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Hybrid reliability analysis for structures with
implicit limit state functions
Wen-Cai Sun1,2, Zi-Chun Yang1,2, Lei Wang1,2
of High Temperature Structural Composite Materials for Naval Ship,
2College of Power Engineering, Naval University of Engineering,
Wuhan, Hubei, China
E-mail: [email protected]
1Institute
For the implicit limit state equation usually encountered in structural hybrid reliability
analysis, the interval parameters have been assumed to obey conservative distribution and
the SVM is used to construct the hyper surface. A series of state points covering around
the design point can be obtained by Markov Chain simulation. Then, combined with the
structural response values, the training samples for SVM can be obtained. The structural
hybrid reliability can be calculated by the combination of the approximate limit state
equation and Monte Carlo method. Two numerical examples verify the feasibility and
effectiveness of the proposed method.
Keywords: Structural Hybrid Reliability; Implicit Limit State Equation; Support Vector
Machine (SVM); Markov Chain.
1. Introduction
The limit state functions of the complex structures are usually implicit or
difficult to write out the analytic expressions. In addition, these structures
commonly contain many types of uncertain information. At present, the
Response Surface (RS) [1], Artificial Neural Network (ANN) [2] and Support
Vector Machine (SVM) [3] are mostly used for approximately fitting of the
implicit limit state functions. The former two are based on the empirical risk
minimization principle, and have many insurmountable defects in practical
applications. Hurtado explained that the root of the problems of Response
Surface method is rigid non-adaptive regression technique [4]. The topology
structure of ANN is difficult to choose, the generalization ability is poor, the
over-learning and the local optimization of ANN are the main problem of ANN
applied in reliability analysis.
SVM is one of the latest research results in machine learning domain, which
is based on structural risk minimization principle and can avoid the overlearning, local optimization and dimension disaster [5]. Rocco and Moreno used
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SVM to assess the reliability of network systems [6]. Hurtado and Alvarez
considered structural reliability problem as pattern recognition problem and
analyzed the structural reliability using SVM and stochastic finite element [3].
Support Vector Regression machine (SVR) is developed by Vapnik based on
SVM theory, and is a new algorithm for nonlinear regression estimation [7].
So far, hybrid reliability researches for implicit limit state equations are
rarely reported. A new type of probabilistic-interval hybrid reliability model was
established in literature [8], and the solving method was derived based on failure
integral formula. The problem of implicit limit state is studied in this paper
based on the reliability model in literature [8]. Firstly, the sample extraction
strategy is researched based on Markov chain technique [9–11]. The interval
variables are dealt as conservative distribution assumption, and the optimal
importance sampling function is used as the stationary distribution of the
Markov chain. The state points around the design point and the alternative points
are used together with the structural response values constituting the training
sample set. The SVR algorithms for probabilistic-interval hybrid reliability
analysis are proposed, and the limit state equation with structural risk
minimization can be obtained. The hybrid reliability can be obtained through
random sampling of probabilistic and interval variables. Two examples are
given to verify the methods in this paper.
2. The Structural Hybrid Reliability Method
Assume the structural limit state equation is
M  G( X ,Y )  0
(1)
where X=[X1,X2,…,Xn] and Y=[Y1,Y2,…,Ym] are probabilistic variable vector
and interval variable vector, respectively. Assume the joint probability density
function of X1,X2,…,Xn and Y1,Y2,…,Ym is fX,Y(x,y) and the integral formula of
structural failure probability can be written as
Pf 
  f
X ,Y
( x , y )dx1  dxn dy1  dym
G ( x , y )0

  f
X
( x ) fY ( y )dx1  dxn dy1  dym
(2)
G ( x , y ) 0


    fY ( y )     f X ( x )dx1  dxn dy1  dym

 G ( x , y ) 0

    Pf ( y ) fY ( y )dy1  dym  E[ Pf ( y )]

where Ω is the variable region of the interval vector Y, G(x,y)≤0 is the
failure region of the structure, Pf(y) is the structural failure probability under the
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realization value of Y. Eq. (2) shows that the structural failure degree is the
mean value of the failure function with Y as its independent variable in the
region of interval variables.
For the case of independent interval variables, the failure integral formula is
equivalent to
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Pf 
n
1
V
1 n
lim  Pf ( yi )  lim  Pf ( yi )
V n  i 1
n n  n i 1
(3)
where yi denotes the ith sample point of the interval variable vector
according to uniform distribution.
For the case of relevant interval variables, the failure integral can be written
as
Pf     Pf ( y ) fY ( y )dy1  dym

