CN109284545B - Optimal condition important sampling method-based structural failure probability solving method - Google Patents

Optimal condition important sampling method-based structural failure probability solving method Download PDF

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CN109284545B
CN109284545B CN201811029879.7A CN201811029879A CN109284545B CN 109284545 B CN109284545 B CN 109284545B CN 201811029879 A CN201811029879 A CN 201811029879A CN 109284545 B CN109284545 B CN 109284545B
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failure probability
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CN109284545A (en
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王攀
岳珠峰
肖思男
谭世旺
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Northwestern Polytechnical University
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Abstract

The invention provides a method for solving the structure failure probability based on the optimal condition important sampling method, which divides the distribution interval of each input variable into continuous but non-intersected subintervals and carries out the failure probability delta of each input variable i Making an estimate, selecting δ i The input variable reaching the maximum value is taken as a condition variable; under the condition of known condition variables, respectively estimating the variance of the estimated value of the failure probability
Figure DDA0001789440130000011
And variance of optimal estimated value of failure probability
Figure DDA0001789440130000012
Judgment of
Figure DDA0001789440130000013
Whether significantly less than
Figure DDA0001789440130000014
If not, then
Figure DDA0001789440130000015
As a final estimate; if yes, adjusting the division of the input variable distribution interval, and re-estimating
Figure DDA0001789440130000016
And
Figure DDA0001789440130000017
iterate until
Figure DDA0001789440130000018
Is not significantly less than
Figure DDA0001789440130000019
The method can improve the efficiency of solving the failure probability of the structure on the premise of ensuring the calculation precision, reduces the calculation cost, has good applicability, and can be widely applied to the calculation of various reliability problems in engineering.

Description

Optimal condition important sampling method-based structural failure probability solving method
Technical Field
The invention relates to the technical field of structural reliability analysis and optimization design, in particular to a method for solving the structural failure probability based on an optimal condition important sampling method.
Background
Structural reliability characterizes the ability of a structure to perform a specified function under specified conditions. A key problem in structural reliability analysis is solving the failure probability P (F) of a structure, which is essentially solving a high-dimensional integral problem of the joint probability density function of all random input variables in the failure domain { X: G (X) ≦ 0}, which can be expressed as
P f ＝Prob[G(X)≤0]＝∫ G(X)≤0 f(X)dX (1)
Wherein, X = (X) 1 ,…,X d ) For a d-dimensional random input variable, F (X) represents a joint probability density function of the input random variable, G (X) represents a functional function of the structure, and F = { X: G (X) ≦ 0} represents a structure failure.
Many very sophisticated methods have been developed for solving the failure probability of structures, which can meet different types of engineering requirements, after decades of research by numerous scholars. However, the traditional solving method has low calculation efficiency and high calculation cost, and a plurality of limit conditions exist in the application process. For example, when the importance sampling method is applied to estimate the failure probability of a structure, the information of the failure domain needs to be known so as to be able to reasonably select the importance sampling density function, and the importance sampling method also has difficulty in estimating the failure probability in a high-dimensional situation.
Disclosure of Invention
The invention aims to provide a method for solving the structural failure probability based on an optimal condition important sampling method, which solves the problems of high calculation cost and low efficiency of the conventional method for solving the structural failure probability.
According to one aspect of the present invention, there is provided a method for solving a structural failure probability, including:
taking variables associated with structural failure as analysis samples, each as an input variable, according to a sampling density function h X (x) Extracting N CIS A sample, dividing the distribution interval of each input variable into m continuous but disjoint sub-intervals A k (k＝1,2,…m)；
Delta for each input variable using significant sampling i Estimate all delta i The estimated values are compared and are selected so that delta i Taking an input variable with an estimated value reaching a maximum value as a condition variable; delta. For the preparation of a coating i The estimated values of (c) are:
Figure BDA0001789440110000021
at known condition variables and corresponding interval numbers m and A k Under the condition (2), the variance of the estimated failure probability values is estimated respectively
Figure BDA0001789440110000022
And variance of optimal estimated value of failure probability
Figure BDA0001789440110000023
Comparison of
Figure BDA0001789440110000024
And
Figure BDA0001789440110000025
judgment of
Figure BDA0001789440110000026
Whether significantly less than
Figure BDA0001789440110000027
If not, then
Figure BDA0001789440110000028
As a final estimate; if yes, adjusting the distribution interval division of the input variable, and re-estimating the failure probability
Figure BDA0001789440110000029
And
Figure BDA00017894401100000210
iterate until
Figure BDA00017894401100000211
Is not significantly less than
Figure BDA00017894401100000212
In an exemplary embodiment of the invention, the variance of the failure probability estimate
Figure BDA00017894401100000213
Calculated according to the following formula:
Figure BDA00017894401100000214
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00017894401100000215
N k the number of samples corresponding to the subinterval.
