CN113886972A - Efficient self-adaption method based on Kriging model - Google Patents

Efficient self-adaption method based on Kriging model Download PDF

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CN113886972A
CN113886972A CN202111241391.2A CN202111241391A CN113886972A CN 113886972 A CN113886972 A CN 113886972A CN 202111241391 A CN202111241391 A CN 202111241391A CN 113886972 A CN113886972 A CN 113886972A
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sample
samples
sample set
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kriging model
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张玉刚
宋坤苓
宋笔锋
喻天翔
申林杰
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Northwestern Polytechnical University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/15—Vehicle, aircraft or watercraft design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/02—Reliability analysis or reliability optimisation; Failure analysis, e.g. worst case scenario performance, failure mode and effects analysis [FMEA]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention relates to a high-efficiency self-adaptive method based on a Kriging model, which is mainly realized by three steps: the method comprises the following steps that firstly, a certain number of samples are generated through a Latin hypercube method to construct an initial Kriging model; secondly, screening a certain number of candidate samples from the candidate samples through a current Kriging model, and determining two samples; and thirdly, updating the candidate sample and the experimental design sample, reconstructing a Kriging model, judging whether the requirements are met, if not, jumping to the second step, and if so, replacing the original model with the current Kriging model to perform reliability analysis. According to the method, the number of the sample points near the extreme state is continuously increased through self-adaptive iteration, and the added samples are determined through maximizing the minimum distance, so that the samples updated through iteration are approximately and uniformly distributed near the extreme state, the sample points are more fully utilized, and the reliability analysis result is more accurate.

Description

Efficient self-adaption method based on Kriging model
The application is a divisional application with the application number of 201810109316.2, application date of 2018, 02 and 05, and the title of 'a high-efficiency self-adaptive method for reliability analysis of lock mechanism of airplane door'.
Technical Field
The invention belongs to the field of reliability analysis and design, and particularly relates to a high-efficiency self-adaptive method based on a Kriging model.
Background
The mechanical problems in engineering often involve more and more complex calculations, the evaluation of failure probability may require very time-consuming calculations, and how to minimize the number of calculations for calling a mechanical structure or mechanism numerical model while ensuring certain result accuracy becomes an important problem to be solved urgently. At present, a common method for solving the problem is to use a surrogate model to replace an original engineering model with a large calculation amount to evaluate the failure probability of the model, and the common reliability evaluation methods based on the surrogate model include a response surface method, a neural network method, a support vector machine method, a Kriging method and the like. Because the Kriging model not only has local and global statistical properties, but also requires a small sample size for constructing the model, the application of the Kriging model in the field of reliability analysis and design is more and more extensive.
From the beginning of the 20 th century, the mathematical idea of Kriging appeared, the idea was applied to practical work only by a geologist D.G.Krigie in France in the 50 th century, and then the Kriging technology is applied in many fields, and the Kriging technology is used as a semi-parameterized interpolation technology and is well developed and optimized in decades. The basic principle of the Kriging technology is that the prediction of Kriging at a certain point requires the information of known points around the certain point to be used for estimating the unknown information by means of linear combination of information weighting in a certain range of the point.
The method for evaluating the reliability problem by applying the Kriging model has the characteristics of simple calculation and strong universality, and is suitable for calculating the complicated and high-dimensional reliability analysis and reliability optimization problems. However, the Kriging model is applied to reliability analysis, and generally, measures are adopted that a series of representative sample points are constructed by an experimental design method, and then the Kriging model is constructed to replace an original implicit and complex analysis model to perform reliability analysis. However, the Kriging model constructed by this method has the following two defects: 1) unreasonable points may exist in the determination of point taking modes or point taking quantity in experimental design, so that the accuracy and precision of the constructed Kriging model are insufficient; 2) because the position of the extreme state of the analyzed model in the design space cannot be determined, the position of the point is generally global during experimental design, and the constructed Kriging model can meet the global fitting accuracy but cannot meet the fitting accuracy near the extreme state of the structure, so that the reliability analysis result has larger deviation. Therefore, a reliability analysis method which is not sensitive to the point taking mode or the number of the points in the experimental design, can improve the local fitting precision near the limit state and is easy to apply is needed in engineering reliability analysis and optimization.
