Publications

Reliability analysis of structures by iterative improved response surface
method
Somdatta Goswami, Shyamal Ghosh, Subrata Chakraborty ⇑
Department of Civil Engineering, Indian Institute of Engineering Science and Technology Shibpur, Howrah, India
a r t i c l e i n f o
Article history:
Received 12 June 2015
Received in revised form 3 February 2016
Accepted 3 February 2016
Keywords:
Reliability
Structures
Moving least-squares method
Iteration
Improved response surface
a b s t r a c t
The moving least-squares method (MLSM) is a more accurate approach compare to the least-squares
method (LSM) based approach in approximating implicit response of structure. The advantage of
MLSM over LSM is explored to reduce the number of iterations required to obtain the updated centre
point of design of experiment (DOE) to construct the final response surface for efficient reliability analysis
of structures. The initial response surface is constructed based on a simplified DOE with mean values of
the random variables as the centre point and updated successively to obtain the improved response sur-
face. The reliability of structure is evaluated using this final response surface. The basis of the efficiency of
the proposed method hinges on the use of simplified DOE instead of computationally involved full facto-
rial design to achieve desired accuracy. As MLSM is more accurate compare to LSM in evaluating response
surface polynomial, the centre point obtained is expected to be more accurate during iterations. Thus, the
number of iteration in the update procedure will reduce and the accuracy of computed reliability will also
improve. The improved performance of the proposed approach with regard to efficiency and accuracy is
elucidated with the help of three numerical examples.
Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
Uncertainty in the parameters characterising the mechanical
behaviour of a system and loads acting on it calls for reliability
analysis. The reliability assessments require computation of prob-
ability of failure. Several methods are available to estimate the
probability of failures. Those can be classified into two groups:
analytical and simulation based methods. In the first group, one
can find the well-known second moment based First Order Relia-
bility Method (FORM) and Second Order Reliability Method. In
the second group, the methods based on Monte Carlo Simulation
(MCS) can be found. All such methods are now established and
well documented in numerous texts [1–5]. These reliability analy-
sis methods involve repetitive evaluations of performance function
and it can be carried out directly so long structural response is pos-
sible to obtain explicitly. When a closed-form expression of the
performance function is available, the number of performance
function calls does not play an important role. On the contrary,
when a finite element (FE) model is involved to obtain the struc-
tural responses, each performance function evaluation may require
enormous computational time, especially when complex nonlinear
constitutive behaviours are involved [6]. Thus, assessing the relia-
bility of a complex structure requires a transaction between the
reliability algorithms and numerical methods used to model the
mechanical behaviour of the system.
The second moment based algorithms require computation of
gradients and hessians of performance function. For implicit per-
formance function, finite difference methods are usually adopted
for approximating the gradients of the performance functions. This
requires a large number of numerical computations. Furthermore,
the second moment methods cannot always provide desired accu-
racy, particularly when the levels of uncertainty in the parameters
are relatively large. Whereas in direct MCS approach, repeated
evaluation of performance function involves large number of exe-
cutions of the FE model of a structure. Thus, development of
approach requiring fairly low computational time becomes impor-
tant; particularly for safety assessments of large complex systems.
Various techniques requiring fewer samples such as the impor-
tance sampling, directional simulation, antithetic varieties etc.
are proposed in the literatures [4,7,8]. Though the number of
samples involved in such techniques is lower than that required
by the former one, still it remains important especially when anal-
ysis of complex structures by FE model is involved [9]. Further-
more, the mechanical model and the reliability evaluation
algorithm need to be merged together for the safety assessment
http://dx.doi.org/10.1016/j.strusafe.2016.02.002
0167-4730/Ó 2016 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.
E-mail address: schak@civil.iiests.ac.in (S. Chakraborty).
Structural Safety 60 (2016) 56–66
Contents lists available at ScienceDirect
Structural Safety
journal homepage: www.elsevier.com/locate/strusafe

of complex problems. For such merging, the structural FE models
are often constructed by using commercial FE software [10]. Thus,
during each simulation one needs to run the FE software which
limits the popular applications of FE method for practical struc-
tural reliability analysis problem. Therefore, alternative techniques
for efficient computation of response of complex structures by
overcoming aforementioned drawbacks while retaining the accu-
racy is of paramount importance to structural engineering commu-
nity. Response surface method (RSM) based metamodelling
technique has emerged as an effective solution to such problems.
The RSM represents a convenient way to achieve a balance
between the number of execution of the FE model and the accuracy
of computed reliability. The present study deals with efficient eval-
uation of reliability of structures in the framework of iterative RSM.
2. Response surface method and reliability of complex system
RSM is one of the most sparking developments in structural
reliability analysis. It is highly suitable for the case where the
closed form expression for the performance function is not known
and need to be evaluated by numerical methods such as FE
method. The use of RSM, originally proposed by Box and Wilson
[11], is still subject of further researches [12–14]. The develop-
ments that have taken place in the field can be studied under three
subheads: (i) the studies dealing with experimental design in the
physical space or in the standard normal space for a better control
of the distance between numerical experiments [15–21], (ii) the
studies related to the forms of response surface functions i.e. use
of polynomial response surface like simple linear and quadratic
response surface [15–17], use of neural network [10] etc. and (iii)
the studies on various methods of design of experiment (DOE)
[15–18]. The application of RSM for structural reliability analysis
was first proposed by Faravelli [22] and the subsequent works
[15–18] set the tone for its application for reliability analysis of
large and complex structural system. For constructing response
surface various DOE are used e.g. saturated design (SD), factorial
design, central composite design (CCD) etc. It is well known that
the accuracy of evaluating a performance function and its gradient
largely depends on the capability of a metamodel to capture the
nonlinearity and local variation of response behaviours. Though,
the CCD approximates far better compared to other designs, it
requires enormous input information involving structural response
analysis compared to other experimental design approaches. Thus,
in order to construct an efficient response surface requiring less
number of FE executions is much desirable for large complex struc-
tures characterised by too many input variables.
The position of the sample points, the type of polynomial
response and its performance is the subject of investigation by sev-
eral authors and is still under discussion. It is well known that in
reliability evaluation of structure using RSM, the centre point of
design to construct the response surface should be close to the
most probable failure point (MPFP) so that the response surface
obtained includes most of the failure region with sufficient accu-
racy. In fact, an improved RSM has been suggested based on this
fact [15] and studied the effectiveness of estimating reliability of
structures using simple SD based RSM. Rajashekhar and Elling-
wood [18] further suggested improvement by repeated application
of the approach to obtain a converged centre point to construct the
final response surface. Allaix and Carbone [23] used an iterative
strategy to determine a response surface that is capable to fit the
limit state function in the neighbourhood of the design point.
The locations of the sample points are decided according to the
importance sensitivity of each random variable to evaluate the free
parameters of the response surface. These studies applied the
least-squares method (LSM) to evaluate the unknown coefficients
of response surface polynomial. The level of accuracy possible to
achieve and the number of iterations required to obtain the con-
verged centre point of design depends on the type of problems,
the nature of variation of the responses and their sensitivities with
respect to uncertain parameters. The number of iterations required
may be large in many cases; thereby reducing the efficiency of the
RSM significantly. But considering the fact that the moving least-
squares method (MLSM) is much more accurate in evaluating
response surface polynomial [24], the centre point obtained by
the MLSM is expected to be more accurate during iterations. It
has been applied to update a linear response surface successively
by adding one new experimental point to the previous set of exper-
imental points [25]. However, starting with linear response surface
and adding only one extra point during iteration may not be accu-
rate enough. The present study tries to explore the advantage of
the MLSM based RSM compare to LSM based RSM to reduce the
number of iterations to obtain the converged centre point of the
DOE to construct the final response surface so that reliability anal-
ysis can be performed on this final response surface more effi-
ciently. The contribution aims in proposing an efficient strategy
for reliability analysis of structures following the basic iterative
algorithm of Rajashekhar and Ellingwood [18] by judicious applica-
tion of the MLSM based RSM instead of conventionally adopted
LSM based RSM. The basis of the efficiency hinges on the use of a
simplified DOE instead of computationally involved full factorial
design usually sought to achieve desired level of accuracy. The ini-
tial response surface is constructed based on the SD considering
the mean values of the random variables as the centre point.
Now this centre point is updated using Bucher and Bourgund
[15] algorithm until convergence. However, the response surface
polynomial coefficients are obtained by MLSM to obtain the con-
verged centre point of the DOE to construct the final response sur-
face. The reliability of the structure is evaluated based on this final
response surface. The improved performance of the proposed
approach with regard to efficiency and accuracy is elucidated with
the help of three numerical examples.
3. LSM based improved response surface for reliability analysis
The RSM is a set of mathematical and statistical techniques
designed to obtain a better understanding about the overall
response by DOE and subsequent analysis of experimental data.
The method primarily uncovers analytically complicated or impli-
cit relationship between several inputs and desired output through
empirical models (non-mechanistic) in which the response func-
tion is replaced by a simple function (often polynomial) that is fit-
ted to data at a set of carefully selected points (referred as DOE). In
a sense, RSM is a system identification procedure, in which a trans-
fer function relating the input parameters (loading and structural
system parameters) to the output parameters (response in terms
of displacements, stress, etc.) is obtained in a suitable way. The
basic procedure to obtain a response surface that will serve as a
surrogate for the FE model requires calculating the predicted val-
ues of the response features at various sample points selected as
per the experimental design.
If there are n response values yi corresponding to n numbers of
observed data, xij (denotes ith observation of the input variable xj in
a design), the relationship between the response and the input
variables can be expressed as:
y ¼ Xb þ ey ð1Þ
In the above multiple non-linear regression model X, y, b and ey
are the design matrix containing the input data from the design,
the response vector, unknown co-efficient vector and error vector,
S. Goswami et al. / Structural Safety 60 (2016) 56–66 57