In an exemplary embodiment of the invention, the variance of the optimal estimate of the failure probability is determined by a variance of the optimal estimate of the failure probability
Figure BDA00017894401100000216
Calculated according to the following formula:
Figure BDA00017894401100000217
wherein σ k Is estimated as
Figure BDA00017894401100000218
y r,k R =1, \8230, N, which is a sample of the response function k ，k＝1,…,m，y r,k By corresponding sample x to each subinterval r,k Bringing in g (x) to obtain y = g (x) = I F (x)f X (x)/h X (x)，N k The number of samples corresponding to the subinterval.
In an exemplary embodiment of the present invention, the distribution interval dividing method of the adjustment input variable includes: estimating the optimal value of the number of samples corresponding to the subintervals according to the following formula
Figure BDA0001789440110000031
Then according to
Figure BDA0001789440110000032
Adjusting the number of samples N corresponding to the subinterval k ，
Figure BDA0001789440110000033
In an exemplary embodiment of the invention, the distribution interval of each input variable is divided into m consecutive but disjoint sub-intervals A k (k =1,2, \ 8230/; m), an equiprobable partition method was adopted.
In an exemplary embodiment of the invention, the solution method is used for solving the failure probability of the I-beam structure.
In an exemplary embodiment of the invention, the input variables include i-beam waist height h, leg width b, waist thickness T, average leg thickness a, bending moment M experienced by the front axle, and torque T experienced by the front axle.
The solution method of the present invention is implemented by increasing the variance of the random variable by combining the total expectation and the all-around difference formula over the subinterval with the significant sampling method to produce a conditional significant sampling method based on the total expectation and the all-around difference formula over the subinterval. On one hand, the efficiency of solving the structure failure probability can be improved on the premise of ensuring the calculation precision, and the calculation cost is reduced; on the other hand, for the problem of high-dimensional reliability with more variables, the method has good applicability, calculation efficiency and precision, greatly improves the feasibility of calculation of the high-dimensional reliability problem, and can be widely applied to the calculation of various reliability problems in engineering.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed.
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The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and together with the description, serve to explain the principles of the invention. It is obvious that the drawings in the following description are only some embodiments of the invention, and that for a person skilled in the art, other drawings can be derived from them without inventive effort.
FIG. 1 is a schematic cross-sectional view of a dangerous front axle of an automobile according to an embodiment of the present invention;
fig. 2 is a comparison of variation coefficients of estimated failure probabilities obtained by three sampling methods with the number of samples.
Detailed Description
Example embodiments will now be described more fully. Example embodiments may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of example embodiments to those skilled in the art.
The terms "a," "an," "the," "said," and "at least one" are used to indicate the presence of one or more elements/components/parts/etc.; the terms "comprising" and "having" are intended to be inclusive and mean that there may be additional elements/components/etc. other than the listed elements/components/etc.
In order to verify the correctness and rationality of the structural failure probability solving method based on the optimal condition important sampling method, firstly, a total expectation formula, an all-round difference formula and the important sampling method are introduced. If Y is a integrable random variable (satisfying E (| Y |) < ∞) and X is any random variable in the same probability space, then the following fully-expected formula exists:
E(Y)＝E X (E(Y|X)) (2)
wherein E (-) denotes the desired operation, E X (. Cndot.) represents the expectation with respect to a random variable X. The all-desired formula indicates that the conditionally desired desire for Y is equal to the unconditional desire for Y given the condition of X.
