CN111523203B - Structure reliability high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling - Google Patents

Abstract

The invention provides a high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling, which comprises the following steps of: firstly, the method comprises the following steps: determination of the structural Primary random variable X = (X) ₁ ,X ₂ ,...,X _n ) And converting the original random variable into a random variable Y = (Y) in a standard normal space ₁ ,Y ₂ ,...,Y _n ) The corresponding extreme state function is converted to g (Y) (conversion is not required if the extreme state function is implicit); II, secondly: the iteration number l =1, and a checking point y is determined ^*(1) And a radius beta of a beta sphere (a hypersphere with an origin as a sphere center and a reliability index beta as a radius) ⁽¹⁾ (ii) a Thirdly, the method comprises the following steps: generating sample points subject to truncation importance distribution by using screening method

Fourthly, the method comprises the following steps: calculating failure probability estimates

Fifthly: calculating variance

And coefficient of variation

Sixthly, the method comprises the following steps: updating the checking point and the radius of the beta sphere; seventhly, the method comprises the following steps: judgment of

(epsilon is a preset precision requirement) or not; if the precision is insufficient, l = l +1, and the step three is carried out until the precision requirement is met.

Description

Structure reliability high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling

Technical Field

The invention relates to a high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling, and belongs to the technical field of structural reliability analysis.

Background

The calculation of failure probability is an important subject of structural reliability analysis. The Monte Carlo method has more accurate calculation result, can also be used for the situation that the extreme state equation is more complex, and has wider application. However, to ensure sufficient accuracy, a large number of samples are required in a practical engineering problem with a low probability of failure. The important sampling method is a method for improving sampling efficiency on the premise of ensuring calculation accuracy, and the basic idea is to construct an important sampling density function to enable the extracted samples to fall into a failure domain more, so that failure probability is improved.

On the basis of the traditional important sampling method, some scholars propose a truncation important sampling method. The method further reduces the sampling area, thereby improving the sampling efficiency. However, the following disadvantages still exist in the existing truncation significant sampling method: (1) Under the condition of unknown check points (or reliability indexes), the method cannot be independently used, and the application range of the method is limited. (2) If the order moment method is adopted to determine the checking point and the reliability index, because the error of the order moment method is larger when the extreme state function is nonlinear, an ideal approximate result of the checking point and the reliability index cannot be obtained. If the obtained reliability index is too small, the high efficiency of the algorithm is difficult to embody; if the obtained reliability index is too large, the sample point originally in the failure domain is erroneously determined to be in the security domain, so that the calculation result is inaccurate.

Disclosure of Invention

The invention aims to provide a high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling aiming at the defects of the truncation important sampling method in the prior art, so that the calculation efficiency is improved on the premise of ensuring the precision when the structural reliability is analyzed.

In order to achieve the purpose, the invention adopts the following technical scheme:

the invention provides a high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling, which comprises the following steps of:

the method comprises the following steps: determination of the structural Primary random variable X = (X) ₁ ,X ₂ ,...,X _n ) And converting the original random variable into a random variable Y = (Y) in a standard normal space ₁ ,Y ₂ ,...,Y _n ) The corresponding extreme state function is converted to g (Y) (no conversion is needed if the extreme state function is implicit);

step two: the iteration number l =1, and a checking point y is determined ^*(1) And a radius beta of a beta sphere (a hypersphere with an origin as a sphere center and a reliability index beta as a radius) ⁽¹⁾ ；

Step three: generating sample points subject to truncation importance distribution by using screening method

Step four: calculating failure probability estimates

Step five: calculating variance

And coefficient of variation

Step six: updating the checking point and the radius of the beta sphere;

step seven: judgment of

(epsilon is a preset precision requirement) or not; if the precision is insufficient, l = l +1, and the step three is carried out until the precision requirement is met.

