An efficient algorithm for calculating Profust failure probability

Xiaobo ZHANG,Zhenzhou LYU,Kaixuan FENG,Chunyan LING

School of Aeronautics,Northwestern Polytechnical University,Xi'an 710072,China

KEYWORDS Failure probability;Fuzzy-state assumption;General performance function;Kriging model;Profust reliability;Reliability

Abstract For efficiently estimating the Profust failure probability based on probability input variables and fuzzy-state assumption,a General Performance Function(GPF)expression is established under the strict mathematical derivation for the Profust reliability model.By constructing the GPF,the calculation of the Profust failure probability can be transformed into the calculation of the traditional failure probability.Then various existing methods for the traditional failure probability can be used to estimate the Profust failure probability.Due to the high efficiency of the Adaptive Kriging(AK)model and the universality of the Monte Carlo Simulation(MCS),AK inserted MCS(abbreviated as AK-MCS)has been proven to be an efficient method for estimating the failure probability.Therefore,the AK-MCS combined with the GPF(abbreviated as AK-MCS+GPF)is proposed for estimating Profust failure probability.The proposed method greatly reduces the computational cost while ensuring the accuracy.Finally,four examples are given to validate the proposed AK-MCS+GPF.The results of the examples show the rationality and the efficiency of the proposed AK-MCS+GPF.

1.Introduction

By considering random input variables,reliability analysis devotes to analyze the failure probability of the structure.1-5The traditional reliability model is based on the binary-state assumption, which shows that there is a clear boundary between the failure state and the safety state.However,the boundary between‘‘safety”and‘‘failure”may be often not clear for the gradual failure in practical engineering problems.There is a fuzzy state between the safety state and the failure state.When the structure is under fuzzy state,the relevant outputs belong to the safety sate or failure state at a certain membership degree.

The concept of fuzzy reliability was proposed and developed by several authors.1,6-9Under the probability input variables assumption, Cai et al.6-9established the Profust reliability model and the Probist reliability model respectively based on the fuzzy-state assumption and the binary-state assumption respectively.The Probist model is based on probability input variables and the binary-state assumption,while the Profust model is based on probability input variables and the fuzzy-state assumption.

This paper mainly concerns with efficient method for estimating Profust failure probability.The fuzziness of the state can be described by the membership function of the performance function to the fuzzy failure domain.Profust failure probability is defined as the integral of the Probability Density Function(PDF)of the performance function multiplied by the membership function.10,11The most common method for estimating Profust failure probability is the direct Monte Carlo Simulation(MCS).MCS is accurate,simple and easy to implement.The result of MCS can be used as the reference to verify the accuracy of the new method.However,the obvious disadvantage of the MCS is low computational efficiency,and it is unaffordable for the engineering application.Refs.10-15 proposed some methods to estimate Profust failure probability.However,these methods need to evaluate a large number of the performance functions to achieve high accuracy,and the computational efficiency is still low.In this paper,starting from the definition of Profust failure probability,a new equivalent expression for calculating Profust failure probability is obtained by the strict mathematical derivation,and the concept of the General Performance Function(GPF)is proposed.Then,the Profust failure probability is converted into the traditional failure probability by means of the GPF.The Adaptive Kriging(AK)model combined with MCS(abbreviated as AK-MCS)16was proposed by Echard,it combines the high efficiency of the AK model with the universality of the MCS method to estimate the traditional failure probability.AKMCS only needs a small number of performance function evaluations to estimate the failure probability with high precision,and it greatly improves the computational efficiency compared with direct MCS.This paper combines the AK-MCS method with the established GPF to estimate the Profust failure probability and forms AK-MCS+GPF method.

The rest of the paper is organized as follows.In Section 2,the definition of Profust failure probability is introduced first.Then a new equivalent expression of the Profust failure probability is obtained through the strict mathematical derivation,and the corresponding concept of the GPF is proposed.In Section 3,after the basic principle of the AK-MCS is briefly described,the steps of AK-MCS+GPF are given for estimating the Profust failure probability.Four examples are used to verify the rationality and high efficiency of AK-MCS+GPF method in Section 4. Finally, conclusions are drawn in Section 5.

