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Reliability Engineering and System Safety 157 (2017) 152–165
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Reliability Engineering and System Safety
http://d0951-83
n Corr
journal homepage: www.elsevier.com/locate/ress
LIF: A new Kriging based learning function and its application tostructural reliability analysis
Zhili Sun a, Jian Wang a,n, Rui Li a, Cao Tong b
a School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, Chinab Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China
a r t i c l e i n f o
Article history:Received 31 March 2016Received in revised form1 August 2016Accepted 10 September 2016Available online 13 September 2016
Keywords:Structural reliabilityKriging meta-modelLearning functionDesign of experimentLeast improvement function
x.doi.org/10.1016/j.ress.2016.09.00320/& 2016 Elsevier Ltd. All rights reserved.
esponding author.
a b s t r a c t
The main task of structural reliability analysis is to estimate failure probability of a studied structuretaking randomness of input variables into account. To consider structural behavior practically, numericalmodels become more and more complicated and time-consuming, which increases the difficulty of re-liability analysis. Therefore, sequential strategies of design of experiment (DoE) are raised. In this re-search, a new learning function, named least improvement function (LIF), is proposed to update DoE ofKriging based reliability analysis method. LIF values how much the accuracy of estimated failure prob-ability will be improved if adding a given point into DoE. It takes both statistical information provided bythe Kriging model and the joint probability density function of input variables into account, which is themost important difference from the existing learning functions. Maximum point of LIF is approximatelydetermined with Markov Chain Monte Carlo(MCMC) simulation. A new reliability analysis method isdeveloped based on the Kriging model, in which LIF, MCMC and Monte Carlo(MC) simulation are em-ployed. Three examples are analyzed. Results show that LIF and the new method proposed in this re-search are very efficient when dealing with nonlinear performance function, small probability, compli-cated limit state and engineering problems with high dimension.
& 2016 Elsevier Ltd. All rights reserved.
1. Introduction
For a given mechanical structure with M-dimension input
random variable ⎡⎣ ⎤⎦( )= …X X X X, , M1 2T, its performance function
( ( ))xG divides the variable space into two domains, i.e. the safedomain ( ( ) > )xG 0 and the failure domain ( ( ) ≤ )xG 0 . The boundaryof them is called the limit state ( ( ) = )xG 0 . The failure probability ofthe structure is defined as
∫= ( ) ( ) ( )≤ x x xP I f d 1f G 0
where ( )xf is the joint probability density function of X and( )≤ xIG 0 is a failure indicator function. The fundamental task of
structural reliability analysis is to perform the integration ap-proximately. In engineering, the performance function of a givenmechanical structure is nonlinear and almost impossible to get itsexplicit formulation generally. A numerical method is usually usedto obtain the structural output. Therefore, several structural re-liability analysis methods are developed to estimate the prob-ability of failure.
Monte Carlo simulation (MCS) [1], estimating Pf with failure
rate of random samples, is the most robust method. However, itneeds to call the performance function (or numerical model) toomany times. Some variance reduction techniques, e.g. line sam-pling (LS) [2,3], subset simulation (SS) [4–7], and importancesampling (IS) [6,8–10], are proposed to reduce the number of callsto the performance function and relieve the pressure of compu-tation. At the same time, to consider structural behavior practicallyand improve their accuracy, numerical models are becoming moreand more complicated, which increases the difficulty of reliabilityanalysis. These variance reduction techniques cannot satisfy thecomputational requirements either. Compared with MCS and itsimproved methods, the first and second order reliability method(FORM and SORM) perform structural reliability analysis withmuch fewer samples. They are based on the knowledge of designpoint and Taylor expansion that ignores higher order terms,therefore their accuracy can hardly be guaranteed especially whendealing with complex nonlinear structures.
Within recent years, different kinds of meta-models [3,11] areproposed to replace the target performance function. Their em-phasis is to fit the limit state well for only the sign of a predictedstructural response influences the estimate of Pf directly. Amongmeta-models, polynomial response surface [11–17] is the mostwidely used one in engineering. The number of terms for a poly-nomial response surface method grows dramatically as polynomial
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165 153
order and the dimension of input variable increase especiallywhen cross terms are taken into account. Sparse polynomial[15,18,19] is adopted to overcome this situation. Kang et al. [20],Roussouly et al. [15], Blatman and Sudret [18] raise methods fordetermining significant terms of a sparse polynomial, and applytheir methods to structural reliability analysis. These methodsobtain a compromise between complexity and nonlinearity ofmeta-models. Compared with polynomial models, machinelearning methods (neural network [21,22], support vector ma-chine (SVM) [23–25], Multi-Layer Perceptrons [26] and so on) aremore suitable to matching performance function with highlynonlinear input-output relationships. Kriging-based methods[27,28] are also widely used in reliability analysis [9,29–34] andGlobal Optimization [35,36]. As an exact interpolation technique,the Kriging model, including a regression part and a stochasticprocess, combines fitting of a polynomial and correlation analysisof residual error. It not only predicts the structural response of apoint but also provides the local uncertainty measure (the so-called Kriging variance). Therefore, the Kriging model is employedin this research.
Besides meta-model formulation, the design of experiments(DoE) also has huge influence on the convergence rate and theaccuracy of reliability analysis [12,15,24,29,37,38]. An excellentstrategy of DoE leads the process of reliability analysis convergingquickly and provides high accuracy at the same time [31,34,38–40]. Various kinds of sequential DoE based on the Kriging modelhave been proposed to improve the accuracy of structural relia-bility analysis and reduce the number of calls to the real perfor-mance function [8,29,32,38,41,42]. Kriging based sequential DoEhas drawn more and more attention because it is often active andcan update itself by adding new sample point based on the sta-tistical information provided by the Kriging model. Despite thedifferences of strategies of DoE, their main idea and steps can besummarized as follows:
1. According to the statistical information provided by the Krigingmodel, define a learning function which can reflect how close apoint is to the limit state ( ( ) = )xG 0 , the potential of a pointcrossing the predicted limit state ( ^( ) = )xG 0 or some otherproperties.
