A primary concern in practical engineering design is ensuring high system reliability throughout a product's lifecycle, which is subject to time-variant operating conditions and component deteriorations. Thus, the capability of dealing with time-dependent probabilistic constraints in reliability-based design optimization (RBDO) is of vital importance in practical engineering design applications. This paper presents a nested extreme response surface (NERS) approach to efficiently carry out time-dependent reliability analysis and determine the optimal designs. This approach employs the kriging model to build a nested response surface of time corresponding to the extreme value of the limit state function. The efficient global optimization (EGO) technique is integrated with the NERS approach to extract the extreme time responses of the limit state function for any given system design. An adaptive response prediction and model maturation (ARPMM) mechanism is developed based on the mean square error (MSE) to concurrently improve the accuracy and computational efficiency of the proposed approach. With the nested response surface of time, the time-dependent reliability analysis can be converted into the time-independent reliability analysis, and existing advanced reliability analysis and design methods can be used. The NERS approach is compared with existing time-dependent reliability analysis approaches and integrated with RBDO for engineered system design with time-dependent probabilistic constraints. Two case studies are used to demonstrate the efficacy of the proposed NERS approach.

References

References
1.
Youn
,
B. D.
,
Choi
,
K. K.
, and
Du
,
L.
,
2005
, “
Enriched Performance Measure Approach for Reliability-Based Design Optimization
,”
AIAA J.
,
43
(
4
), pp.
874
884
.10.2514/1.6648
2.
Youn
,
B.
, and
Wang
,
P.
,
2009
, “
Complementary Interaction Method (CIM) for System Reliability Assessment
,”
J. Mech. Des.
,
131
(
4
), p.
041004(15)
.10.1115/1.3086794
3.
Noh
,
Y.
,
Choi
,
K. K.
,
Lee
,
I.
,
Gorsich
,
D.
, and
Lamb
,
D.
,
2011
, “
Reliability-Based Design Optimization With Confidence Level for Non-Gaussian Distributions Using Bootstrap Method
,”
J. Mech. Des.
,
133
(
9
), p.
091001(12)
.10.1115/1.4004545
4.
Wang
,
P.
,
Hu
,
C.
, and
Youn
,
B. D.
,
2011
, “
A Generalized Complementary Intersection Method for System Reliability Analysis and Design
,”
J. Mech. Des.
,
133
(
7
), p.
071003(13)
.10.1115/1.4004198
5.
Du
,
X.
, and
Chen
,
W.
,
2004
, “
Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design
,”
J. Mech. Des.
,
126
(
2
), pp.
225
233
.10.1115/1.1649968
6.
Youn
,
B.
,
Hu
,
C.
, and
Wang
,
P.
,
2011
, “
Resilience-Driven System Design for Complex Engineered Systems
,”
J. Mech. Des.
,
133
(
10
), p.
101011(15)
.10.1115/1.4004981
7.
Liu
,
D.
, and
Peng
,
Y.
,
2012
, “
Reliability Analysis by Mean-Value Second-Order Expansion
,”
J. Mech. Des.
,
134
(
6
), p.
061005(8)
.10.1115/1.4006528
8.
Youn
,
B. D.
,
Choi
,
K.
, and
Du
,
L.
,
2005
, “
Adaptive Probability Analysis Using an Enhanced Hybrid Mean Value Method
,”
Struct. Multidiscip. Optim.
,
29
(
2
), pp.
134
148
.10.1007/s00158-004-0452-6
9.
Du
,
X.
,
2012
, “
First Order Reliability Method With Truncated Random Variables
,”
J. Mech. Des.
,
134
(
9
), p.
091005(9)
.10.1115/1.4007150
10.
Rahman
,
S.
, and
Xu
,
H.
,
2004
, “
A Univariate Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics
,”
Probab. Eng. Mech.
,
19
(
4
), pp.
393
408
.10.1016/j.probengmech.2004.04.003
11.
Youn
,
B. D.
,
Xi
,
Z.
, and
Wang
,
P.
,
2008
, “
Eigenvector Dimension Reduction (EDR) Method for Sensitivity-Free Probability Analysis
,”
Struct. Multidiscip. Optim.