If X and Y are random variables within the same probability space, and Y has a finite variance, then the following full-variance equation:
V(Y)＝E X (V(Y|X))+V X (E(Y|X)) (3)
wherein V (·) represents a variance operation, V X (. Cndot.) represents variance with respect to a random variable X.
The following introduces and demonstrates the fully expected and fully differenced formulas over subintervals. Suppose that the distribution interval of the random variable X is [ b ] 1 ,b 2 ]Will [ b ] be 1 ,b 2 ]Partitioning into m contiguous but disjoint sub-intervals, i.e. A k ＝[a k-1 ,a k ]，k＝1,2,…,m，a 0 ＝b 1 ，a m ＝b 2 . There is thus a fully desired formula and a fully differenced formula over the following subintervals:
Figure BDA0001789440110000041
Figure BDA0001789440110000042
wherein the content of the first and second substances,
Figure BDA0001789440110000051
indicating expectations regarding sub-intervals，
Figure BDA0001789440110000052
Indicating the variance for the subintervals.
The proof of the formulae (4) and (5) is as follows:
the outer layer on the right side of equation (2) is desirably written in the form of an integral, so that the following equation can be obtained
Figure BDA0001789440110000053
Wherein, F X (x) As a function of the probability distribution of the random variable X, f X (x) As a function of the probability density of the random variable X,
Figure BDA0001789440110000054
denotes the sub-interval A k Probability density function of internal random variable X, order
Figure BDA0001789440110000055
p k Indicates that the random variable X is in the subinterval A k Inner probability, thereby obtaining
Figure BDA0001789440110000056
In the sub-interval A k Using the total expectation formula (2) internally, thereby obtaining
Figure BDA0001789440110000057
Thereby obtaining formula (4).
From the variance definition, the variance of Y can be rewritten as follows
V(Y)＝E(Y-E(Y)) 2 ＝E(Y 2 )-[E(Y)] 2 (9)
Further, according to the formula (4), a compound having a structure represented by the formula
Figure BDA0001789440110000058
Figure BDA0001789440110000059
Bringing it into formula (9) can give
Figure BDA00017894401100000510
Thus, equation (5) is obtained, and therefore the total expectation equation and the total difference equation in the subinterval are satisfied.
The important sampling method can enable more samples to fall into a failure domain by introducing an important sampling density function h (X), thereby improving the sampling efficiency. The expression for the probability of failure can be expressed using the important sampling theory as:
Figure BDA0001789440110000061
wherein R is d Representing a d-dimensional input variable space, I F (X) represents a failure domain indication function. When x ∈ { x | G (x) ≦ 0}, I F (X) =1, when X ∈ { X | G (X)>0, I F (X) =0. Where G (x) is a limit state equation, i.e., a function, used to determine whether to fail.
According to the significant sampling density function h X (x) Generating N IS A sample x i (i＝1,2,…,N IS ) Further, the estimated expression of the failure probability P (F) can be expressed as
Figure BDA0001789440110000062
Since all samples are independently and identically distributed, an estimate of the probability of failure
Figure BDA0001789440110000064
Can be expressed as
Figure BDA0001789440110000063
In the conventional significant sampling method, a significant sampling density function is generally constructed by shifting a sampling center to a design point, so that more failure samples can be obtained. For linear function, this method can make 50% of samples fall into failure domain, and the key of this method for constructing important sampling density function is to find design point. However, the search for the design point is usually complex, and especially for a complex implicit function, it often needs to spend a lot of cost to search for the design point.
The invention is realized by increasing the variance of the random variable, and the total expectation and the all-round difference formula on the subinterval are combined with the important sampling method to generate the condition important sampling method based on the total expectation and the all-round difference formula on the subinterval. The method does not need to search for a design point, so that additional calculation cost is not needed.
Assume an input random variable X i Has a distribution interval of [ b 1 ,b 2 ]Will be [ b ] 1 ,b 2 ]Partitioning into m contiguous but disjoint sub-intervals, i.e. A k ＝[a k-1 ,a k ]，k＝1,2,…,m，a 0 ＝b 1 ，a m ＝b 2 . According to equation (4), the failure probability in equation (11) can be expressed as:
Figure BDA0001789440110000071
wherein the content of the first and second substances,
Figure BDA0001789440110000072
is the important sampling density function h X (x) The edge probability density function of (2).
