The performance of a multidisciplinary system is inevitably affected by various sources of uncertainties, usually categorized as aleatory (e.g., input variability) or epistemic (e.g., model uncertainty) uncertainty. In the framework of design under uncertainty, all sources of uncertainties should be aggregated to assess the uncertainty of system quantities of interest (QOIs). In a multidisciplinary design system, uncertainty propagation (UP) refers to the analysis that quantifies the overall uncertainty of system QOIs resulting from all sources of aleatory and epistemic uncertainty originating in the individual disciplines. However, due to the complexity of multidisciplinary simulation, especially the coupling relationships between individual disciplines, many UP approaches in the existing literature only consider aleatory uncertainty and ignore the impact of epistemic uncertainty. In this paper, we address the issue of efficient uncertainty quantification of system QOIs considering both aleatory and epistemic uncertainties. We propose a spatial-random-process (SRP) based multidisciplinary uncertainty analysis (MUA) method that, subsequent to SRP-based disciplinary model uncertainty quantification, fully utilizes the structure of SRP emulators and leads to compact analytical formulas for assessing statistical moments of uncertain QOIs. The proposed method is applied to a benchmark electronic packaging design problem. The estimated low-order statistical moments of the QOIs are compared to the results from Monte Carlo simulations (MCSs) to demonstrate the effectiveness of the method. The UP result is then used to facilitate the robust design optimization of the electronic packaging system.

References

References
1.
Shubin
,
G. R.
,
1994
, “
Optimization Problem Formulation for Multidisciplinary Design
,”
Proceedings of the Conference Inverse Problems and Optimal Design in Industry
, Philadelphia, July 8–10, Vieweg+Teubner Verlag, Springer, Wiesbaden, pp.
213
216
.
2.
Alexandrov
,
N. M.
, and
Lewis
,
R. M.
,
2000
, “
Algorithmic Perspectives on Problem Formulations in MDO
,”
AIAA
Paper No. 2000-4719.
3.
Belytschko
,
T.
,
1980
, “
Fluid-Structure Interaction
,”
Comput. Struct.
,
12
(
4
), pp.
459
469
.
4.
McNamara
,
J. J.
, and
Friedmann
,
P. P.
,
2011
, “
Aeroelastic and Aerothermoelastic Analysis in Hypersonic Flow: Past, Present, and Future
,”
AIAA J.
,
49
(
6
), pp.
1089
1122
.
5.
Yao
,
W.
,
Chen
,
X. Q.
,
Luo
,
W. C.
,
van Tooren
,
M.
, and
Guo
,
J.
,
2011
, “
Review of Uncertainty-Based Multidisciplinary Design Optimization Methods for Aerospace Vehicles
,”
Prog. Aerosp. Sci.
,
47
(
6
), pp.
450
479
.
6.
Matthies
,
H. G.
,
2007
, “
Quantifying Uncertainty: Modern Computational Representation of Probability and Applications
,”
Extreme Man-Made and Natural Hazards in Dynamics of Structures
,
Springer
, Dordrecht,
the Netherlands
, pp.
105
135
.
7.
Kiureghian
,
A. D.
, and
Ditlevsen
,
O.
,
2009
, “
Aleatory or Epistemic? Does It Matter?
Struct. Saf.
,
31
(
2
), pp.
105
112
.
8.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2001
, “
Bayesian Calibration of Computer Models
,”
J. R. Stat. Soc., Ser. B
,
63
(3), pp.
425
464
.
9.
Zaman
,
K.
, and
Mahadevan
,
S.
,
2013
, “
Robustness-Based Design Optimization of Multidisciplinary System Under Epistemic Uncertainty
,”
AIAA J.
,
51
(
5
), pp.
1021
1031
.
10.
Helton
,
J. C.
,
Johnson
,
J. D.
,
Sallaberry
,
C. J.
, and
Storlie
,
C. B.
,
2006
, “
Survey of Sampling-Based Methods for Uncertainty and Sensitivity Analysis
,”
Reliab. Eng. Syst. Saf.
,
91
(
10–11
), pp.
1175
1209
.
11.
Landau
,
D. P.
, and
Binder
,
K.
,
2005
,
A Guide to Monte Carlo Simulations in Statistical Physics
,
2nd ed.
,
Cambridge University Press
,
New York
.
12.
Robert
,
C.
, and
Casella
,
G.
,
1999
,
Monte Carlo Statistical Methods
,
Springer
,
New York
.
13.
Thoft-Cristensen
,
P.
, and
Baker
,
M. J.
,
1982
,
Structural Reliability Theory and Its Applications
,
Springer
,
New York
.
14.
Green
,
L. L.
,
Lin
,
H.-Z.
, and
Khalessi
,
M. R.
,
2002
, “
Probabilistic Methods for Uncertainty Propagation Applied to Aircraft Design
,”
AIAA
Paper No. 2002.3140.
15.
Cao
,
H.
, and
Duan
,
B.
,
2004
, “
Uncertainty Analysis for Multidisciplinary Systems Based on Convex Models
,”
AIAA
Paper No. 2004-4504.
16.
Du
,
X.
