This paper proposes to apply the convolution integral method to the novel second-order reliability method (SORM) to further improve its computational efficiency. The novel SORM showed better accuracy in estimating the probability of failure than conventional SORMs by utilizing a linear combination of noncentral or general chi-squared random variables. However, the novel SORM requires significant computational time when integrating the linear combination to calculate the probability of failure. In particular, when the dimension of performance functions is higher than three, the computational time for full integration increases exponentially. To reduce this computational burden for the novel SORM, we propose to obtain the distribution of the linear combination using the convolution and to use the distribution for the probability of failure estimation. Since it converts an N-dimensional full integration into one-dimensional integration, the proposed method is computationally very efficient. Numerical study illustrates that the accuracy of the proposed method is almost the same as the full integral method and Monte Carlo simulation (MCS) with much improved efficiency.

References

1.
Hasofer
,
A. M.
, and
Lind
,
N. C.
,
1974
, “
Exact and Invariant Second-Moment Code Format
,”
J. Eng. Mech. Div.
,
100
(
1
), pp.
111
121
.http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0021808
2.
Tu
,
J.
,
Choi
,
K. K.
, and
Park
,
Y. H.
,
2001
, “
Design Potential Method for Robust System Parameter Design
,”
AIAA J.
,
39
(
4
), pp.
667
677
.
3.
Breitung
,
K.
,
1984
, “
Asymptotic Approximations for Multinormal Integrals
,”
J. Eng. Mech.
,
110
(
3
), pp.
357
366
.
4.
Hohenbichler
,
M.
, and
Rackwitz
,
R.
,
1988
, “
Improvement of Second-Order Reliability Estimates by Importance Sampling
,”
J. Eng. Mech.
,
114
(
12
), pp.
2195
2199
.
5.
Tvedt
,
L.
,
1990
, “
Distribution of Quadratic Forms in Normal Space-Application to Structural Reliability
,”
J. Eng. Mech.
,
116
(
6
), pp.
1183
1197
.
6.
Adhikari
,
S.
,
2004
, “
Reliability Analysis Using Parabolic Failure Surface Approximation
,”
J. Eng. Mech.
,
130
(
12
), pp.
1407
1427
.
7.
Rubinstein
,
R. Y.
,
1981
,
Simulation and the Monte Carlo Method
,
Wiley, New York
.
8.
Lin
,
C. Y.
,
Huang
,
W. H.
,
Jeng
,
M. C.
, and
Doong
,
J. L.
,
1997
, “
Study of an Assembly Tolerance Allocation Model Based on Monte Carlo Simulation
,”
J. Mater. Process. Technol.
,
70
(
1
), pp.
9
16
.
9.
Denny
,
M.
,
2001
, “
Introduction to Importance Sampling in Rare-Event Simulations
,”
Eur. J. Phys.
,
22
(
4
), p.
403
.
10.
Walker
,
J.
,
1986
, “
Practical Application of Variance Reduction Techniques in Probabilistic Assessments
,”
Second International Conference on Radioactive Waste Management
, Winnipeg, MB, Canada, Sept. 7–11, pp. 517–521.https://inis.iaea.org/search/search.aspx?orig_q=RN:22005292
11.
Rao
,
B. N.
, and
Chowdhury
,
R.
,
2008
, “
Probabilistic Analysis Using High Dimensional Model Representation and Fast Fourier Transform
,”
Int. J. Comput. Methods Eng. Sci. Mech.
,
9
(
6
), pp.
342
357
.
12.
Balu
,
A. S.
, and
Rao
,
B. N.
,
2013
, “
High Dimensional Model Representation for Structural Reliability Bounds Estimation Under Mixed Uncertainties
,”
Int. J. Struct. Eng.
,
4
(
3
), pp.
251
272
.
13.
Lee
,
I.
,
Choi
,
K. K.
,
Du
,
L.
, and
Gorsich
,
D.
,
2008
, “
Inverse Analysis Method Using MPP-Based Dimension Reduction for Reliability-Based Design Optimization of Nonlinear and Multi-Dimensional Systems
,”
Comput. Methods Appl. Mech. Eng.
,
198
(
1
), pp.
14
27
.
14.
Hu
,
C.
,
Youn
,
B. D.
, and
Yoon
,
H.
,
2013
, “
An Adaptive Dimension Decomposition and Reselection Method for Reliability Analysis
,”
Struct. Multidiscip. Optim.
,
47
(
3
), pp.
423
440
.
15.
Noh
,
Y.
,
Choi
,
K. K.
, and
Lee
,
I.
,
2009
, “
Reduction of Ordering Effect in Reliability-Based Design Optimization Using Dimension Reduction Method
,”
AIAA J.
,
47
(
4
), pp.
994
1004
.
16.
Bae
,
H. R.
, and
Alyanak
,
E.
,
2016
, “
Sequential Subspace Reliability Method With Univariate Revolving Integration
,”
AIAA J.
,
54
(
7
), pp.
2160
2170
.
17.
Balu
,
A. S.
, and
Rao
,
B. N.
,
2015
, “
Confidence Bounds on Failure Probability Using MHDMR
,”
Advances in Structural Engineering
, Springer, India, pp.
2515
2524
.
18.
Kang
,
S.
,
Park
,
J.
, and
Lee
,
I.
,
2017
, “
Accuracy Improvement of the Most Probable Point-Based Dimension Reduction Method Using the Hessian Matrix
,”
Int. J. Numer. Methods Eng.
,
111
(
3
), pp.
203
217
.
19.
Lee
,
I.
,
Noh
,
Y.
, and
Yoo
,
D.
,
2012
, “
A Novel Second-Order Reliability Method (SORM) Using Noncentral or Generalized Chi-Squared Distributions
,”
ASME J. Mech. Des.
,
134
(
10
), p.
100912
.
20.
Lim
,
J.
,
Lee
,
B.
, and
Lee
,
I.
,
2014
, “
Second‐Order Reliability Method‐Based Inverse Reliability Analysis Using Hessian Update for Accurate and Efficient Reliability‐Based Design Optimization
,”
Int. J. Numer. Methods Eng.
,
100
(
10
), pp.
773
792
.
21.
Madsen
,
H. O.
,
Krenk
,
S.
, and
Lind
,
N. C.
,
2006
,
Methods of Structural Safety
,
Courier Corporation
, New York.
22.
Grinstead
,
C. M.
, and
Snell
,
J. L.
,
2012
,
Introduction to Probability
,
American Mathematical Society
,
Providence, RI
.
23.
Rosenblatt
,
M.
,
1952
, “
Remarks on a Multivariate Transformation
,”
Ann. Math. Stat.
,
23
(
3
), pp.
470
472
.
24.
Rahman
,
S.
, and
Wei
,
D.
,
2006
, “
A Univariate Approximation at Most Probable Point for Higher-Order Reliability Analysis
,”
Int. J. Solids Struct.
,
43
(
9
), pp.
2820
2839
.
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