Wherein in step one, the step of converting the original random variable into a random variable Y in a standard normal space = (Y) ₁ ,Y ₂ ,...,Y _n ) ", it does the following:

in the formula (I), the compound is shown in the specification,

is the mean value of the random variable and,

is the standard deviation of the random variable;

wherein the "extreme state function" in step one is converted to g (Y) "by:

will be provided with

And substituting the original limit state function to obtain the converted limit state function.

Wherein "determining the checking point y" described in the second step ^*(1) And beta sphere radius beta ⁽¹⁾ ", it does the following:

determination of checking points by order moment method

Calculating the radius of the beta sphere in a standard normal space

A smaller beta value can also be preset, and the average value point is taken as the initial checking point.

Wherein the generation of sample points subject to truncation of the significance distribution by means of the screening method described in step three

", it does the following:

firstly, taking checking points

For sampling the gravity center, important sampling is performed to obtain a group of random numbers

And screening the obtained sample points, if the sample points can simultaneously meet the following requirements:

and

the sample points are described as being subject to the truncation importance distribution and are recorded as

Wherein the failure probability estimate is calculated as described in step four

", which is done as follows:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) The total number of samples; s is the number of iterations; n is a radical of hydrogen ^(l) For each iteration to fall within the region omega ₂ The number of sample points (division of area range is shown in fig. 2); i (·) is an illustrative function; f (-) is a probability density function; h is a total of ^(l) (. Cndot.) represents the significant sample density function in the l-th iteration.

Note: falls into the region omega ₁ And Ω ₃ The sample(s) can be judged to be within the safe domain without being substituted into the solution of the extreme state function g (Y).

Wherein "calculating the variance" described in step five

And coefficient of variation

", which is done as follows:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) The total number of samples; s is the number of iterations; n is a radical of hydrogen ^(l) For each iteration to fall within the region omega ₂ The number of sample points (division of area range is shown in fig. 2); i (-) is an illustrative function; f (-) is a probability density function; h is ^(l) (. Cndot.) represents the significant sample density function in the l-th iteration; n = N ⁽¹⁾ +N ⁽²⁾ +...+N ^(s) To fall into the region Ω ₂ The total number of sample points;

is an estimated value of the failure probability;

in the formula (I), the compound is shown in the specification,

is the variance;

is an estimate of the probability of failure.

Wherein, the method for updating the checking point and the beta sphere radius in the step six is as follows:

if any sample point falls into the failure domain, recording the sample point corresponding to the maximum probability density value in the failure domain

Comparative inequality

Gamma is generally 1.02, if yes, the sample point corresponding to the minimum limit state value in the safety domain is recorded

β ^(l+1) ＝β ^**(l) (ii) a If not, y ^*(l+1) ＝y ^*(l) ，β ^(l+1) ＝β ^(l) ；

If no sample point falls into the failure domain, recording the sample point corresponding to the minimum limit state value in the safety domain

β ^(l+1) ＝β ^**(l) 。

The schematic diagram of the checking points and the update of the beta sphere radius is shown in figure 3.

The invention has the beneficial effects that:

(1) The method of the invention adopts the mode of beta sphere and section double truncation in single iteration to reduce the important sampling area, reduce the calling times of the extreme state function, and has small calculated amount and high sampling efficiency.

(2) The method can give a smaller beta value in advance in the first iteration and continuously adjust according to the sample point in the subsequent iteration, thereby solving the problem that the traditional truncation important sampling method can not be independently used under the condition of unknown check points (or reliability indexes).

(3) The method of the invention continuously adjusts the checking points in each iteration, thereby being capable of rapidly approaching to the vicinity of the real design checking points, being more accurate compared with the traditional truncation important sampling method which adopts a moment method and other positioning checking points, thereby increasing the occurrence probability of sample points which have great contribution to failure probability in effective sampling and accelerating the convergence speed of failure probability calculation.

(4) The method of the invention continuously adjusts the radius of the beta sphere in each iteration, thereby being capable of rapidly approaching to the real beta value and leading the accuracy of the calculation result to be higher.