2.Definition of Profust failure probability and its equivalent expression by GPF

2.1.Definition of Profust failure probability

In the traditional reliability analysis based on the binary-state assumption,the state of the structure is clearly classified as the safety one and the failure one.g(X)is a performance function related to the n-dimensional probability input variables denoted by.The following Eq.(1)is called the limit state equation.

g(X)=0 is the boundary between the failure state and the safety state,and it is shown in Fig.1.

The traditional failure probability Pfunder the binary-state assumption is defined by

where xk(k=1,2,...,N)are N samples generated according to fX(x);Nfis the number of samples falling into F.

The coefficient of variation Covofcan be estimated by

where Var[·]is the variance operator.

When the function of the structure is gradually degraded,the fuzziness of the state shown in Fig.2 can describe gradual failure well.In the fuzzy domain,the structure belongs neither to failure nor to safety completely,but belongs to failure or safety with a certain membership level.Denote the fuzzy failure domain asand the membership function of g( x)belonging tocan be used to describe the fuzzy failure state.

Common types of membership function of g(x)to fuzzy failure stateinclude Linear typeNormal typeand Cauchy typeThe corresponding expressions to these common memberships and their images(Fig.3)are given as follows:

Fig.1 Binary-state assumption.

Fig.2 Fuzzy-state assumption.

where a1and a2,b1and b2,and c1and c2are respectively the position and shape parameters of the Linear,Normal and Cauchy membership functions derived from the statistical data by the expert.

Under the probability input and fuzzy state assumption,the Profust failure probabilitycan be defined by2

Using the mean of the sample to estimate expectation,an estimatebased on the MCS is shown in

where xi(i=1,2,...,N)are N samples generated according to fX(x).The coefficient of variation ofcan be derived as

Since the guaranty of the law of large number,the convergent solution shown in Eq.(9)can be used as the reference to verify the accuracy of the new method.

2.2.GPF

The Profust failure probability defined by Eq.(8)can also be succinctly expressed as the integral form of the PDF fG(g)of the performance function multiplied by the membership function u~F(g).6

Fig.4 Linear-type membership function.

Taking the linear membership function as an example,when g(x)＜a1,u~F(g)=1,and when g(x)＞a2,u~F(g)=0,while a1＜g(x)＜a2,0 ≤u~F(g)≤1.It can be known from the membership function in Fig.4 that u~F(g)is monotonic when a1＜g(x)＜a2.In general,most of the membership function has the monotonicity similar to that of Fig.4 or has the segmented monotonicity.We will use the linear membership function as an example to derive the equivalent expression of the Profust failure probability in Eq.(11).By segmenting the integral domain in Eq.(11),the equivalent expression for Profust failure probability can be obtained by

Since g(x)＜a1, u~F(g)=1, and g(x)＞a2, u~F(g)=0,Eq.(12)can be further deduced into

where FG(g)is the Cumulative Distribution Function(CDF)of the performance function.

Fig.3 Three types of membership functions.

Substitute Eq.(15)into Eq.(13),and Eq.(16)is obtained.

Introduce an auxiliary random variable Xn+1following a standard normal distribution,and then CDF Φ(xn+1)of Xn+1obeys the uniform distribution within the interval[0,1].Because λ ∈[0,1],we can assume λ=Φ(xn+1)and substitute λ=Φ(xn+1)into Eq.(16).Therefore,the equivalent expression of the Profust failure probability can be obtained by

where φ(xn+1)is the PDF of the introduced auxiliary Xn+1.is the JPDF of the extended random input variable

Then the new performance functiondefining the integral domain in Eq.(17)is defined as the GPF,and it is shown as

From Eq.(17),it can be seen that by defining the GPFdue to the introduction of the auxiliary Xn+1in Eq.(18),the Profust failure probabilitycan be expressed as the traditional failure probability with respect to the GPF as shown in

Introduce the indicator functionof the clear failure domaindefined by the GPF in the extended input variable spaceandis defined as

In fact,the auxiliary variable Xn+1can also follow the nonnormal distribution,such as uniform distribution,lognormal distribution,etc.The distribution type of Xn+1does not affect the accuracy of the proposed method.This paper takes standard normal variables as an example.