,
37
(
1
), pp.
13
28
.10.1007/s00158-007-0210-7
12.
Hazelrigg
,
G. A.
,
1998
, “
A Framework for Decision-Based Engineering Design
,”
J. Mech. Des.
,
120
(
4
), pp.
653
658
.10.1115/1.2829328
13.
Zhang
,
R.
, and
Mahadevan
,
S.
,
2000
, “
Model Uncertainty and Bayesian Updating in Reliability-Based Inspection
,”
Struct. Saf.
,
22
(
2
), pp.
145
160
.10.1016/S0167-4730(00)00005-9
14.
Youn
,
B. D.
, and
Wang
,
P.
,
2008
, “
Bayesian Reliability-Based Design Optimization Using Eigenvector Dimension Reduction (EDR) Method
,”
Struct. Multidiscip. Optim.
,
36
(
2
), pp.
107
123
.10.1007/s00158-007-0202-7
15.
Wang
,
P.
,
Youn
,
B. D.
,
Xi
,
Z.
, and
Kloess
,
A.
,
2009
, “
Bayesian Reliability Analysis With Evolving, Insufficient, and Subjective Data Sets
,”
J. Mech. Des.
,
131
(
11
), p.
111008
(
11
).10.1115/1.4000251
16.
Currin
,
C.
,
Mitchell
,
T.
,
Morris
,
M.
, and
Ylvisaker
,
D.
,
1991
, “
Bayesian Prediction of Deterministic Functions, With Applications to the Design and Analysis of Computer Experiments
,”
J. Am. Stat. Assoc.
,
86
(
416
), pp.
953
963
.10.1080/01621459.1991.10475138
17.
Li
,
J.
,
Chen
,
J.
, and
Fan
,
W.
,
2007
, “
The Equivalent Extreme-Value Event and Evaluation of the Structural System Reliability
,”
Struct. Saf.
,
29
(
2
), pp.
112
131
.10.1016/j.strusafe.2006.03.002
18.
Chen
,
J. B.
, and
Li
,
J.
,
2007
, “
The Extreme Value Distribution and Dynamic Reliability Analysis of Nonlinear Structures With Uncertain Parameters
,”
Struct. Saf.
,
29
(
2
), pp.
77
93
.10.1016/j.strusafe.2006.02.002
19.
Li
,
J.
, and
Mourelatos
,
Z. P.
,
2009
, “
Time-Dependent Reliability Estimation for Dynamic Problems Using a Niching Genetic Algorithm
,”
J. Mech. Des.
,
131
(
7
), p.
071009
.10.1115/1.3149842
20.
Lutes
,
L. D.
, and
Sarkani
,
S.
,
2009
, “
Reliability Analysis of Systems Subject to First-Passage Failure
,” NASA Technical Report No. NASA/CR-2009-215782.
21.
Kuschel
,
N.
, and
Rackwitz
,
R.
,
2000
, “
Optimal Design Under Time-Variant Reliability Constraints
,”
Struct. Saf.
,
22
(
2
), pp.
113
127
.10.1016/S0167-4730(99)00043-0
22.
Li
,
C.
, and
Der Kiureghian
,
A.
,
1995
, “
Mean Out-Crossing Rate of Nonlinear Response to Stochastic Input
,”
Proceedings of ICASP-7
,
Balkema
,
Rotterdam
, pp.
295
302
.
23.
Schrupp
,
K.
, and
Rackwitz
,
R.
,
1988
, “
Out-crossing Rates of Marked Poisson Cluster Processes in Structural Reliability
,”
Appl. Math. Model.
,
12
(
5
), pp.
482
490
.10.1016/0307-904X(88)90085-6
24.
Breitung
,
K.
,
1994
, “
Asymptotic Approximations for the Crossing Rates of Poisson Square Waves
,”
NIST Special Publication SP
, pp.
75
75
.
25.
Singh
,
A.
,
Mourelatos
,
Z. P.
, and
Li
,
J.
,
2010
, “
Design for Lifecycle Cost Using Time-Dependent Reliability
,”
J. Mech. Des.
,
132
(
9
), p.
091008
.10.1115/1.4002200
26.