(5) The method of the invention is scientific, has good manufacturability and has wide popularization and application value.

Drawings

FIG. 1 is a flow chart of the method of the present invention.

Fig. 2, a schematic diagram of region range division.

Fig. 3, schematic diagram of the verification points and the radius update of the beta sphere.

Fig. 4 is a model diagram of the mechanics of the tie rod of example 1.

Figure 5, example 2 cantilever beam structure with concentrated force schematic.

Fig. 6, example 3 schematic view of an internally pressurized cylindrical container.

Detailed Description

Example 1: a symmetric hollow circular tube pull rod is shown in figure 4 by a mechanical model diagram, and the ultimate equation of state is

Wherein R is the strength of the pull rod, Q is the load on the pull rod, d ₁ Is the outer diameter of the cross section, d ₀ The inner diameters of the sections are in normal distribution, and the parameters are shown in Table 1.

Table 1 example 1 random variable parameters

The invention discloses a high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling, which is shown in figure 1 and comprises the following steps:

the method comprises the following steps: in this example, there are four random variables, and the distribution type and parameters are given.

Normalizing the original random variable standard:

the extreme state equation after standard normalization is

Step two: the iteration number l =1, and the checking point y is preset in the example ^*(1) = (0,0,0,0), beta sphere radius beta ⁽¹⁾ ＝1

Step three: generating sample points that are subject to a truncated importance distribution

The sample points that resulted in a distribution that obeyed truncation of the significance are shown in table 2, programmed with MATLAB software.

Table 2 example 1 first iteration sample points

Step four: calculating failure probability estimates

Ω ₂ The division of the area range is schematically shown in figure 2;

step five: calculating variance

And coefficient of variation

Step six: updating the checking points and the radius of the beta sphere

y ^*(2) ＝(-1.7323,1.9311,-0.9873,0.3970)

β ⁽²⁾ ＝2.8040

The checking point and beta sphere radius updating process is schematically shown in FIG. 3;

step seven: first iteration without judgment

(in this example, ε =10 ^-3 ) And directly entering second iteration, wherein l =2, and turning to the third step until the precision requirement is met.

The convergence condition is reached through 3 iterations in this example, and the result of the iterative calculation of the probability of failure is shown in table 3.

Table 3 example 1 iterative calculation of failure probability

Table 4 shows the comparison of the method of the present invention with other methods, and the calculated failure probability is compared with the above method, and the result of 1000000 times of sampling by the monte carlo method is used as an accurate solution, and the relative error is calculated.

Table 4 comparison of calculation results of each method in example 1

Method	Probability of failure	Number of samples	Relative error (%)
				Monte Carlo method	0.004761	1000000	——
Beta sphere truncation important sampling method	0.004645	3000	2.4365
				The method of the invention	0.004798	3000	0.7771

As can be seen from table 4, compared with the beta sphere truncation important sampling method, the method provided by the present invention can be independently used under the condition of unknown check points (or reliability indexes), has a wider application range, and has a more accurate calculation result under the same sampling times, which indicates that the method of the present invention can effectively solve the problem of failure probability calculation, and has a higher advantage in practical engineering application.

Example 2: a cantilever beam with concentrated force is shown in figure 5 in its configuration and loaded condition. The elastic modulus E, the section moment of inertia I and the applied load force P are independent normal random variables, and the parameters are shown in a table 5; length L =5m is constant.

Table 5 example 2 random variable parameters

And (3) establishing a limit state equation by considering the maximum deformation of the cantilever beam:

g＝EI-78.125P＝0

the invention relates to a high-efficiency failure probability calculation method based on self-adaptive double-truncation important samples, which is shown in figure 1 and comprises the following steps:

the method comprises the following steps: in this example, there are three random variables, the distribution type and parameters are given.