    Pf ( y ) fY1 ( y1 ) fY2 |Y1 ( y2 | y1 ) fYm |Y1 ,Y2 ,,Ym1 ( ym | y1 , y2 ,  , ym 1 )dy1  dym
(4)

 E[ Pf ( y )]  lim
n 
1 n
 Pf ( y j )
n j 1
where yj denotes the jth sample point of the interval variable vector
according to conditional probability.
If yk denotes the kth sample point of the interval variable vector, the Monte
Carlo calculation formula of structural reliability is
Pr  1  Pf  1 
1
N
N
 P (y
f
k
)
(5)
k 1
where N denote the simulation times.
3. Analysis Method for Implicit Limit State Problems
3.1. The sample extraction strategy based on Markov Chain
To ensure the accuracy of structural reliability analysis, the limit state surface
should have high accuracy near the design point. So, the samples that support
the surface should be distributed around the design point, where it is the region
of greater contribution to structural failure.
If the optimal importance sampling function is used as the stationary
distribution of Markov chains, the Markov chain state points can be well
covered around the structural design point. Then, a good approximation of the
limit state equation can be obtained through SVM training.
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When probability variables and interval variables all exist in the structure,
the method in section 1 is equivalent to probability reliability analysis with the
interval variables according to uniform distribution. The assumption for interval
variables is proved to be conservative in many studies. So, this method is
reasonable in theory, and partial to safety in application. In order to facilitate the
representation, the random variables and interval variables are uniformly
denoted as X=[X1,X2,…,Xn](n is the dimension of structural variables). The joint
probability density function of X is f(x) and the conditional probability density
function of the variables in failure region F is
f  x | F  I F ( x ) f ( x ) Pf
(6)
where IF(x) is the indicator function of F. If x∈F, IF(x)=1, otherwise,
IF(x)=0. Pf denotes the structural failure degree.
Eq. (6) is the optimal importance sampling function in reliability
calculation. Take it as the stationary distribution of Markov chain, and the
obtained state points can be well distributed around the structural design point.
For convenience, make
q( x )  I F ( x ) f ( x )
(7)
First, take one point in the failure region as the initial point according to
engineering experience or numerical method. Then, on the basic of the j-1th
state point x(j-1), generate the alternative point ε according to the n-dimensional
proposed distribution f   | x  j 1 and calculate the ratio as



r  q    q x ( j 1)

(8)
Then, the jth state point denoted as x(j)=[x1(j),x2(j),…,xn(j)]T of Markov chain
can be obtained according to Metropolis criterion, i.e. make x(j)=ε with
probability min{1,r} and x(j)=x(j-1) with probability 1-min{1,r}. Repeat the above
steps, and the state points denoted as x(1),x(2),---,x(M) can be obtained whose
stationary distribution is the optimal importance sampling function f(x|F).
The proposed distribution f   | x  j 1 has a great influence on the
robustness of Metropolis algorithm. f   | x  j 1 must be a density function
with symmetric distribution, and the algorithm is more robust for bell shaped
probability density function than super cube shaped one. In this paper, n
dimension Gauss distribution is chosen as the proposed distribution as follows

