Statistical sensitivity analysis (SSA) is playing an increasingly important role in engineering design, especially with the consideration of uncertainty. However, it is not straightforward to apply SSA to the design of complex engineering systems due to both computational and organizational difficulties. In this paper, to facilitate the application of SSA to the design of complex systems especially those that follow hierarchical modeling structures, a hierarchical statistical sensitivity analysis (HSSA) method containing a top-down strategy for SSA and an aggregation approach to evaluating the global statistical sensitivity index (GSSI) is developed. The top-down strategy for HSSA is introduced to invoke the SSA of the critical submodels based on the significance of submodel performances. A simplified formulation of the GSSI is studied to represent the effect of a lower-level submodel input on a higher-level model response by aggregating the submodel SSA results across intermediate levels. A sufficient condition under which the simplified formulation provides an accurate solution is derived. To improve the accuracy of the GSSI formulation for a general situation, a modified formulation is proposed by including an adjustment coefficient (AC) to capture the impact of the nonlinearities of the upper-level models. To improve the efficiency, the same set of samples used in submodel SSAs is used to evaluate the AC. The proposed HSSA method is examined through mathematical examples and a three-level hierarchical model used in vehicle suspension systems design.

1.
Wagner
,
T. C.
, 1993, “
A General Decomposition Methodology for Optimal System Design
,” Ph.D. thesis, University of Michigan, Ann Arbor, MI.
2.
Michelena
,
N. F.
, and
Papalambros
,
P. Y.
, 1995, “
Optimal Model-Based Decomposition of Powertrain System Design
,”
ASME J. Mech. Des.
1050-0472,
117
(
4
), pp.
499
505
.
3.
Krishnamachari
,
R. S.
, and
Papalambros
,
P. Y.
, 1997, “
Optimal Hierarchical Decomposition Synthesis Using Integer Programming
,”
ASME J. Mech. Des.
1050-0472,
119
(
4
), pp.
440
447
.
4.
Lee
,
J. K.
, and
Lim
,
Y. H.
, 2004, “
Hierarchical Modeling and Simulation Environment for Intelligent Transportation Systems
,”
Simulation
0037-5497,
80
(
2
), pp.
61
76
.
5.
Gentil
,
S.
, and
Montmain
,
J.
, 2004, “
Hierarchical Representation of Complex Systems for Supporting Human Decision Making
,”
Adv. Eng. Inf.
1474-0346,
18
(
3
), pp.
143
159
.
6.
Hao
,
S.
,
Liu
,
W. K.
,
Moran
,
B.
,
Vernerey
,
F.
, and
Olson
,
G. B.
, 2004, “
Multi-Scale Constitutive Model and Computational Framework for the Design of Ultra-High Strength, High Toughness Steels
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
193
, pp.
1865
1908
.
7.
Vernerey
,
F. J.
, 2006, “
Multi-Scale Mechanics of Microstructured Materials
,” Ph.D. thesis, Northwestern University, Evanston, IL.
8.
Gall
,
K.
, and
Horstemeyer
,
M. F.
, 2000, “
Integration of Basic Materials Research Into the Design of Cast Components by a Multi-Scale Methodology
,”
ASME J. Eng. Mater. Technol.
0094-4289,
122
(
3
), pp.
355
362
.
9.
Choi
,
H.-J.
,
Allen
,
J. K.
,
Rosen
,
D.
,
McDowell
,
D. L.
, and
Mistree
,
F.
, 2005, “
An Inductive Design Exploration Method for the Integrated Design of Multi-Scale Material and Products
,”
Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
,
Long Beach, CA
, Sept. 24–28.
10.
Panchal
,
J. H.
,
Choi
,
H.-J.
,
Shepherd
,
J.
,
Allen
,
J. K.
,
McDowell
,
D. L.
, and
Mistree
,
F.
, 2005, “
A Strategy for Simulation-Based Design of Multiscale, Multi-Functional Products and Associated Design Processes
,”
Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
,
Long Beach, CA
, Sept. 24–28.
11.
Kokkolaras
,
M.
,
Mourelatos
,
Z. P.
, and
Papalambros
,
P. Y.
, 2006, “
Design Optimization of Hierarchically Decomposed Multilevel Systems Under Uncertainty
,”
ASME J. Mech. Des.
1050-0472,
128
(
2
), pp.
503
508
.
12.
Kim
,
H. M.
,
Michelena
,
N. F.
,
Papalambros
,
P.
, and
Jiang
,
T.
, 2003, “
Target Cascading in Optimal System Design
,”
ASME J. Mech. Des.
1050-0472,
125
(
3
), pp.
474
480
.
13.
Liu
,
H.
,
Chen
,
W.
,
Kokkolaras
,
M.
,
Papalambros
,
P. Y.
, and
Kim
,
H. M.
, 2005, “
Probabilistic Analytical Target Cascading: A Moment Matching Formulation for Multilevel Optimization under Uncertainty
,”
Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
,
Long Beach, CA
, Sept. 24–28.
14.
Du
,
X.
, and
Chen
,
W.
, 2005, “
Collaborative Reliability Analysis under the Framework of Multidisciplinary Systems Design
,”
Optim. Eng.
1389-4420,
6
(
1
), pp.
63
84
.
15.
Saltelli
,
A.
,
Chan
,
K.