Normalizing the original random variable standard:

the extreme state equation after standard normalization is (5Y) ₁ +20)(Y ₂ +10)-7.8125(2.5Y ₃ +8)＝0

Step two: the iteration number l =1, and the checking point y is preset in the example ^*(1) = 0, beta sphere radius beta ⁽¹⁾ ＝1

Step three: generating sample points that are subject to a truncated importance distribution

The sample points that resulted in a distribution that obeyed truncation of the significance are shown in table 6, programmed with MATLAB software.

Table 6 example 2 first iteration sample points

Step four: calculating failure probability estimates

Ω ₂ The division of the area range is schematically shown in figure 2;

step five: calculating variance

And coefficient of variation

Step six: updating the checking points and the radius of the beta sphere

y ^*(2) ＝(-2.1304,-0.9784,1.0031)

β ⁽²⁾ ＝2.5499

The checking point and beta sphere radius updating process is schematically shown in FIG. 3;

step seven: first iteration need not judge

(in this example,. Epsilon = 10) ^-3 ) And directly entering second iteration, wherein l =2, and turning to the third step until the precision requirement is met.

The convergence condition is reached through 2 iterations in this example, and the result of the iterative calculation of the probability of failure is shown in table 7.

Table 7 example 2 iterative calculation of failure probability

Table 8 shows the comparison of the method of the present invention with other methods, and the calculated failure probability is compared with the calculation results of the above method, and the relative error is calculated using the result of 1000000 times of sampling by the monte carlo method as an accurate solution.

Table 8 comparison of the calculation results of the methods of example 2

Method	Probability of failure (10-3)	Number of samples	Relative error (%)
				Monte Carlo method	5.9630	1000000	——
Beta sphere truncation important sampling method	6.0517	2000	1.4875
				The method of the invention	5.9540	2000	0.1509

As can be seen from table 8, compared with the beta sphere truncation important sampling method, the method provided by the present invention can be independently used under the condition of unknown check points (or reliability indexes), has a wider application range, and has a more accurate calculation result at the same sampling times, which indicates that the method of the present invention can effectively solve the problem of failure probability calculation, and has a higher advantage in practical engineering application.

Example 3: a cylindrical container with internal pressure is shown in FIG. 6, and 15MnV is used as the material. Inner diameter D, inner pressure P, wall thickness t and yield strength σ _s Is independent of normalThe random variables, parameters are shown in Table 9.

Table 9 example 3 random variable parameters

For conventional internally-pressurized cylindrical thin-walled containers, subjected to two-way, i.e. axial, stresses

Stress in circumferential direction

Radial stress S _r 0. According to the first strength theory (maximum principal stress theory), the ultimate state equation of the internal pressure cylinder can be obtained as follows:

g＝σ _s -S _eq ＝0

in the formula, S _eq For equivalent stress, in this example

The invention discloses a high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling, which is shown in figure 1 and comprises the following steps:

the method comprises the following steps: in this example, there are four random variables, and the distribution type and parameters are given.

Normalizing the original random variable standard:

the extreme state equation after standard normalization is

Step two: the iteration number l =1, and the checking point y is preset in the example ^*(1) = (0,0,0,0), beta sphere radius beta ⁽¹⁾ ＝1

Step three: generating sample points that are subject to a truncated importance distribution

The sample points that resulted in a obedient truncation of the significance distributions are shown in table 10, programmed with MATLAB software.

Table 10 example 3 first iteration sample points

Step four: calculating failure probability estimates

Ω ₂ The division of the area range is schematically shown in figure 2;

step five: calculating variance

And coefficient of variation

Step six: updating checking point and beta sphere radius

y ^*(2) ＝(-2.3146,1.5738,1.1962,-0.7278)

β ⁽²⁾ ＝3.1297

The checking point and beta sphere radius updating process is schematically shown in FIG. 3;

step seven: first iteration need not judge

(in this example,. Epsilon = 10) ^-4 ) And directly entering second iteration, wherein l =2, and turning to the third step until the precision requirement is met.