According to the significant sampling density function h X (x) Samples of the input variables are taken and compared with a subinterval A k The corresponding sample is denoted x r,k (r＝1,…,N k ) I.e. x r,k Falls within the sub-interval A k ，N k Is a sub-interval A k Corresponding toNumber of samples, whereby the total number of samples is
Figure BDA0001789440110000073
Order to
Figure BDA0001789440110000074
Then psi k Can be expressed as
Figure BDA0001789440110000075
Estimate psi k The variance of (A) can be expressed as
Figure BDA0001789440110000076
Thus, the estimated value of the failure probability in equation (14) can be expressed as
Figure BDA0001789440110000077
Estimated value
Figure BDA0001789440110000078
Can be expressed as
Figure BDA0001789440110000079
In general, the subinterval A k Corresponding number of samples N k And P k Proportional ratio, i.e. N k ＝P k N CIS (k =1, \8230;, m). In combination with formula (5), we can obtain the following formula
Figure BDA00017894401100000710
Comparing the formula (17) with the formula (13)
Figure BDA0001789440110000081
Is not negative, it can be known that when N is CIS ≥N IS When the temperature of the water is higher than the set temperature,
Figure BDA0001789440110000082
to reduce to the maximum extent
Figure BDA0001789440110000083
Should be selected such that
Figure BDA0001789440110000084
The input variable that reaches the maximum value is taken as the condition variable. Will be provided with
Figure BDA0001789440110000085
Is regarded as to
Figure BDA0001789440110000086
Is an estimation of the variance sensitivity index of the response function, and can be estimated simultaneously only by using the sample for estimating the failure probability
Figure BDA0001789440110000087
Without adding additional computational cost. The estimation process is as follows:
let y = g (x) = I F (x)f X (x)/h X (x) Sample x corresponding to each subinterval r,k (r＝1,…,N k K =1, \8230;, m) is substituted into g (x) to obtain a corresponding response function with a sample y r,k (r＝1,…,N k K =1, \ 8230;, m). Thereby, the device
Figure BDA0001789440110000088
Is estimated as
Figure BDA0001789440110000089
Figure BDA00017894401100000810
Is estimated as
Figure BDA00017894401100000811
Figure BDA00017894401100000812
Is estimated as
Figure BDA00017894401100000813
In combination with the above, an optimal condition significant sampling method is proposed. When the subinterval and the total number of samples are given, the variance of the estimated failure probability values obtained by the condition-critical sampling method can be minimized by adjusting the number of samples corresponding to each subinterval.
As shown in equation (16), the variance of the failure probability estimate may be expressed as
Figure BDA00017894401100000814
Wherein the content of the first and second substances,
Figure BDA0001789440110000091
in order to make the variance of the failure probability estimate
Figure BDA0001789440110000092
To achieve the minimum, the following optimization problems need to be solved:
Figure BDA0001789440110000093
the optimization problem in the formula (22) is solved by adopting a Lagrange multiplier method, and the content is as follows:
first, the following Lagrangian function is defined
Figure BDA0001789440110000094
Where λ is the lagrange multiplier. Let equation (23) have partial derivatives equal to 0 with respect to all variables, the following equation can be obtained
Figure BDA0001789440110000095
From the first m equations in equation (24), the
Figure BDA0001789440110000096
k =1, \8230;, m, which is then substituted into the last equation in equation (23) can result in
Figure BDA0001789440110000097
Thereby obtaining the optimal number of samples corresponding to the subinterval as
Figure BDA0001789440110000098
At this time, the corresponding failure probability estimation value is recorded as
Figure BDA0001789440110000099
The variance is
Figure BDA00017894401100000910
According to equation (25), in order to determine the optimal value of the number of samples corresponding to the subinterval, P needs to be estimated first k And σ k 。
Figure BDA00017894401100000911
It can be estimated efficiently by a numerical integration method. Alternatively, P can be modeled by a Monte Carlo simulation k And (6) estimating. First according to
Figure BDA00017894401100000912
Extracting variable X i N samples, and then the statistics of the number of the samples falling into the subinterval a k The number of samples in the column is denoted as N k And thus P k Can be expressed as
Figure BDA0001789440110000101
The process does not need to calculate the function, and does not cause large extra calculation cost. Sigma k The estimation can be performed according to equation (19), and the samples used are still the samples used for estimating the failure probability, so that no additional calculation cost is added.