The advantage of Eq.(19)is that it can not only employ MCS for estimating the Profust failure probability,but also adopt the existing efficient calculation methods of traditional failure probability such as Line Sampling(LS)method,Subset Simulation(SS)method,Important Sampling(IS)method,Directional Sampling(DS)method,AK-MCS method,etc.

3.Solution of Profust failure probability by AK-MCS based on GPF

The AK-MCS method16was proposed by Echard,and it combines the high efficiency of the AK model17,18with the universality of the MCS to calculate the traditional failure probability.In the following part,based on the principle of AK model and AK-MCS,the solution steps of the Profust failure probability by AK-MCS based on GPF are presented in detail.

3.1.U learning function

The basic theory of the Kriging model is given in Appendix.In general,the Kriging model constructed by the training set selected by a Design Of Experiment(DOE)process cannot meet the accuracy requirements,so it needs a corresponding update iteration strategy and convergence criterion to ensure that the converged Kriging model has satisfactory approximate accuracy.For different approximate goals,the strategy to update the Kriging model is different.For the Profust failure probability studied in this paper,its failure domain is defined by the GPF.The main purpose of constructing the Kriging model of the GPF is to identify the value of the indicator function of the failure domain at each sample in Eq.(21).In other words,the established Kriging model needs to identify the sign of the GPF at each sample.The exact value of the GPF at the sample is not the focus for estimatingTherefore,we choose the U learning function to iteratively select the new training point for updating the Kriging model so as to correctly identify the sign of the GPF at the sample.

The U learning function is most commonly used.It is shown in

The U learning function is a good measure of the probability that the sign of the performance function of the sample is misjudged.of the prediction point is larger(i.e.,is farther from the boundaryand σ^ykis smaller(it represents that the Kriging model is more accurate),the value of the U learning function is larger,and the probability that the sign of the performance function at the sample is misjudged is smaller.When the lower bound of the U learning function is 2,the probability of misjudging the prediction sign is Φ(-2 )=0.023.Therefore,the candidate sample with the smallest U learning function is selected as a new training point to update the Kriging model step by step,and the U learning function of all the samples greater than 2 is taken as the convergence condition.

3.2.Steps of AK-MCS+GPF method

According to the basic principles of the Kriging model and the nature of the U learning function,the steps for calculating Profust failure probability by the AK-MCS+GPF method constructed in this paper are as follows,and the flowchart is shown in Fig.5.

Step 1 Generate N-size sampleaccording to the JPDFof the input variableand store them in the sample pool S.

Step 2 Randomly select N 1 samples (Ref. 16 suggests 12 samples)from S as an initial training set,and calculate the performance function values according to the constructed GPF.

Step 3 Construct an initial Kriging model by training sets.

Step 4 Calculate the U learning function of each candidate sample point in S by the current Kriging model.

Step 5 Search a new training pointin S with the smallest U learning function,i.e.,

Step 7 Calculate Profust failure probability estimate

Fig.5 Flowchart of AK-MCS+GPF method.

