This paper presents a probabilistic framework to include the effects of both aleatory and epistemic uncertainty sources in coupled multidisciplinary analysis (MDA). A likelihood-based decoupling approach has been previously developed for probabilistic analysis of multidisciplinary systems, but only with aleatory uncertainty in the inputs. This paper extends this approach to incorporate the effects of epistemic uncertainty arising from data uncertainty and model errors. Data uncertainty regarding input variables (due to sparse and interval data) is included through parametric or nonparametric distributions using the principle of likelihood. Model error is included in MDA through an auxiliary variable approach based on the probability integral transform. In the presence of natural variability, data uncertainty, and model uncertainty, the proposed methodology is employed to estimate the probability density functions (PDFs) of coupling variables as well as the subsystem and system level outputs that satisfy interdisciplinary compatibility. Global sensitivity analysis (GSA), which has previously considered only aleatory inputs and feedforward or monolithic problems, is extended in this paper to quantify the contribution of model uncertainty in feedback-coupled MDA by exploiting the auxiliary variable approach. The proposed methodology is demonstrated using a mathematical MDA problem and an electronic packaging application example featuring coupled thermal and electrical subsystem analyses. The results indicate that the proposed methodology can effectively quantify the uncertainty in MDA while maintaining computational efficiency.

References

References
1.
Cramer
,
E. J.
,
Dennis
,
J. E.
,
Frank
,
P. D.
, and
Shubin
,
G. R.
,
1993
, “
Problem Formulation for Multidisciplinary Optimization Problem Formulation for Multidisciplinary Optimization
,”
SIAM J. Optim.
,
4
(
4
), pp.
754
776
.10.1137/0804044
2.
Ilan
,
K.
,
Steve
,
A.
,
Robert
,
B.
,
Peter
,
G.
, and
Ian
,
S.
,
1994
, “
Multidisciplinary Optimization Methods for Aircraft Preliminary Design
,”
AIAA
Paper No. 4325.
3.
Belytschko
,
T.
,
1980
, “
Fluid–Structure Interaction
,”
Comput. Struct.
,
12
(
4
), pp.
459
469
.10.1016/0045-7949(80)90121-2
4.
Thornton
,
E. A.
,
1992
, “
Thermal Structures: Four Decades of Progress
,”
J. Aircr.
,
29
(
3
), pp.
485
498
.10.2514/3.46187
5.
Wieting
,
A. R.
,
Dechaumphai
,
P.
,
Bey
,
K. S.
,
Thornton
,
E. A.
, and
Morgan
,
K.
,
1991
, “
Application of Integrated Fluid-Thermal-Structural Analysis Methods
,”
Thin Walled Struct.
,
11
(
1–2
), pp.
1
23
.10.1016/0263-8231(91)90008-7
6.
Sobieszczanski-Sobieski
,
J.
, and
Haftka
,
R.
,
1997
, “
Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments
,”
Struct. Optim.
,
14
(
1
) pp.
1
23
.10.1007/BF01197554
7.
Felippa
,
C. A.
,
Park
,
K. C.
, and
Farhat
,
C.
,
2001
, “
Partitioned Analysis of Coupled Mechanical Systems
,”
Comput. Methods Appl. Mech. Eng.
,
190
(
24–25
), pp.
3247
3270
.10.1016/S0045-7825(00)00391-1
8.
Michler
,
C.
, and
Hulshoff
,
S.
,
2004
, “
A Monolithic Approach to Fluid–Structure Interaction
,”
Comput. Fluids
,
33
(
5–6
), pp.
839
848
.10.1016/j.compfluid.2003.06.006
9.
Haldar
,
A.
, and
Mahadevan
,
S.
,
2000
,
Probability, Reliability, and Statistical Methods in Engineering Design
,
Wiley
,
New York
.
10.
Sankararaman
,
S.
, and
Mahadevan
,
S.
,
2011
, “
Likelihood-Based Representation of Epistemic Uncertainty due to Sparse Point Data and/or Interval Data
,”
Reliab. Eng. Syst. Saf.
,
96
(
7
), pp.
814
824
.10.1016/j.ress.2011.02.003
11.
Agarwal
,
H.
,
Renaud
,
J. E.
,
Preston
,
E. L.
, and
Padmanabhan
,
D.
,
2004
, “
Uncertainty Quantification Using Evidence Theory in Multidisciplinary Design Optimization
,”
Reliab. Eng. Syst. Saf.
,
85
(
1–3
), pp.
281
294
.10.1016/j.ress.2004.03.017
12.
Helton
,
J. C.
,
Johnson
,
J. D.
,
Oberkampf
,
W. L.
, and
Storlie
,
C. B.
,
2007
, “
A Sampling-Based Computational Strategy for the Representation of Epistemic Uncertainty in Model Predictions With Evidence Theory
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
37–40
), pp.