2. Minimize (or maximize) the learning function in a randomsample set or the whole variable space, and find the minimum(or maximum) point.
3. The minimum (or maximum) point is the best next one. Andadd it to the DoE.
The learning function plays a critical role during the DoE up-dating. An inefficient learning function may slow down the pro-gress of reliability analysis, or evenworse. Many learning functionshave been proposed from different perspectives. Expected feasi-bility function (EFF), developed by Bichon et al. [29], searches forpoints in the vicinity of the limit state over the variable space. Themaximum point of EFF is added into the DoE step by step so that^( ) =xG 0 converges to ( ) =xG 0. EFF roughly estimates the failureprobability very quickly. The learning function U, proposed byEchard et al. [34], focuses on the probability of misclassificationmade by the Kriging model on the sign of ( )xG . Just like discussedin [41], U gives more weight to points in the vicinity of the pre-dicted limit state rather than the Kriging variance, which is itsmain difference from EFF. Lv et al. [41] and Yang et al. [42,43]present two new learning functions which are named H and ex-pected risk function (ERF) respectively. H is based on informationentropy, and its principle is similar to EFF's. Just like the expectedimprovement function (EIF) [35] values how much the objectivefunction would be improved at a point than the current optimum,
ERF can identify the risk that the sign of a point is wronglypredicted.
In this research, a new learning function named least im-provement function (LIF) is proposed and applied to structuralreliability analysis. According to the statistical information pro-vided by the Kriging model, the probability of making a mistake onthe sign of any point in the variable space can be measured, based
on which the uncertainty of predicted failure probability ( ^ )Pf isdefined. LIF is constructed to approximately measure how much
the uncertainty of P̂f will be improved if adding a given point intoDoE. The main difference between LIF and other learning functionsis that it takes both Kriging statistical information and the prob-ability density function into account. LIF searches the point that
influences the accuracy of P̂f most rather than the one whose signis uncertain most. The optimal point of LIF is determined withMCMC simulation. An active learning reliability method is raisedby employing the Kriging model and LIF. MCS is adopted to esti-
mate P̂f and its uncertainty.The remainder of this paper is organized as follows. Section 2
introduces the Kriging model and MC simulation briefly. Section 3presents the least improvement function in detail and analyzes it,which is followed by the proposed reliability analysis method insection 4. Three examples are employed in Section 5 to illustratethe efficiency of LIF and the proposed method. Section 6 is theconclusion.
2. Kriging model for structural reliability analysis
The Kriging model, a nonlinear interpolation meta-model de-veloped for geostatistics by Matheron [27], consists of two parts,i.e. a linear regression model and a random function. It is supposedthat the performance function ( ( ))xG can be denoted as
∑ ββ( ) = ( ) + ( ) = ( ) + ( )( )=
x x x g x xG g z z2h
p
h hT
1
The first term of Eq. (2) is a realization of a regression function.( )xgh ( )= …h p1, 2, , is the basis regression function and its order
is set to be one in this research. The second term of Eq. (2) is aGaussian stochastic process whose mean is zero and the covar-iance of random process ( )xz is
θσ[ ( ) ( )] = ( ) ( )x x x xz z RCov , , ; 3i j i j2
where s2 is the variance of the Gaussian process, and ( )θx xR , ;i j is
the correlation coefficient between ( )xz i and ( )xz j with parameterθ. The most widely used correlation function is the Gaussian cor-relation function, whose form is
⎡⎣⎢⎢
⎤⎦⎥⎥( )∏θ θ( ) = − −
( )=
x xR x x, ; exp4
i jm
M
m im
jm
1
2
where xim is mth component of xi.
Given an initial DoE ⎡⎣ ⎤⎦= …S x x x, ,DoE N1 2 0, and the correspond-
ing structural response ⎡⎣ ⎤⎦= …Y y y y, ,DoE N1 2
T
0, the unknown para-
meters β, s2, θ in Eqs. (2)–(4) can be estimated by maximizing thelikelihood function,
⎡⎣ ⎤⎦( )θ σ( ) = − (^ ) + ( ) ( )RL Nmax ln ln det 502
where
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165154
( )β β β
θ
σ̂ = ( − ) ( − ) = ( )
= [ ( ) ( ) … ( )] = ( )
− − − −
×
Y G R Y G G R G G R YG
g x g x g x R x x
N
R
1
, , , , ;N i j N N
2
0
T 1 T 1 1 T 1
1 2T
0 0 0
Let β̂ , σ̂2, θ̂ denote the optimum estimation of β, σ2, θ respectively
gotten from Eq. (5). Now consider the linear combination pre-dictor,
^( ) = ( ) ( )x c x YG 6T
The error of Eq. (6) is
β^( ) − ( ) = ( ) − ( ) = ( ) − + ( ( ) − ( )) ^x x c x Y x c x Z G c x g xG G G zT T T T
where
= ( … )Z z z z, , , N1 2T
0
To minimize the mean square error (MSE) with unbiased estima-tion of y, the result is
β γμ ( ) = ^( ) = ( )^ + ( ) ( )x x g x r xG 7GT T
⎜ ⎟⎛⎝
⎞⎠( )σ σ( ) = + ( ) ( ) − ( ) ( )
( )− − −x u x G R G u x r x R r x1
8G2 2 T T 1 1 T 1
where
⎡⎣ ⎤⎦γ β θ θ= ( − ^) ( ) = ( ^) … ( ^)
( ) = ( ) − ( )
−
−
R Y G r x x x x x
u x G R r x g x
R R, ; , , , ;N1
1
T 1
0
Reference [44] gives the detailed derivation of Eqs. (7) and (8).According to the Gaussian process regression theory, the structuralresponse of x is subject to a normal distribution,
( )μ σ( ) ∼ ( ) ( ) ( )x x xG N , 9G G2
The estimation of the failure probability of the structural modelmentioned in Section 1 is
∫^ = ( ) ( ) ( )^≤ x x xP I f d 10f G 0
It is very time consuming to perform the multiple integration ofEq. (10) directly when >M 3. Therefore, MCS and some variancereduction techniques (IS, LS and so on) are often adopted to
compute P̂f approximately. As the most robust method, MCS isused in this research. According to the law of large numbers,
∑^ ≈ ( )( )=
^< xPN
I1
11f
MC i
N
G MC i1
0 ,
MC
where NMC is the number of random samples and
( )= …x i N1, 2MC i MC, is a sequence of i.i.d. random samples drawn
from distribution ( )xf . So ( ) ∼ ( ^ )^< xI B P1,G MC i f0 , and ( )^< xIG MC i0 , is a
sequence of i.i.d. random samples. The coefficient of variation of P̂f
is calculated as
δ =( ^ )
^ =− ^
^( )
P
P
P
N P
var 1
12MC
f
f
f
MC f
3. Least improvement function
In this section, a new learning function named least improve-ment function (LIF) is proposed. First of all, some widely refer-enced learning functions are introduced briefly. Then the
uncertainty of P̂f is measured based on the statistical informationprovided by the Kriging model, following which the new learningfunction LIF is constructed and analyzed.