After the optimal value of the number of samples corresponding to the subinterval is determined, the number of samples corresponding to the subinterval can be adjusted, and the failure probability is re-estimated, so that the variance of the estimated value is minimum. However, this process requires computation of the function, which introduces additional computational cost. In fact, P is estimated k And σ k Then, the variance of the optimal estimated value of the failure probability can be estimated according to the formula (26), and the formula (26) can be compared with the formula (16), if so
Figure BDA0001789440110000102
Compared with
Figure BDA0001789440110000103
If the number of the samples is reduced significantly, the probability of failure can be estimated again by adjusting the number of the samples corresponding to each subinterval, and if the number of the samples is reduced significantly, the probability of failure can be estimated again by adjusting the number of the samples corresponding to each subinterval
Figure BDA0001789440110000104
Compared with
Figure BDA0001789440110000105
Without a significant reduction, the probability of failure does not have to be re-estimated.
According to the above analysis, the method for solving the structure failure probability based on the optimal condition importance sampling method of the present invention may include the following steps:
s1, taking variables associated with structural failure as analysis samples, taking each variable as an input variable, and extracting N CIS A sample, dividing the distribution interval of each input variable into m continuous but disjointSub-interval A of k (k＝1,2,…m)；
S2, adopting an important sampling method to measure delta of each input variable i Estimate all delta i The estimated values are compared and are selected so that delta i Taking the input variable with the estimation value reaching the maximum value as a condition variable; delta i The estimated values of (c) are:
Figure BDA0001789440110000106
step S3, the condition variables and the corresponding interval numbers m and A are known k Under the condition (2), the variance of the estimated failure probability values is estimated respectively
Figure BDA0001789440110000107
And variance of optimal estimated value of failure probability
Figure BDA0001789440110000108
Comparison
Figure BDA0001789440110000109
And
Figure BDA00017894401100001010
judgment of
Figure BDA00017894401100001011
Whether or not it is significantly less than
Figure BDA00017894401100001012
If not, then
Figure BDA00017894401100001013
As a final estimate; if yes, the distribution interval division of the input variables is adjusted, and the failure probability is re-estimated
Figure BDA00017894401100001014
And
Figure BDA00017894401100001015
iterate until
Figure BDA00017894401100001016
Is not significantly less than
Figure BDA00017894401100001017
The following describes in detail the steps of the method for solving the probability of structural failure based on the optimal condition importance sampling method according to the embodiment of the present invention:
in step S1, the sampling may take a variety of sampling forms, and in this exemplary embodiment, according to a sampling density function h X (x) To extract a sample. When dividing samples, m and A k The method is artificially defined, and can be divided according to equal probability for convenient division, and at the moment, the subintervals corresponding to all variables are determined; of course, the probability division may be unequal, and in this case, the number of subintervals of each variable needs to be artificially determined.
Thus, step S1 may comprise the following sub-steps:
step S1.1, sample extraction
Determining variables associated with structural failure as analysis samples according to a sampling density function h X (x) Extracting N CIS A sample
Figure BDA0001789440110000111
All variables associated with structural failure are included in each sample as random input variables for subsequent calculations.
Step S1.2, sample division
Dividing the distribution interval of each input variable into m consecutive but disjoint sub-intervals A k (k＝1,2,…m)，N k Is a sub-interval A k Corresponding number of samples.
In step S2, the purpose of this step is to determine the condition variable, in which case the variance of the failure probability can be minimized. This step may include the following sub-steps:
step S2.1, define δ i ：
Figure BDA0001789440110000112
d is the number of structural variables associated with the structural failure to be solved.