3980
3998
.10.1016/j.cma.2006.10.049
13.
Du
,
L.
,
Choi
,
K. K.
,
Youn
,
B. D.
, and
Gorsich
,
D.
,
2006
, “
Possibility-Based Design Optimization Method for Design Problems With Both Statistical and Fuzzy Input Data
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
928
935
.10.1115/1.2204972
14.
Zhang
,
X.
, and
Huang
,
H.-Z.
,
2009
, “
Sequential Optimization and Reliability Assessment for Multidisciplinary Design Optimization Under Aleatory and Epistemic Uncertainties
,”
Struct. Multidiscip. Optim.
,
40
(
1–6
), pp.
165
175
.10.1007/s00158-008-0348-y
15.
Zadeh
,
L. A.
,
2002
, “
Toward a Perception-Based Theory of Probabilistic Reasoning With Imprecise Probabilities
,”
J. Sta. Plan. Inference
,
105
(
1
),
233
264
.
16.
Ferson
,
S.
,
Kreinovich
,
V.
,
Ginzburg
,
L.
,
Myers
,
D. S.
, and
Sentz
,
K.
,
2003
, “
Constructing Probability Boxes and Dempster-Shafer Structures
,” Technical Report No. SAND2002-4015.
17.
Matsumura
,
T.
, and
Haftka
,
R. T.
,
2013
, “
Reliability Based Design Optimization Modeling Future Redesign With Different Epistemic Uncertainty Treatments
,”
ASME J. Mech. Des.
,
135
(
9
), p.
091006
.10.1115/1.4024726
18.
Zaman
,
K.
,
Rangavajhala
,
S.
,
McDonald
,
M. P.
, and
Mahadevan
,
S.
,
2011
, “
A Probabilistic Approach for Representation of Interval Uncertainty
,”
Reliab. Eng. Syst. Saf.
,
96
(
1
), pp.
117
130
.10.1016/j.ress.2010.07.012
19.
Gu
,
X.
,
Renaud
,
J. E. E.
,
Batill
,
S. M. M.
,
Brach
,
R. M. M.
, and
Budhiraja
,
A. S.
,
2000
, “
Worst Case Propagated Uncertainty of Multidisciplinary Systems in Robust Optimization
,”
Struct. Optim.
,
20
(
3
), pp.
190
213
.10.1007/s001580050148
20.
Kokkolaras
,
M.
,
Mourelatos
,
Z. P.
, and
Papalambros
,
P. Y.
,
2006
, “
Design Optimization of Hierarchically Decomposed Multilevel Systems Under Uncertainty
,”
ASME J. Mech. Des.
,
128
(
2
), pp.
503
508
.10.1115/1.2168470
21.
Liu
,
H.
,
Chen
,
W.
,
Kokkolaras
,
M.
,
Papalambros
,
P. Y.
, and
Kim
,
H. M.
,
2006
, “
Probabilistic Analytical Target Cascading: A Moment Matching Formulation for Multilevel Optimization Under Uncertainty
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
991
1000
.10.1115/1.2205870
22.
Du
,
X.
, and
Chen
,
W.
,
2005
, “
Collaborative Reliability Analysis Under the Framework of Multidisciplinary Systems Design
,”
Optim. Eng.
,
6
(
1
), pp.
63
84
.10.1023/B:OPTE.0000048537.35387.fa
23.
Mahadevan
,
S.
, and
Smith
,
N.
,
2006
, “
Efficient First-Order Reliability Analysis of Multidisciplinary Systems
,”
Int. J. Reliab. Saf.
,
1
(
1
), pp.
137
154
.10.1504/IJRS.2006.010694
24.
Sankararaman
,
S.
, and
Mahadevan
,
S.
,
2012
, “
Likelihood-Based Approach to Multidisciplinary Analysis Under Uncertainty
,”
ASME J. Mech. Des.
,
134
(
3
), p.
031008
.10.1115/1.4005619
25.
Rebba
,
R.
,
Mahadevan
,
S.
, and
Huang
,
S.
,
2006
, “
Validation and Error Estimation of Computational Models
,”
Reliab. Eng. Syst. Saf.
,
91
(
10–11
), pp.
1390
1397
.10.1016/j.ress.2005.11.035
26.
Mahadevan
,
S.
, and
Liang
,
B.
,
2011
Error and Uncertainty Quantification and Sensitivity Analysis in Mechanics Computational Models
,”
Int. J. Uncertainty Quantif.
,
1
(
2
), pp.
147
161
.10.1615/IntJUncertaintyQuantification.v1.i2.30
27.
Kennedy
,
M. C.
, and
O’Hagan
,
A.
,
2001
, “
Bayesian Calibration of Computer Models
,”
J. R. Stat. Soc.
,
63
(
3
), pp.
425
464
.10.1111/1467-9868.00294
28.