Disclosure of Invention
The invention provides a set of feasible and effective reliability analysis method aiming at the reliability analysis and optimization problem, in particular to the reliability analysis and design link in the research and development process. The method aims to overcome the phenomenon that the accuracy of the Kriging model is poor due to the fact that the existing reliability analysis method based on the Kriging model unreasonably selects sample points in the experimental design, and the distribution quantity and the uniform distribution performance of the experimental sample points in the vicinity of the limit state are improved, so that a more accurate reliability analysis result is obtained.
To sum up, the invention provides a high-efficiency self-adaptive reliability analysis method based on a Kriging model, which is mainly realized by three steps: the method comprises the following steps that firstly, a certain number of samples are generated through a Latin hypercube method to construct an initial Kriging model; secondly, screening a certain number of candidate samples from the candidate samples through a current Kriging model, and then determining two samples through a maximum and minimum distance criterion; and thirdly, updating the candidate sample and the experimental design sample, reconstructing a Kriging model, judging whether the requirements are met, if not, jumping to the second step, and if so, replacing the original model with the current Kriging model to perform reliability analysis.
Specifically, the invention of the present invention comprises the following detailed steps:
1) determining design variables and extreme state functions: determining the design variable x ═ x (x) for the problem being treated1,x2,…,xn) The method comprises the steps of establishing a limit state function G (x) by using a functional characteristic quantity H and a failure criterion I, wherein n represents the number of design variables.
2) Determining a design space: determining the upper and lower limits L of each design variable according to the distribution type and design requirement of the design variablesiAnd Ui(i 1, 2.. times.n), i.e., determining the design space, the upper and lower limits for normal design variables can be determined according to the "3 σ principle", i.e., L, in generali＝μi-3σiAnd Ui＝μi+3σiIn which μiIs the mean value, σ, of the variableiIs the standard deviation of the variables.
3) Generating a candidate sample set and a test sample set: respectively generating candidate sample sets X containing T uniform samples by utilizing uniform sampling in a design spaceCAnd test sample set XTSample number T is suggested
Figure BDA0003319664380000031
Wherein the symbol [ ·]"is the ceiling operator.
4) Create initial DOE and construct Kriging model: generation of N using Latin hypercube method0The samples form an experimental design sample setXDIn which N is0Calling the extreme state function G (x) to calculate N0Function values of samples, which constitute a sample point set { (X, G (X)) | X ∈ XDI.e. the initial DOE. The number of times z of initializing the structural model is 1, and the Kriging model is constructed by utilizing the sample point set
Figure BDA0003319664380000032
5) Constructing a verification sample set and calculating a classification index of the samples: from test sample set XTScreening out the nearest Kriging model
Figure BDA0003319664380000033
Front T of extreme state0Each sample constitutes a verification sample set XVI.e. by
Figure BDA0003319664380000034
Front T nearest 00A sample, wherein T0＝[T/100]. Will verify the sample set XVAccording to which the sample
Figure BDA0003319664380000035
The positive and negative conditions of the function value are divided into +1 and-1, and the classification index is given according to the formula (1)
Figure BDA0003319664380000036
And (7) assigning values.
Figure BDA0003319664380000037
6) Screening the first set of candidate samples and determining the first sample: using the current Kriging model
Figure BDA0003319664380000038
Analyzing a current candidate sample set XCScreening out the top T closest to the limit state0Using the samples as a first candidate sample set XFC. Calculating X by equations (2) to (4)FCMiddle sample and current experimental design sampleCollection XDMaximum minimum distance L of middle sample1max-minThen X is determined using equation (5)FCEqual to "maximum minimum distance" L1max-minAll of the samples of (1). In the methods presented herein, a first k is selected that is not 01(i) Value of corresponding XFCIs the first sample x1Join to current XDIn (1).
Figure BDA0003319664380000041
Figure BDA0003319664380000042
Figure BDA0003319664380000043
Figure BDA0003319664380000044
7) Constructing a second set of candidate samples and determining a second sample: in the first candidate sample set XFCSelect all and samples x1Constructing a second candidate sample set X by corresponding samples with different signs of function value signsSCI.e. by
Figure BDA0003319664380000045
Figure BDA0003319664380000046
Calculating X by equations (6) to (8)SCMiddle sample and current experimental design sample set XDMaximum minimum distance L of middle sample2max-minThen X is determined using equation (9)SCEqual to "maximum minimum distance" L2max-minAll of the samples of (1). In the methods presented herein, a first k is selected that is not 02(i) Value of corresponding XSCIs the second sample x2Join to current XDIn (1).
