8300DEST_PATH_IMAGE137
defined as

Figure 592865DEST_PATH_IMAGE138
Figure 592865DEST_PATH_IMAGE138

其中，

Figure 980115DEST_PATH_IMAGE139
，变量
Figure 195196DEST_PATH_IMAGE140
，具体的概率分布情况如表2所示。in,
Figure 980115DEST_PATH_IMAGE139
,variable
Figure 195196DEST_PATH_IMAGE140
, and the specific probability distribution is shown in Table 2.

Figure 18795DEST_PATH_IMAGE141
Figure 18795DEST_PATH_IMAGE141

10）根据步骤1所确定的变量的概率密度分布函数

Figure 407051DEST_PATH_IMAGE142
，利用蒙特卡洛抽样方法，抽取
Figure 632496DEST_PATH_IMAGE143
个候选样本点，组成候选样本点集
Figure 330063DEST_PATH_IMAGE144
；本实施例2中，抽取
Figure 906537DEST_PATH_IMAGE145
个候选样本点；10) According to the probability density distribution function of the variable determined in step 1
Figure 407051DEST_PATH_IMAGE142
, using Monte Carlo sampling, to extract
Figure 632496DEST_PATH_IMAGE143
candidate sample points to form a candidate sample point set
Figure 330063DEST_PATH_IMAGE144
; In the present embodiment 2, extract
Figure 906537DEST_PATH_IMAGE145
candidate sample points;

11）根据步骤1所确定的变量，采用拉丁超立方方法在各变量的取值范围

Figure 36167DEST_PATH_IMAGE146
内抽取
Figure 381698DEST_PATH_IMAGE147
个初始随机样本点，组成初始的训练集
Figure 898DEST_PATH_IMAGE148
，并令
Figure 2352DEST_PATH_IMAGE149
用于记录迭代次数；11) According to the variables determined in step 1, use the Latin hypercube method in the value range of each variable
Figure 36167DEST_PATH_IMAGE146
internal extraction
Figure 381698DEST_PATH_IMAGE147
initial random sample points to form the initial training set
Figure 898DEST_PATH_IMAGE148
, and let
Figure 2352DEST_PATH_IMAGE149
Used to record the number of iterations;

其中

Figure 217564DEST_PATH_IMAGE150
表示标准正态分布的累积分布函数，而
Figure 683181DEST_PATH_IMAGE151
表示为变量的累积分布函数的逆函数；本实施例中，抽取
Figure 676544DEST_PATH_IMAGE152
个初始随机样本点；in
Figure 217564DEST_PATH_IMAGE150
represents the cumulative distribution function of the standard normal distribution, and
Figure 683181DEST_PATH_IMAGE151
is expressed as the inverse function of the cumulative distribution function of the variable; in this example, the extraction
Figure 676544DEST_PATH_IMAGE152
initial random sample points;

12）根据训练集

Figure 696453DEST_PATH_IMAGE153
，获取
Figure 230202DEST_PATH_IMAGE154
对应的结构的功能函数的函数值，并构建代理模型
Figure 802523DEST_PATH_IMAGE155
；在构建代理模型时，可以采用现有的代理模型方法进行构建，例如多项式响应面、多项式混沌展开、神经网络、支持向量机以及Kriging模型等等。代理模型方法等为现有的方法，本处不在赘述。在本实施例中分别采用高斯过程回归神经网络与RBF神经网络方法构建代理模型，以表示本发明适用性强，且可以适用于非kriging模型的代理模型方法；12) According to the training set
Figure 696453DEST_PATH_IMAGE153
,Obtain
Figure 230202DEST_PATH_IMAGE154
The function value of the function function of the corresponding structure, and build the surrogate model
Figure 802523DEST_PATH_IMAGE155
; When constructing a surrogate model, existing surrogate model methods can be used for construction, such as polynomial response surface, polynomial chaotic expansion, neural network, support vector machine and Kriging model, etc. The proxy model method and the like are existing methods, which will not be repeated here. In this embodiment, the Gaussian process regression neural network and the RBF neural network method are respectively used to construct the surrogate model to indicate that the present invention has strong applicability and can be applied to the surrogate model method of the non-kriging model;