3.1. Some widely used learning functions
EFF, developed by Bichon et al. [29], searches for points in thevicinity of the limit state over the variable space, which is definedas
( )∫ ( ) ( )ε( ) = − | ¯( ) − ( )| ( ) ( )( )ε
ε
¯ ( )−
¯ ( )+
( )x x x xEFF G G f G Gx d13x
x
xG
G
G
where ¯ ( )xG is set to be 0 in structural reliability analysis and ε isproportional to σ ( )xG . The value of ε and the integral result of Eq.(13) are detailed in [29]. It is obvious that EFF becomes larger
when x is near the predicted limit state or ^( )xG has more un-certainty. It can be treated as an expectation that point x is locatedin the domain of ε ε− < ( ) <xG .
The learning function U, proposed by Echard et al. [34], focuseson the probability of misclassification made by the Kriging modelon the sign of ( )xG . U is constructed as
μσ
( ) =| ( )|
( ) ( )x
x
xU
14G
G
According to Eq. (9), Φ( − ( ))xU is the probability of point x
crossing ^( ) =xG 0 and deteriorating the accuracy of P̂f .Eqs. (15) and (16) are two learning functions present by Lv et al.
[41] and Yang et al. [43,42] respectively.
∫ ( ) ( ) ( )( ) = − ( ) ( ) ( )( )ε
ε
− ( ) ( )x x x xH f G f G Gln d15x xG G
⎧⎨⎪⎪
⎩⎪⎪
∫∫
( ) ( )
( ) ( )( ) =
( ) ( ) ( ) ^( ) <
( ) ( ) ( ) ^( ) ≥( )
+∞
( )
−∞
( )
xx x x x
x x x xERF
G f G G G
G f G G G
d 0
d 016
x
x
G
G
0
0
3.2. The least improvement function
Comparing Eq. (1) with Eq. (10), it is easy to conclude that the
sign of ^( )xG affects the accuracy of P̂f directly, and ( )xf weights
how important x is to the accuracy of P̂f . The probability of makinga wrong prediction about the sign of ( )xG is Φ( − ( ))xU (Eq. (9)). So
the uncertainty or accuracy of P̂f can be measured as
∫ ( )Φ= − ( ) ( ) ( )x x xU fUF d 17
which is named as the uncertainty function (UF) of P̂f . It is obvious
that as UF converges to 0, the estimated failure probability ( P̂f )tends to Pf :
^ =→
P Plim f fUF 0
Taking [8,34] as reference, the sign of ( )xG is seen as confirmed if
( ) >xU 2. The domain of { }( ) >x xU 2 contributes little to UF,therefore Eq. (17) can be approximated as
∫ ( )Φ≈ − ( ) ( )( )( )≤
x x xU fUF d18xU 2
The principle of the new learning function proposed in thisresearch is to diminish UF by adding new point into DoE andupdating the Kriging model sequentially. To approximately valuehow much a given point x0 in the variable space but not included
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165 155
in the current DoE can diminish UF, two hypotheses are formed.
1. Add x0 into the current DoE with its performance function value( )xG 0 and reconstruct the Kriging model. It is worthy to em-
phasize that ( )xG 0 here refers to the normal distribution variabledefined by Eq. (9). The updated U and UF are denoted as U0 andUF0 respectively. If ( )xG 0 is not exactly equal to 0, there is apositive constant e0 for the studied structure and the currentKriging model.
( ) ( ) ( ) ( )− < > =x x x x x xIf r then U where r e G, 2,0 0 0 0 0 0
2. As a result of adding x0 into the current DoE with ( )xG 0 , theuncertainty of the sign for points out of the spherical neigh-borhood of x0 does not become worse as a whole, which is tosay
∫ ∫( ) ( )Φ Φ− ( ) ( ) ≥ − ( ) ( )( )∥ − ∥> ( ) ∥ − ∥> ( )
x x x x x xU f U fd d19x x x x x xr r
00 0 0 0
The first hypothesis can be interpreted as the way that bene-fitting from adding x0 into the current DoE and reconstructing theKriging model, the sign of ( )xG is almost sure if x is located in thespherical neighborhood of x0. And the radius of the sphericalneighborhood is proportional to | ( )|xG 0 . It is not always true inmathematics, but it is strict and can be proved in theory if
( ) δ>xG 0 0, where δ0 is a fixed positive number. The second oneindicates that the uncertainty of the sign for points out of thespherical neighborhood of x0 does not become worse in general ifits real response is calculated. It is reasonable in engineering andtheory, although authors cannot prove it for now.