respectively. The above generalised expression of response surface
can be mathematically expressed for typical quadratic polynomial
basis as following:
y ¼ b0 þ
Xk
i¼1
bixi þ
Xk
i¼1
Xk
j¼1
bijxixj ð2Þ
The LSM of estimation technique is traditionally applied to
obtain the unknown polynomial coefficient by minimising the
error norm defined as:
eðbÞ ¼
Xn
i¼1
yi À b0 À
Xk
i¼1
bixi À
Xk
i¼1
Xk
j¼1
bijxixj
!2
¼ ðy À XbÞT
ðy À XbÞ ð3Þ
And the least squares estimate of b is obtained as,
b ¼ ½XT
XŠ
À1
fXT
yg ð4Þ
It may be noted here that based on the least square regression
using equal weightage of all the data points, the polynomial coef-
ficients obtained are constant. Thus, one best possible response
surface is obtained. Now response at any desired point (x) can be
readily obtained using Eq. (2). As the coefficients remain constants
for any position x where one needs to evaluate the response, this is
considered to be a global approximation. To fit an accurate model
within reasonable time, it is required that the initial input data (X
and y) are selected judiciously. As already discussed, based on the
fact that the centre point for DOE to construct the response surface
for reliability evaluation of structure should be close to the limit
state, Bucher and Bourgund [15] proposed a procedure to improve
the response surface approximation in the context of reliability
analysis. This concept is the basis of the adaptive RSM based effi-
cient reliability analysis algorithm proposed in the present study.
Thus, the fundamental of the above mentioned approach is first
briefly discussed in the following sections for effective presenta-
tion of the proposed algorithm.
In the iterative scheme of Bucher and Bourgund [15], the mean
value of each random variable is initially selected as the centre
point. Accordingly the performance function g(x) is first approxi-
mated by a response surface using simple DOE scheme about the
mean values of the random variables, x. The function gðxÞ is used
along with the information of the mean and standard deviation
of the random variables to obtain an estimate of the design point
XD. It is not intended for use as the design point for reliability esti-
mate; rather it is used to obtain a new centre point for updating
the responses surface. Once xD is found, g (xD) is evaluated and
the new centre point xM for interpolation is chosen on a straight
line from the mean vector x to xD as following:
xM ¼ x þ ðxD À xÞ
gðxÞ
gðxÞ À gðxDÞ
if gðxDÞ P gðxÞ ð5aÞ
xM ¼ xD þ ðx À xD1
Þ
gðxDÞ
gðxDÞ À gðxÞ
if gðxDÞ gðxÞ ð5bÞ
Now a new DOE is performed around this new centre point xM
to obtain an updated response surface to approximate the perfor-
mance function. This strategy ensures that the new centre point
is closer to the exact limit state g(x) = 0. The basis of obtaining
the improved centre point for next DOE is depicted in Fig. 1. The
first approximation of the response surface based on the mean val-
ues of the random variables as the centre point of the design is con-
ceptually shown in the left side of the figure. Once xD is obtained
from Eq. (5), the improved responses surface approximation is
now obtained considering this xD as the centre point of the design
(a point closer to the MPFP) as shown in the right side of the figure.
Rajashekhar and Ellingwood [18] further suggested repeated appli-
cation of the above scheme to obtain an improved response surface
for approximating the performance function until a pre-selected
convergence criterion of ðxMiþ1
À xMi
Þ=xMi
6 e is satisfied, e can be
considered to be |0.05|. More details may be seen in Refs. [15,18].
Note that the approach is intended for applying comparatively
simple DOE scheme for efficient reliability evaluation. For example,
if one uses factorial design then 4k + 4 functions evaluations and if
saturated design is used then 2k + 1 functions evaluations are
involved, k is number of random variables involved in the perfor-
mance function.
4. MLSM based proposed efficient reliability evaluation
algorithm
The level of accuracy possible to achieve and number of itera-
tion required to obtain a converged response surface by LSM based
RSM as discussed in the previous section depends on the type of
problems. For example, for a highly non-linear problem and or a
system having very low probability of failure, the number of itera-
tions could be very high. Thus, even for a simpler DOE scheme e.g.
saturated design, the efficient evaluation of reliability of large com-
plex system as envisaged may be sacrificed because each iteration
needs to perform a fresh design with 2k + 1 more functions evalu-
ations i.e. FE executions. In this regard, it is important to note that
the original algorithm [15] and the iterative algorithm [18] to
obtain an improved response surface use the LSM in the evaluation
of the polynomial coefficient, b .
As already discussed, the conventional response surface using
LSM is based on the global approximation of scattered position
data yielding one approximation curve for entire domain. It is well
known that the LSM is one of the major causes of large approxima-
tion errors [24]. This is due to the fact that the efficiency of global
approximations depends largely on the selection of the basis func-
tions which should be chosen to resemble as closely as possible to
actual variation of the response within the domain. But, such a
selection is not straightforward [26] e.g. typically in LSM the basis
functions are fixed second order polynomial. The MLSM [24–29]
circumvents such problems by establishing a local approximation
for b(x) around each point in the desired domain through a varying
weight function with respect to the position of the approximation.
The term ‘desired domain’ means the region around the calculation
point (x) where responses are required to evaluate. Size of this
influencing region is controlled by the weight function. The weight,
associated with a particular sampling point x, decays as a point x
moves away from it. The introduction of the dependence on the
weight results in reducing the approximation error at each point
by performing a weighted local averaging of the information
obtained by the support points that are closer to it. This leads to
a smaller dependence of the fit on the type of basis functions used
[30–32]. In essence, the MLSM based RSM can be considered as a
weighted LSM that has varying weight functions with respect to
the position of approximation. There exist separate approximation
functions for each calculation point of interest. The desired points
of interest depend on the purpose for which response surface is
used. As already mentioned, in view of reliability analysis of struc-
tures, the desired domain should be as close as possible to the
MPFP so that the response surface obtained includes most of the
failure region with sufficient accuracy.
The relationship between the response and the input variables
in MLSM are recast as:
y ¼ XbðxÞ þ ey ð6Þ
58 S. Goswami et al. / Structural Safety 60 (2016) 56–66