Saltelli
,
A.
,
Ratto
,
M.
,
Andres
,
T.
,
Campolongo
,
F.
,
Cariboni
,
J.
,
Gatelli
,
D.
,
Saisana
,
M.
, and
Tarantola
,
S.
,
2008
,
Global Sensitivity Analysis: The Primer
,
Wiley-Interscience
,
Hoboken, NJ
.10.1002/9780470725184
29.
McKeeman
,
W. M.
,
1962
, “
Algorithm 145: Adaptive Numerical Integration by Simpson’s Rule
,”
Commun. ACM
,
5
(
12
), p.
604
.10.1145/355580.369102
30.
Mahadevan
,
S.
, and
Rebba
,
R.
,
2006
, “
Inclusion of Model Errors in Reliability-Based Optimization
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
936
944
.10.1115/1.2204973
31.
Chen
,
W.
,
Baghdasaryan
,
L.
,
Buranathiti
,
T.
, and
Cao
,
J.
,
2004
, “
Model Validation via Uncertainty Propagation and Data Transformations
,”
AIAA J.
,
42
(
7
), pp.
1403
1415
.10.2514/1.491
32.
Sankararaman
,
S.
,
Ling
,
Y.
,
Shantz
,
C.
, and
Mahadevan
,
S.
,
2011
, “
Inference of Equivalent Initial Flaw Size Under Multiple Sources of Uncertainty
,”
Int. J. Fatigue
,
33
(
2
), pp.
75
89
.10.1016/j.ijfatigue.2010.06.008
33.
Huang
,
S.
,
Mahadevan
,
S.
, and
Rebba
,
R.
,
2007
, “
Collocation-Based Stochastic Finite Element Analysis for Random Field Problems
,”
Probab. Eng. Mech.
,
22
(
2
), pp.
194
205
.10.1016/j.probengmech.2006.11.004
34.
Clarke
,
S. M.
,
Griebsch
,
J. H.
, and
Simpson
,
T. W.
,
2005
, “
Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses
,”
ASME J. Mech. Des.
,
127
(
6
), pp.
1077
1087
.10.1115/1.1897403
35.
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
.10.1115/1.2429697
36.
Hombal
,
V.
, and
Mahadevan
,
S.
,
2011
, “
Bias Minimization in Gaussian Process Surrogate Modeling for Uncertainty Quantification
,”
Int. J. Uncertainty Quantif.
,
1
(
4
), pp.
321
349
.10.1615/Int.J.UncertaintyQuantification.2011003343
37.
Zhu
,
P.
,
Zhang
,
S.
, and
Chen
,
W.
,
2014
, “
Multi-Point Objective-Oriented Sequential Sampling Strategy for Constrained Robust Design
,”
Eng. Optim.
, pp.
1
21
.10.1080/0305215X.2014.887705
38.
Pearson
,
E.
,
1938
, “
The Probability Integral Transformation for Testing Goodness of Fit and Combining Independent Tests of Significance
,”
Biometrika
,
30
(
1–2
), pp.
134
148
.10.1093/biomet/30.1-2.134
39.
Sankararaman
,
S.
, and
Mahadevan
,
S.
,
2012
, “
Separating the Contributions of Variability and Parameter Uncertainty in Probability Distributions
,”
Reliab. Eng. Syst. Saf.
,
112
(
4
), pp.
187
199
.10.1016/j.ress.2012.11.024
40.
Padula
,
S.
,
Alexandrov
,
N.
, and
Green
,
L.
,
1996
, “
MDO Test Suite at NASA Langley Research Center
,” AIAA Paper, (96-4028).
41.
Rangavajhala
,
S.
,
Sura
,
V.
,
Hombal
,
V.
, and
Mahadevan
,
S.
,
2011
Discretization Error Estimation in Multidisciplinary Simulations
,”
AIAA J.
,
49
(
12
), pp.
2673
2683
.10.2514/1.J051085
42.
Liu
,
Y.
,
Chen
,
W.
,
Arendt
,
P.
, and
Huang
,
H.-Z.
,
2011
, “
Toward a Better Understanding of Model Validation Metrics
,”
ASME J. Mech. Des.
,
133
(
7
), p.
071005
.10.1115/1.4004223
43.
Ling
,
Y.
, and
Mahadevan
,
S.
,
2013
, “
Quantitative Model Validation Techniques: New Insights
,”
Reliab. Eng. Syst. Saf.
,
111
, pp.
217
231
.10.1016/j.ress.2012.11.011
44.
Rangavajhala
,
S.
,
Liang
,
C.
, and
Mahadevan
,
S.
,
2012
, “
Design Optimization Under Aleatory and Epistemic Uncertainties
,”
Proceedings of 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
, Indianapolis, IN, Sept.17–19.10.2514/6.2012-5665
You do not currently have access to this content.