According to the hypotheses above and Eqs. (17)–(19), UF di-minishes as a result of x0 added into DoE, which is constructed as
∫ ∫∫
∫∫∫
∫∫
∫
( ) ( )( )
( )
( )
( )
( )
( )
( )
Φ Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
− = − ( ) ( ) − − ( ) ( )
= − ( ) ( )
− − ( ) ( )
+ − ( ) ( )
− − ( ) ( )
≥ − ( ) ( )
− − ( ) ( )
≈ − ( ) ( )( )
∥ − ∥< ( )
∥ − ∥< ( )
∥ − ∥≥ ( )
∥ − ∥≥ ( )
∥ − ∥< ( )
∥ − ∥< ( )
∥ − ∥< ( )
x x x x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
U f U f
U f
U f
U f
U f
U f
U f
U f
UF UF d d
d
d
d
d
d
d
d20
x x x
x x x
x x x
x x x
x x x
x x x
x x x
r
r
r
r
r
r
r
0 0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
The integrand of the last term in Eq. (20) is approximately treatedas a constant function because the domain of integration is notlarge and the integrand is continuous. Therefore,
∫ ( ) ( )Φ Φ− ( ) ( ) ≈ − ( ) ( ) ( )( )∥ − ∥< ( )
x x x x x xU f k U f rd21x x xr
M0 0 0 0
0 0
where k0 is related to the dimension of basic variable(M). Ac-cording to the first hypothesis, ( )xr 0 is proportional to | ( )|xG 0 . Then,
( ) ( )Φ Φ− ≥ − ( ) ( ) ( ) = − ( ) ( )
| ( )| ( )
x x x x x
x
k U f r k e U f
G
UF UF
22
M M
M
0 0 0 0 0 0 0 0 0
0
Taking statistical information Eq. (9) provided by the Krigingmodel into account, the expectation improvement of (UF-UF0) is
( )( ) ( )Φ− ≥ − ( ) ( ) | ( )| ( )x x xE k e U f E GUF UF 23M M
0 0 0 0 0 0
For a given studied structure and a Kriging model, k0 and e0 areconstant and have no relationship with x0, so they have no in-fluence on the maximum point of the right side of the inequalitysign in Eq. (23). The lower limit of E(UF-UF0) is proportional to
( )( )Φ − ( ) ( ) | ( )|x x xU f E GM0 0 0 . Therefore, this research proposes it as a
new learning function:
( )( )Φ( ) = − ( ) ( ) | ( )| ( )x x x xU f E GLIF 24M
0 0 0 0
LIF is the abbreviation of least improvement function. One canunderstand Eq. (24) in an intuitive way. If point x0 is added to thecurrent DoE, one can almost confirm the signs of all points near x0.In another word, a spherical neighborhood is confirmed as a partof the safe domain or the failure domain after DoE includes x0. Thevolume of the spherical neighborhood is proportional to | ( )|xGM
0 ,and ( )Φ − ( ) ( )x xU f0 0 approximately quantifies how important apoint in the spherical neighborhood is to the current UF.
( )Φ − ( ) ( )| ( )|x x xU f GM0 0 0 is approximately proportional to the de-
cease of UF because of x0. As ( )xG 0 is a normal distribution variablebefore calling to the performance function to calculate it, one canonly measure the expectation of ( )Φ − ( ) ( )| ( )|x x xU f GM
0 0 0 , which isthe LIF(x0) defined in Eq. (24).
According to Eq. (9), ( )xG 0 can be treated as a normal dis-tribution variable, and its mean value and variance are μ ( )xG and
σ ( )xG2 respectively. if M is an even number, the last term of Eq. (24)
which is a mathematical expectation of | ( )|xGM0 is calculated as,
⎛
⎝⎜⎜
⎞
⎠⎟⎟
( )
( )
( )
∫
∫
∫
( )
( )
∑
∑
π σσ
σπ
σ
π
σ
| ( )| = ( )
( )−
( ) − ( )( ) ( )
= ( ) + ( ) −
= ( ) ( )
− = ( )
+ ( ) ( )( − )!!( )
−∞
+∞
−∞
+∞
=
−
−∞
+∞
=
−
x x
x
x xx x
x x
x x
x
x x
E G G
G uG
t u t t
C u
t t t u
C u m
12
exp2
d
12
exp /2 d
12
exp /2 d
2 125
M M
G
GG
G GM
m
M
Mm
GM m
Gm
mGM
m
M
Mm
GM m
Gm
0 0
0
0 02
20 0
0 02
00 0
20
1
/22 2
02
0
where
σ=
( ) − ( )( )
( − )!!= · ·…·( − )x x
xt
G um m2 1 1 3 2 1G
G
0 0
0
Otherwise, if M is an uneven number, ( )| ( )|xE GM0 is calculated as,
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
( )
( )
( )
∫
∫
∫
∫
( )
( )
( )
∑
π σ σ
π σ
σ
π σ
π σ
| ( )| = | ( )|( )
−( ) − ( )
( )
( ) = | ( )|( )
−( ) − ( )
( )( )
= ( ) + ( ) −
= ( ) ( )
−( )
σ
σ
−∞
+∞
+∞
−( )( )
+∞
=
−
−( )( )
+∞
x xx
x x
x
x xx
x x
xx
x x
x x
E G GG u
G G
G uG
t u t t
C u
t t t
12
exp2
d 21
2
exp2
d
2/ exp /2 d
2/
exp /2 d26
xx
xx
M M
G
G
G
M
G
G
G
u G GM
m
M
Mm
GM m
Gm
um
0 00
0 02
20
00
00
0 02
20
0
0 02
00 0
2
GG
GG
00
00
Fig. 1. Illustration of LIF with 2-dimension input variable.