13）利用步骤4所建立的代理模型，结合步骤2的候选样本点集

Figure 497946DEST_PATH_IMAGE156
进行蒙特卡洛数值仿真，获取当前迭代步的代理模型预估的结构的失效概率
Figure 473992DEST_PATH_IMAGE157
。利用蒙特卡洛方法进行结构可靠性分析为成熟方法，这里不再阐述；13) Using the surrogate model established in step 4, combined with the candidate sample point set in step 2
Figure 497946DEST_PATH_IMAGE156
Perform a Monte Carlo numerical simulation to obtain the failure probability of the structure estimated by the surrogate model of the current iteration step
Figure 473992DEST_PATH_IMAGE157
. The use of Monte Carlo method for structural reliability analysis is a mature method, and will not be described here;

14）判断失效概率

Figure 811433DEST_PATH_IMAGE158
与上一次迭代的结果
Figure 720483DEST_PATH_IMAGE159
的相对误差是否小于
Figure 55649DEST_PATH_IMAGE160
，即：14) Judge failure probability
Figure 811433DEST_PATH_IMAGE158
with the result of the previous iteration
Figure 720483DEST_PATH_IMAGE159
Is the relative error less than
Figure 55649DEST_PATH_IMAGE160
,which is:

Figure 66462DEST_PATH_IMAGE161
；
Figure 66462DEST_PATH_IMAGE161
;

如果满足收敛要求，则获取最终的代理模型和最终预估的失效概率；如果不满足收敛要求，则进行下一步，进行自适应序列加点；另外，当

Figure 942014DEST_PATH_IMAGE162
时，跳过这一步，进行下一步；If the convergence requirements are met, the final surrogate model and the final estimated failure probability are obtained; if the convergence requirements are not met, proceed to the next step and add points to the adaptive sequence; in addition, when
Figure 942014DEST_PATH_IMAGE162
, skip this step and go to the next step;

本实施例2中，设置

Figure 174412DEST_PATH_IMAGE163
；In this embodiment 2, set
Figure 174412DEST_PATH_IMAGE163
;

15）利用学习函数LF对候选样本点集

Figure 211638DEST_PATH_IMAGE164
进行数值仿真，并获取最新的样本点
Figure 224594DEST_PATH_IMAGE165
，并更新训练集
Figure 887525DEST_PATH_IMAGE166
；其中学习函数LF的具体表达如下：15) Use the learning function LF to analyze the candidate sample point set
Figure 211638DEST_PATH_IMAGE164
Run numerical simulations and get the latest sample points
Figure 224594DEST_PATH_IMAGE165
, and update the training set
Figure 887525DEST_PATH_IMAGE166
; The specific expression of the learning function LF is as follows:

Figure 974430DEST_PATH_IMAGE167
Figure 974430DEST_PATH_IMAGE167

其中，

Figure 182557DEST_PATH_IMAGE168
表示点
Figure 151650DEST_PATH_IMAGE169
对于训练集
Figure 369005DEST_PATH_IMAGE170
中各点欧式距离的平均值，
Figure 123465DEST_PATH_IMAGE171
表示点
Figure 768073DEST_PATH_IMAGE169
对于训练集中
Figure 427725DEST_PATH_IMAGE172
各点欧式距离的最小值。in,
Figure 182557DEST_PATH_IMAGE168
Representation point
Figure 151650DEST_PATH_IMAGE169
for the training set
Figure 369005DEST_PATH_IMAGE170
The mean value of the Euclidean distance of each point in the
Figure 123465DEST_PATH_IMAGE171
Representation point
Figure 768073DEST_PATH_IMAGE169
for the training set
Figure 427725DEST_PATH_IMAGE172
The minimum value of the Euclidean distance for each point.