Figure BDA0003319664380000047
Figure BDA0003319664380000048
Figure BDA0003319664380000049
Figure BDA00033196643800000410
8) Update DOE and remove corresponding samples: calling limit state function G (x) to calculate sample x1And x2Then update the sample point set { (X, G (X)) | X ∈ XDI.e. update the DOE. Then, the candidate sample set XCNeutral and x1And x2The corresponding sample is deleted.
9) Reconstructing a Kriging model and calculating the number of misclassifications: and (3) enabling the model construction time z to be z +1, then reconstructing the Kriging model by using the current DOE sample point set, and calculating the verification sample set X by using the reconstructed Kriging modelVSample function values in
Figure BDA0003319664380000051
Dividing into +1 and-1 according to the positive and negative conditions of function value, and giving classification index according to formula (1)
Figure BDA0003319664380000052
Assigning value, and calculating misclassification quantity by formula (10)
Figure BDA0003319664380000053
Namely, the Kriging model at the z-th time is opposite to the Kriging model at the z-1 th time for verifying the sample set XVThe misclassification quantity of (a), misclassification index used for reflecting the two-time modelThe degree of difference in the failure boundaries.
Figure BDA0003319664380000054
10) Judging the output stability of the model: calculating the stability index of the Kriging model at the z-th time through the formula (11)
Figure BDA0003319664380000055
It is used for representing the quantity of misclassifications of the Kriging model of the z-th time
Figure BDA0003319664380000056
Whether or not the allowable value N is satisfiedmis0In general, Nmis0＝[T0×5％]. Model convergence stability index to reduce the effects of contingencies
Figure BDA0003319664380000057
The model misclassification quantity is defined as the condition that the number of the model misclassifications of two continuous times does not exceed the allowable value, and is obtained by calculation of a formula (12). If it is not
Figure BDA0003319664380000058
Considering that the constructed Kriging model is stable, the algorithm immediately proceeds to step 11), otherwise, the algorithm proceeds back to step 5).
Figure BDA0003319664380000059
Figure BDA00033196643800000510
11) Generate Monte Carlo sample set and evaluate failure probability: generation of N by Monte Carlo random samplingmcsThe samples constitute a Monte Carlo sample set XMCSNumber of samples NmcsThe value of (b) is not specifically required, 10000 is taken as an initial value, and then the function value is calculated by using the Kriging model constructed at the last time, and the formula (13) and the formula (c) are used14) Assessing failure probability
Figure BDA00033196643800000511
Figure BDA00033196643800000512
Figure BDA0003319664380000061
12) Judging whether the failure probability estimation value is stable: the variation coefficient cov of failure probability is calculated by the formula (15) and is used for judging the Monte Carlo sample number NmcsWhether the estimated probability of failure is sufficient. If cov is less than its allowable value Δ cov (typically 5%), the calculated probability of failure
Figure BDA0003319664380000062
The final result is obtained, and the algorithm is ended; otherwise, let Nmcs＝10×NmcsAnd then, the step 11) is carried out.
Figure BDA0003319664380000063
The method has the advantages that the number of initially selected sample points is small, the number of sample points near the extreme state is continuously increased through self-adaptive iteration, the added samples are determined through the maximized minimum distance, the samples updated through iteration are approximately and uniformly distributed near the extreme state, the sample points are more fully utilized, and under the condition of the same number of experimental samples, the obtained Kriging model is more accurately classified, so that the reliability analysis result is more accurate; the method is easy to program, simple and feasible, and is suitable for the field of engineering reliability analysis and optimization design with huge calculation amount, such as the reliability optimization design of complex multi-body dynamic mechanical mechanisms and the multidisciplinary reliability analysis and optimization design of complex engineering systems of aircrafts, automobiles, ships and the like.