Table 1Results of example 1.
Method Ncall ( )^ −P 10f3 δ ( )%Pf
ΔPf(%)
MCS 106 4.416 1.50 –
AK-MCSþU 126 4.416 – <0.1AK-MCSþEFF 124 4.416 – <0.1
The proposed method 6þ31 4.31 2.02 −2.46þ21 4.27 2.05 −3.36þ20 4.54 2.41 2.86þ30 4.52 2.35 2.46þ30 4.45 2.16 0.815þ24 4.34 2.48 −1.715þ25 4.33 2.08 −1.930þ21 4.30 2.02 −2.630þ18 4.38 2.14 −0.8
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165156
where
σ=
( ) − ( )( )
x xx
tG uG
G
0 0
0
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜⎜
⎞⎠⎟⎟( )
( )
∫
∫σ σ
− = −( )( )
−( )( )
+ ( − ) −( )
σ
σ
−( )( )
+∞ −
−( )( )
+∞−
xx
xx
t t tu u
m t t t
exp /2 d exp2
1 exp /2 d27
xx
xx
um G
G
mG
G
um
2 0
0
1 20
20
2 2
GG
GG
00
00
⎛⎝⎜⎜
⎞⎠⎟⎟( )∫
σ− = −
( )( ) ( )σ−
( )( )
+∞ xx
t t tu
exp /2 d exp2 28
xx
uG
G
22
02
0GG
00
⎛⎝⎜⎜
⎛⎝⎜
⎞⎠⎟⎞⎠⎟⎟( )∫ π Φ
σ− = − −
( )( ) ( )σ−
( )( )
+∞ xx
t tu
exp /2 d 2 129
xx
uG
G
2 0
0GG
00
For any x in the variable space, LIF(x) can be obtained by Eqs.(24)–(29):
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎛⎝⎜⎜
⎞⎠⎟⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟( )∫
( )
( )
∑
∑
( )
Φ
σ
Φ π σ
( ) =
− ( ) ( ) ( )
+ ( ) ( )( − )!!
− ( ) ( ) ( ) ( )
−σ
=
−
=
−
−( )( )
+∞
30
x
x x x
x x
x x x x
U f u
C u m
M
U f C u
t t t
M
LIF
2 1
is even
2/
exp /2 d
is uneven
xx
GM
m
M
Mm
GM m
Gm
m
M
Mm
GM m
Gm
uGG
m
1
/22 2 2
0
00
2
The larger LIF(x) is, the more x can diminish UF. Then we can addthe maximum point of LIF into the DoE and reconstruct the cor-
responding Kriging model. As a result the uncertainty of P̂f con-verges to zero. Repeat the above process until a certain conditionof convergence is satisfied.
Considering an analytical example with 2-dimension inputvariable [15,20,45], its performance function is
( ) = ( + ) − ( + ) − ( )xG x xexp 0.4 7 exp 0.3 5 200 311 2
It is assumed that the input random variable ⎡⎣ ⎤⎦=X X X,1 2T
issubject to standard bivariate normal distribution and X1 and X2 areindependent. The limit state is derived as
⎡⎣ ⎤⎦( )= ( + ) + − ( )x x2.5 ln exp 0.3 5 200 7 321 2
Generate 20 random samples with Latin hypercube sampling(LHS), which is the initial DoE. Fig. 1 illustrates the LIF. This 3-Dplot announces that LIF may have more than one local maximumpoint for a given Kriging model, and points far away from thepredicted limit state are with LIF values close to zero generally. Thelatter coincides with the conclusion that the domain ( ) <xU 2contributes most of the UF. Therefore, to search the maximumpoint of LIF(x) or determine the best next point, one just needs tosearch the domain ( ) <xU 2 and avoid local maximum points as faras possible. The MCMC method performs the generation of con-ditional samples efficiently. This characteristic is suitable for gen-erating random points from the conditional distribution
( | ( ) < )x xf U 2 . The main reason that the MCMC simulation isadopted rather than an optimization algorithm is that the resultfrom the latter depends on the initial value seriously. The proce-dure of determining the best point is summarized as follows:
Step 1: Generate NUF conditional random points with the MCMCmethod from the conditional distribution ( | ( ) < )x xf U 2 .
The sample of points is denoted by { = … }x n N, 1, 2, ,nUF, UF .NUF is set to be 104 in this research.
Step 2: Determine the best next point among{ = … }x n N, 1, 2, ,nUF, UF . Compute LIF( x nUF, ) according to Eq.(30). And the point that maximizes LIF is defined as the bestnext point.
= { ( ) = … }x x n Nargmax LIF , 1, 2, ,nbest UF, UF
4. An adaptive reliability analysis method
In this section, a new adaptive reliability analysis method isproposed based on the Kriging model. In the method, candidatepoints of the best next one are refreshed with the MCMC methodevery iteration, and LIF developed in Section 3 is adopted to de-termine the best next one among NUF candidates. The main stepsof the proposed method are summarized as follows:
Step 1: Produce initial DoE with LHS method and call the perfor-mance function to evaluate the structural response of pointsin the initial DoE. It is supposed that the input vector X issubject to multivariate normal distribution with zero meanand unit variance, which is reasonable in engineering be-cause most of random vectors can be transformed to multi-variate normal distribution exactly or approximately. Theupper and lower bounds of LHS are set to be 5 and �5
Fig. 2. The convergence procedure of predicted limit state (example 1). Baby blue lines in each sub graph are ^( ) =xG 0 while light black ones are ( ) =xG 0 The numbers of DoEin these four subgraphs are 6, 16, 26 and 37 respectively ( =N 60 ). (For interpretation of the references to color in this figure caption, the reader is referred to the web versionof this paper.)