更进一步，在候选样本点集

Figure 183191DEST_PATH_IMAGE173
中，使得学习函数
Figure 307005DEST_PATH_IMAGE174
最小的点将被自适应序列选择为新的样本点，具体的数学表达如下：Further, in the candidate sample point set
Figure 183191DEST_PATH_IMAGE173
, so that the learning function
Figure 307005DEST_PATH_IMAGE174
The smallest point will be selected as a new sample point by the adaptive sequence, and the specific mathematical expression is as follows:

Figure 325777DEST_PATH_IMAGE175
Figure 325777DEST_PATH_IMAGE175

16）将

Figure 256080DEST_PATH_IMAGE176
并入训练集
Figure 815237DEST_PATH_IMAGE170
，更新
Figure 793558DEST_PATH_IMAGE177
，令
Figure 779968DEST_PATH_IMAGE178
，回到步骤4；16) will
Figure 256080DEST_PATH_IMAGE176
merge into the training set
Figure 815237DEST_PATH_IMAGE170
,renew
Figure 793558DEST_PATH_IMAGE177
,make
Figure 779968DEST_PATH_IMAGE178
, go back to step 4;

图5表示了采用高斯过程回归神经网络作为代理模型方法进行自适应序列加点预测失效概率收敛过程，而图6表示为采用RBF神经网络作为代理模型方法进行自适应序列加点预测失效概率收敛过程。更为详细的结果表示在表3中。Figure 5 shows the convergence process of using Gaussian process regression neural network as the surrogate model method to predict the failure probability of adaptive sequence adding points, while Figure 6 shows the convergence process of using RBF neural network as the surrogate model method to predict the failure probability of adaptive sequence adding points. More detailed results are shown in Table 3.

Figure 414212DEST_PATH_IMAGE179
Figure 414212DEST_PATH_IMAGE179

根据图5、图6与表3可以得知，采用本发明方法结合不同的代理模型，在利用LF学习函数进行自适应序列加点迭代，并满足收敛条件后，与蒙特卡洛仿真方法对比，能够精确高效地对结构进行可靠性分析。According to Fig. 5, Fig. 6 and Table 3, it can be known that using the method of the present invention in combination with different surrogate models, using the LF learning function to perform adaptive sequence adding point iteration, and after satisfying the convergence conditions, compared with the Monte Carlo simulation method, it can be Perform reliability analysis of structures accurately and efficiently.

Claims (5)