Andrieu-Renaud
,
C.
,
Sudret
,
B.
, and
Lemaire
,
M.
,
2004
, “
The PHI2 Method: A Way to Compute Time-Variant Reliability
,”
Reliab. Eng. Syst. Saf.
,
84
(
1
), pp.
75
86
.10.1016/j.ress.2003.10.005
27.
Rackwitz
,
R.
,
1998
, “
Computational Techniques in Stationary and Non-Stationary Load Combination—A Review and Some Extensions
,”
J. Struct. Eng.
,
25
(
1
), pp.
1
20
.
28.
Sudret
,
B.
,
2008
, “
Analytical Derivation of the Out-Crossing Rate in Time-Variant Reliability Problems
,”
Struct. Infrastruct. Eng.
,
4
(
5
), pp.
353
362
.10.1080/15732470701270058
29.
Zhang
,
J.
, and
Du
,
X.
,
2011
, “
Time-Dependent Reliability Analysis for Function Generator Mechanisms
,”
J. Mech. Des.
,
133
(
3
), p.
031005(9)
.10.1115/1.4003539
30.
Du
,
X.
,
2012
, “
Toward Time-Dependent Robustness Metrics
,”
J. Mech. Des.
,
134
(
1
), p.
011004
.10.1115/1.4005445
31.
Son
,
Y. K.
, and
Savage
,
G. J.
,
2007
, “
Set Theoretic Formulation of Performance Reliability of Multiple Response Time-Variant Systems Due to Degradations in System Components
,”
Quality Reliab. Eng. Int.
,
23
(
2
), pp.
171
188
.10.1002/qre.783
32.
Hagen
,
O.
, and
Tvedt
,
L.
,
1991
, “
Vector Process Out-Crossing as Parallel System Sensitivity Measure
,”
J. Eng. Mech.
,
117
(
10
), pp.
2201
2220
.10.1061/(ASCE)0733-9399(1991)117:10(2201)
33.
Breitung
,
K.
,
1988
, “
Asymptotic Crossing Rates for Stationary Gaussian Vector Processes
,”
Stochastic Process. Appl.
,
29
(
2
), pp.
195
207
.10.1016/0304-4149(88)90037-3
34.
Xu
,
H.
, and
Rahman
,
S.
,
2005
, “
Decomposition Methods for Structural Reliability Analysis
,”
Probab. Eng. Mech.
,
20
(
3
), pp.
239
250
.10.1016/j.probengmech.2005.05.005
35.
Youn
,
B. D.
, and
Xi
,
Z.
,
2009
, “
Reliability-Based Robust Design Optimization Using the Eigenvector Dimension Reduction (EDR) Method
,”
Struct. Multidiscip. Optim.
,
37
(
5
), pp.
475
492
.10.1007/s00158-008-0239-2
36.
Xu
,
H.
, and
Rahman
,
S.
,
2004
, “
A Generalized Dimension-Reduction Method for Multidimensional Integration in Stochastic Mechanics
,”
Int. J. Numer. Methods Eng.
,
61
(
12
), pp.
1992
2019
.10.1002/nme.1135
37.
Jones
,
D. R.
,
Schonlau
,
M.
, and
Welch
,
W. J.
,
1998
, “
Efficient Global Optimization Of Expensive Black-Box Functions
,”
J. Global Optim.
,
13
(
4
), pp.
455
492
.10.1023/A:1008306431147
38.
Schonlau
,
M.
,
1997
, “
Computer Experiments and Global Optimization
,” Ph.D. dissertation,
University of Waterloo
,
Waterloo, Ontario, Canada
.
39.
Stuckman
,
B. E.
,
1988
, “
A Global Search Method for Optimizing Nonlinear Systems
,”
IEEE Trans. Syst
.,
Man Cybern.
,
18
(
6
), pp.
965
977
.10.1109/21.23094
40.
Žilinskas
,
A.
,
1992
, “
A Review of Statistical Models for Global Optimization
,”
J. Global Optim.
,
2
(
2
), pp.
145
153
.10.1007/BF00122051
41.
Koehler
,
J.
, and
Owen
,
A.
,
1996
, “
Computer Experiments
Handbook of Statistics, 13: Design and Analysis of Experiments
,
S.