The convergence condition is achieved through 3 iterations in this example, and the calculation result of the iteration of the failure probability is shown in table 11.

Table 11 example 3 iterative calculation of failure probability

Table 12 shows the comparison of the method of the present invention with other methods, and the calculated failure probability is compared with the above method, and the result of 1000000 times of sampling by the monte carlo method is used as an accurate solution, and the relative error is calculated.

Table 12 comparison of calculation results of example 3 methods

Method	Probability of failure (10) -4 )	Number of samples	Relative error (%)
				Monte Carlo method	4.8000	1000000	——
Beta sphere truncation important sampling method	4.5763	4500	4.6604
				The method of the invention	4.8165	4500	0.3438

It can be seen from table 12 that, compared with the beta sphere truncation important sampling method, the method provided by the present invention can be independently used under the condition of unknown check point (or reliability index), the application range is wider, and the calculation result is more accurate under the same sampling times, which shows that the method of the present invention can effectively solve the problem of failure probability calculation, and has higher advantages in practical engineering application.

Claims (24)

1. A high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling aims at a hollow circular tube pull rod with a symmetrical structure and has a limit state equation of

Wherein R is the strength of the pull rod, Q is the load on the pull rod, d ₁ Is the outer diameter of the cross section, d ₀ The inner diameters of the sections are in normal distribution, and the parameters are shown in table 1;

TABLE 1

Variables of Mean value Standard deviation of R(MPa) 400 11 Q(N) 170000 2600 d ₁ (mm) 35 0.175 d ₀ (mm) 25 0.125

The method is characterized by comprising the following steps:

the method comprises the following steps: four random variables are provided, and distribution types and parameters are given;

normalizing the original random variable standard:

the extreme state equation after standard normalization is

Step two: the iteration number l =1 and a preset check point y ^*(1) = (0,0,0,0), beta sphere radius beta ⁽¹⁾ ＝1；

Step three: generating sample points which are subject to truncation of the important distribution;

programming by using MATLAB software to obtain sample points which obey truncation important distribution, wherein the sample points are shown in a table 2;

TABLE 2

Step four: calculating a failure probability estimation value;

step five: calculating variance

And coefficient of variation

Step six: updating the checking point and the radius of the beta sphere;

y ^*(2) ＝(-1.7323,1.9311,-0.9873,0.3970)

β ⁽²⁾ ＝2.8040

step seven: first iteration need not judge

Directly entering second iteration, wherein l =2, and turning to the third step until the precision requirement is met;

the convergence condition is reached through 3 iterations, and the result of the iterative calculation of the failure probability is shown in table 3;

TABLE 3

2. The method for calculating the high-efficiency failure probability based on the adaptive double-truncation important sampling as claimed in claim 1, wherein the method comprises the following steps: in the first step of the method,

in the formula (I), the compound is shown in the specification,

is the average of the random variables and is,

is the standard deviation of the random variable.

3. The method for calculating the high-efficiency failure probability based on the adaptive double truncated important samples as claimed in claim 1, characterized in that: in step one, the

And substituting the original limit state function to obtain the converted limit state function.

4. The method for calculating the high-efficiency failure probability based on the adaptive double-truncation important sampling as claimed in claim 1, wherein the method comprises the following steps: in step two, a checking point y is determined ^*(1) And beta sphere radius beta ⁽¹⁾ ", do the following:

determination of checking points by order moment method

Calculated in standard normal spaceRadius of interrue beta sphere

5. The method for calculating the high-efficiency failure probability based on the adaptive double-truncation important sampling as claimed in claim 1, wherein the method comprises the following steps: in step three, the procedure is as follows:

first, taking the checking points

For sampling the center of gravity, important sampling is performed to obtain a group of random numbers

And screening the obtained sample points, if the sample points can simultaneously meet the following requirements:

and

then the sample points are described as being subject to truncation importance distribution, and is recorded as

6. The method for calculating the high-efficiency failure probability based on the adaptive double truncated important samples as claimed in claim 1, characterized in that: in step four, the procedure is as follows:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) Total number of samples; s is the number of iterations; n is a radical of hydrogen ^(l) For each iteration to fall within the region omega ₂ The number of sample points; i (·) is an illustrative function; f (·)) Is a probability density function; h is a total of ^(l) (. Cndot.) represents the significant sampling density function in the l-th iteration.