Re-calling Eq. (3) the modified error norm Ly(x) can be defined
as the sum of the weighted errors to obtain the unknown coeffi-
cients bðxÞ,
Min: LyðxÞ ¼
Xn
iÀ1
Wie2
i ¼ T
WðxÞ ¼ ðY À XbðxÞÞT
WðxÞðY À XbðxÞÞ
which yields bðxÞ ¼ ½XT
WðxÞXŠ
À1
XT
WðxÞy
ð7Þ
In the above equation, W(x) is the diagonal matrix of the weight
function and it depends on the location of the associated approxi-
mation point of interest (x); n is the total number of sample points.
W(x) can be obtained by utilising the weighting function such as
constant, linear, quadratic, higher order polynomials, exponential
functions etc. [24] as described below,
where d is the distance of the point where the approximate response
is required (x) to the origin of the approximating domain (i.e. the
centre point of DOE) and RI is the radius of the sphere of influence.
The MLSM based response surface as presented above is obtained
by putting more importance to the data points around the point
where response is sought. This is based on the fact that as the dis-
tance of any data points increases from the calculation point, its
importance in the approximation function should reduce. In fact
beyond certain size of the influencing region data are not included.
This is controlled by the radius of influencing region RI. The accuracy
of the MLSM based RSM largely depends on the weight function W
(x). This function should prioritize support points that are close to
the interpolation point, and should vanish beyond the influence
radius RI. However, an appropriate support size RI should be selected
at any point x so that a sufficient number of neighbouring
supporting points are included to avoid singularity in the solution
for b(x) i.e. RI should include certain least number of data points
[28]. More details on this issue and the use of weight function, size
of influencing region can be found elsewhere [24,28,33,34].
The weighting function as described by Eq. (8) has its maximum
value of 1.0 at a normalised distance = 0.0 and 0.0 (the minimum
value) outside of a normalised distance = 1.0. The function
decreases smoothly from 1.0 to 0.0. The exponential form of the
weight function is used in the present numerical study. Eventually,
the weight matrix W(x) can be constructed by using the weighting
function in the diagonal terms as bellow:
WðxÞ ¼
wðx À x1Þ 0 Á Á Á 0
0 wðx À x2Þ Á Á Á 0
Á Á Á Á Á Á Á Á Á Á Á Á
0 0 Á Á Á wðx À xnÞ
2
6
6
6
4
3
7
7
7
5
ð9Þ
By minimising the modified error norm Ly(x), the coefficients b
(x) can be obtained from Eq. (7) which are no longer remain con-
stants; rather it changes as the position x (where one wants to
evaluate the value of the response surface) changes. This is inter-
preted as approximation in local domain and the moving process
performs a global approximation over the entire design domain.
As the MLSM is much more accurate in evaluating response surface
polynomial than the LSM, the centre point obtained through Eq. (5)
is expected to be more accurate during iteration. Therefore, it is
expected that the number of iteration to obtain the converged
response surface will be reduced. Also the accuracy of computed
reliability using response surface model obtained by the proposed
approach will improve. This is due to the fact that for reliability
analysis one needs to evaluate the approximate limit state function
i.e. the response surface as obtained by the RSM. Now, as the MLSM
wðx À xIÞ ¼ wðdÞ ¼
if ðd=RI 1Þ;
Constant 1
Linear 1 À d=RI
Quadratic 1 À ðd=RIÞ
2
Polynomial 1 À 6ðd=RIÞ
2
þ 8ðd=RIÞ
3
À 3ðd=RIÞ
4
Exponential expðÀd=RIÞ
8

:
if ðd=RI 1Þ wðdÞ ¼ 0
8

:
ð8Þ
Fig. 1. The strategy of updating response surface in the context of RSM based reliability analysis.

based approach yield more accurate responses than the LSM based
response surface, the reliability results obtained through MLSM
based approach will be more accurate. The essential steps required
for estimating reliability by the proposed algorithm is depicted in
Fig. 2 through a flow chart.
5. Numerical study
The results of numerical study are presented in this section to
elucidate the proposed efficient MLSM based RSM for reliability
analysis of structure involving implicit performance function. In
doing so, the improved capability of approximating response of
structure by the proposed MLSM based approach over the conven-
tional LSM based approach is first demonstrated in example one.
The improved efficiency and accuracy possible to achieve by the
proposed approach for reliability analysis of structure is studied
in the next two examples. For all the three examples, the initial
design is constructed considering the centre point at the mean val-
ues (xi) of the input variables (xi) and at axial point xi ¼ xi Æ hiri,
where hi is a positive integer (as axial points) and ri is the standard
deviation of the ith input variable. For each input variable three
axial points (hi = 1, 2, 3 etc.) are considered on each axis. The
DOE is termed as redundant design. For adding each level of data
one needs to add 2n more data points. However, when n is large,
this number will be much less than that required by CCD.
5.1. Example 1: sphere under internal pressure
The example concerns with the approximation of Von-Mises
stress of a hollow sphere subjected to internal pressure. The equiv-
alent Von-Misses stress (req) can be approximated as
reqðr ¼ r0Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðr2
r þ r2
h À 2rrrh
q
¼ ðjrr À rhjÞ ¼
3p0
2
Â
r3
1
r3
1 À r3
0
ð10Þ
where r0 and r1 is the internal and external radius, respectively, p0 is
the internal pressure, rr and rh are the radial and circumferential
stress, respectively. The details of the statistical properties of the
various random variables involve in the problem are depicted in
Fig. 2. The flow chart of the proposed MLSM based iterative RSM for reliability evaluation.
Table 1
The characteristics of the various random variables of the hollow sphere.
Random
variables
Probability density
function
Mean
value
Standard
deviation
Pressure (p0) Lognormal 120 MPa 18 MPa
Int. radius (r0) Lognormal 50 mm 3.75 mm
Ext. radius (r1) Lognormal 100 mm 7.5 mm
Fig. 3. The comparison of response predicted by the LSM and MLSM based RSM.

Table 1. This example problem has been taken up to demonstrate
the advantages of the MLSM based RSM for response approximation
otherwise closed form solution is already available and one really
does not require applying RSM for this.
To demonstrate the effectiveness of the MLSM based RSM in
approximating the response of the sphere, the variations of the
mean Von-Misses stress with number of simulation as obtained
by both the LSM and MLSM based response surfaces and direct
MCS are shown in Fig. 3. For comparative study 106
numbers of
random samples for each variable are generated using the associ-
ated known pdf. These are used as the input in the direct MCS to
obtain the exact response using Eq. (10) and also to predict the
response from the approximate response surfaces. It is evident
from the figure that the SD-LSM cannot predict the actual response
with sufficient accuracy. As expected it is evident from Fig. 3 that
the CCD-MLSM gives the best approximations of the response.
Though, the SD-MLSM approximates reasonably; there is scope of
further improvement of the prediction as notable variations are
observed with the direct MCS results.
Furthermore, to study the effectiveness of the MLSM based
RSM, various statistical metrics i.e. the Root Mean Square Error
(RMSE), the co-efficient of determination (R2
) and the average pre-
diction error (em) are computed to study the performance of both
the RSMs. The use of these statistics is quite common in the liter-
atures to study the suitability of various metamodels which are
readily computed from the following equations [35,36]:
RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Xp
i¼1
ð^yi À yiÞ
2
=p
v
u
u
t ; R2
¼
Pp
i¼1ð^yi À yiÞ
2
Pp
i¼1ðyi À yiÞ2
and
em ¼
Pp
i¼1 100 j^yiÀyij
yi