f * ( |x (j -1) ) 
1
 2 
n/2



1/ 2
89
e



1
  x  j 1
2

T

 1   x  j 1

(9)
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where  is n  n dimension covariance matrix. The standard deviation of
the proposed distribution has influence on the distance of the next sample and
the current sample, and the convergence rate of Markov chain is affected too.
The reduced variance strategy is applied in this paper written as
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 ij   j i0 (  1, i  1, 2,, n, j  1, 2,, M )
(10)
where  ij is the variance of the ith component generating the jth state point,
 is the initial variance of the ith component and can be the original variance.
Thus, the region of greater contribution to structural failure can be quickly
explored in the early stage of Markov chain simulation. Then, more samples can
be centered around the design point or the limit state surface in later stage.
To make full use of the information generated in Markov chain simulation,
the alternative sample point and the state point will be used together, and form
the training samples of SVM with the corresponding output index (functional
values). Thus, the fitting accuracy of the limit state equation will be improved.
0
i
3.2. Algorithm of SVM prediction
a) In Markov chain simulation, the state samples, alternative samples and their
corresponding output value are saved.
b) Eliminate repeat samples of the state points, and merge the other state
points and the alternative samples. Form the training samples
T=(X×Y)l={(x1,y1),…,(xl,yl)}T together with the output index, where xi ∈Rn,
Y={y1,y2,…,yl} is the output values. In addition, it can be known from the
Markov chain simulation that l=M.
c) Choose the insensitive coefficient, penalty parameter and the kernel
function K(·,·).
d) Construct the following convex quadratic programming problem:
 l

  ( i   i )( j   j )( ( xi )γ ( x j ))
 min

2l
R
 i , j 1

l
l



    ( i   i )   yi ( i   i )