Drawings
FIG. 1 is a flow chart of an efficient adaptive method for mechanism reliability analysis according to the present invention FIG. 2 is a schematic diagram of a lock mechanism composition
FIG. 3 is a force diagram of the lock mechanism
1-a lock body; 2-a piston; 3-a rocker arm; 4-a piston rod; 5-a latch hook connecting rod; 6-latch hook
Detailed Description
The embodiments are described in detail below with reference to the accompanying drawings. Taking the problem of reliability of the unlocking function of a lock mechanism of a cabin door of an airplane as an example, the proposed self-adaption method and the Kriging method based on direct Latin hypercube experimental design are compared and researched. In the calculation example, the reliability results obtained by the Kriging model and the original model through the same Monte Carlo sample calculation are compared to verify the practicability and the efficiency of the Kriging model constructed by the method in reliability estimation.
The invention provides a high-efficiency self-adaptive reliability analysis method based on a Kriging model, which is implemented by combining a flow chart shown in figure 1 and comprising the following specific implementation steps:
1) determining design variables and extreme state functions: determining the design variable x ═ x (x) for the problem being treated1,x2,…,xn) The method comprises the steps of establishing a limit state function G (x) by using a functional characteristic quantity H and a failure criterion I, wherein n represents the number of design variables.
2) Determining a design space: determining the upper and lower limits L of each design variable according to the distribution type and design requirement of the design variablesiAnd Ui(i 1, 2.. times.n), i.e., determining the design space, the upper and lower limits for normal design variables can be determined according to the "3 σ principle", i.e., L, in generali＝μi-3σiAnd Ui＝μi+3σiIn which μiIs the mean value, σ, of the variableiIs the standard deviation of the variables.
3) Generating a candidate sample set and a test sample set: respectively generating candidate sample sets X containing T uniform samples by utilizing uniform sampling in a design spaceCAnd test sample set XTNumber of samplesT advice fetch
Figure BDA0003319664380000071
Wherein the symbol [ ·]"is the ceiling operator.
4) Create initial DOE and construct Kriging model: generation of N using Latin hypercube method0Each sample constitutes an experimental design sample set XDIn which N is0Calling the extreme state function G (x) to calculate N0Function values of samples, which constitute a sample point set { (X, G (X)) | X ∈ XDI.e. the initial DOE. The number of times z of initializing the structural model is 1, and the Kriging model is constructed by utilizing the sample point set
Figure BDA0003319664380000072
5) Constructing a verification sample set and calculating a classification index of the samples: from test sample set XTScreening out the nearest Kriging model
Figure BDA0003319664380000073
Front T of extreme state0Each sample constitutes a verification sample set XVI.e. by
Figure BDA0003319664380000074
Front T nearest 00A sample, wherein T0＝[T/100]. Will verify the sample set XVAccording to which the sample
Figure BDA0003319664380000075
The positive and negative conditions of the function value are divided into +1 and-1, and the classification index is given according to the formula (1)
Figure BDA0003319664380000076
And (7) assigning values.
Figure BDA0003319664380000081
6) Screening the first set of candidate samples and determining the first sample: using the current Kriging model
Figure BDA0003319664380000082
Analyzing a current candidate sample set XCScreening out the top T closest to the limit state0Using the samples as a first candidate sample set XFC. Calculating X by equations (2) to (4)FCMiddle sample and current experimental design sample set XDMaximum minimum distance L of middle sample1max-minThen X is determined using equation (5)FCEqual to "maximum minimum distance" L1max-minAll of the samples of (1). In the methods presented herein, a first k is selected that is not 01(i) Value of corresponding XFCIs the first sample x1Join to current XDIn (1).
Figure BDA0003319664380000083
Figure BDA0003319664380000084
Figure BDA0003319664380000085
Figure BDA0003319664380000086
7) Constructing a second set of candidate samples and determining a second sample: in the first candidate sample set XFCSelect all and samples x1Constructing a second candidate sample set X by corresponding samples with different signs of function value signsSCI.e. by
Figure BDA0003319664380000087
Figure BDA0003319664380000088
Calculating X by equations (6) to (8)SCMiddle sample and current experimentDesign sample set XDMaximum minimum distance L of middle sample2max-minThen X is determined using equation (9)SCEqual to "maximum minimum distance" L2max-minAll of the samples of (1). In the methods presented herein, a first k is selected that is not 02(i) Value of corresponding XSCIs the second sample x2Join to current XDIn (1).