1.一种面向多种代理模型的结构可靠性分析自适应加点方法，其包括以下步骤：1. A structural reliability analysis self-adaptive method for adding points for multiple surrogate models, comprising the following steps: 1)指定待分析结构的功能函数
Figure 401770DEST_PATH_IMAGE001
，确定功能函数的变量
Figure 703438DEST_PATH_IMAGE002
及其概率分布信息；1) Specify the functional function of the structure to be analyzed
Figure 401770DEST_PATH_IMAGE001
, which determines the variables of the function function
Figure 703438DEST_PATH_IMAGE002
and its probability distribution information; 2)根据步骤1）所确定的变量的概率密度分布函数
Figure 798433DEST_PATH_IMAGE003
，抽取
Figure 513449DEST_PATH_IMAGE004
个候选样本点，组成候选样本点集
Figure 86512DEST_PATH_IMAGE005
；2) According to the probability density distribution function of the variable determined in step 1)
Figure 798433DEST_PATH_IMAGE003
, extract
Figure 513449DEST_PATH_IMAGE004
candidate sample points to form a candidate sample point set
Figure 86512DEST_PATH_IMAGE005
; 3)根据步骤1）所确定的变量，采用拉丁超立方法在各变量的取值范围
Figure 85430DEST_PATH_IMAGE006
内抽取
Figure 351326DEST_PATH_IMAGE007
个初始随机样本点，组成初始的训练集
Figure 225741DEST_PATH_IMAGE008
，并令
Figure 461551DEST_PATH_IMAGE009
用于记录迭代次数；3) According to the variables determined in step 1), use the Latin hyper-dimension method in the value range of each variable
Figure 85430DEST_PATH_IMAGE006
internal extraction
Figure 351326DEST_PATH_IMAGE007
initial random sample points to form the initial training set
Figure 225741DEST_PATH_IMAGE008
, and let
Figure 461551DEST_PATH_IMAGE009
Used to record the number of iterations; 其中
Figure 941074DEST_PATH_IMAGE010
表示标准正态分布的累积分布函数，而
Figure 174609DEST_PATH_IMAGE011
表示为变量的累积分布函数的逆函数；in
Figure 941074DEST_PATH_IMAGE010
represents the cumulative distribution function of the standard normal distribution, and
Figure 174609DEST_PATH_IMAGE011
is expressed as the inverse function of the cumulative distribution function of the variable; 4)根据训练集
Figure 614949DEST_PATH_IMAGE008
，获取
Figure 529815DEST_PATH_IMAGE012
对应的结构的功能函数的函数值，并构建代理模型；4) According to the training set
Figure 614949DEST_PATH_IMAGE008
,Obtain
Figure 529815DEST_PATH_IMAGE012
The function value of the function function of the corresponding structure, and build the proxy model; 5)利用步骤4所建立的代理模型，结合步骤2）的候选样本点集
Figure 926161DEST_PATH_IMAGE013
进行数值仿真，获取当前迭代步的代理模型预估的结构的失效概率
Figure 658494DEST_PATH_IMAGE014
；5) Using the surrogate model established in step 4, combined with the candidate sample point set of step 2)
Figure 926161DEST_PATH_IMAGE013
Carry out numerical simulation to obtain the failure probability of the structure estimated by the surrogate model of the current iteration step
Figure 658494DEST_PATH_IMAGE014
; 6)判断失效概率
Figure 179605DEST_PATH_IMAGE015
与上一次迭代的结果
Figure 960479DEST_PATH_IMAGE016
的相对误差是否小于
Figure 791425DEST_PATH_IMAGE017
，而
Figure 304446DEST_PATH_IMAGE018
为一足够小的常数，收敛条件为：6) Judging the probability of failure
Figure 179605DEST_PATH_IMAGE015
with the result of the previous iteration
Figure 960479DEST_PATH_IMAGE016
Is the relative error less than
Figure 791425DEST_PATH_IMAGE017
,and
Figure 304446DEST_PATH_IMAGE018
is a sufficiently small constant, the convergence condition is:
Figure 703067DEST_PATH_IMAGE019
Figure 703067DEST_PATH_IMAGE019
 如果满足收敛要求，则获取最终的代理模型和最终预估的失效概率；If the convergence requirements are met, the final surrogate model and the final estimated failure probability are obtained; 如果不满足收敛要求或者
Figure 22053DEST_PATH_IMAGE020
，则进行下一步，进行自适应序列加点；If convergence requirements are not met or
Figure 22053DEST_PATH_IMAGE020
, then proceed to the next step to add points to the adaptive sequence; 7)利用
Figure 65095DEST_PATH_IMAGE021
学习函数对候选样本点集
Figure 889963DEST_PATH_IMAGE022
进行数值仿真，并获取最新的样本点
Figure 713562DEST_PATH_IMAGE023
，并更新训练集
Figure 508343DEST_PATH_IMAGE024
；其中学习函数
Figure 61684DEST_PATH_IMAGE021
的具体表达如下：7) Utilize
Figure 65095DEST_PATH_IMAGE021