p
ð11Þ
In the above equations, p is the total numbers of samples (taken
as 106
for the present numerical computation), yi is the actual
response obtained by the direct MCS for the ith sample point.
The responses are evaluated at each of the simulated random sam-
ple points by both the RSMs and termed as the predicted responses
^yi in Eq. (11). It is to be noted that the MLSM based approach uses
the weight factor to obtain the response surface for each calcula-
tion of ^yi i.e. b(x) is different for each simulated sample. However,
when ^yi is calculated by the LSM based approach, b remains same
for each calculation i.e. there will be only one response surface for
all the sample points. The results of statistical tests for both the SD
and CCD based response surfaces are shown in Table 2. As
expected, it can be noted from the table that lesser RMSE and em
values and higher R2
value are attained by the CCD based approach
for both the response surfaces. Though, CCD gives very accurate
results yet the applicability of SD seems to be relevant considering
the fact that CCD requires 2n
+ 2n + 1 design points whereas the
later requires 2n + 1 points for n number of variables and for each
added level one needs to add 2n extra data points. However, when
the number of finite element runs required for large complex
structures for example for the third numerical problem having
21 variables one needs N ¼ 221
þ 2 Â 21 þ 1 ¼ 20; 97; 195 runs
to generate response surface by CCD and its use becomes imprac-
tical. In the present example, four random variables require only
25 runs by the CCD and seem to be not a computationally challeng-
ing task. This example problem has been taken up for demonstrat-
ing the applicability of the proposed approach to improve the
accuracy of response prediction. Though, the SD-MLSM provides
better approximations compared to SD-LSM, however the differ-
ence with CCD is still notable and warns the use of SD based
RSM for reliability evaluations. Thus, an adaptive RSM is attempted
in the present study following the iteration strategy [18] using
MLSM based approach instead of LSM based approach. The effec-
tiveness of the proposed approach is studied in the next two
examples.
5.2. Example 2: a three dimensional truss
The example problem considered is a space truss having nine
bar members as shown in fig. 4. A vertical load of 2 kN is applied
at all nodes of the truss. The horizontal load F is acting in the
x-direction at the top three nodes of the truss which is considered
to be random. The statistical properties of the various random vari-
ables involve in the problem are depicted in Table 3. The model
consists of 15 nodes and 39 element groups. The reliability is com-
puted with respect to the following implicit performance function,
gðXÞ ¼ UA À U0 ð12Þ
where UA is the displacement at the top corner point A of the truss
and U0 is the allowable displacement. The LSM and MLSM based
approaches are now applied to obtain the updated response surface
for reliability analysis. The LSM based update is exactly the same as
proposed by Rajashekhar and Ellingwood [18]; whereas the MLSM
based approach uses the weight factor to obtain the response sur-
face during iteration. Figs. 5a–5d shows the comparison of conver-
gence of reliability index by the proposed MLSM and the
conventional LSM based RSM approach to that of obtained by the
direct MCS study for various combination of allowable displace-
ment (defining safety), thickness of truss members and horizontal
load F at the top nodes. The probability of failure (pf) is estimated
by the direct MCS considering 106
random response samples
obtained by the FE analysis. The MCS based reliability index is then
obtained as: b = /À1
(1 À pf). It can be noted from these plots that
the reliability results obtained by the MLSM based updated
approach with fewer iterations are very close to the reliability
indices computed by the direct MCS. Whereas, a significant differ-
ence is noted for the computed reliability results by the LSM based
updated approach to that of obtained by the direct MCS approach
even after several iterations.
The accuracy of reliability index prediction by the proposed
MLSM based RSM are further compared for different allowable dis-
placement, for different thickness of the truss members and for dif-
ferent horizontal forces in Tables 4–6, respectively. The reliabilities
are evaluated based on the final converged response surface. The
results of the direct MCS are also shown in the same table to study
the level of accuracy possible to achieve by different approaches.
As expected, the reliability evaluated considering the converged
response surface by simulation approach is better than that of
obtained by the FORM algorithm. The improvement in the reliabil-
ity results by the proposed approach over the conventional LSM
based approach is noted for all cases when compare to the reliabil-
ity results obtained by the direct MCS study. However, the
improvement in the computed reliability by the proposed MLSM
based approach is significant with respect to that of achieved by
the LSM based approach for relatively larger values of the
Table 2
The various statistical indices as obtained by the LSM and MLSM based RSM.
RMSE (%) R2
em (%)
SD Level 1 LSM 81 0.724 76
MLSM 13.89 0.925 13
Level 2 LSM 13.40 0.9743 12.5
MLSM 12.21 0.9841 10.3
Level3 LSM 7.8 0.984 10.1
MLSM 6.3 0.9861 8.5
CCD LSM 3.4 0.990 3.18
MLSM 2.1 0.994 2.05

reliability index. In this regard it may be pointed out that usually
the probability of failure of typical structural engineering problems
is quite low i.e. corresponding to larger values of reliability index.
Thus, the proposed MLSM based iterative RSM is expected to be
useful compare to LSM based RSM for safety evaluation of struc-
tures involving implicit performance function. The observation
with FORM results are similar; off course as expected the level of
accuracy is less than the simulation based LSM and MLSM results.
In this regard, it is important to note here that the reliability
indices are computed by considering as much iteration required
for obtaining a converged response surface. However, more study
is felt essential in this regard to fully explore the advantage of
the proposed MLSM based iterative algorithm for nonlinear and
time dependent reliability analysis problem.
Fig. 4. The space truss: (a) 3D view, (b) top view.
Fig. 5b. The comparison of convergence of reliability (U0 = 16 mm, D = 140 mm,
t = 4 mm, F = 8kN).
Fig. 5a. The comparison of convergence of reliability (U0 = 14 mm, D = 140 mm,
t = 4 mm, F = 8kN).
Table 3
The statistical properties of the various random variables of the truss.
Random variables Probability density function Mean value COV (%)
Young’s modulus (E in kN/m) Normal 2.1 Â 108
10
Horizontal load (F in kN) Normal 8 20
Diameter of the truss members (D in mm) Uniform 140 5
Thickness of the truss members (t in mm) Uniform 4 5

5.3. Example 3: a five storey frame structure
The second example considered is a five storeyed frame
structure as shown in Fig. 6. The frame structure consists of
thirty-five members. Each frame member is modelled by a two
noded finite element i.e. a line element having two nodes. Each
node has 3 degrees of freedom and the element stiffness matrix
for each member is of size 6 Â 6. The variables Ei, Ai and Ii represent
the Young’s modulus, cross section area and moment of inertia of
the ith element, respectively. The variables Fj are the horizontal
loads at various nodes as shown in the figure. The statistical prop-
erties of twenty-one random variables are detailed in Table 7. The
reliability is computed with respect to the implicit performance
function as, gðXÞ ¼ Dx À U0, Dx is the displacement at top right
node as shown in Fig. 6 and U0 is the allowable displacement.
Figs. 7a and 7b shows the convergence rate of the computed
reliability index for maximum allowable displacement of 50 mm
and 60 mm, respectively. For parametric study, the joint loads
are varied by multiplying Fj by a load factor. The load factor is con-
sidered to be 1.0 to develop these plots. Figs. 7c and 7d further
depict the convergence rate of estimated reliability index for load
factor of 0.9 and 1.1, respectively. The maximum allowable dis-
placement is taken as 50 mm. Finally, Figs. 8a and 8b compares
the accuracy of the predicted reliability results for varying allow-
able displacement and load factor. The results depict the level of
accuracy possible to achieve by the proposed MLSM bases
approach compare to the LSM based approach after obtaining con-
verged centre point based on which final response surface is
constructed.
Based on the detailed study of the numerical results of the
example problems it can be readily observed that the rate of con-
vergence towards the direct MCS value is not only fast but the
accuracy is also far better by the proposed MLSM based approach,
making it more efficient for updating the response surface itera-
tively for final reliability evaluation. In this regard it may be
pointed out here that the numbers of iterations for different cases
are not fixed values by either RSMs. It has been generally noted
that 3–4 iterations are sufficient in most of the cases to obtain a
converged centre point by the MLSM based approach whereas 8–
10 iterations are necessary by the LSM approach. Even in some
cases it did not converged after 10 iterations by the LSM based
approach. This clearly shows the effectiveness of the prosed MLSM
based iterative scheme to obtaining response surface for reliability
analysis of structure having implicit limit state function. The obser-
vation on convergence is consistent for all the examples studied
and also for a wide range of variation of different parameters
involve in the respective problems. It is important to note that
for achieving fast convergence, the accuracy of estimated reliability
is not sacrificed when compared with the reliability results
obtained by the direct MCS, making it more efficient for RSM based
reliability evaluation. As expected, the accuracy of the estimated
reliability is always better by the MCS performed on the updated
response surface obtained by the LSM or the MLSM approaches
when compare with the corresponding second moment based reli-
ability results. However, it has been generally observed that the
reliability result obtained by the proposed MLSM based approach
are always close to the reliability results obtained by the direct
MCS based study than that of obtained by usually adopted LSM
based RSM.
It may be mentioned here that the value of ‘h’ has large effect on
accuracy and it has been suggested that with the improvement of
response surface in each iteration, progressively smaller values of h
Fig. 5d. The comparison of convergence of reliability (U0 = 10 mm, D = 140 mm,
t = 4.5 mm, F = 8kN).
Fig. 5c. The comparison of convergence of reliability (U0 = 10 mm, D = 140 mm,
t = 4 mm, F = 8kN).
Table 5
The comparison of reliability indices for different thickness.
Thickness of the
truss members (t)
Reliability index (D = 140 mm, force = 8 kN,
U0 = 10 mm)
LSM-
FORM
LSM-
simulation
MLSM-
FORM
MLSM
simulation
Direct
MCS
4 1.314 1.239 1.312 1.269 1.28
5 2.494 2.279 2.071 2.079 2.08
6 3.488 3.021 2.707 2.719 2.71
Table 4
The comparison of reliability indices for different allowable displacement.
Allowable
displacement (U0)
Reliability index (D = 140 mm, t = 4 mm force = 8kN
LSM-
FORM
LSM-
simulation
MLSM-
FORM
MLSM
simulation
Direct
MCS
10 1.314 1.239 1.312 1.269 1.28
14 3.171 2.781 2.520 2.514 2.51
16 3.984 3.314 3.011 3.03 3.00