7. The method for calculating the high-efficiency failure probability based on the adaptive double-truncation important sampling as claimed in claim 1, wherein the method comprises the following steps: in step five, the variance is calculated

And coefficient of variation

", do the following:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) Total number of samples; s is the number of iterations; n is a radical of ^(l) For each iteration to fall within the region omega ₂ The number of sample points; i (-) is an illustrative function; f (-) is a probability density function; h is a total of ^(l) (. H) represents the significant sampling density function in the l-th iteration; n = N ⁽¹⁾ +N ⁽²⁾ +...+N ^(s) To fall into the region omega ₂ The total number of sample points;

is an estimated value of the failure probability;

in the formula (I), the compound is shown in the specification,

is the variance;

is an estimated failure probability.

8. The method for calculating the high-efficiency failure probability based on the adaptive double-truncation important sampling as claimed in claim 1, wherein the method comprises the following steps: in step six, updating the checking point and the beta sphere radius as follows:

if any sample point falls into the failure domain, recording the sample point corresponding to the maximum probability density value in the failure domain

Comparison inequality beta ^*(l) ＞γβ ^(l) And if the minimum limit state value is satisfied, recording a sample point corresponding to the minimum limit state value in the safety domain

β ^(l+1) ＝β ^**(l) (ii) a If not, y ^*(l+1) ＝y ^*(l) ，β ^(l+1) ＝β ^(l) ；

If no sample point falls into the failure domain, recording the sample point corresponding to the minimum limit state value in the safety domain

β ^(l+1) ＝β ^**(l) 。

9. A high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling aims at a cantilever beam with a concentrated force; the elastic modulus E, the section moment of inertia I and the applied load force P are independent normal random variables, and the parameters are shown in a table 4; length L =5m is constant;

TABLE 4

Variables of Mean value Standard deviation of E(kN/m ² ) 2×10 ⁷ 0.5×10 ⁷ I(m ⁴ ) 10 ^-4 0.1×10 ^-4 P(kN) 8 2.5

And (3) establishing a limit state equation by considering the maximum deformation of the cantilever beam:

g＝EI-78.125P＝0

the method is characterized by comprising the following steps:

the method comprises the following steps: three random variables are provided, and distribution types and parameters are given;

normalizing the original random variable standard:

the extreme state equation after standard normalization is (5Y) ₁ +20)(Y ₂ +10)-7.8125(2.5Y ₃ +8)＝0

Step two: the iteration number l =1 and a preset check point y ^*(1) = (0,0,0), beta sphere radius beta ⁽¹⁾ ＝1；

Step three: generating sample points which are subject to truncation of the important distribution;

using MATLAB software to program, obtaining sample points which obey truncation important distribution as shown in Table 5;

TABLE 5

Step four: calculating a failure probability estimation value;

step five: calculating variance

And coefficient of variation

Step six: updating the checking point and the radius of the beta sphere;

y ^*(2) ＝(-2.1304,-0.9784,1.0031)

β ⁽²⁾ ＝2.5499

step seven: first iteration without judgment

Directly entering second iteration, wherein l =2, and turning to the third step until the precision requirement is met;

the convergence condition is achieved through 2 iterations, and the failure probability iterative computation result is shown in table 6;

TABLE 6

10. The method for calculating the high efficiency failure probability based on the adaptive double truncated important samples according to claim 9, is characterized in that: in the first step, the first step is carried out,

in the formula (I), the compound is shown in the specification,

is the average of the random variables and is,

is the standard deviation of the random variable.