Fig. 3. Graphs of P̂f and UF/ P̂f with =N 60 . Imaginary lines in (a) and (b) correspond to the referenced probability of failure ( × −4.416 10 3) and condition of convergence(0.05) respectively.
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165 157
respectively. The number of points in the initial DoE is illu-strated in detail in Section 5.
Step 2: Construct the Kriging model based on the current DoEaccording to the theory in Section 2. DACE [28,46–48], atoolbox about Kriging in MATLAB, is used to build the Kriging
model and predict performance values and their variances inthe variable space.
Step 3: Estimate P̂f and UF with MCS. The number of randomsamples (NMC) is set to make sure the coefficient of variationof P̂f no larger than 0.03, which is to say
Fig. 4. The illustration of the difference of the proposed method from AK-MCSþU and AK-MCSþEFF. Red lines denote the predicted limit states and green lines are the reallimit state. The number of points in DoE is the same for each row (26 for the first row, 46 for the second and 86 for the third). (For interpretation of the references to color inthis figure caption, the reader is referred to the web version of this paper.)
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165158
δ ≤ ( )0.03 33MC
The estimation of UF is denoted as
( )∑ Φ≈ − ( )( )=
xN
UUF1
34MC n
N
n1
MC
Step 4: Decide whether P̂f and UF satisfy the condition of con-
vergence. As analyzed in Section 3, P̂f converges to Pf when
UF is small enough. According to the definition of UF (Eq.(17)), it can be easily proved that UF is a upper limit of the
absolute error of P̂f relative to Pf in the sense of mathematic
expectation.
(| ^ − |) ≤E P P UFf f
Therefore, the condition of convergence in this research is setas
^ ≤ ( )PUF/ 0.05 35f
Step 5: If P̂f and UF satisfy Eq. (35), end the iteration, and P̂f is theestimation of the studied structure. Otherwise continue theiterative process.
Step 6: Determine the best next point. As mentioned in Section 4,MCMC simulation is adopted to determine the maximumpoint of LIF approximately. Generate NUF random sampleswith MCMC method in the domain of integration of Eq. (18).And the maximum point of LIF for { }= …x n N, 1, 2,nUF, UF istreated as the best next point. Add the best next point intoDoE and evaluate its structural response. Return to step 2.
5. Academic validation
In this section, two analytical examples and a truss structureare employed to validate the efficiency of LIF and the proposed
Fig. 5. Graphs of P̂f and UF/ P̂f with =N 150 in (a) (b) and =N 300 in (c) (d).
Table 2Results of example 2.
Method Ncall ( )^ −P 10f2 δ ( )%Pf
ΔPf(%)
MCS ×6 104 7.34 1.51 –
AK-MCSþU 6þ363 7.28 2.07 0.82K-MCSþEFF 6þ370 7.23 2.21 1.5
AK-MCSþLIF 6þ285 7.22 2.39 −1.66þ275 7.33 2.47 −0.16þ280 7.19 2.32 −2.0
The proposed method 6þ305 7.27 2.42 −1.06þ295 7.38 2.46 0.56þ295 7.32 2.33 −0.320þ295 7.39 2.37 0.720þ280 7.31 2.37 −0.420þ290 7.18 2.19 −2.240þ275 7.24 2.32 −1.440þ260 7.24 2.08 −1.440þ285 7.44 2.35 1.4
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165 159
reliability analysis method.
5.1. Example 1: a series system with two-dimension input variable
The first analytical example already adopted in [49,5,34] is aseries system including four branches. Its performance function isdefined as
⎧
⎨⎪⎪⎪
⎩⎪⎪⎪
⎫
⎬⎪⎪⎪
⎭⎪⎪⎪
( ) =
+ ( − ) − ( + )
+ ( − ) + ( + )
( − ) +
( − ) + ( )
xG
x x x x
x x x x
x x
x x
min
3 0.1 / 2 ;
3 0.1 / 2 ;
6/ 2 ;
6/ 2 ; 36
1 22
1 2
1 22
1 2
1 2
2 1
where the input variable ⎡⎣ ⎤⎦=X X X,1 2Tis subject to standard nor-
mal distribution and X1 and X2 are mutually independent.The proposed method described in Section 4 is applied to this
example. All the results are listed in Table 1. To demonstrate theefficiency of LIF and the method, results from [34] are also sum-marized. The information provided in Table 1 includes the numberof calls to the performance function(Ncall), the estimation of failure
probability (P̂f ) and its corresponding coefficient of variation ( )δPf,
the relative error ( ΔPf) of P̂f compared with reference failure
probability.Fig. 2 shows the procedure that predicted limit state converges
to the real one ( )=N 60 . From Fig. 2, the initial predicted limit state
does not tally with the real, which is the same to the initial P̂f
(Fig. 3a) and UF/ P̂f (Fig. 3b). As the number of DoE grows, thingschange to much better (Figs. 2b, c and 3). Fig. 3a indicates that theproposed method can roughly estimate Pf with about 20 points,whose accuracy is enough for engineering. Other points are used
to make P̂f and UF satisfy the condition of convergence (Eq. (35)).When the process converges, the predicted limit state is close tothe real in the area of interest. Although they do not coincide with
each other in the whole space, the accuracy of P̂f is enough.Fig. 4 compares different methods in terms of the procedure of
predicted limit states converging to the real, in which AK-MCSþU,
Fig. 6. Comparisons among AK-MCSþU, AK-MCSþLIF and the proposed method in terms of predicted limit states with the same number of DoE points for example 2. Ncall isset to be 100, 200, and 300 respectively.
Fig. 7. Comparisons of different methods in terms of P̂f and P̂UF/ f for example 2.