,
Guo
,
J.
, and
Beeram
,
H.
,
2008
, “
Sequential Optimization and Reliability Assessment for Multidisciplinary Systems Design
,”
Struct. Multidiscip. Optim.
,
35
(
2
), pp.
117
130
.
17.
Guo
,
J.
, and
Du
,
X. P.
,
2010
, “
Reliability Analysis for Multidisciplinary Systems With Random and Interval Variables
,”
AIAA J.
,
48
(
1
), pp.
82
91
.
18.
Gu
,
X.
, and
Renaud
,
J. E.
,
2002
, “
Implementation Study of Implicit Uncertainty Propagation (IUP) in Decomposition-Based Optimization
,”
AIAA
Paper No. 2002-5416.
19.
Gu
,
X. S.
,
Renaud
,
J. E.
, and
Penninger
,
C. L.
,
2006
, “
Implicit Uncertainty Propagation for Robust Collaborative Optimization
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
1001
1013
.
20.
Du
,
X.
, and
Chen
,
W.
,
2002
, “
Collaborative Reliability Analysis for Multidisciplinary Systems Design
,”
AIAA
Paper No. 2002-5474.
21.
Du
,
X.
, and
Chen
,
W.
,
2005
, “
Collaborative Reliability Analysis Under the Framework of Multidisciplinary Systems Design
,”
Optim. Eng.
,
6
(
1
), pp.
63
84
.
22.
Xiong
,
F.
,
Yin
,
X.
,
Chen
,
W.
, and
Yang
,
S.
,
2010
, “
Enhanced Probabilistic Analytical Target Cascading With Application to Multi-Scale Design
,”
Eng. Optim.
,
42
(
6
), pp.
581
592
.
23.
Sankararaman
,
S.
, and
Mahadevan
,
S.
,
2012
, “
Likelihood-Based Approach to Multidisciplinary Analysis Under Uncertainty
,”
ASME J. Mech. Des.
,
134
(
3
), p.
031008
.
24.
Liang
,
C.
, and
Mahadevan
,
S.
,
2013
, “
Stochastic Multidisciplinary Analysis With High Dimensional Coupling
,”
Tenth World Congress on Structural and Multidisciplinary Optimization
, Orlando, FL, May 19–24.
25.
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
.
26.
Zhang
,
S. L.
,
Zhu
,
P.
,
Chen
,
W.
, and
Arendt
,
P.
,
2013
, “
Concurrent Treatment of Parametric Uncertainty and Metamodeling Uncertainty in Robust Design
,”
Struct. Multidiscip. Optim.
,
47
(
1
), pp.
63
76
.
27.
Gu
,
X.
,
Renaud
,
J. E.
,
Batill
,
S. M.
,
Brach
,
R. M.
, and
Budhiraja
,
A. S.
,
2000
, “
Worst Case Propagated Uncertainty of Multidisciplinary Systems in Robust Design Optimization
,”
Struct. Multidiscip. Optim.
,
20
(
3
), pp.
190
213
.
28.
Du
,
X.
, and
Chen
,
W.
,
2000
, “
An Efficient Approach to Probabilistic Uncertainty Analysis in Simulation-Based Multidisciplinary Design
,”
AIAA
Paper No. 2000-0423.
29.
Du
,
X.
, and
Chen
,
W.
,
2000
, “
Methodology for Managing the Effect of Uncertainty in Simulation-Based Design
,”
AIAA J.
,
38
(
8
), pp.
1471
1478
.
30.
Du
,
X.
, and
Chen
,
W.
,
2002
, “
Efficient Uncertainty Analysis Methods for Multidisciplinary Robust Design
,”
AIAA J.
,
40
(
3
), pp.
545
552
.
31.
Jiang
,
X. M.
, and
Mahadevan
,
S.
,
2009
, “
Bayesian Hierarchical Uncertainty Quantification by Structural Equation Modeling
,”
Int. J. Numer. Methods Eng.
,
80
(
6–7
), pp.
717
737
.
32.
Jiang
,
X. M.
, and
Mahadevan
,
S.
,
2009
, “
Bayesian Structural Equation Modeling Method for Hierarchical Model Validation
,”
Reliab. Eng. Syst. Saf.
,
94
(
4
), pp.
796
809
.
33.
Sankararaman
,
S.
,
McLemore
,
K.
,
Mahadevan
,
S.
,
Bradford
,
S. C.
, and
Peterson
,
L. D.
,
2013
, “
Test Resource Allocation in Hierarchical Systems Using Bayesian Networks
,”
AIAA J.
,
51
(
3
), pp.
537
550
.
34.
Allaire
,
D.
,
He
,
Q.
,
Deyst
,
J.
, and
Willcox
,
K.
,
2012
, “
An Information-Theoretic Metric of System Complexity With Application to Engineering System Design
,”
ASME J. Mech. Des.
,
134
(
10
), p.
100906
.
35.
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
.
36.
Conti
,
S.
,
Gosling
,
J. P.
,
Oakley
,
J. E.
, and
O'Hagan
,
A.
,
2009
, “
Gaussian Process Emulation of Dynamic Computer Codes
,”
Biometrika
,
96
(
3
), pp.
