A new metamodeling approach is proposed to characterize the output (response) random process of a dynamic system with random variables, excited by input random processes. The metamodel is then used to efficiently estimate the time-dependent reliability. The input random processes are decomposed using principal components, and a few simulations are used to estimate the distributions of the decomposition coefficients. A similar decomposition is performed on the output random process. A Kriging model is then built between the input and output decomposition coefficients and is used subsequently to quantify the output random process. The innovation of our approach is that the system input is not deterministic but random. We establish, therefore, a surrogate model between the input and output random processes. To achieve this goal, we use an integral expression of the total probability theorem to estimate the marginal distribution of the output decomposition coefficients. The integral is efficiently estimated using a Monte Carlo (MC) approach which simulates from a mixture of sampling distributions with equal mixing probabilities. The quantified output random process is finally used to estimate the time-dependent probability of failure. The proposed method is illustrated with a corroding beam example.

References

References
1.
Singh
,
A.
,
Mourelatos
,
Z. P.
, and
Li
,
J.
,
2010
, “
Design for Lifecycle Cost Using Time-Dependent Reliability
,”
ASME J. Mech. Des.
,
132
(
9
), p.
091008
.
2.
Andrieu-Renaud
,
C.
,
Sudret
,
B.
, and
Lemaire
,
M.
,
2004
, “
The PHI2 Method: A Way to Compute Time-Variant Reliability
,”
Reliab. Eng. Saf. Syst.
,
84
(
1
), pp.
75
86
.
3.
Rice
,
S. O.
,
1944
, “
Mathematical Analysis of Random Noise
,”
Bell. Syst. Tech. J.
,
23
(
3
), pp.
282
332
. [Republished in: Wax, N., ed., 1954, Selected Papers on Noise and Stochastic Processes, Dover, New York.]
4.
Zhang
,
J.
, and
Du
,
X.
,
2011
, “
Time-Dependent Reliability Analysis for Function Generator Mechanisms
,”
ASME J. Mech. Des.
,
133
(
3
), p.
031005
.
5.
Singh
,
A.
, and
Mourelatos
,
Z. P.
,
2010
, “
On the Time-Dependent Reliability of Non-Monotonic, Non-Repairable Systems
,”
SAE
Paper No. 2010-01-0696.
6.
Hu
,
Z.
, and
Du
,
X.
,
2012
, “
Time-Dependent Reliability Analysis by a Sampling Approach to Extreme Values of Stochastic Processes
,”
ASME
Paper No. DETC2012-70132.
7.
Hu
,
Z.
,
Li
,
H.
,
Du
,
X.
, and
Chandrashekhara
,
K.
,
2012
, “
Simulation-Based Time-Dependent Reliability Analysis for Composite Hydrokinetic Turbine Blades
,”
Struct. Multidiscip. Optim.
,
47
(
5
), pp.
765
781
.
8.
Li
,
J.
, and
Mourelatos
,
Z. P.
,
2009
, “
Time-Dependent Reliability Estimation for Dynamic Problems Using a Niching Genetic Algorithm
,”
ASME J. Mech. Des.
,
131
(
7
), p.
071009
.
9.
Singh
,
A.
, and
Mourelatos
,
Z. P.
,
2010
, “
Time-Dependent Reliability Estimation for Dynamic Systems Using a Random Process Approach
,”
SAE
Paper No. 2010-01-0644.
10.
Singh
,
A.
,
Mourelatos
,
Z. P.
, and
Nikolaidis
,
E.
,
2011
, “
Time-Dependent Reliability of Random Dynamic Systems Using Time-Series Modeling and Importance Sampling
,”
SAE
Paper No. 2011-01-0728.
11.
Au
,
S. K.
, and
Beck
,
J. L.
,
2003
, “
Subset Simulation and Its Application to Seismic Risk Based on Dynamic Analysis
,”
J. Eng. Mech.
,
129
(
8
), pp.
901
917
.
12.
Beck
,
J. L.
, and
Au
,
S. K.
,
2002
, “
Bayesian Updating of Structural Models and Reliability Using Markov Chain Monte Carlo Simulation
,”
J. Eng. Mech.
,
128
(
4
), pp.
380
391
.
13.
Ching
,
J.
,
Beck
,
J. L.
, and
Au
,
S. K.
,
2005
, “
Reliability Estimation for Dynamic Systems Subject to Stochastic Excitation Using Subset Simulation With Splitting
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
12–16
), pp.
1557
1579
.
14.
Wang
,
Z.
,
Mourelatos
,
Z. P.
,
Li
,
J.
,
Singh
,
A.
, and
Baseski
,
I.
,
2013
, “
Time-Dependent Reliability of Dynamic Systems Using Subset Simulation With Splitting Over a Series of Correlated Time Intervals
,”
ASME
Paper No. DETC2013-12257.
15.
Cressie
,
N.
,
1993
,
Statistics for Spatial Data
,
Wiley
,
New York
.
16.
Sacks
,
J.
,
Welch
,
W. J.
,
Mitchell
,
J. J.
, and
Wynn
,
H. P.
,
1989
, “
Design and Analysis of Computer Experiments
,”
Stat. Sci.
,
4
(
4
), pp.
409
435
.
17.
Lophaven
,
S. N.
,
Nielsen
,
H. B.
, and
Sondergaard
,
J.
,
2002
, “
DACE: A MATLAB Kriging ToolBox, Version 2, Informatics and Mathematical Modeling
,” Technical University of Denmark, Technical Report No. IMM-TR-2002-12.
18.
Wang
,
G. G.
, and
Shan
,
S.
,
2007
, “
Review of Metamodeling Techniques in Support of Engineering Design Optimization
,”
ASME J. Mech. Des.