11. The method for calculating the high efficiency failure probability based on the adaptive double truncated important samples according to claim 9, characterized in that: in step one, the

And substituting the original limit state function to obtain the converted limit state function.

12. A radical as claimed in claim 9The high-efficiency failure probability calculation method for the self-adaptive double-truncation important sampling is characterized by comprising the following steps of: in step two, a checking point y is determined ^*(1) And beta sphere radius beta ⁽¹⁾ ", do the following:

determination of checking points by order moment method

Calculating the radius of the beta sphere in a standard normal space

13. The method for calculating the high efficiency failure probability based on the adaptive double truncated important samples according to claim 9, characterized in that: in step three, the procedure is as follows:

firstly, taking checking points

For sampling the gravity center, important sampling is performed to obtain a group of random numbers

And screening the obtained sample points, if the sample points can meet the following requirements:

and

then the sample points are described as being subject to truncation importance distribution, and is recorded as

14. The method for calculating the high efficiency failure probability based on the adaptive double truncated important samples according to claim 9, characterized in that: in step four, the procedure is as follows:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) The total number of samples; s is the number of iterations; n is a radical of hydrogen ^(l) For each iteration to fall within the region omega ₂ The number of sample points; i (-) is an illustrative function; f (-) is a probability density function; h is a total of ^(l) (. Cndot.) represents the significant sample density function in the l-th iteration.

15. The method for calculating the high efficiency failure probability based on the adaptive double truncated important samples according to claim 9, characterized in that: in step five, the variance is calculated

And coefficient of variation

", do the following:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) The total number of samples; s is the number of iterations; n is a radical of ^(l) For each iteration to fall within the region omega ₂ The number of sample points; i (-) is an illustrative function; f (-) is a probability density function; h is a total of ^(l) (. Cndot.) represents the significant sample density function in the l-th iteration; n = N ⁽¹⁾ +N ⁽²⁾ +...+N ^(s) To fall into the region Ω ₂ The total number of sample points;

is an estimated value of the failure probability;

in the formula (I), the compound is shown in the specification,

is the variance;

is an estimated failure probability.

16. The method for calculating the high efficiency failure probability based on the adaptive double truncated important samples according to claim 9, is characterized in that: in step six, the checking point and the beta sphere radius are updated as follows:

if any sample point falls into the failure domain, recording the sample point corresponding to the maximum probability density value in the failure domain

Comparison of inequality beta ^*(l) ＞γβ ^(l) And if the minimum limit state value is satisfied, recording a sample point corresponding to the minimum limit state value in the safety domain

β ^(l+1) ＝β ^**(l) (ii) a If not, y ^*(l+1) ＝y ^*(l) ，β ^(l+1) ＝β ^(l) ；

If no sample point falls into the failure domain, recording the sample point corresponding to the minimum limit state value in the safety domain

β ^(l+1) ＝β ^**(l) 。

17. A high-efficiency failure probability calculation method based on self-adaptive double-truncation important sampling is characterized in that for a certain internal-pressure cylindrical container, 15MnV is used as a material; inner diameter D, inner pressure P, wall thickness t and yield strength σ _s Is an independent normal random variable, and the parameters are shown in a table 7;

TABLE 7

Variables of Mean value Standard deviation of σ _s (MPa) 392 31.4 P(MPa) 20 2.4 D(mm) 460 7.0 t(mm) 19 0.8

The cylindrical, thin-walled, internally-pressurized container being subjected to bidirectional, i.e. axial, stresses

Stress in circumferential direction

Radial stress S _r ≈0；

The ultimate equation of state for the internal pressure cylinder is obtained according to the first intensity theory as:

g＝σ _s -S _eq ＝0

in the formula, S _eq In order to be an equivalent stress,

the method is characterized by comprising the following steps:

the method comprises the following steps: four random variables are provided, and distribution types and parameters are given;

normalizing the original random variable standard:

the extreme state equation after standard normalization is

Step two: the iteration number l =1 and a preset checking point y ^*(1) = (0,0,0,0), beta sphere radius beta ⁽¹⁾ ＝1；