663
676
.
37.
Conti
,
S.
, and
O'Hagan
,
A.
,
2010
, “
Bayesian Emulation of Complex Multi-Output and Dynamic Computer Models
,”
J. Stat. Plann. Inference
,
140
(
3
), pp.
640
651
.
38.
Higdon
,
D.
,
Kennedy
,
M.
,
Cavendish
,
J. C.
,
Cafeo
,
J. A.
, and
Ryne
,
R. D.
,
2004
, “
Combining Field Data and Computer Simulations for Calibration and Prediction
,”
SIAM J. Sci. Comput.
,
26
(
2
), pp.
448
466
.
39.
Higdon
,
D.
,
Gattiker
,
J.
,
Williams
,
B.
, and
Rightley
,
M.
,
2008
, “
Computer Model Calibration Using High-Dimensional Output
,”
J. Am. Stat. Assoc.
,
103
(
482
), pp.
570
583
.
40.
Arendt
,
P. D.
,
Apley
,
D. W.
, and
Chen
,
W.
,
2012
, “
Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability
,”
ASME J. Mech. Des.
,
134
(
10
), p.
100908
.
41.
Arendt
,
P. D.
,
Apley
,
D. W.
,
Chen
,
W.
,
Lamb
,
D.
, and
Gorsich
,
D.
,
2012
, “
Improving Identifiability in Model Calibration Using Multiple Responses
,”
ASME J. Mech. Des.
,
134
(
10
), p.
100909
.
42.
Jiang
,
Z.
,
Chen
,
W.
,
Fu
,
Y.
, and
Yang
,
R.-J.
,
2013
, “
Reliability-Based Design Optimization With Model Bias and Data Uncertainty
,”
SAE Int. J. Mater. Manuf.
,
6
(
3
), pp.
502
516
.
43.
Rasmussen
,
C. E.
, and
Williams
,
C. K. I.
,
2006
,
Gaussian Processes for Machine Learning
,
The MIT Press
, Cambridge, MA.
44.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2000
, “
Predicting the Output From a Complex Computer Code When Fast Approximations Are Available
,”
Biometrika
,
87
(
1
), pp.
1
13
.
45.
Hasselman
,
T. K.
,
Yap
,
K.
,
Lin
,
C.-H.
, and
Cafeo
,
J. A.
,
2005
, “
A Case Study in Model Improvement for Vehicle Crashworthiness Simulation
,”
23rd International Modal Analysis Conference
(
IMAC-XXIII
), Orlando, FL, Jan. 31–Feb. 3.
46.
Chen
,
W.
,
Xiong
,
Y.
,
Tsui
,
K. L.
, and
Wang
,
S.
,
2008
, “
A Design-Driven Validation Approach Using Bayesian Prediction Models
,”
ASME J. Mech. Des.
,
130
(
2
), p.
021101
.
47.
Qian
,
P. Z. G.
, and
Wu
,
C. F. J.
,
2008
, “
Bayesian Hierarchical Modeling for Integrating Low-Accuracy and High-Accuracy Experiments
,”
Technometrics
,
50
(
2
), pp.
192
204
.
48.
Wang
,
S. C.
,
Chen
,
W.
, and
Tsui
,
K. L.
,
2009
, “
Bayesian Validation of Computer Models
,”
Technometrics
,
51
(
4
), pp.
439
451
.
49.
Bayarri
,
M. J.
,
Berger
,
J. O.
,
Paulo
,
R.
,
Sacks
,
J.
,
Cafeo
,
J. A.
,
Cavendish
,
J.
,
Lin
,
C. H.
, and
Tu
,
J.
,
2007
, “
A Framework for Validation of Computer Models
,”
Technometrics
,
49
(
2
), pp.
138
154
.
50.
Liu
,
F.
,
Bayarri
,
M. J.
,
Berger
,
J. O.
,
Paulo
,
R.
, and
Sacks
,
J.
,
2008
, “
A Bayesian Analysis of the Thermal Challenge Problem
,”
Comput. Methods Appl. Mech. Eng.
,
197
(
29–32
), pp.
2457
2466
.
51.
Salas
,
A. O.
,
1995
, “
MDO Test Suite
,” University at Buffalo, Buffalo, NY, http://www.eng.buffalo.edu/Research/MODEL/mdo.test.orig/class2prob3.html
52.
Renaud
,
J.
, and
Gabriele
,
G.
,
1994
, “
Approximation in Nonhierarchic System Optimization
,”
AIAA J.
,
32
(
1
), pp.
198
205
.
53.
Padula
,
S. L.
,
Alexandrov
,
N.
, and
Green
,
L. L.
,
1996
, “
MDO Test Suite at NASA Langley Research Center
,”
AIAA
Paper No. 96-4028.
54.
Kodiyalam
,
S.
, and
Sobieszczanski-Sobieski
,
J.
,
2001
, “
Multidisciplinary Design Optimization: Some Formal Methods, Framework Requirements, and Application to Vehicle Design
,”
Int. J. Veh. Des.
,
25
(
1–2
), pp.
3
22
.
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