By the important sampling method theory, the samples are utilized
Figure BDA0001789440110000113
Delta for each input variable according to equation (20) i Make an estimation so i The estimated value of (c) is:
Figure BDA0001789440110000114
wherein, I F (X) represents a fail domain indicator function, when X ∈ { X | G (X) ≦ 0}, I ∈ { X | G (X) ≦ 0}, and F (X) =1, when X ∈ { X | G (X)>0, I F (X)＝0。f X (x) Is a probability density function of a random variable X. h is X (x) Is an important sampling density function. X i Is a random variable.
Figure BDA0001789440110000115
Is the important sampling density function h X (x) The edge probability density function of (2).
Step S2.2, the estimation of step 2.1 is performed for each input variable and all deltas calculated are i The magnitudes of the estimates are compared and are selected so that delta i The input variable at which the estimated value reaches the maximum value is taken as a condition variable.
In step S3, estimation
Figure BDA0001789440110000121
May comprise the following substeps:
step S3.1.1, in the known condition variables and the corresponding interval numbers m and A k Under the condition (1), estimating the failure probability according to the equation (15),
Figure BDA0001789440110000122
wherein the content of the first and second substances,
Figure BDA0001789440110000123
x r,k for samples corresponding to each subinterval, r =1, \ 8230;, N k ，k＝1,…,m，N k Is a sub-interval A k The corresponding number of samples.
Step S3.1.2, estimating the corresponding variance according to the formula (16)
Figure BDA0001789440110000124
Figure BDA0001789440110000125
In step S3, estimation is performed
Figure BDA0001789440110000126
May comprise the following substeps:
step S3.2.1, in the known condition variable and corresponding interval number m and A k Using the sample under the conditions of
Figure BDA0001789440110000127
Estimate σ from (19) k (k＝1,2,…m)
Defining:
Figure BDA0001789440110000128
then sigma k Is estimated as
Figure BDA0001789440110000129
In the formula, y r,k Is a sample of the response function, r =1, \8230;, N k ，k＝1,…,m，y r,k By corresponding samples x for each subinterval r,k Bringing in g (x) to obtain y = g (x) = I F (x)f X (x)/h X (x)。
Step S3.2.2, estimating the optimal estimated value of the failure probability according to the formula (26)
Figure BDA00017894401100001210
Variance of (2)
Figure BDA00017894401100001211
Figure BDA00017894401100001212
In step S3, the distribution interval dividing method for adjusting the input variable may be: estimating the optimal value of the number of samples corresponding to the subinterval according to the following formula
Figure BDA00017894401100001213
Then according to
Figure BDA00017894401100001214
Adjusting the number of samples N corresponding to the subinterval k ，
Figure BDA0001789440110000131
The input variable can be purposefully adjusted through the step, and the problems of uncertainty and large operation quantity caused by random adjustment are avoided.
In the above embodiments, those skilled in the art can know that the execution sequence is not unique, for example, the estimation in step 3
Figure BDA0001789440110000132
And
Figure BDA0001789440110000133
the order of (a) may be reversed.
The structural failure probability solving method of the present invention is described in detail below with reference to specific examples, and the present embodiment is described by taking an i-beam shown in fig. 1 as an example.
Fig. 1 shows a dangerous section occurring in an i-beam structure, where the maximum normal and shear stresses can be expressed as σ and τ = T/W, respectively ρ Wherein M and T represent respectively the bending moment and the torque to which the front axle is subjected, W x And W ρ The section coefficients and the polar section coefficients are expressed separately, and they can be expressed as:
Figure BDA0001789440110000134
wherein h represents the h-beam waist height, b represents the h-beam leg width, t represents the h-beam waist thickness, and a represents the average leg thickness.
According to the static strength failure criterion of the structure, the following function can be constructed
Figure BDA0001789440110000135
Wherein σ s Expressing the yield limit of the static strength of the structure, sigma can be obtained according to the material property of the front axle structure s ＝460MPa。
The geometrical dimensions a, b, T and h of the dangerous section of the I-shaped beam and the bending moment M and the torque T of the front axle are taken as mutually independent random input variables (d = 6), the geometrical dimensions a, b, T and h and the bending moment M and the torque T are subjected to normal distribution, and the distribution parameters are shown in a table 1.