Ghosh
and
C. R.
Rao
, eds., pp.
261
308
,
Elsevier
,
Amsterdam
.
42.
Sacks
,
J.
,
Welch
,
W. J.
,
Mitchell
,
T. J.
, and
Wynn
,
H. P.
,
1989
, “
Design and Analysis of Computer Experiments
,”
Stat. Sci.
,
4
(
4
), pp.
409
423
.10.1214/ss/1177012413
43.
Mockus
,
J.
,
Tiesis
,
V.
, and
Zilinskas
,
A.
,
1978
, “
The Application of Bayesian Methods for Seeking the Extreme
,”
Towards Global Optimization
, Vol. 2,
L. C. W.
Dixon
and
G. P.
Szego
, eds., pp.
117
129
.
44.
Haftka
,
R. T.
, and
Watsonft
,
L. T.
,
1999
, “
Response Surface Models Combining Linear and Euler Aerodynamics for Supersonic Transport Design
,”
J. Aircraft
,
36
(
1
), pp.
75
86
.10.2514/2.2415
45.
Madsen
,
J. I.
,
Shyy
,
W.
, and
Haftka
,
R. T.
,
2000
, “
Response Surface Techniques for Diffuser Shape Optimization
,”
AIAA J.
,
38
(
9
), pp.
1512
1518
.10.2514/2.1160
46.
Welch
,
W. J.
,
Buck
,
R. J.
,
Sacks
,
J.
,
Wynn
,
H. P.
,
Mitchell
,
T. J.
, and
Morris
,
M. D.
,
1992
, “
Screening, Predicting, and Computer Experiments
,”
Technometrics
,
34
(
1
), pp.
15
25
.10.2307/1269548
47.
Wang
,
G. G.
,
Dong
,
Z.
, and
Aitchison
,
P.
,
2001
, “
Adaptive Response Surface Method—A Global Optimization Scheme for Approximation-Based Design Problems
,”
Eng. Optim.
,
33
(
6
), pp.
707
733
.10.1080/03052150108940940
48.
Keane
,
A. J.
, and
Nair
,
P. B.
,
2005
,
Computational Approaches for Aerospace Design
,
John Wiley & Sons, Ltd.
,
West Sussex
, p.
582
.
49.
Simpson
,
T. W.
,
Mauery
,
T. M.
,
Korte
,
J. J.
, and
Mistree
,
F.
,
1998
, “
Comparison of Response Surface and Kriging Models for Multidisciplinary Design Optimization
,” AIAA, Paper No. 98, 4758(7).
50.
Paciorek
,
C. J.
,
2003
, “
Nonstationary Gaussian Processes for Regression and Spatial Modelling
,” Ph.D. dissertation,
Carnegie Mellon University
,
Pittsburgh, PA
.
51.
Farhang-Mehr
,
A.
, and
Azarm
,
S.
,
2005
, “
Bayesian Metamodeling of Engineering Design Simulations: A Sequential Approach With Adaptation to Irregularities in the Response Behavior
,”
Int. J. Numer. Methods Eng.
,
62
(
15
), pp.
2104
2126
.10.1002/nme.1261
52.
Qin
,
S.
, and
Cui
,
W.
,
2003
, “
Effect of Corrosion Models on the Time-Dependent Reliability of Steel Plated Elements
,”
Marine Struct.
,
16
(
1
), pp.
15
34
.10.1016/S0951-8339(02)00028-X
53.
Madsen
,
H. O.
,
Krenk
,
S.
, and
Lind
,
N. C.
,
2006
,
Methods of Structural Safety
,
Dover Publications
,
New York
.
54.
Choi
,
H. G. R.
,
Park
,
M. H.
, and
Salisbury
,
E.
,
2000
, “
Optimal Tolerance Allocation With Loss Functions
,”
J. Manuf. Sci. Eng.
,
122
(
3
), pp.
529
535
.10.1115/1.1285918
55.
Xue
,
W.
, and
Pyle
,
R.
,
2004
, “
Optimal Design of Roller One Way Clutch for Starter Drives
,” SAE Technical Paper No. 2004-01-1151.
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