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165160
AKþMCSþEFF and the proposed method are contained. Thenumber of initial DoE is set to be 6 for all methods, and subgraphsin a same row holds the same number of DoE. Fig. 4 clarifies thatthe proposed method is able to identify the area of importancequickly and approximate the target limit state roughly, which
distinguishes it from other methods. One can also realize that theaccuracy of methods mentioned in Fig. 4 is no much difference ifthere are enough points in DoE (subgraphs of the third row).
To test how much the number of initial DoE can influence theprocess of convergence, different values of N0 are adopted.
Fig. 8. Comparison between AK-MCSþLIF and the proposed method with =N 60 . The first two sub graphs, (a) and (b), reflect the convergence procedure of AK-MCSþLIF,and the latter two, (c) and (d), correspond to the proposed method.
Fig. 9. Graphs of P̂f and UF/ P̂f with =N 200 in (a), (b) and =N 400 in (c), (d).
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165 161
Fig. 10. The truss structure (unit m).
Table 3Distribution information of the truss structure.
Variable Distribution Mean Standard deviation
( )−P P N1 6 Gumbel ×5 104 ×7.5 103
( )A m12 Lognormal × −2 10 3 × −2 10 4
( )A m22 Lognormal × −1 10 3 × −1 10 4
( )E Pa1 Lognormal ×2.1 1011 ×2.1 1010
( )E Pa2 Lognormal ×2.1 1011 ×2.1 1010
Table 4Results of example 3.
Method Ncall ( )^ −P 10f5 δ ( )%Pf
ΔPf(%)
IS ×5 105 3.45 – –
AK-MCSþU >200 3.30 2.46 −4.4AK-MCSþEFF >200 3.27 2.47 −5.27AK-MCSþLIF 11þ110 3.31 2.51 −4.06The proposed method 15þ130 3.49 2.40 1.2
15þ120 3.48 2.45 0.930þ130 3.50 2.06 1.430þ140 3.41 2.45 −1.250þ120 3.55 2.31 −2.950þ100 3.39 2.04 −1.7
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165162
According to Fig. 5a and c, increasing the number of initial DoE
improves the quality of initial P̂f , which results from good initialpredicted limit state. However it does not fasten the convergenceobviously. On the contrary, it may negatively affect the number ofcalls to the performance function (Table 1). The comparison ofdifferent number of initial DoE is also detailed in the followingexamples.
5.2. Example 2: modified Rastrigin function
This two-dimension analytical example already analyzed in[9,34,49] consists of a modified Rastrigin function. The perfor-mance function is
( )∑ π( ) = − − ( )( )=
xG x x10 5cos 237m
m m1
22
The input variable ⎡⎣ ⎤⎦=X X X,T
1 2 is subject to standard normaldistribution and X1 and X2 are mutually independent. The limitstate of Eq. (37) is much more complicated than the one in Section5.1 (Fig. 2). In this research, this example is employed to demon-strate the efficiency of LIF and the proposed method and illustratethe different between some other Kriging based active methodsand the proposed one.
AK-MCSþLIF and the proposed method are applied to this
example and both methods are run 3 times with =N 60 to testtheir stability. AK-MCSþLIF means that the method raised by [34]is adopted but LIF is employed as the learning function. AK-MCSþLIF here is to demonstrate the outperformance of LIF in theframework of AK-MCS and compare the proposed method to AK-MCS with the same learning function. Results are summarized inTable 2. It is easily noticed that numbers of calls to the perfor-mance function for AK-MCSþLIF and the proposed method arelower. Fig. 6 lists the comparisons among predicted limit statesfrom AK-MCSþU, AK-MCSþLIF and the proposed method. Fig. 7compares AK-MCSþU, AK-MCSþEFF, AK-MCSþLIF and the pro-posed method, in which ×2 105 given random points are em-
ployed. Figs. 7(a) and 7(b) show the procedures of P̂f converging to
Pf and the decrease of P̂UF/ f respectively.Both Figs. 6 and 7 validate the efficiency of LIF developed in this
research. Compared with U and EFF, LIF is more adept in re-
cognizing points that are helpful to improve the accuracy of P̂f orthe Kriging model, which benefits the rough estimate of Pf. AK-MCSþLIF and the proposed method are very similar to each otherin the sense of accuracy. From Fig. 8 and Table 2, it is easily noticedthat to satisfy the condition of convergence defined by Eq. (35) theproposed method needs more points than AK-MCSþLIF in general
and P̂f from the latter is very steady. The main difference betweenthem is that the former determines the best next point from agiven sample of points along the procedure if the initial randompoints are enough, while the proposed method generates randomcandidates with MCMC method every iteration. Fig. 9 shows the
graphs of predicted failure probability and the corresponding P̂UF/ f
with N0 set to be 20 and 40, whose results are from the proposedmethod and also listed in Table 2.
5.3. Example 3: truss structure with 10 dimensions
This truss structure has already been studied in [15,18]. Asshown in Fig. 10, it contains 11 horizontal bars and 12 sloping ones.A1, E1 denote horizontal bars cross section and Youngs modulirespectively, while A2, E2 denote sloping bars'. 6 loads, from P1 toP6, are applied on nodes of horizontal bars. These ten randomvariables are independent, and their distribution information islisted in Table 3.