, and
Scott
,
E. M.
, 2000,
Sensitivity Analysis
,
Wiley
,
New York
.
16.
Sobol
,
I. M.
, 1993, “
Sensitivity Analysis for Nonlinear Mathematical Models
,”
Math. Model. Comput. Exper.
,
1
, pp.
407
414
.
17.
Saltelli
,
A.
, and
Bolado
,
R.
, 1998, “
An Alternative Way to Compute Fourier Amplitude Sensitivity Test (FAST)
,”
Comput. Stat. Data Anal.
0167-9473,
26
(
4
), pp.
445
460
.
18.
Fang
,
S. F.
,
Gertner
,
G. Z.
,
Shinkareva
,
S.
,
Wang
,
G. X.
, and
Anderson
,
A.
, 2003, “
Improved Generalized Fourier Amplitude Sensitivity Test (FAST) for Model Assessment
,”
Stat. Comput.
0960-3174,
13
(
3
), pp.
221
226
.
19.
Liu
,
H.
,
Chen
,
W.
, and
Sudjianto
,
A.
, 2006, “
Relative Entropy Based Method for Global and Regional Sensitivity Analysis in Probabilistic Design
,”
ASME J. Mech. Des.
1050-0472,
128
(
2
), pp.
326
336
.
20.
Cramer
,
E. J.
,
Dennis
Jr,
J. E.
,
Frank
,
P. D.
,
Lewis
,
R. M.
, and
Shubin
,
G. R.
, 1994, “
Problem Formulation for Multidisciplinary Optimization
,”
SIAM J. Optim.
1052-6234,
4
(
4
), pp.
754
776
.
21.
Wehrhahn
,
E.
, 1991, “
Hierarchical Sensitivity Analysis of Circuits
,”
IEEE International Symposium on Circuits and Systems
, pp.
864
867
.
22.
Noor
,
A. K.
,
Starnes
,
J. H.
, and
Peters
,
J. M.
, 2000, “
Uncertainty Analysis of Composite Structures
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
185
(
2–4
), pp.
413
432
.
23.
Sobieszczanski-Sobieski
,
J.
,
Bloebaum
,
C.
, and
Hajela
,
P.
, 1991, “
Sensitivity of Control-Augmented Structure Obtained by a System Decomposition Method
,”
AIAA J.
0001-1452,
29
(
2
), pp.
264
270
.
24.
McVeigh
,
C.
,
Vernerey
,
F.
,
Liu
,
W. K.
, and
Brinson
,
L. C.
, 2006, “
Multiresolution Analysis for Material Design
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
195
(
37–40
), pp.
5053
5076
.
25.
Kim
,
H. M.
,
Kumar
,
D. K. D.
,
Chen
,
W.
, and
Papalambros
,
P.
, 2006, “
Target Exploration for Disconnected Feasible Regions in Enterprise-Driven Multilevel Product Design
,”
AIAA J.
0001-1452,
44
(
1
), pp.
67
77
.
26.
Mcrae
,
G. J.
,
Tilden
,
J. W.
, and
Seinfeld
,
J. H.
, 1982, “
Global Sensitivity Analysis—a Computational Implementation of the Fourier Amplitude Sensitivity Test (Fast)
,”
Comput. Chem. Eng.
0098-1354,
6
(
1
), pp.
15
25
.
27.
Sobol
,
I. M.
, 2001, “
Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates
,”
Math. Comput. Simul.
0378-4754,
55
(
1–3
), pp.
271
280
.
28.
Chen
,
W.
,
Jin
,
R.
, and
Sudjianto
,
A.
, 2005, “
Analytical Variance-Based Global Sensitivity Analysis in Simulation-Based Design under Uncertainty
,”
ASME J. Mech. Des.
1050-0472,
127
(
5
), pp.
875
886
.
29.
Homma
,
T.
, and
Saltelli
,
A.
, 1996, “
Importance Measures in Global Sensitivity Analysis of Nonlinear Models
,”
Reliab. Eng. Syst. Saf.
0951-8320,
52
(
1
), pp.
1
17
.
30.
Jansen
,
M. J. W.
, 1996, “
Winding Stairs Sample Analysis program WINDINGS 2.0
,” Agricultural University of Wageningen.
31.
Clark
,
C. E.
, 1961, “
Importance Sampling in Monte Carlo Analyses
,”
Oper. Res.
0030-364X,
9
(
5
), pp.
603
620
.
32.
Hocevar
,
D. E.
,
Lightner
,
M. R.
, and
Trick
,
T. N.
, 1983, “
A Study of Variance Reduction Techniques for Estimating Circuit Yields
,”
IEEE Trans. Comput.-Aided Des.
0278-0070,
2
(
3
), pp.
180
192
.
33.
Frey
,
D. D.
,
Reber
,
G.
, and
Lin
,
Y.
, 2005, “
A Quadrature-Based Sampling Technique for Robust Design with Computer Models
,”
Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
,
Long Beach, CA
, Sept. 24–28.
34.
Wu
,
J. C. F.
, and
Hamada
,
M.
, 2000,
Experiments: Planning, Analysis, and Parameter Design Optimization
,
Wiley
,
New York
.
35.
Tamhane
,
A. C.
, and
Dunlop
,
D. D.
, 2000,
Statistics and Data Analysis from Elementary to Intermediate
,
Prentice-Hall
,
Upper Saddle River, NJ
.
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