Step three: generating sample points which obey the truncated important distribution;

using MATLAB software to program, the sample points that obey the truncated significant distribution are obtained as shown in Table 8;

TABLE 8

Step four: calculating a failure probability estimation value;

step five: calculating variance

And coefficient of variation

Step six: updating the checking point and the radius of the beta sphere;

y ^*(2) ＝(-2.3146,1.5738,1.1962,-0.7278)

β ⁽²⁾ ＝3.1297

step seven: first iteration need not judge

Directly entering second iteration, wherein l =2, and turning to the third step until the precision requirement is met;

the convergence condition is achieved through 3 iterations, and the failure probability iterative computation result is shown in table 9;

TABLE 9

18. The method of claim 17 for efficient failure probability computation based on adaptive double truncated key samples, wherein: in the first step, the first step is carried out,

in the formula (I), the compound is shown in the specification,

is the mean value of the random variable and,

is the standard deviation of the random variable.

19. The method of claim 17, wherein the method comprises the following steps: in step one, the

And substituting the original limit state function to obtain the converted limit state function.

20. The method of claim 17 for efficient failure probability computation based on adaptive double truncated key samples, wherein: in step two, a checking point y is determined ^*(1) And beta sphere radius beta ⁽¹⁾ ", do the following:

determination of checking points by order moment method

Calculating the radius of the beta sphere in a standard normal space

21. The method of claim 17, wherein the method comprises the following steps: in step three, the method is as follows:

firstly, taking checking points

For sampling the center of gravity, important sampling is performed to obtain a group of random numbers

And screening the obtained sample points, if the sample points can meet the following requirements:

and

then the sample points are described as being subject to truncation importance distribution, and is recorded as

22. The method of claim 17 for efficient failure probability computation based on adaptive double truncated key samples, wherein: in step four, the procedure is as follows:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) Total number of samples; s is the number of iterations; n is a radical of ^(l) For each iteration to fall within the region omega ₂ The number of sample points; i (·) is an illustrative function; f (-) is a probability density function; h is a total of ^(l) (. Cndot.) represents the significant sampling density function in the l-th iteration.

23. An efficient failure based on adaptive double truncated significant sampling as claimed in claim 17The probability calculation method is characterized in that: in step five, the variance is calculated

And coefficient of variation

", do the following:

wherein M = M ⁽¹⁾ +M ⁽²⁾ +...+M ^(s) The total number of samples; s is the number of iterations; n is a radical of hydrogen ^(l) For each iteration to fall within the region omega ₂ The number of sample points; i (·) is an illustrative function; f (-) is a probability density function; h is ^(l) (. H) represents the significant sampling density function in the l-th iteration; n = N ⁽¹⁾ +N ⁽²⁾ +...+N ^(s) To fall into the region Ω ₂ The total number of sample points;

is an estimated value of the failure probability;

in the formula (I), the compound is shown in the specification,

is the variance;

is an estimated failure probability.

24. The method of claim 17, wherein the method comprises the following steps: in step six, updating the checking point and the beta sphere radius as follows:

if any sample point falls into the failure domain, recording the sample point corresponding to the maximum probability density value in the failure domain

Comparison of inequality beta ^*(l) ＞γβ ^(l) And if the minimum limit state value is satisfied, recording a sample point corresponding to the minimum limit state value in the safety domain

β ^(l+1) ＝β ^**(l) (ii) a If not, y ^*(l+1) ＝y ^*(l) ，β ^(l+1) ＝β ^(l) ；

If no sample point falls into the failure domain, recording the sample point corresponding to the minimum limit state value in the safety domain

β ^(l+1) ＝β ^**(l) 。

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