From the flowchart (Fig. 5) of the AK-MCS+GPF method for estimating Profust failure probability,it can be seen that Profust failure probability can be transformed into a traditional failure probability with the clear boundaries through the derivation process of the GPF.In the AK-MCS+GPF method,by iteratively updating the Kriging model and employing the convergent Kriging model to predict the failure samples in the sample pool of the extended input variable space,the number of evaluating the GPF can be greatly reduced,and the computational cost of estimating the Profust failure probability is greatly reduced.Moreover,by using the U learning function greater than 2 as the convergence criterion,the probability that the Kriging model misjudges the failure sample in S can be controlled to be less than Φ(-2 )=0.023,so that the proposed algorithm can keep the calculation accuracy of MCS.In summary,the reason that the AK-MCS+GPF method has high accuracy and efficiency in estimating the Profust failure probability is realized by adopting the following three strategies:(1)Construction of the GPF,(2)Inserting AK model instead of the actual GPF into the MCS process to recognize failure samples from the sample pool generated by MCS,(3)Using U learning function to control the probability of misjudging the failure samples.

4.Example analysis

Four examples are employed to verify the rationality and efficiency of the proposed AK-MCS+GPF.The results based on the MCS method shown in Eq.(9)can be used as the reference to verify the accuracy of the new method.The subset simulation method is applied to estimate Profust failure probability in Ref.10,and it is marked as SS.The SS method is used as the comparison to verify the efficiency of the new method. IS+GPF denotes the Important Sampling (IS)method combined with the GPF.Ncalldenotes the number of evaluating the performance function,and Cov denotes coefficient of variation.

4.1.Example 1:Exponential performance function

Consider a performance function g(X)=exp(0.2X1+1.4)-X2-0.5,where the basic variables X1and X2are independent and obey the standard normal distribution,i.e.Xi～N(0,1)(i=1,2).According to Eq. (18),the GPF is constructed aswhere X3～N(0,1).The results of Profust failure probabilities calculated by different methods are listed in Table 1.

Example 1 is an exponential performance function.The parameters of linear,normal and Cauchy membership functions are listed respectively in Table 1.It can be seen from Table 1 that the numbers of evaluating the GPF for the IS+GPF method are 5000,5000 and 8000 respectively corresponding to the linear,normal and Cauchy membership function,and those for the SS method10are 3×104,3×104and 3×104respectively.Obviously,compared with the SS method,the IS+GPF method has higher computational efficiency under the same computational accuracy. The AK-MCS+GPF method combines the high efficiency of the AK model with the universality of the MCS.From Table 1,it can be seen that the Ncallof AK-MCS+GPF method is less than 100,so the computational efficiency of AK-MCS+GPF method is better than those of IS+GPF method and SS method.Therefore,the AK-MCS+GPF method has the highest computational efficiency under the same computational accuracy in the listed methods,and can significantly reduce the computational cost.

4.2.Example 2:Creep-fatigue failure model19

Based on the experimental data,the performance function of the nonlinear creep fatigue is as follows:

Table 2 Distribution parameters of input variables for Example 2.

From Table 3,it can also be seen that the proposed GPFbased method established in this paper is more efficient than SS method.At the same level of accuracy as the MCS,the AK-MCS+GPF method greatly improves computational efficiency. It demonstrates that AK-MCS+GPF method has the high efficiency of the AK model and the universal applicability of the MCS,and the proposed method is applicable to different distribution types of the random input.

4.3.Example 3:an automobile front axle20

In the automobile engineering,the front axle is utilized to support the weight of the front part of the vehicle.Nowadays,theI-beam structure is popular in the design of front axle due to its high bend strength and light weight.As shown in Fig.6,the dangerous location is in cross-section of the I-beam part.The maximum normal stress and shear stress are σmax=M/Wxand τmax=T/Wρ,respectively,where M,T,Wxand Wρa(bǔ)re the bending moment,torque,section factor and polar section factor,respectively.Wxand Wρcan be estimated by the following equations:

Table 1 Results offor Example 1.

Table 1 Results offor Example 1.

Table 3 Results of for Example 2.

Table 3 Results of for Example 2.

Fig.6 Automobile front axle structure.

To check the strength of front axle,the performance function in the Probist model is expressed as

where σsis the yield strength.According to the material property of the front axle,σs=580 MPa.The geometry variables of I-beam including a,B,C and h shown in Fig.6 and the load variables including M and T are mutually independent with normal distribution.The distribution parameters are listed in Table 4.Also three types of membership functions are selected.Profust failure probabilities calculated using various methods are shown in Table 5.