should be taken [18]. But, exactly what will be the magnitude of h
is not very clear. The present choice of h value yields the response
surface with sufﬁcient accuracy by the MLSM based updating
procedure as may be seen from the plots. But, the same is not true
in case of LSM based approach. In Figs. 5 and 7, though the
response surface converged in case of MLSM based approach
within few iterations; this is not the case for LSM based approach
Fig. 6. The ﬁve storied frame.
Fig. 7a. The comparison of convergence of reliability (U0 = 50 mm).
Fig. 7b. The comparison of convergence of reliability (U0 = 60 mm).
Table 6
The comparison of reliability indices for different horizontal forces.
Horizontal force (F) Reliability index (D = 140 mm, t = 4 mm U0 = 10 mm)
LSM-FORM LSM-simulation MLSM-FORM MLSM-simulation Direct MCS
6 2.892 2.715 2.478 2.611 2.61
7 1.857 1.920 1.858 1.899 1.88
8 1.314 1.239 1.312 1.269 1.28

in many cases even after 10 iterations. In this regard it is to be
noted that the present algorithm needs the evaluation of reliability
index and associated design point xd in each iteration to update the
centre point using Eq. (5) to obtain the improved response surface
successively. Thus, one should use suitable algorithm for comput-
ing reliability index and associated design point. If the level of
uncertainty is high for any of the important random variables
involved in the performance function, then one should use the
MCS approach instead of FORM to evaluate these quantities. As it
is well known that FORM is based on ﬁrst order perturbation which
is valid only for comparatively small uncertainty of random vari-
ables. However, this aspect needs further study; particularly the
number of iterations to get converged centre point in case of larger
level of uncertainty.
6. Summary and conclusions
An MLSM based adaptive RSM method is presented to update
the centre point of design iteratively to construct a much improved
response surface efﬁciently for reliability assessment of large com-
plex system. The improved capability of approximating responses
by the MLSM based RSM over LSM based RSM is clearly noted in
Fig. 8a. The comparison of reliability for different allowable displacement (load
factor 1.0).
Table 7
The statistical properties of various random parameters involve in the frame problem.
Variable Distribution Unit Mean Standard deviation
F1 Gumbel max kN 160.1448 32.0290
F2 Gumbel max kN 106.764 21.3528
F3 Gumbel max kN 85.41 17.0820
EB Normal kN/m2
23796360 2379636
EC Normal kN/m2
21737520 2173752
I1 Normal m4
0.0021 0.0002
I2 Normal m4
0.0036 0.0004
I3 Normal m4
0.0067 0.0007
I4 Normal m4
0.0076 0.0008
I5 Normal m4
0.0038 0.0004
I6 Normal m4
4.99EÀ03 4.99EÀ04
I7 Normal m4
4.68EÀ03 4.68EÀ04
I8 Normal m4
8.24EÀ03 8.24EÀ04
A1 Normal m2
0.10375 5.19EÀ03
A2 Normal m2
0.125 6.25EÀ03
A3 Normal m2
0.1725 8.63EÀ03
A4 Normal m2
0.18 9.00EÀ03
A5 Normal m2
0.15 7.50EÀ03
A6 Normal m2
0.165 8.25EÀ03
A7 Normal m2
0.18 9.00EÀ03
A8 Normal m2
0.195 9.75EÀ03
Fig. 7c. The comparison of convergence of reliability (load factor = 0.9).
Fig. 7d. The comparison of convergence of reliability (load factor = 1.1).
Fig. 8b. The comparison of reliability for different load factor (allowable displace-
ment 50 mm).

the example one. The various statistical metrics compute also con-
firms the observations. It is noted from the last two examples that
the MLSM needs less iteration compare to the LSM to obtain the
converged centre point to construct the final response surface.
Even it is noticed that the LSM did not convergences after several
iterations in some cases. This definitely indicates the effectiveness
of the proposed MLSM based iterative scheme in obtaining
improved response surface for reliability analysis of structure hav-
ing implicit limit state function. However, the numbers of itera-
tions for different cases are not fixed values by either approaches
and it is problem dependent. It has been generally observed that
the level of accuracy possible to achieve to compute reliability of
the considered structures by the proposed approach are always
better compare to that of obtain by the LSM based approach. Thus,
the proposed MLSM based adaptive RSM could be a potential algo-
rithm for reliability analysis of large complex structures. However,
the present numerical study has been performed for linear static
problems only and needs more study for reliability of structures
involving nonlinear and time dependent reliability analysis prob-
lem to fully explore the advantage of the proposed approach.
Acknowledgement
The financial support received in TSD Scheme No. DST/TSG/
STS/2012/45, 21.1.13 from the DST, Govt. of India in connection
with this work is gratefully acknowledged.
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International Conference on Structural Engineering and Mechanics
December 20-22, 2013, Rourkela, India
ADAPTIVE RESPONSE SURFACE METHOD IN STRUCTURAL RESPONSE
APPROXIMATION UNDER UNCERTAINTY
Somdatta Goswami
1
Subrata Chakraborty
2
Shymal Ghosh
3
1
PhD Student, Dept.t of Civil Engineering, Bengal Engineering and Science University, Shibpur, Howrah, India.
Email: somdatta89@gmail.com
2
Professor, Department of Civil Engineering, Bengal Engineering and Science University, Shibpur, Howrah,
India. Email: schak@civil.becs.ac.in
3
PhD Student, Dept.t of Civil Engineering, Bengal Engineering and Science University, Shibpur, Howrah, India.
Email: gh13shyamal@gmail.com
ABSTRACT:
Repeated evaluation of structural response is required in reliability analysis, optimization, and model updating
due to iterative nature of computational algorithms. If the structure of interest is represented by a complex finite
element (FE) model, large number of computations involved can rule out many approaches due to the expense
of carrying out many runs involving considerable computational time. Furthermore, the structural FE models are
often constructed by using commercial FE software and during each iteration one needs to run the FE software
which limits the popular applications of FE method in practice. Hence, alternative techniques for efficient
computation of response of complex structures by overcoming aforementioned drawbacks while retaining the
accuracy is of paramount importance to structural engineering community. Response Surface Method (RSM)
has emerged as an effective solution to such problems. For large complex structures characterized by large
number of input variables, in order to make an efficient response surface requiring less number of FE executions
is very much desirable. In this regard, the Moving Least Square Method (MLSM), a local approximation
approach fits the response curve more accurately than Least Square Method (LSM), basically a global
approximation method. The focus of the present work is on the application of adaptive RSM for efficient
computation of structural response. For this, RSM is proposed using redundant design (i.e. SD with 2 or 3 levels)
in the framework of MLSM.
KEYWORDS: Complex Structure, Response Surface Method, Moving Least Square Method
1. INTRODUCTION
The reliability analysis, optimization, and model updating of structures typically needs to evaluate structural
response repeatedly due to iterative nature of most of the computational algorithms. Nowadays, the mechanical
response of structures is obtained with much improved numerical modelling for better accuracy. Among techniques
that can perform such modelling, the finite element (FE) method is probably the most used due to its ability to deal
with complex geometries and highly nonlinear constitutive behaviours. Thus, when the structure of interest is
represented by a complex FE model, large number of computations involved can rule out many algorithm for
reliability analysis, optimization, model updating etc. due to the expense of carrying out many runs involving
considerable computational time. Furthermore, the structural FE models are often constructed by using various
commercial FE software like ANSYS, ABAQUS, NASTRAN etc. and during each iteration one needs to run the
FE software which limits the popular applications of FE method for the above mentioned applications in
practice. Hence, alternative techniques for efficient computation of response of complex structures by
overcoming such drawbacks while retaining the accuracy are of paramount importance to structural engineering
community. Response Surface Method (RSM) has emerged as an effective solution to such problems.