The deflection of node E in Fig. 10 denoted by ( )s x , is the re-sponse of the structure for it is larger than any other nodes. Thethreshold of | ( )|s x is 0.14 m in accordance with [15,18]. Hence theperformance function of the studied structure is
( ) = − | ( )| ( )x s xG 0.14 38
According to [18], the referential failure probability ( PfREF) is
3.45�10�5, which is from IS with 500,000 simulations.Apply the proposed method to the truss structure, and run it
several times with different numbers of initial DoE. Table 4 sum-
marizes the results. Fig. 11 presents graphs of P̂f and P̂UF/ f . From
Fig. 11, P̂f converges to Pf with desirable accuracy quickly. However,does not satisfy the condition of convergence until the number ofDoE is about 150. Similar with Fig. 7, Fig. 12 shows the comparisonamong AK-MCSþU, AK-MCSþEFF, AK-MCSþLIF and the proposedmethod. One can notice that methods with LIF are efficient. AK-MCSþLIF and the proposed method make little distinction. How-ever, it takes more than 80 h to perform AK-MCSþLIF, while theproposed method needs less than 10 h. AK-MCSþU and AK-MCSþEFF are approximately equal to AK-MCSþLIF in the sense ofrunning time. The main reason is that the failure of the studiedstructure is a rare event ( × −3.45 10 5). The number of randompoints for AK-MCS methods is larger than 40 million. As the scale
Fig. 11. Graphs of P̂f and P̂UF/ f with different N0.
Fig. 12. Comparisons of different methods in terms of P̂f and P̂UF/ f for the truss structure.
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165 163
of DoE grows, these methods are much more time-consumingthan the proposed method to determine a best next point.
6. Conclusion
In this research, a new learning function named least im-provement function and a reliability analysis method are pro-posed. Compared with existing learning functions, LIF takes thejoint probability density function of input variables into account
and is used to approximately indicate how much the accuracy of P̂f
will be improved in the sense of mathematical expectation ifadding a given point into the DoE. It obtains a compromise amongthe Kriging variance, estimated response and joint probability
density function. Based on the Kriging model and LIF, a new re-liability analysis method is developed. In the method, MCMC isused to generate conditional random points as candidates of thebest next one. The proposed learning function LIF quantifies howmuch a given point x can improve the accuracy of the Krigingmodel and determines the best next one. Three examples areanalyzed. Results show that LIF and the proposed method are veryefficient when dealing with nonlinear performance function, smallprobability, complicated limit state and engineering problemswith high dimension. By analyzing example 2 and example 3, theconclusion is reached that AK-MCSþLIF and the proposed methodperform similarly in the sense of accuracy and the number of callsto the target performance function. The proposed method is moreefficient in determining the best next point when dealing with
Z. Sun et al. / Reliability Engineering and System Safety 157 (2017) 152–165164
problems with a high dimension input vector and a rare event tofailure. From Tables 1, 2 and 4, larger number of initial DoE is likelyto result in larger number of calls to performance function for theproposed learning function and method, especially when theperformance function is high nonlinear or with many input vari-ables. The condition of convergence in the proposed method is notreligious in theory. Integrating all results into account, it is obviousthat the condition of convergence defined by Eq. (35) is too rig-
orous. As mentioned in Section 3.1, UF measures the accuracy of P̂f
or its corresponding Kriging model. Actually, UF is an upper limit
of the error of P̂f relative to Pf in the sense of mathematical ex-pectation. It is a difficult task to estimate how much UF is larger
than the real error of P̂f , which is various with problems andKriging models. Therefore, the proposed condition of convergencebehaves differently for the studied examples. The accuracy mea-
surement of P̂f or the Kriging model benefits not only the timelytermination of reliability analysis procedure but also the im-provement of the Kriging model or the efficient DoE strategy.Finding a reasonable condition of convergence or a more accurate
measurement of P̂f for Kriging based reliability analysis method isan important part of our future work.
Acknowledgment
The study was funded by National Science and TechnologyMajor Project of China (Grant No. 2013ZX04011-011) and theFundamental Research Funds for the Central Universities of China(Grant No. N140306004). Their financial supports are gratefullyacknowledged.
Analysis of the uncertainty of Eq. (34)
To simplify the description, a new symbol UFMC is introduced:
⎛⎝⎜⎜
⎞⎠⎟⎟
⎛
⎝⎜⎜⎜
⎛⎝⎜⎜
⎞⎠⎟⎟
⎞
⎠⎟⎟⎟
( )( ) ( )
( ) ( )
( )
∑
∑ ∑
∑
Φ Φ
Φ Φ
Φ
( ) = − ( ) = − ( )
≈ − ( ) − − ( )
< − ( )
=
= =
=
x x
x x
x
NU
NU
N NU
NU
NU
var UF1
var1
var
1 1 1
12
MCMC n
N
nMC
MC MC n
N
nMC n
N
n
MC n
N
n
21
1
2
1
2
21
MC
MC MC
MC
Hence,
( )∑ Φ( ) < − ( )( )=
xN
Uvar UF1
2 39MC
MC n
N
n1
MC
The number of random points for MCS (NMC) satisfies Eq. (33):
− ^
^ ≤P
N P
10.03f
MC f
Therefore,
^ ≥− ^
≈( )
N PP1
0. 031111
40MC ff2
If UFMC meets the condition of convergence defined by Eq. (35),then,
( ) ( )Φ Φ^ =
∑ − ( )
∑ ( )=
∑ − ( )^ ≤
( )
=
= ^<
=x
x
x
P
NU
NI
U
N P
UF1
10.05
41
MC
f
MCnN
n
MCiN
G MC i
nN
n
MC f
1
1 0 ,
1
MC
MC
MC
According to the central limit theorem,
( < + ( ) ) ≈P UF UF 1.65 var UF 0.95MC MC
Integrating Eqs. (39)–(41),
( )∑ Φ( )
^ = ^ − ( )
≤^
^ <
=
xP N P
U
N P
N P
1.65 var UF 1.65
2
1.65 0.05
20.0078
MC
f MC f n
N
n
MC f
MC f
1
MC
Then,
⎛
⎝⎜⎜
⎞
⎠⎟⎟^ < ≥
( )P
P
UF0.058 0.95
42f
It can be concluded that the real value of UF is less than P̂0.058 f
with confidence coefficient of 95% at least if the estimate of UF(UFMC) satisfies Eq. (35).
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