1
1
i
i

l


()
s.t.  ( i   i )  0;0   i  C , i  1,Λ, l

i 1
(11)
e) Solve the convex quadratic programming problem in d), and the
Lagrange multiplier and offset b can be obtained such as
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 ()  (1 , 1 ,  2 ,  2 , ,  l ,  l  )T
(12)
l
b  y j   ( i   i ) K ( xi , x j )   , j  (0, C )
(13)
i 1
or
l
(14)
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b  y k   ( i   i ) K ( xi , xk )   , k  (0, C )
i 1
f) Construct the regression forecasting model of SVM as
l
y  g ( x )   ( i   i ) K ( xi , x)  b
(15)
i 1
g) Test the accuracy of training using test samples, and adjust the precision
parameter, penalty parameter and the kernel function parameter.
h) For the interval variables, extract N samples according to the principle of
uniform distribution. For probability variables, extract N samples according to
their probability distribution respectively. Then, N group forecasting samples
can be obtained.
i) Take the output function of the SVM as the explicit approximation of the
implicit limit state function. The structural hybrid reliability can be obtained
through predicting the response values of samples and Monte Carlo simulation.
4. Example
4.1. Example 1
The limit state equation of a cylindrical pressure vessel for strength failure is
M  K Ic  0.975 PD a / Q / t  0
where the physical meanings and the statistic features of all variables are
listed in Table 1.
Table 1 The physical meanings and statistical features of all variables in Example 1.
Physical
quantity
mean
value
standard
deviation
Fracture
toughness
KIc /MPam1/2
Working
pressure
P /MPa
124
5.88
18.6
0.3933
91
Internal diameter
D/mm
Wall
thickness
t/mm
Crack
depth
a/mm
1500±30
5±0.6
3±0.5
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Take the results of direct Monte Carlo method as the baseline of comparison
with the method proposed in this paper. The failure degree is 9.31×10-3 through
ten million sampling.
Use Markov chain method in this paper and extract 100 training samples.
Then, use SVM to make regression training, search optimal parameters and then
verify the training accuracy under different number of samples. In this example,
radial basis kernel function is used. The penalty factor C and insensitive
coefficient ε is +∞ and 0.01, respectively. The relative error of the reliability
with σ is shown in Fig.1. After many times parameter testing, the error reaches
the minimum value when σ=56. Fix the model parameters and reduce the
training samples to verify the accuracy. The variation of the relative error with
sample size is shown in Fig. 2.
2
0
relative error(%)
-2
-4
*
N =100 C=+∞ ε=0.01
-6
-8
-10
-12
-14
-16
10
20
30
40
50
60
kernel function parameterσ
Fig. 1 The variation of relative error with σ.
0.10
0.05
σ=56 C=+∞ ε=0.01
0.00
relative error(%)
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-0.05
-0.10
-0.15
-0.20
-0.25
-0.30
-0.35
50
60
70
80
sample size N
90
100
*
Fig. 2 The variation of relative error with sample size.
It is shown in Fig. 1 that the relative error can be controlled lower than
0.1% after parameter adjustment, and the result can keep stable when σ is within
a certain range.
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From Fig. 2, the accuracy is improved with the increasing of the training
sample size. In addition, when the number of training sample is 50, the relative
is -0.28%, which has enough accuracy. So, the excellent properties of SVM for
small sample size problem are improved. Moreover, the sample extraction
strategy can provide excellent training samples for SVM, and these samples can
be well centered around the structural design point or the limit state surface. The
excellent training samples play an importance role in the performance and
accuracy of SVM.
4.2. Example 2
The limit state function for fracture failure of the turbine disc of a certain type of
aircraft engine [12] is
M  sS 
C 2
 2  2 J
2
where the variables’ physical meanings are as follows
σs–strength limit; S–section area; J–moment of inertia of cross section;
ρ–mass density; w–angular velocity; C–coefficient.
The statistical properties are listed in Table 3.
Table 3 The statistical properties of the basic variables.
σs /MPa
n/(r/min)
C
mean value
960
12150
5.3682
standard deviation
65.1
134
0.1134
Variable
3
variable
ρ/(kg/ m )
mean value
8140
standard deviation
165
2
-3
S/m (10 )
J/m4(10-4)
[6.283 2,6.726 4]
[1.144 7,1.284 7]
0.00
-0.04
*
relative error(%)
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N =100 C=+∞ ε =0.01
-0.08
-0.12
-0.16
-0.20
20
25
30
35
40
kernel function parameterσ
Fig. 3 The variation of relative error with σ.
93
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-0.015
σ=44 C=+∞ ε=0.01
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relative error(%)
-0.020
-0.025
-0.030
-0.035
-0.040
40
50
60
70
80
90
100
*
sample size N
Fig. 4 The relative errors under different number of samples.
The structural failure degree simulated by direct Monte Carlo is 1.3×
10 (ten million times). The methods presented in this paper including sample
extraction strategy based on Markov chain and SVM are used to reconstitute the
limit state equation. Extract 100 training samples according to the above
method, choose radial basis kernel function and take the penalty factor C and
insensitive coefficient as +∞ and 0.01, respectively. The variation of the
reliability’s relative error with the kernel function parameter σ is shown in
Fig. 3. Through many debugging, the relative error reached the minimum value
–0.017% when σ=44. Fix the model parameters, and the relative error under
different number of samples is shown in Fig. 4.
It can be shown in Fig. 3 that the relative error of reliability can keep stable
with kernel function parameter σ in a large range. The absolute value of relative
error can be controlled lower than 0.1%. Fig. 4 shows that high accuracy can be
obtained when the sample size is 40 and the excellent properties of SVM for
small sample size problem and the progressiveness of the sample extraction
strategy proposed in this paper are improved.
The reasons of the high accuracy of the proposed method can be summed as
follows: ① SVM technology is based on structural risk minimization principle
and has excellent learning ability for small samples and generalization
performance. ② The training samples simulated by Markov chain can be well
distributed nearing the structural design point and the limit state surface, so there
are more information including in these samples about the structural failure or
the limit state. ③ The alternative samples and the state samples are used
together, so the information generated in sample simulation are fully used. ④
Different from the classification technology, the SVM regression technology
applied the information besides the classification mark, i.e. the corresponding
function value of these samples, so the information can be used more fully.
-4
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5. Conclusion
The limit state equations of complex engineering structures are commonly
implicit. In addition, complex structures mostly have many variables with
different types of uncertainty due to the lack of history and test information for
partial variables. In view of the implicit limit state function in structural hybrid
reliability analysis, a method based on Markov chain sample simulation strategy
and SVM regression is proposed in this paper. A serial of state points distributed
around the design point can be obtained through Markov chain simulation.
Together with the alternative samples and the corresponding response values,
the training sample set of SVM can be formed. Then, the effective
approximation of the limit state equation can be got through SVM training. The
structural hybrid reliability can be calculated combining with the Monte Carlo
method.
Acknowledgments
This research is supported by the National Natural Science Foundation of China
(51509254) and the Youth Science Funds of Naval University of Engineering
(HGDQNJJ13013/HGDQNEQJJ15009).
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