The RSM is one of the most sparking developments in structural reliability analysis. It is highly suitable for the
case where the performance function has no known closed form expression and need to be evaluated by
numerical methods such as FE method. The use of RSM, originally proposed by Box and Wilson [1], is still
subjected to further researches [2-4]. The developments that has been taken place in the field of RSM can be
studied under three subheads: (i) Studies dealing with experimental design in physical space or in the standard
normal space for a better control of the distance between numerical experiments [5-10], (ii) Studies dealing with
the forms of RS functions i.e. use of polynomial RS like simple linear RS [5], a quadratic RS without mixed
terms [4,6], use of neural network [8] etc. and (iii) studies on various methods of experimental design in RSM
[4-7]. The application of RSM in structural reliability was first proposed by Faravelli [11] and the subsequent
works [4-7] set the tone for application of RSM for reliability analysis of large and complex structural system.
For constructing the RS various design of experiment (DOE) are used e.g. Saturated Design (SD), Factorial
Design (FD), Central Composite Design (CCD) etc.
It is well known that the accuracy of evaluating a performance function and its gradient largely depends on the
capability of a meta-model to capture the nonlinearity and local variation of behaviours. Though, the CCD
approximates very well compared to other designs, it requires enormous input structural response information
compared to other DOE approaches. Thus, in order to make an efficient RS requiring less number of FE
executions is very much desirable for large complex structures characterized by large number of input variables.
In this regard, the Moving Least Square Method (MLSM), a local approximation approach fits the response
curve more accurately than Least Square Method (LSM), basically a global approximation method [12-14]. The
focus of the present work is on the application of adaptive RSM for efficient computation of structural response.
For this, RSM is proposed using redundant design (i.e. SD with 2 or 3 levels) in the framework of MLSM.
Numerical study is performed to study the efficiency and accuracy of the proposed algorithm compared to the
CCD based RSM and direct simulation results.
2. RESPONSE SURFACE METHOD
The RSM is a set of mathematical and statistical techniques designed to gain a better understanding about the
overall response by DOE experiments and subsequent analysis of experimental data. The method primarily
uncovers analytically complicated or an unknown relationship between several inputs and desired output
through empirical models (non-mechanistic) in which the response function is replaced by a simple function
(often polynomial) that is fitted to data at a set of carefully selected points (referred as DOE), normally obtained
from experimental investigation or numerical simulation. In a sense, RSM is a system identification procedure,
in which a transfer function relating the input parameters (loading and system conditions) to the output
parameters (response in terms of displacements, stress, etc.) is obtained in a suitable way. To create a response
surface that will serve as a surrogate for the FE simulation model, the basic process consists of calculating
predicted values of the response features at various sample points in the parameter space by performing an
experiment at each of those points.
2.1 Design of Experiments
The DOE to select different sample points in the variable space is very important in the RSM application as it
will directly influence the efficiency and the accuracy of the response surface model fitting procedure. There are
various sample point selection methods, developed suitably for different types of RSM applications. The
popularly used methods in structural reliability analysis are the Saturated Design (SD), the Central Composite
Design (CCD) and the Factorial Design (FD). In SD the points chosen are the mean values of the response
variable xi (as centre point) and at axial points, xi = ±ℎ , where ℎ is a positive integer. The points chosen
for a type-3 two variables RSM are shown in figure 1a, hi are considered to be 1.0 in figure 1. The corner, axial
and centre points are selected in the CCD for fitting a second order response surface model. As it requires a
relatively large number of sample points, the CCD method is generally chosen only in the later stage of RSM
application when the number of important variables is reduced reasonably. The CCD consists of a complete 2k
factorial design (factorial design is detailed in next paragraph), centre point and two axial points on the axis of
each input variable at a distance of γ from the design centre, where γ = 2 in order to make the design cyclic.

For example, the sample points for a Type 3 two variables model are shown in figure 1.b. In between the SD and
the CCD methods, there is another popular design method known as the FD method (Figure 1.c), which selects
the corner points only. The full FD method generates q sample values for each co-ordinate, thus producing a
total of sample points for k variables. The 2k factorial points (i.e. q=2) for a sample of two variable problem
are presented in figure 3. It can be noted that even for moderate values of q and k this may become prohibitively
expensive (e.g. for q=3 and k=5, number of minimum sample points required is 243). However, for lower values
of q and k the method is quite effective (e.g. for q=2 and k=2, number of minimum sample points required is 4).
Figure 1.a The SD for two
variables RSM
Figure 1.b The CCD for two variables
RSM
Figure 1.c The FD for two
variables RSM problem
2.2. Conventional LSM based Response Surface Method
If there are n response values yi corresponding to n numbers of observed data, xij (denotes i-th observation of the
input variable xj in a DOE), the relationship between the response and the input variables can be expressed by
Eq. 1.
y+ εy = Xβ (1)
In the above multiple non-linear regression model X, y, β and εy are the design matrix containing the input
data from the DOE, the response vector, the unknown co-efficient vector and the error vector, respectively.
Typically, the quadratic polynomial form used in the RSM is as following:
k k k
i i ij i j
i i j
x x x∑ ∑ ∑
= = =
β + β + βy = 0
1 1 1
(2)
The LSM of estimation technique is usually applied to obtain the unknown polynomial coefficient by
minimizing the error norm defined as in Eq. 3.
( ) ( )
2
n k k k T
i i i ij i j
i i i j
L y x x x )∑ ∑ ∑ ∑
= = = =
 
= −β − β − β = β β 
 
y - X y - X0
1 1 1 1
(3)
And the least squares estimate of β is obtained as,
{ }T T−
 β =
 
X X X y
1
(4)
Once the polynomial coefficientsβ are obtained from the above equation, the response y can be readily
evaluated for any set of input parameters. To fit an accurate model within reasonable time, it is required that the
initial input data (X and y) are selected judiciously. The main disadvantages of applying LSM to determine the

coefficients of approximating polynomials to obtain a suitable response surface are limitations in the shapes that
linear models can assume over long ranges, possibly poor extrapolation properties, and sensitivity to outliers. As
the explanatory variables become extreme, the outputs of the linear model will also always more extreme. This
means that linear models may not be effective for extrapolating the results of a process for which data cannot be
collected in the region of interest. Of course extrapolation is potentially dangerous regardless of the model type.
Finally, while the method of least squares often gives optimal estimates of the unknown parameters, it is very
sensitive to the presence of unusual data points in the data used to fit a model. One or two outliers can sometimes
seriously skew the results of a least squares analysis. This makes model validation, especially with respect to
outliers, critical to obtaining sound answers to the questions motivating the construction of the model. Moreover,
LSM is global approximation of the data. One of the limitations of using least squares method in analysis is that
outliers, which are significantly bad observations, can skew the results because they have more impact. This
impact is because the square of a number grows large faster than the number. It is better to reject the outliers
using some other method prior to using least squares on the remaining data. Of course, this must be
substantiated because rejecting data otherwise is bad practice.
2.3 MLSM Based Proposed Response Surface Method
The fundamental concepts of the LSM and MLSM based RSM construction is explained considering figure 3.
The DOE points (or the scattered input data points) are denoted by the dots. The LSM yields a global
approximation, which is represented by the dotted curve. But, such predicted responses fail to capture the actual
trend of the responses within a local domain which is pointed out by the large circles. On the other hand, owing
to in-built weight factors, the MLSM produces, basically a local approximation approach and can capture the
zone-wise variations of the actual responses. Considering the scattered data as a whole, only one approximation
curve can be obtained by the LSM based prediction. On the contrary, with the MLSM based approach, there
exist separate approximation functions for each calculation point of interest. Therefore, the coefficients of the
response surface model are not constants but are varying with respect to the position of the desired point of
interest. This locally weighted approximation can be obtained from the consideration of the effective data near
the desired location and the data is weighted according to the distance from this location. The concepts of the
LSM and the MLSM based RSM approximations.
Figure 2 The concept of LSM and MLSM based
RSM approximations
Figure 3 The space truss: (a) 3D view, (b) side view and
(c) top view
In essence, the MLSM based RSM can be considered as a weighted least squares method that has varying
weight functions with respect to the position of approximation. Re-calling equation (4) the modified error norm
Ly(x) can be defined as the sum of the weighted errors,