Figure BDA0003319664380000089
Figure BDA00033196643800000810
Figure BDA00033196643800000811
Figure BDA0003319664380000091
8) Update DOE and remove corresponding samples: calling limit state function G (x) to calculate sample x1And x2Then update the sample point set { (X, G (X)) | X ∈ XDI.e. update the DOE. Then, the candidate sample set XCNeutral and x1And x2The corresponding sample is deleted.
9) Reconstructing a Kriging model and calculating the number of misclassifications: and (3) enabling the model construction time z to be z +1, then reconstructing the Kriging model by using the current DOE sample point set, and calculating the verification sample set X by using the reconstructed Kriging modelVSample function values in
Figure BDA0003319664380000092
Dividing into +1 and-1 according to the positive and negative conditions of function value, and giving classification index according to formula (1)
Figure BDA0003319664380000093
Assigning value, and calculating misclassification quantity by formula (10)
Figure BDA0003319664380000094
Namely, the Kriging model at the z-th time is opposite to the Kriging model at the z-1 th time for verifying the sample set XVThe misclassification index is used for reflecting the difference degree of the failure boundaries of the two models.
Figure BDA0003319664380000095
10) Judging the output stability of the model: calculating the stability index of the Kriging model at the z-th time through the formula (11)
Figure BDA0003319664380000096
It is used for representing the quantity of misclassifications of the Kriging model of the z-th time
Figure BDA0003319664380000097
Whether or not the allowable value N is satisfiedmis0In general, Nmis0＝[T0×5％]. Model convergence stability index to reduce the effects of contingencies
Figure BDA0003319664380000098
The model misclassification quantity is defined as the condition that the number of the model misclassifications of two continuous times does not exceed the allowable value, and is obtained by calculation of a formula (12). If it is not
Figure BDA0003319664380000099
Considering that the constructed Kriging model is stable, the algorithm immediately proceeds to step 11), otherwise, the algorithm proceeds back to step 5).
Figure BDA00033196643800000910
Figure BDA00033196643800000911
11) Generate Monte Carlo sample set and evaluate failure probability: generation of N by Monte Carlo random samplingmcsThe samples constitute a Monte Carlo sample set XMCSNumber of samples NmcsThe value of (A) is not particularly required, 10000 is taken as an initial value, then the function value is calculated by using the Kriging model constructed at the last time, and the failure probability is evaluated by the formula (13) and the formula (14)
Figure BDA0003319664380000101
Figure BDA0003319664380000102
Figure BDA0003319664380000103
12) Judging whether the failure probability estimation value is stable: the variation coefficient cov of failure probability is calculated by the formula (15) and is used for judging the Monte Carlo sample number NmcsWhether the estimated probability of failure is sufficient. If cov is less than its allowable value Δ cov (typically 5%), the calculated probability of failure
Figure BDA0003319664380000104
The final result is obtained, and the algorithm is ended; otherwise, let Nmcs＝10×NmcsAnd then, the step 11) is carried out.
Figure BDA0003319664380000105
Note: in the proposed method, unspecified parameters are set as follows: initial misclassification number
Figure BDA0003319664380000106
Figure BDA0003319664380000107
Initial stability index
Figure BDA0003319664380000108
Initial model convergence stability index
Figure BDA0003319664380000109
As shown in fig. 2 and fig. 3, the lock mechanism mainly includes 6 components, namely, a lock body 1, a piston 2, a rocker arm 3, a piston link 4, a latch hook link 5, and a latch hook 6. All hinges in the lock mechanism are revolute pairs (total 6, R)0，R1，R2，R3，R4And R5) The rocker arm is connected with the latch hook through a spring. In the unlocking process, due to the influence of some internal factors in the lock mechanism, the resistance is possibly overlarge, the lock mechanism cannot be smoothly unlocked under the condition of the existing maximum driving force, the unlocking function of the lock mechanism is disabled, and then the unlocking function of the lock mechanism is not provided with the failure domain D with insufficient driving forceFIs composed of
DF＝{x|F＞[F]}
Where F is the actual driving force required by the piston, which is assumed to vary linearly with time, and returns to a smaller value after the latch hook is opened, [ F ] is the maximum driving force that can be provided by the piston, [ F ] ═ 900N. The mechanical parameters which have influence on the unlocking function of the lock mechanism and are obtained by carrying out dynamic analysis on the lock mechanism comprise: the damping coefficient of the piston motion, the maximum contact pressure between the locking hook and the locking ring, the friction coefficient between the locking hook and the locking ring, the unhooking angle between the locking hook and the locking ring, the rigidity coefficient of the spring and the maximum contact angle between the locking hook and the locking ring in the opening process of the locking mechanism. Assuming that all input variables obey an independent normal distribution, table 1 lists their distribution parameters.