TABLE 1 distribution parameters of input random variables in the front axle construction of a motor vehicle
Figure BDA0001789440110000136
Step S1, determining a sample
Extracting total sample size, namely extracting 2000 samples of the geometrical sizes a, b, T and h of random input variables of the I-beam and the bending moment M and the torque T suffered by the front axle respectively to form corresponding 2000 groups of samples, namely N CIS ＝2000。
The distribution interval of all input variables is divided equally probabilistically into 20 sub-intervals, i.e. m =20.
Step S2, determining condition variables
Estimate delta for each input variable i The results are shown in Table 2
TABLE 2 calculation of δ i Is estimated by
Figure BDA0001789440110000141
As can be seen from Table 2, δ corresponds to the variable T i Has the largest value, and therefore T is selected as the conditioning variable.
The final estimated value can be obtained by the method of the foregoing step S3.
In order to verify the effectiveness of the method of the present invention, the optimal condition-critical sampling of the present invention was compared with the conventional critical sampling method, the condition-critical sampling method. In order to compare the estimation results and the robustness of the three sampling methods, the three sampling methods are repeated 200 times respectively, the average value is taken as the estimation value of the failure probability, and the corresponding coefficient of variation is estimated at the same time, and the results are shown in table 3. It can be seen that the estimated values of the failure probability obtained by the three methods are very close, but the variation coefficient of the estimated value of the failure probability obtained by the optimal condition important sampling method is smaller than that of the estimated values of the failure probability obtained by the other two methods, so that the optimal condition important sampling method has the best robustness. In order to compare the convergence rates of the three sampling methods more clearly, the coefficients of variation of the three sampling methods for different sample numbers were calculated, and the results are shown in fig. 2. It can be seen that the optimal conditional significant sampling method has the highest convergence rate, followed by the conditional significant sampling method, which has the lowest convergence rate. It can also be seen that the number of samples required for optimal subsampling is minimal if the coefficient of variation of the failure probability estimate is to be guaranteed to a given level.
Table 3 estimated values of failure probability and their coefficients of variation
Figure BDA0001789440110000142
For the i-beam structure, the six variables of a, b, T, h, M and T are considered in the above embodiment, and the determination of the random variable is mainly determined comprehensively according to various factors influencing the structural failure, so the variable can be replaced or increased or decreased, in this example, if the property of the material is considered, the elastic modulus or poisson's ratio of the material can be added as an input variable, and the calculation method is not changed.
Similarly, if the I-beam structure is replaced by other structures, the structure failure probability can be solved by adopting the same method as long as all factors influencing the structure failure are determined and taken as input variables.
Other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the invention following, in general, the principles of the invention and including such departures from the present disclosure as come within known or customary practice within the art to which the invention pertains. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.

Claims (3)

1. A method for solving the failure probability of a structure based on an optimal condition importance sampling method is characterized in that the method is used for solving the failure probability of an I-beam structure and comprises the following steps:
taking variables related to structural failure as analysis samples, taking each variable as an input variable, wherein the input variables comprise H-beam waist height h, leg width b, waist thickness T, average leg thickness a, bending moment M borne by a front shaft and torque T borne by the front shaft, and extracting N CIS A sample, dividing the distribution interval of each input variable into m continuous but disjoint sub-intervals A k (k＝1,2,...m)；
Delta for each input variable using significant sampling i Estimate all delta i The estimated values are compared and are selected so that delta i Estimate value is reachedTaking the input variable of the maximum value as a condition variable; delta i The estimated value of (c) is:
Figure FDA0003814661990000011