( )
1
2 T
y
1
Min.L x ( )
whichyields
Q
T
i i
i
T
( x ) (
W
x )
ε
−
−
∑
 =
= =
 
= T
W x (Y - xβ) W(x)(Y - xβ
β x W(x)x x W
)
y
ò ò
(5)
In the above equation, W(x) is the diagonal matrix of the weight function and it depends on the location of the
associated approximation point of interest (x); Q is the total number of sample points. W(x) can be obtained by
utilizing the weighting function such as constant, linear, quadratic, higher order polynomials, exponential
functions etc. [23] as described below,
2
2 3 4
Constant 1
Linear 1 /
( / 0), Quadratic 1 ( / )
( ) ( )
Polynomial 1 6( / ) 8( / ) 3( / )
Exponential exp( / )
( / 0) ( ) 0
I
I I
I
I I I
I
I
d R
if d R d R
w x x w d
d R d R d R
d R
if d R w d
 
  −  −
− = =   − + − 
 −

=
(6)
Where, d is the distance of the point where the approximate response is required to the origin of the
approximating domain and IR is the radius of the sphere of influence, chosen as twice the distance between the
centre point and the extreme most experimental point. The value of RI is so chosen in order to secure sufficient
number of neighbouring experimental points so as to avoid singularity. More details about the calculation of RI
can be found elsewhere [12, 14-15]. It can be noted from equation (8) that the weight associated with a
particular sampling point xi decays as the point x moves away from xi. The weighting function has its maximum
value of 1.0 at 0.0 normalized distance and 0.0 (minimum value) outside of 1.0 normalized distance, i.e. w (0.0)
= 1.0, w (d/RI1.0) = 0.0. The function decreases smoothly from 1.0 to 0.0. The exponential form of the weight
function is used in the present numerical study. Eventually, the weight matrix W(x) can be constructed by using
the weighting function in diagonal terms as bellow:
1
2
( ) 0 ... 0
0 ( ) ... 0
( )
... ... ... ...
0 0 ... ( )n
w x x
w x x
W x
w x x
− 
 − =
 
 
− 
(7)
Other form of weight functions are also adopted in the previous study [26]. By minimizing the modified error
norm Ly(x), the coefficients β can be obtained from Eq. 7. It can be noted that the coefficient β is a function of
the location or position x. Thus, the procedure to calculate ‘β’ is a local approximation, and ‘moving’ processes
performs a global approximation over the entire design domain. It is expected that not only the number of
iteration to obtain the converged RS will be reduced but also the accuracy of computed reliability using RS
obtained by the proposed MLSM based approach will be also improve.
3. NUMERICAL STUDY
The example considered is a thirty nine bar truss structure as shown in Figure 3. The height of the structure is 16
m. Its basis is an equilateral triangle of side 6.93 m. The model consists of 15 nodes and 39 elements grouped
into 4 design variables. A vertical load of 2kN is applied at all nodes of the truss. The horizontal load F is acting
at the top three nodes of the truss in the x-direction which is considered to be random. The statistical properties
of various random variables involve in this problem are depicted in Table 1.

Table 1 39-Bar Truss: Characteristics of the Random Variables
Random Variables
Probability Density
Function
Mean Value COV
E(kN/m2
) Young’s Modulus Normal 2.1*108
10%
F(kN) Horizontal Loading Normal 8000 20%
D(mm) Diameter Uniformly Distributed 140 5%
T(mm) Thickness Uniformly Distributed 4 5%
To demonstrate the effectiveness of the MLSM based RSM in approximating the response of structure, the
variation of mean horizontal displacement at node ‘A’ with number of simulation as obtained by the SD based
RSM and CCD based RSM and direct MCS are shown in figure 4 a and b, respectively. Both the LSM and
MLSM based RSM as mentioned in the previous sections are used to approximate the response for comparative
study. The SD method consists of nine data points and the CCD needs twenty five data points for four random
variables considered in the present study. The response evaluations at those points are performed by FE analysis
of space truss using two-nodded truss element having three degrees of freedom at each node. For comparative
study, the responses are also obtained by the direct MCS technique. For this, random samples of the four
random variables are generated following the associated known probabilistic information as furnished in table 1.
Also the same input (as generated for direct MCS) are entered in all the RS models to obtain the approximate
response from the associated RS models. From figure 4 it is evident that the SD-LSM cannot predict the FE
results with sufficient accuracy. Though, the SD-MLSM approximates reasonably; there is scope of further
improvement of the prediction as notable variation with direct MCS results observed. As expected it is evident
from figure 4b that the CCD-MLSM gives the best approximations of responses, but more numbers of FE
response evaluations are required in the DOE.
2000 4000 6000 8000 10000
7.20
7.22
7.24
7.26
7.28
MeanDisplacement(mm)
Number of Simulation
SD_LSM
SD_MLSM
Direct MCS
Figure 4 Comparison of the response predicted by
SD (LSM and MLSM)
2000 4000 6000 8000 10000 12000
7.23
7.24
7.25
7.26
7.27
7.28
CCD_LSM
CCD_MLSM
Direct MCS
Figure 5 Comparison of the response predicted by CCD
(LSM and MLSM)
Now to further improve the SD-MLSM results, the redundant design (RD) are used. Figure 6 shows the
variation of the mean displacement of node A, along the x-axis with number of simulation for DOE considering
level 1, level 1+2 and level 1+2+3 data points considering both LSM and direct MCS. Figure 7 shows the
variation of mean displacement of node A, along the x-axis with the number of simulation for level 1, level 2
and level 3 DOE data for SD-MLSM and direct MCS. From figure.7 it can be noted that if 3 levels of data are
used to construct the RS curve, it better approximates, rather almost close to direct MCS results.

2000 4000 6000 8000 10000
7.20
7.22
7.24
7.26
7.28
7.30
SD1_LSM SD2_LSM
SD3_LSM Direct MCS
Figure 6 Comparison of the response obtained by
different levels of SD-LSM
2000 4000 6000 8000 10000
7.20
7.24
7.28
SD1_MLSM SD2_MLSM
SD3_MLSM Direct MCS
Figure 7 Comparison of the response obtained by
different levels of SD-MLSM
Furthermore, to study the effectiveness of the MLSM based RSM, various statistical metrics i.e. the Root Mean
Square Error (RMSE), the co-efficient of determination (R ) and the average prediction error (εm) usually used
to check the validity of the meta-models are computed for both the LSM and MLSM based RSM [27-28]. These
are briefly discussed in the following. The RMSE is defined as:
2
1
ˆ( ) /
p
i i
i
y y p
=
−∑ , (8)
In the above, ˆiy is the predicted response obtained by the meta-model and iy is the actual response obtained
by the direct MCS for ith
sample point. The total number of sample (p) is taken as one lakh for the present
numerical computation. The co-efficient of determination (R2
) and the average prediction error ( m) is defined as:
2
2 1
2
1
ˆ( )
( )
p
i i
i
p
i i
i
y y
R
y y
=
=
−
=
−
∑
∑
And
1
ˆ| |
(100 )
p
i i
i i
m
y y
y
p
ε =
−
=
∑
(9)
For a good metamodel, these coefficients should be as close as possible to the value of 1.0 and RMSE should be
minimum. The trends of the numerical results as shown in table 2 clearly indicate the superiority the MLSM
based RSM.
Table 2 The performance of LSM and MLSM based RSM
RMSE R2
mε
SD
Level 1
LSM 82.50% 0.654 9.3%
MLSM 26.41% 0.934 2.71%
Level 1+2
LSM 26.28% 0.954 2.67%
MLSM 26.18% 0.956 2.65%
Level 1+2+3
LSM 26.25% 0.957 2.66%
MLSM 26.15% 0.964 2.64%
CCD
LSM 12.1% 0.984 1.96%
MLSM 11.2% 0.986 1.38%