TABLE 1 random variable distribution types and parameters for lock mechanisms
Figure BDA0003319664380000111
The reliability analysis implementation steps of the method are as follows:
1) determining design variables and extreme state functions: the design variable is x ═ x1 x2 x3 x4 x5 x6]Each component of the random variable is subject to independent normal distribution, and the mean value is [ 500075000.2549700058 ]]Standard deviation σ ═ 2001000.02512000.5]。
Lab simulation analysis software is used for establishing a dynamic model of the lock mechanism, and a calculation result F (x) of driving force required by unlocking is obtained, so that a limit state function of
G(x)＝[F]-F(x)
2) Determining a design space: the upper and lower limits L ═ 440072000.17546640056.5, U ═ 560078000.32552760059.5, for each design variable were determined according to the "3 σ rule".
3) Generating a candidate sample set and a test sample set: respectively generating a design space by using uniform sampling
Figure BDA0003319664380000112
Candidate sample set X of individual samplesCAnd test sample set XT。
4) Create initial DOE and construct Kriging model: generation of N using Latin hypercube method0The experiment design sample set X is composed of 3 × 6 and 18 samplesDCalling a limit state function G (X) to calculate function values of the 18 samples to form a sample point set { (X, G (X)) | X ∈ XDAs shown in table 2, i.e. the initial DOE. The number of times z of initializing the structural model is 1, and the Kriging model is constructed by utilizing the sample point set
Figure BDA0003319664380000113
TABLE 2 initial DOE
Figure BDA0003319664380000121
5) Constructing a verification sample set and calculating a classification index of the samples: from test sample set XTScreening out the nearest Kriging model
Figure BDA0003319664380000122
Front T of extreme state0＝[T/100]245 samples constitute a verification sample set XVI.e. by
Figure BDA0003319664380000123
The first 245 samples closest to 0. Will verify the sample set XVAccording to which the sample
Figure BDA0003319664380000124
The positive and negative conditions of the function value are divided into +1 and-1, and the classification index is given according to the formula (1)
Figure BDA0003319664380000125
The assignment, table 3 gives the calculation results when z is 1.
Table 3 validation sample set X when z is 1VCalculation results
Figure BDA0003319664380000126
6) Screening the first set of candidate samples and determining the first sample: using the current Kriging model
Figure BDA0003319664380000127
Analyzing a current candidate sample set XCScreening out the first 245 samples closest to the limit state as a first candidate sample set XFC. Calculating X by equations (2) to (4)FCMiddle sample and current experimental design sample set XDMaximum minimum distance L of middle sample1max-minThen X is determined using equation (5)FCEqual to "maximum minimum distance" L1max-minAll of the samples of (1). Selecting the first k not to be 01(i) Value of corresponding XFCIs the first sample x1Join to current XDIn (1). Table 4 shows the first candidate sample set X when z is 1FCThe result is calculated, and the first sample is
Figure BDA0003319664380000131
Table 4 first candidate sample set X when z is 1FCCalculation results
Figure BDA0003319664380000132
7) Constructing a second set of candidate samples and determining a second sample: in the first candidate sample set XFCSelect all and samples x1Constructing a second candidate sample set X by corresponding samples with different signs of function value signsSCI.e. by
Figure BDA0003319664380000133
Figure BDA0003319664380000134
Calculating X by equations (6) to (8)SCMiddle sample and current experimental design sample set XDMaximum minimum distance L of middle sample2max-minThen X is determined using equation (9)SCEqual to "maximum minimum distance" L2max-minAll of the samples of (1). Selecting the first k not to be 02(i) Value of corresponding XSCIs the second sample x2Join to current XDIn (1). Table 5 gives the second candidate sample set X when z is 1SCThe result is calculated, and the first sample is
Figure BDA0003319664380000135
Second candidate sample set X when Table 5z is 1SCCalculation results
Figure BDA0003319664380000136
Figure BDA0003319664380000141
8) Update DOE and remove corresponding samples: calling limit state function G (x) to calculate sample x1And x2Then update the sample point set { (X, G (X)) | X ∈ XDI.e. update the DOE. Then, the candidate sample set XCNeutral and x1And x2The corresponding sample is deleted. Table 6 shows the updated DOE when z is 1.