wherein X i Representing an extracted variable, F X (x) Probability distribution function, f, representing a random variable X X (x) As a function of the probability density of the random variable X, I F (X) denotes a failure domain indicator function, h X (x) A function representing the significant sample density is shown,
Figure FDA0003814661990000012
to represent
Figure FDA0003814661990000013
An estimate of the variance of (a) is,
Figure FDA0003814661990000014
the response function is represented by a function of the response,
Figure FDA0003814661990000015
Figure FDA0003814661990000016
is the important sampling density function h X (x) Edge probability density function of (2), x i (i＝1,2,...,N IS ) Representing a sample;
at known condition variables and corresponding interval numbers m and A k Under the condition (2), the variance of the estimated failure probability values is estimated respectively
Figure FDA0003814661990000017
And variance of optimal estimated value of failure probability
Figure FDA0003814661990000018
Comparison of
Figure FDA0003814661990000019
And
Figure FDA00038146619900000110
judgment of
Figure FDA00038146619900000111
Whether significantly less than
Figure FDA00038146619900000112
If not, then
Figure FDA00038146619900000113
As a final estimate; if yes, the distribution interval division of the input variables is adjusted, and the failure probability is re-estimated
Figure FDA00038146619900000114
And
Figure FDA00038146619900000115
iterate until
Figure FDA00038146619900000116
Is not significantly less than
Figure FDA00038146619900000117
Estimating
Figure FDA00038146619900000118
May include the following substeps:
at known condition variables and corresponding interval numbers m and A k Under the condition (1), estimating the failure probability according to the equation (15),
Figure FDA0003814661990000021
wherein the content of the first and second substances,
Figure FDA0003814661990000022
x r,k for samples corresponding to each subinterval, r =1 k ，k＝1,...,m，N k Is a sub-interval A k The corresponding number of samples;
the corresponding variance is estimated according to equation (16)
Figure FDA0003814661990000023
Figure FDA0003814661990000024
Wherein the content of the first and second substances,
Figure FDA0003814661990000025
N k is the number of samples corresponding to the sub-interval,
Figure FDA0003814661990000026
is the important sampling density function h X (x) Of the edge probability density function, x i (i＝1,2,...,N IS ) The samples are represented by a representation of the sample,
Figure FDA0003814661990000027
denotes the sub-interval A k Inner part
Figure FDA0003814661990000028
The variance of (a) is determined,
Figure FDA0003814661990000029
denotes the sub-interval A k Inner part
Figure FDA00038146619900000210
In the expectation that the position of the target is not changed,
Figure FDA00038146619900000211
representψ k Is determined by the estimated value of (c),
Figure FDA00038146619900000212
to represent
Figure FDA00038146619900000213
The variance of (a);
estimating
Figure FDA00038146619900000214
May include the following substeps:
at known condition variables and corresponding interval numbers m and A k Using the sample under the conditions of
Figure FDA00038146619900000215
Estimate σ from (19) k (k＝1,2,...m)；
Defining:
Figure FDA00038146619900000216
then σ k Is estimated as (a) being,
Figure FDA00038146619900000217
in the formula, y r,k Is a sample of the response function, r =1 k ，k＝1,...,m，y r,k By corresponding samples x for each subinterval r,k G (x) is taken in, y = g (x) = I F (x)f X (x)/h X (x)；
The optimal estimated value of the failure probability is estimated according to the formula (26)
Figure FDA00038146619900000218
Variance of (2)
Figure FDA00038146619900000219
Figure FDA00038146619900000220
Wherein σ k Is estimated as
Figure FDA00038146619900000221
y r,k As a sample of the response function, r =1 k ，k＝1,...,m，y r,k By corresponding sample x to each subinterval r,k G (x) is taken in, y = g (x) = I F (x)f X (x)/h X (x)，N k Is the number of samples corresponding to the sub-interval,
Figure FDA0003814661990000031
which represents the total number of samples to be taken,
Figure FDA0003814661990000032
2. the method according to claim 1, wherein adjusting the distribution interval division of the input variables comprises:
estimating the optimal value of the number of samples corresponding to the subinterval according to the following formula
Figure FDA0003814661990000033
Then according to
Figure FDA0003814661990000034
Adjusting the number of samples N corresponding to the subinterval k ，
Figure FDA0003814661990000035
Wherein λ is a lagrange multiplier.
3. The method of claim 1, wherein the distribution interval for each input variable is determined by a method of calculating the distribution interval for each input variableDivided into m contiguous but disjoint sub-intervals A k When, an equiprobable partition is adopted, where k =1,2.
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