Learning function pair candidate sample point set
Figure 889963DEST_PATH_IMAGE022
Run numerical simulations and get the latest sample points
Figure 713562DEST_PATH_IMAGE023
, and update the training set
Figure 508343DEST_PATH_IMAGE024
; where the learning function
Figure 61684DEST_PATH_IMAGE021
The specific expression is as follows:
Figure 182087DEST_PATH_IMAGE025
Figure 182087DEST_PATH_IMAGE025
 其中，
Figure 227403DEST_PATH_IMAGE026
表示点
Figure 199776DEST_PATH_IMAGE027
对于训练集
Figure 482990DEST_PATH_IMAGE028
中各点欧式距离的平均值，
Figure 571031DEST_PATH_IMAGE029
表示点
Figure 165961DEST_PATH_IMAGE030
对于训练集
Figure 833703DEST_PATH_IMAGE028
中各点欧式距离的最小值；in,
Figure 227403DEST_PATH_IMAGE026
Representation point
Figure 199776DEST_PATH_IMAGE027
for the training set
Figure 482990DEST_PATH_IMAGE028
The mean value of the Euclidean distance of each point in the
Figure 571031DEST_PATH_IMAGE029
Representation point
Figure 165961DEST_PATH_IMAGE030
for the training set
Figure 833703DEST_PATH_IMAGE028
The minimum value of the Euclidean distance of each point; 8)将
Figure 768161DEST_PATH_IMAGE031
并入训练集
Figure 105732DEST_PATH_IMAGE032
，更新
Figure 63324DEST_PATH_IMAGE033
，令
Figure 331494DEST_PATH_IMAGE034
，回到步骤4）。8) will
Figure 768161DEST_PATH_IMAGE031
merge into the training set
Figure 105732DEST_PATH_IMAGE032
,renew
Figure 63324DEST_PATH_IMAGE033
,make
Figure 331494DEST_PATH_IMAGE034
, go back to step 4). 2.根据权利要求1所述的一种面向结构可靠性分析的序列加点方法，其特征在于：步骤7）引入
Figure 448355DEST_PATH_IMAGE035
为了确保后续自适应的样本点用于改善当前代理模型的极限状态函数，而
Figure 815882DEST_PATH_IMAGE036
则是作为全局考虑因子，增加样本点最不密集处的权重，另一方面，
Figure 323087DEST_PATH_IMAGE037
则是用于避免样本点过分密集；而
Figure 709462DEST_PATH_IMAGE038
能够保证整个迭代过程中，所预估的失效概率能依
Figure 290616DEST_PATH_IMAGE039
收敛，确保迭代过程的鲁棒性。2. A structural reliability analysis-oriented sequence adding method according to claim 1, characterized in that: step 7) introduces
Figure 448355DEST_PATH_IMAGE035
In order to ensure that the sample points of subsequent adaptation are used to improve the limit state function of the current surrogate model, while
Figure 815882DEST_PATH_IMAGE036
It is used as a global consideration factor to increase the weight of the least dense sample points. On the other hand,
Figure 323087DEST_PATH_IMAGE037
It is used to avoid excessively dense sample points; and
Figure 709462DEST_PATH_IMAGE038
It can ensure that the estimated failure probability can be
Figure 290616DEST_PATH_IMAGE039
Convergence to ensure robustness of the iterative process. 3.根据权利要求1或2所述的一种面向多种代理模型的结构可靠性分析自适应加点方法，其特征在于：步骤7）在候选样本点集
Figure 156941DEST_PATH_IMAGE040
中，使得学习函
Figure 213759DEST_PATH_IMAGE041
最小的点将被自适应序列选择为新的样本点。3. A structural reliability analysis adaptive point addition method for multiple surrogate models according to claim 1 or 2, characterized in that: step 7) in the candidate sample point set
Figure 156941DEST_PATH_IMAGE040
, so that the learning function
Figure 213759DEST_PATH_IMAGE041
The smallest point will be selected as a new sample point by the adaptive sequence. 4.根据权利要求3所述的一种面向多种代理模型的结构可靠性分析自适应加点方法，其特征在于：步骤7）在候选样本点集
Figure 761415DEST_PATH_IMAGE042
中，具体的数学表达如下：4. The method for self-adaptive addition of points for structural reliability analysis for multiple surrogate models according to claim 3, characterized in that: step 7) in the candidate sample point set
Figure 761415DEST_PATH_IMAGE042
, the specific mathematical expression is as follows:
Figure 259392DEST_PATH_IMAGE043
。
Figure 259392DEST_PATH_IMAGE043
. 5.根据权利要求4所述的一种面向多种代理模型的结构可靠性分析自适应加点方法，其特征在于：步骤2）所述的抽取个候选样本点，采用蒙特卡洛抽样方法抽取候选样本点。5. The self-adapting method for structural reliability analysis for multiple proxy models according to claim 4, characterized in that: the extraction described in step 2) The candidate sample points are selected by Monte Carlo sampling method.
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