Weighted Linear Response Surface Method For Structural Reliability Analysis
Qing-hui Dai1
1
2
Jun-yuan Fang2
Department of Mechanical Engineering, North China Electric Power University, Baoding China
Department of Mechanical Engineering, North China Electric Power University, Baoding China
( 1dqh6789@126.com,2fangjunyuan@126.com)
Abstract - Response face method is a common method to
do reliability analysis for uncertain expressed complex
structures of limit state function. This thesis introduces a
weighing linear response surface method on the base of
traditional response surface method. This method mainly
chooses good sample points, and gives up bad sample points
by the way of giving the point rational weight number in
order to raise the fit precision. Then the thesis presents the
calculation steps and flow chart of the weighing linear
response surface method. Finally, it inspects and verifies the
superiority of the new method through a count case on fit
precision
Keywords - response face method, weighing, failure
probability
I. INTRODUCTION
FOSM (First Order Second moment Method) and
AFOSM (Advanced First Order Second moment Method)
are the relatively simple method in structural reliability
analysis[1]. There are some reality can’t satisfy the
prerequisite with regard of FOSM and AFOSM, such as,
it can only solve the known problem of the limit state
function. Monte Carlo[2]simulation method can fill the
lack of FOSM and AFOSM, but it’s low efficiency and
very time-consuming, and in some cases there are
convergence problems. The response surface method [15]
overcomes the deficiencies of these methods. In
addition, the response surface method can be directly
combined with the finite element which wears a structural
reliability analysis.
The traditional response surface method [6-8] rechooses
the new sample points each times of iteration, which
makes the nice sample points be ignored and logging in
the bad sample points when fitting the response surface.
At last, it makes the result of decreasing the fitting
precision. This article describes a weighted linear
response surface method, whose principle is giving the
sample a reasonable weight number, choosing the good
sample points closed to the real response surface,
eliminating the bad sample points which is far away from
the real response surface. In this way, the good sample
points can be reused, and the reliability analysis fitting
precision can be put up.
II. THE RESPONSE SURFACE METHOD
A. Reliability and reliability index
Expression of the structure of the limit state function
can be expressed as follows：
 
G X  G  x1 , x2 ,
Where,
x1 , x2 ,
, xn 
(1)
, xn are basic variables, such as
material properties, load parameters, geometric
 X   0 , F is the
corresponding failure domain. When G  X   0 , S is
dimensions, etc. When G
the corresponding failure domain. The probability of
failure represented by Pf :
   
Pf  P F   P G X  0 
Where,

 
 
f x X dx (2)
G X 0
 
f x X is the joint probability density function
of the vector of basic variables X
  x1 , x2 ,
, xn  .
There follows the relationship between the failure
probability Pf and reliability Pr :
Pf  Pr  1
(3)
When the basic variables follow a normal distribution,
there comes the following relationship:
 Pf      

 Pr     
Where,
index
(4)
 is reliability index, the higher the reliability
 is, the higher reliability Pr is.
B. Response surface methods for reliability analysis
The basic idea of response surface method is to select
the polynomial containing parameters to be determined
instead of the real limit state function, and select the
appropriate sample point to obtain the unknown
parameters. Finally, after times of iterations, it ensures the
selection of the polynomial function approximation to the
real limit state function on the probability of failure.
The core issue of response surface method is as
follows:
1) Choosing the form of polynomial function
The real limit state function G
tests-fitting
 
 X  mainly uses the
G X which can specifically express the
function relation. The selection ways of G
 X  are mostly
divided into two ways: linear polynomial:
 
n
G X  b0   bi xi
Where,
(5)
i 1
xi is the i basic variable; bi  i  0,1, 2,
, n
is the undetermined coefficient of the polynomial function.
n is the number of the basic variable.
The quadratic polynomial without cross terms shows
as follows:
 
 
b0
bi  i  0,1, 2,
n
n
i 1
j 1
is
the
constant
(6)
Where,
coefficient;
, n  is one-time items coefficient;
bn j  j  1, 2, , n  is the quadratic term coefficient;
n is the number of the basic variable.
On the circumstance of many based variables,
choosing
 