Table 6 DOE updated when z is 1
Figure BDA0003319664380000142
9) Reconstructing a Kriging model and calculating the number of misclassifications: and (3) enabling the model construction time z to be z +1, then reconstructing the Kriging model by using the current DOE sample point set, and calculating the verification sample set X by using the reconstructed Kriging modelVSample function values in
Figure BDA0003319664380000143
Dividing into +1 and-1 according to the positive and negative conditions of function value, and giving classification index according to formula (1)
Figure BDA0003319664380000144
Assigning value, and calculating misclassification quantity by formula (10)
Figure BDA0003319664380000145
10) Judging the output stability of the model: calculating the stability index of the Kriging model at the z-th time through the formula (11)
Figure BDA0003319664380000146
N in formula (11)mis0＝[T0×5％]13. Calculating the model convergence stability index by equation (12)
Figure BDA0003319664380000147
If it is not
Figure BDA0003319664380000148
Considering that the constructed Kriging model is stable, the algorithm immediately proceeds to step 11), otherwise, the algorithm proceeds back to step 5).
11) Generate Monte Carlo sample set and evaluate failure probability: generation of N by Monte Carlo random samplingmcs(10000 samples are taken as initial values) to form a Monte Carlo sample set XMCSThen, the function value of the Kriging model constructed in the last time is calculated, and the failure probability is evaluated through the formula (13) and the formula (14)
Figure BDA0003319664380000151
12) Judging whether the failure probability estimation value is stable: the coefficient of variation cov of the failure probability is calculated by equation (15), and if cov is less than 5% of its allowable value, the calculated failure probability
Figure BDA0003319664380000152
The final result is obtained, and the algorithm is ended; otherwise, let Nmcs＝10×NmcsAnd then, the step 11) is carried out.
The proposed method yields a stable Kriging model at z 90, in NmcsThe failure probability is obtained when 100000
Figure BDA0003319664380000153
The final result of (1) was 1.031 × 10-2. The number of times of the common calling of the extreme state function G (x) in the calculation process is N0+2 × (z-1) ═ 196, and the final DOE is given in table 7.
Final DOE for the procedure set forth in Table 7
Figure BDA0003319664380000154
To demonstrate the practicality and efficiency of the proposed method, the reliability analysis results and the calculation results of the relative error of reliability of Monte Carlo, Kriging model constructed by direct experimental design (Kriging1) and Kriging model constructed by the proposed method (Kriging2) are listed in Table 8.
TABLE 8 comparison of calculation results
Figure BDA0003319664380000155
Figure BDA0003319664380000161
According to the calculation results in table 8, the relative errors of the calculated reliability of the Kriging model constructed by the direct Latin hypercube experimental design are large, while the relative errors of the Kriging model constructed by the method provided by the invention are obviously low. It is worth noting that the direct latin hypercube design method has a reliability analysis of the constructed Kriging model with a relative error of 2.33% with respect to the Monte Carlo analysis even when 1000 samples are used, whereas the proposed method has a reliability analysis of the Kriging model constructed with only 196 samples (18 initial samples and 196 required for 89 iteration cycles) with a relative error of 0.19% with respect to the Monte Carlo analysis. According to the corresponding result of Kriging1 in Table 8, it can be found that the method for constructing the Kriging model through direct experimental design has certain blindness and instability. Thus, this example fully demonstrates the practicality and efficiency of the proposed method.