,
129
(
4
), pp.
370
380
.
19.
Simpson
,
T. W.
,
Peplinski
,
J. D.
,
Koch
,
P. N.
, and
Allen
,
J. K.
,
2001
, “
Metamodels for Computer-Based Engineering Design: Survey and Recommendations
,”
Eng. Comput.
,
17
(
2
), pp.
129
150
.
20.
Simpson
,
T. W.
,
Lin
,
D. K. J.
, and
Chen
,
W.
,
2001
, “
Sampling Strategies for Computer Experiments: Design and Analysis
,”
Int. J. Reliab. Appl.
,
2
(
3
), pp.
209
240
.
21.
Pacheco
,
J. E.
,
Amon
,
C. H.
, and
Finger
,
S.
,
2003
, “
Bayesian Surrogates Applied to Conceptual Stages of the Engineering Design Process
,”
ASME J. Mech. Des.
,
125
(
4
), pp.
664
672
.
22.
Weiss
,
L. E.
,
Amon
,
C. H.
,
Finger
,
S.
,
Miller
,
E. D.
,
Romero
,
D.
,
Verdinelli
,
I.
,
Walker
,
L. M.
, and
Campbell
,
P. G.
,
2005
, “
Bayesian Computer-Aided Experimental Design of Heterogeneous Scaffolds for Tissue Engineering
,”
Comput.-Aided Des.
,
37
(
11
), pp.
1127
1139
.
23.
Romero
,
D. A.
,
Amon
,
C. H.
, and
Finger
,
S.
,
2006
, “
On Adaptive Sampling for Single and Multi-Response Bayesian Surrogate Models
,”
ASME
Paper No. DETC2006-99210.
24.
Romero
,
D. A.
,
Amon
,
C. H.
,
Finger
,
S.
, and
Verdinelli
,
I.
,
2004
, “
Multi-Stage Bayesian Surrogates for the Design of Time-Dependent Systems
,”
ASME
Paper No. DETC2004-57510.
25.
Wehrwein
,
D.
, and
Mourelatos
,
Z. P.
,
2009
, “
Optimization of Engine Torque Management Under Uncertainty for Vehicle Driveline Clunk Using Time-Dependent Metamodels
,”
ASME J. Mech. Des.
,
131
(
5
), p.
051001
.
26.
Missoum
,
S.
,
2008
, “
Probabilistic Optimal Design in the Presence of Random Fields
,”
Struct. Multidiscip. Optim.
,
35
(
6
), pp.
523
530
.
27.
Ghanem
,
R.
, and
Spanos
,
P. D.
,
1991
,
Stochastic Finite Elements: A Spectral Approach
,
Springer
,
New York
.
28.
Aha
,
D. W.
,
1997
, “
Editorial, Special Issue on Lazy Learning
,”
Artif. Intell. Rev.
,
11
(
1–5
), pp.
1
6
.
29.
Atkeson
,
C. G.
,
Moore
,
A. W.
, and
Schaal
,
S.
,
1997
, “
Locally Weighted Learning
,”
Artif. Intell. Rev.
,
11
(
1–5
), pp.
11
73
.
30.
Birattari
,
M.
,
Bontempi
,
G.
, and
Bersini
,
H.
,
1999
, “
Lazy Learning Meets the Recursive Least-Squares Algorithm
,”
Advances in Neural Information Processing Systems 11
,
M. S.
Kearns
,
S. A.
Solla
, and
D. A.
Cohn
, eds.,
MIT Press
,
Cambridge, MA
, pp.
375
381
.
31.
Singh
,
A.
,
Mourelatos
,
Z. P.
, and
Li
,
J.
,
2009
, “
Design for Lifecycle Cost Using Time-Dependent Reliability
,”
ASME
Paper No. DETC2009-86587.
32.
Bayarri
,
M. J.
,
Walsh
,
D.
,
Berger
,
J. O.
,
Cafeo
,
J.
,
Garcia-Donato
,
G.
,
Liu
,
F.
,
Palomo
,
J.
,
Parthasarathy
,
R. J.
,
Paulo
,
R.
, and
Sacks
,
J.
,
2007
, “
Computer Model Validation With Functional Output
,”
Ann. Stat.
,
35
(
5
), pp.
1874
1906
.
33.
Higdon
,
D.
,
Gattiker
,
J.
,
Williams
,
B.
, and
Rightley
,
M.
,
2008
, “
Computer Model Calibration Using High-Dimensional Outputs
,”
J. Am. Stat. Assoc.
,
103
(
482
), pp.
570
583
.
34.
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
.
35.
Santner
,
T. J.
,
Williams
,
B. J.
, and
Notz
,
W. I.
,
2003
,
The Design and Analysis of Computer Experiments
,
Springer
,
New York
.
36.
Rizzo
,
M. L.
,
2007
,
Statistical Computing With R
,
Chapman and Hall
,
Boca Raton, FL
.
37.
Dalbey
,
K.
, and
Swiler
,
L.
,
2014
, “
Gaussian Process Adaptive Importance Sampling
,”
Int. J. Uncertainty Quantif.
,
4
(
2
), pp.
133
149
.
38.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2001
, “
Bayesian Calibration of Computer Models
,”
J. R. Stat. Soc.: Ser. B (Stat. Methodol.)
,
63
(
3
), pp.
425
464
.
39.
Viana
,
F. A. C.
,
Simpson
,
T. W.
,
Balabanov
,
V.
, and
Toropov
,
V.
,
2014
, “
Metamodeling in Multidisciplinary Design Optimization: How Far Have We Really Come?
,”
AIAA J.
,
52
(
4
), pp.
670
690
.
You do not currently have access to this content.