G X
A. The form of polynomial function
n
G X  b0   bi xi
G X  b0   bi xi   bn j x 2j
Where,
choose an iteration method to meet require of precision of
fitting expression failure probability and real failure
probability.
Linear response surface method is only used on the
circumstance that nonlinearity degree of the structure real
limit state equation is not big. However, most time on
practical engineering applications, the nonlinearity degree
of the real limit state equation is big and coefficient of
variation of the basic variable is small, which can also get
the high precision by using linear response surface
method. As the excellent feature of using less
approximate function of undetermined coefficients and
demanding less sample points, linear response surface
method is wildly used in engineering.
contains quadratic polynomial with
cross terms makes amount of calculation very huge, so it
is not be adopted.
2) The choice of sample points
There are many ways to select the sample points as
usual, such as Bucher design (matrix design), central
composite design, random sampling, two-level factorial
design, etc. Now the finite element soft has added the
function of selecting samples as well. For example,
ANSYS has the three ways to confirm the position of the
sample point: central composite design, matrix design and
custom-design.
3) The method of the polynomial function fitting
Presently, the most common usage of the fitting
method is least squares method. Then there expands some
weighted least square method, in order to fix the weak
point of not distinguishing the good and bad sample
points.
III. WEIGHTED LINEAR RESPONSE SURFACE
METHOD
bi  i  0,1, 2,
(7)
i 1
, n  is undetermined
coefficient, the number of coefficients to be determined
is  n  1 .
B. The determination of the undetermined coefficients in
polynomial
The
vector
b   b0 , b1 , b2 ,
of
undetermined
coefficients
, bn  can be obtained from weighted
least squares method, The number of sample points m has
to be higher than or equal to the number of coefficients of
the response surface
calculate
the
G  X i  i  1, 2,
n  1, Usually, m  2  n  1 . And
corresponding
value
, m of the sample point X i . A is
the sample matrix of
m   n  1 order of m sample
points, which shows bellow:
1


A 1


1

x1 j
xij
xmj
b   ATWA ATW y
x1n 


xin 


x mn 
1
Linear response surface method is the method that
linear polynomial is fitted by limit sate function,
confirming the n  1 undetermined coefficients of linear
polynomial through deterministic test. Finally, it will
(8)
, G  X m   , xij is
the j basic variable value of the i row, W is weight
Where,
y   G  X1  , G  X 2  ,
1) Using G  X   b0 
 X .
n
b x
i i
i 1
to fit the limited
matrix.
state function G
C. selecting the sample points
2) Using Rackwize-Fiessler method to normalize
the un-normalized basic variable.
The physical dimension, strength and deadweight of
general components conforms normal distribution. Also
there are many distributions such as normal distribution,
logarithmic distribution and weibull distribution in
ANSYS.
If the distillation function and density function of
Linear response surface method usually uses the
design method which Bucher came up with in 1990 to
select the sample points. Basically, it selects a center point
first, and then gets the sample points through the distance
deviated along the direction of coordinate axes. This
distance usually is the f times of the standard
deviation  x of the basic variable
i
xi , f is interpolated
coefficient, and most time choosing the value between 1
and 3, generally, 2 or 3 is chosen when the first iterated,
then comes to the value 1. Fig. 1 shows the selecting
sample points principle designed by Bucher in two
dimensional spaces.
x2
x2
G( X )  0
2  f  x2
2
2  f  x2
deviation of the basic variable
xi which equivalents of
xi and follows a normal distribution can be got from
formula 10 and formula 11.