4. CONCLUSIONS
An efficient MLSM based RSM method is investigated for approximating responses of mechanical model;
otherwise typically which need to be obtained by FE method. The approach is intended for response evaluation
of large complex system. Thus, simplified DOE i.e. SD is adopted for efficient computation. The CCD-MLSM
approach gives the best approximation followed by the CCD-LSM, SD-MLSM, SD-LSM respectively in order
of accuracy when a single level is considered in the DOE. The results of SD-MLSM can be improved drastically
if the number of level in the DOE is increased. Though, the CCD gives very accurate results yet applicability of
SD seems to be relevant considering the fact that CCD requires 2k
+2k+1 design points whereas SD requires
2k+1 points. Thus, the number of FE runs required for large complex structures (e.g. for 21 variables one needs
more than twenty lakhs FE runs to generate response surface by CCD) makes CCD impractical. It has been
found that the MLSM based RSM captures the trend of the responses, obtained by the direct MCS based results,
much more efficiently and more accurately. The statistical error metrics, i.e. RMSE, R2
and mε are also found
to be better. By studying the various norms in table 2 one can easily realise that though the SD-MLSM provides
better values compared to SD-LSM, particularly when number of levels are increased. Though, the SD-MLSM
approximates reasonably; the difference with CCD is still notable and there is scope of further improvement of
the prediction. This can be possibly attempted considering specific application of responses. However, this
needs more study.
REFERENCES
[1] Box, G.E. P. and Wilson, K. B. (1954). The exploration and exploitation of response surfaces: some general
considerations and examples’, Biometrics 10, 16–60.
[2] Das, P. K. and Zheng Y. (2000). Improved response surface method and its application to stiffened plate
reliability analysis. Engineering Structures; 22(5): 544–551.
[3] Roussouly, N.; Petitjean, F.; Salaun M. (2013). A new adaptive response surface method for reliability
analysis.Probabilistic Engineering Mechanics; 32:103-115.
[4] Khuri A. I. and Mukhopadhyay, S. (2010). Response surface methodology, WIREs Computational Statistics;
2; 128–149.
[5] Bucher, C. G. and Bourgund, U. (1990). A fast and efficient response surface approach for structural
reliability problems. Structural Safety; 7: 57-66.
[6] Kim, S. H. and Na, S. W. (1997). Response surface method using vector projected sampling points.
Structural Safety; 19:3-19.
[7] Liu, Y. W. and Moses, F. (1994). A sequential response surface method and its application in the reliability
analysis of aircraft structural systems. Structural Safety; 16:39-46.
[8] Rajashekhar, M. R. and Ellingwood, B. R. (1993). A new look at the response surface approach for
reliability analysis, Structural Safety; 12:205-220.
[9] Lemaire, M.; Mohamed, A.; Flore`s, M. O. (1997). The use of finite element codes for the reliability of
structural systems. In: 7th
IFIP WG 7.5 Working Conf on Reliability and Optimisation of Structural Systems,
Boulder, CO 2-4 April, 1996. Elsevier (Pergamon), 223-30.
[10]Enevolsen, I.; Faber, M. H.; Sorensen, J. D. (1994). Adaptive response surface technique in reliability
estimation. In: Schueller, Shinozuka, Yao, eds. Structural safety and reliability:1257–64.
[11]Lemaire, M. (1998). Finite element and reliability: combined methods by response surface. In:
Frantziskonis GN. Ed., Probabilities and materials tests, models and applications for the 21st century,
NATO ASI series 3 high technology; 46. Kluwer Academic, 317-31.
[12]Faravelli, L. (1989). Response surface approach for reliability analyses. Journal of Engineering Mechanics
ASCE; 115(2); 2763-2781.

[13]Kim, C.; Wang, S.; Choi, K. K. (2005). Efficient response surface modeling by using moving least-squares
method and sensitivity. AIAA Journal; 43(1): 2404-2411.
[14]Bhattacharjya, S. and Chakraborty, S. (2011). Robust optimization of Structures subjected to stochastic
earthquake excitation under limited information on system parameters uncertainty, Engineering
Optimization; 43(12):1311-1330.
[15]Taflanidis, A. A.; Cheung S.H. (2012). Stochastic sampling using moving least squares response surface
approximations. Probabilistic Engineering Mechanics 2012; 28: 216-224.

Second International Conference on Vulnerability and Risk Analysis and Management (ICVRAM2014)
13 - 16 July 2014, University of Liverpool, UK
Michael Beer, Ivan S.K. Au Jim W. Hall (editors)
ADAPTIVE RESPONSE SURFACE METHOD BASED
EFFICIENT MONTE CARLO SIMULATION
SOMDATTA GOSWAMI 1
and SUBRATA CHAKRABORTY2
1
Department of Civil Engineering, Bengal Engineering Science University, Shibpur INDIA
E-mail: somdatta89@gmail.com
2
Department of Civil Engineering, Bengal Engineering Science University, Shibpur INDIA
E-mail: schakbec@yahoo.co.in, schak@civil.becs.ac.in
The reliability evaluation of complex structure requires a deal between reliability analysis
algorithms and numerical methods used to model the mechanical behavior. The reliability
evaluation can be carried out directly so long the structural response is possible to obtain
explicitly. Because a closed-form expression of the performance function can be easily
obtained and the number of performance function calls does not play an important role. On
the contrary, when a finite element (FE) model is involved, each performance function
computation may require a high computational cost, especially for large complex system.
Furthermore, the mechanical model and the reliability method need to be merged together for
such complex problem. For this, the structural models, often constructed by commercial FE
software needs to run the software during each simulation. This limits the popular
applications of FE method for practical structural reliability analysis problem. Hence,
alternative techniques for efficient computation of response of complex structures by
overcoming the aforementioned drawbacks while retaining the accuracy is of paramount
importance to structural engineering community. Response Surface Method (RSM) has
emerged as an effective solution to such problems. The accuracy of evaluating the
performance function largely depends on the capability of a response surface (RS) to capture
the variation of behaviors. Though, the Central Composite Design (CCD) approximates very
well compared to other designs, it requires enormous input structural response information
compared to simpler design approaches. Thus, for large complex structures characterized by
large number of input random variables, in order to make an efficient response surface
requiring less number of FE executions is very much desirable. The present study deals with
the reliability evaluation of structure by efficient adaptive RSM based Monte Carlo
Simulation (MCS) technique.
It is well known that in the reliability evaluation of structure using RSM based MCS, the
center point of DOE to construct the RS should be close to the most probable failure point so
that the RS obtained includes most of the failure region. In fact, based on this fact, an
adaptive interpolation method was proposed by Bucher and Burgound (1990). Rajashekar
and Ellingwood (1993) further suggested repeated application of the approach to obtain a
converge center point to construct the final RS. Generally, the RSM used in these studies is
based on global approximation of scatter position data, obtained by using the least-squares
method (LSM). However, the LSM is one of the major sources of error in prediction by the
RSM. The moving least-squares method (MLSM), basically a local approximation approach,
is found to be more efficient in this regard (Kim et al. 2005). In the present study, an

Michael Beer, Ivan S.K. Au Jim W. Hall (editors)
2
improved RS is obtained by applying MLSM. The initial RS is constructed considering the
mean values of the random variables as the center point. Now this center point is updated
using Bucher and Burgound (1990) algorithm until convergence. Once converged RS is
obtained, direct MCS is performed to obtain the reliability of the structure. For comparative
study, the procedure of MCS is also performed using LSM based RS generation approach.
The reliability evaluation algorithm is elucidated through numerical study. Typical results
of comparative study of reliability of a hollow sphere subjected to internal pressure are
presented in table 1 and Figure 2. The limit state function considered for reliability analysis
is the elastic-plastic failure i.e. Von-Mises stress reaching the allowable yield stress at a point
of the sphere. It is important to note that though simple saturated design (SD) method is
used, the reliability predictions are quite close to the direct MCS based results. Furthermore,
the number of iterations required by the MLSM based approach is far less compare to
theLSM based approach. Thus, efficient as well more accurate estimation of reliability of
structure is possible by the proposed approach. Further details of numerical study including
results of complex problem will be presented in the final paper.
Table 1 Comparison of probability of failures for different nominal yield stress
Probability of Failure
Yield
stress
LSM MLSM
Direct
MCS
280 0.056 0.053 0.054
300 0.032 0.028 0.029
320 0.016 0.012 0.013
90 95 100 105 110 115
0.00
0.02
0.04
0.06
0.08
0.10
ProbabilityofFailure
Internal Pressure
LSM
MLSM
Direct MCS
Figure 1 Comparison of probability of failures obtained by the proposed approach with
increasing internal pressure
Keywords: Reliability, Simulation, Complex Structure, Response Surface, Moving Least Square
Method
References
1. C.G. Bucher and U. Bourgund, Structural Safety, 7, 57–66, 1990.
2. M. R. Rajashekhar and B. R. Ellingwood, Structural Safety, 2, 205–20, 1993..
3. C. Kim, S. Wang and K. K. Choi, AIAA J., 43(1), 2404-2411, 2005.

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