The above embodiments are only preferred embodiments of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are also within the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (8)

1. The efficient adaptive method based on the Kriging model is characterized by comprising the following steps of:
step 1, determining a design variable x and a limit state function G (x);
step 2, determining a design space;
step 3, generating a candidate sample set XCAnd test sample set XT；
Step 4, establishing an initial DOE and constructing a Kriging model;
step 5, constructing a verification sample set and calculating classification indexes of samples in the verification sample set;
step 6, screening the first candidate sample set XFCAnd determines the first sample x1；
Step 7, constructing a second candidate sample set XSCAnd determining a second sample x2；
Step 8, updating the DOE and removing the corresponding sample;
step 9, reconstructing a Kriging model and calculating the number of misclassifications;
step 10, judging the output stability of the model, if so, executing step 11, otherwise, returning to execute step 5;
step 11, generating a Monte Carlo sample set and evaluating failure probability;
step 12, judging whether the failure probability estimated value is stable, if so, ending, otherwise, resetting the number of samples in the Monte Carlo sample set, and returning to execute the step 11;
the step 2 specifically comprises: determining the lower limit L of each design variable according to the distribution type and the design requirement of the design variablesiAnd upper limit UiWherein i is 1, 2.. times, n, and then determining a design space; l isi＝μi-3σiAnd Ui＝μi+3σiIn which μiIs the mean value, σ, of the variableiIs the standard deviation of the variables;
the step 5 specifically includes: from test sample set XTScreening out the nearest Kriging model
Figure FDA0003319664370000011
Front T of extreme state0Each sample constitutes a verification sample set XVWherein T is0＝[T/100]Will verify the sample set XVAccording to which the sample
Figure FDA0003319664370000012
The positive and negative conditions of the function value are divided into +1 and-1, and the classification index is given according to the formula (1)
Figure FDA0003319664370000013
Assigning;
Figure FDA0003319664370000021
2. the method according to claim 1, wherein step 1 specifically comprises: determining the design variable x ═ x (x) for the problem being treated1,x2,…,xn) Establishing a limit state function G (x) by the functional characteristic quantity H and the failure criterion I, wherein n is the number of design variables.
3. The method according to claim 1, wherein step 3 specifically comprises: respectively generating candidate sample sets X containing T uniform samples by utilizing uniform sampling in a design spaceCAnd test sample set XT。
4. The method according to claim 1, wherein the step 4 specifically comprises: generating N by adopting Latin hypercube method0Each sample constitutes an experimental design sample set XDIn which N is0Calling a limit state function G (x) to calculate said N0Function values of samples, which constitute a sample point set { (X, G (X)) | X ∈ XDObtaining an initial DOE, initializing the model construction times z to be 1, and constructing a Kriging model by using the sample point set
Figure FDA0003319664370000022
5. The method according to claim 1, wherein the step 6 specifically comprises: using the current Kriging model
Figure FDA0003319664370000023
Analyzing a current candidate sample set XCScreening out the top T closest to the limit state0Using the samples as a first candidate sample set XFC；
Calculating X by equations (2) to (4)FCMiddle sample and current experimental design sample set XDMaximum minimum distance L of middle sample1max-minThen X is determined using equation (5)FCEqual to "maximum minimum distance" L1max-minSelecting the first k not to be 01(i) Value of corresponding XFCIs the first sample x1Join to current XDPerforming the following steps;
Figure FDA0003319664370000024
Figure FDA0003319664370000025
Figure FDA0003319664370000026
Figure FDA0003319664370000027
6. the method according to claim 1, wherein the step 7 specifically comprises: in the first candidate sample set XFCSelect all and samples x1Constructing a second candidate sample set X by corresponding samples with different signs of function value signsSCWherein
Figure FDA0003319664370000031
Calculating X by equations (6) to (8)SCZhongxiao sample andpre-experimental design sample set XDMaximum minimum distance L of middle sample2max-minThen X is determined using equation (9)SCEqual to "maximum minimum distance" L2max-minSelecting the first k not to be 02(i) Value of corresponding XSCIs the second sample x2Join to current XDPerforming the following steps;
Figure FDA0003319664370000032
Figure FDA0003319664370000033
Figure FDA0003319664370000034
Figure FDA0003319664370000035
7. the method according to claim 1, wherein the step 8 specifically comprises: calling limit state function G (x) to calculate sample x1And x2Then update the sample point set { (X, G (X)) | X ∈ XDAnd updating the DOE.
8. The method according to claim 1, wherein the step 9 specifically comprises: and (3) enabling the model construction time z to be z +1, then reconstructing the Kriging model by using the current DOE sample point set, and calculating the verification sample set X by using the reconstructed Kriging modelVSample function values in
Figure FDA0003319664370000036
According to the positive and negative conditions of function valueThe conditions are classified into "+ 1" and "-1", and are indexed according to the formula (1)
Figure FDA0003319664370000037
Assigning value, and calculating misclassification quantity by formula (10)
Figure FDA0003319664370000038
Figure FDA0003319664370000039
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