G( X )  0
x1
x1
 
 xi   1 Fi  X i* 

(10)
 f  X 
*
i
i
(11)
Where,  and  are the distillation function and density
function which follow the standard normal distribution,
b. Meet the accuracy requirements
Fig.1 the selecting sample points principle designed by Bucher in two
dimensional spaces
D. The confirmation of the weight matrix W
W is the n  n diagonal matrix of weights, wi is
X i* is the pre-choose point which is supposed to be mean
value as usual.
3) Choosing a way for taking the sample point, and
the procedure comes to the follows:
① The first iteration of the sampling center point is
denoted by X
variables
weight factor. Then here comes the constructor method of
diagonal matrix of weights[9-10].
 ybest  min G  X i  ,  i  1, 2,

 w  ybest
 i GX 
i



 w1




W 





w
m


, n  , the mean value and standard
 xi  X i*   xi   1 Fi  X i* 
1  f  x1 1 1  f  x
1
a. The first iteration
fi  xi  i  1, 2,
G( X )  0
G( X )  0
Fi  xi  and
basic variable xi on limit state function is
, m
*1
, Computing the mean value of the basic
   1 , 2 , , n  , give as
X *(1)    ( 1 , 2 ,
, n )
(12)
n  1 sample points: compare the absolute
value of the i sample point's limit state function G  X i 
② Select
and
(9)
E. the calculation procedure of weighted linear response
surface method
j 's G  X j  , j  n  2 , Then select the sample
point with small absolute value, and get rid of the sample
point with large absolute value.
4) Determine the diagonal matrix of weights W from
Eq. (9).
5) The obtained Coefficient matrix
Eq. (8), then gets
 
G X
k 
b
is used in
k 
from Eq. (7), where, k is the
k times iteration.
6) Apply the FOSM/AFOSM method to obtain the
design point
X D k  of the k times iteration, then obtain
the reliability index
 
G X

k 
 k 
or failure probability
k
Pf  of
IV. EXAMPLE
Here comes the limit state function:
.
 
G X  exp  0.2 x1  1.4   x2
7) Compute the value of the limit state function

G X D k  at the point X D k  . Determine the new central
point X
* k 1
of the
k  1 times iteration from Eq. (13)
Where，the basic variables
xi  i  1, 2  follow a
standard normal distribution,
Xi 
* k 1
 Xi   
*k
X
Where,
k 
Di
X
* k 
i
G


X  GX  
G X * k 
*k
(13)
TABLE I
in order to ensure that the new center point close to the
true limit state equation.
  k     k 1   ,
then stop, where  is a given reference accuracy.
Calculation of weighted linear response surface
method given below flowchart:
Calculated result
Method
f
AFOSM
Traditional
linear
response
surface
method
weighted
linear
response
surface
method
Start
Weather basic variables are normal
distribution or not
no
Use R-F method to turn basic variables
into equivalence normal distribution
yes
 
N  0,1 , We regard
the result of AFOSM as the exact solution. Compare the
result of traditional linear response surface method and
the weighted linear response surface method with the
result of AFOSM. The results show in Table I.
k
D
X i* k 1 is the i coordinate value. This strategy is
8) Repeat the previous step until
xi
n
G X  b0   bi xi
i 1
Obtain n+1 sample point by Bucher
The number
of
sampling
β
Pf
(×10-4)
Errors
(%)
3.350
4.045
0
1
2
3
24
24
24
3.430
3.430
3.532
3.015
3.015
2.061
25.464
25.464
49.048
1
2
3
24
24
18
3.3450
3.3450
3.350
4.046
4.046
4.037
0.025
0.025
0.198
We can be seen from the table I of results. The
calculated results of traditional linear response surface
method produce large errors ，the calculated results of
weighted linear response surface method produce very
small errors. Besides, when the interpolation coefficient
f changes, the variation range of the calculated results is
smaller than the results of the traditional linear response
surface method.
Obtain diagonal matrix of weights W
Calculating
Obtain
 
b according to Eqs (8), getting G X
 k  and
XD
k 
  k  using FOSM/AFOSM
  k     k 1  
no
yes
Obtain new central point X * k 1 of (k+1) times
iteration based on Eqs. (13)
Output

Pf  1    
k

Finish
Fig.2 Weighted linear response surface method for computing flow chart
V. CONCLUSION
The traditional response surface method re-choose the
new sample points in each iteration, which makes the nice
sample points be ignored and login the bad sample points
when fitting the response surface. At last, it makes the
result of decreasing the fitting precision. After introduce
weight matrix ， the calculated results of
weighted
linear response surface method produce very small
errors，and more stable.
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