Table 4 Distribution parameters of input variables for Example 3.

It can be seen from Table 5 that the AK-MCS+GPF method still has high accuracy and efficiency for the engineering example.

4.4.Example 4:a headless rivet model21

In the aircraft industry,assembling the widely used sheet metal parts is usually through riveting.This example adopts a headless rivet model.In the riveting process,there are many factors affecting the quality of rivets.Squeeze stress is one of the main factors.The rivet will fail when the squeeze stress exceeds its limitation.In this paper,a simplified riveting process with headless rivet is constructed.The riveting process contains two stages21shown in Fig.7.

Table 5 Results of for Example 3.

Table 5 Results of for Example 3.

Fig.7 Riveting process.

The maximum squeeze stress for a certain riveting process can be obtained as21

where d is the rivet diameter of state A,h is the rivet length of state A,and D0is the rivet diameter of state B.t denotes the whole thickness of two sheets.K represents the strength coefficient.The height of the driven rivet head is H=2.2 mm.The strain hardening exponent of this material is nSHE=0.15.

The ultimate squeeze strength is σsq=582 MPa.When the maximum squeeze stress exceeds the ultimate squeeze strength,the failure of the rivet will happen.Thus,the following performance function can be obtained:

Table 6 Distribution parameters of input variables in Example 4.

The distribution parameters of input variables are illustrated in Table 6(Cov denotes the coefficient of variation).This example also selects three types of membership functions.The results of the different methods are shown in Table 7.

Headless rivets have a safety-failure transition state in addition to the safety state and failure state.Therefore,it is necessary to introduce fuzzy failure domain to describe this state.It can be seen from Table 7 that the AK-MCS+GPF method established in this paper is the most efficient among the four methods under the condition of ensuring accuracy.

5.Conclusions

In this paper,by introducing an auxiliary input,the GPF expression is established under the strict mathematical derivation for the Profust reliability model.Thus,the calculation of Profust failure probability is converted into the calculation of traditional failure probability.From the contents of Sections 2 and 3,it can be seen that the AK-MCS+GPF method established in this paper is mainly composed of the following three key parts:(1)derive the GPF by the introduced auxiliary input;(2)insert AK model instead of GPF into MCS process to identify failure samples;(3)use U learning function to control the probability of misjudging failure samples.The proposed method has the high efficiency of the AK model and the universality of the MCS.Four examples indicate that the AK-MCS+GPF method can achieve the same accuracy as the MCS method but with a smaller number of evaluating per-formance function.In addition,through the introduction of GPF,various methods of traditional failure probability calculation (line sampling method, subset simulation method,important sampling method,etc.)can also be used to calculate Profust failure probability.

Table 7 Results of for Example 4.

Table 7 Results of for Example 4.

Acknowledgments

This work was supported by the National Natural Science Foundation of China(Nos.NSFC 51475370 and 51775439).

Appendix.Basic principle of Kriging model17,18

where R(xi,xj)is the correlation function.Several models exist to define the correlation function.In this paper,the anisotropic Gaussian model is selected.It can be expressed as

where θ=[θ1,θ2,...,θn]Tis the required parameter vector,anddenotes the kth dimension component of the training point xi.For the vector of regression coefficients β and the Gaussian process variance σ2,their estimated values can be obtained from the training points,as shown in Eqs.(A4)and(A5).

where F is the matrix of regression model,which consists of the vectors of the regression basis function f(xi)for all training points,and R is the matrix of correlation between each pair of points.

The unknown parameter in the model is θ=[θ1,θ2,...,θn]T,which can be obtained by maximum likelihood estimation as follows:

For any unknown point(prediction point)x,the Kriging model can give its optimal unbiased estimationand prediction varianceas shown in Eqs.(A7)and(A8).

where r(x)is the vector formed by the correlation function between the prediction point and the known training points.