Response surface methods or metamodels are commonly used to approximate large computationally expensive engineering systems. Five response surface methods are studied: Stepwise Regression, Moving Least Square, Kriging, Multiquadric, and Adaptive and Interactive Modeling System. A real-world frontal impact design problem is used as an example, which is a complex, highly nonlinear, transient, dynamic, large deformation finite element model. To study the accuracy of the metamodel, the optimal Latin Hypercube Sampling method is used to distribute the sampling points uniformly over the entire design space. The Root Mean Square Error (RMSE) is used as the error indicator. Convergence rate, widely used in the arena of the finite element method for evaluating new element’s performance, was exploited in this vehicle impact example.

1.
Craig
,
K. J.
,
Stander
,
N.
,
Dooge
,
D. A.
, and
Varadappa
,
S.
, 2005, “
Automotive Crashworthiness Design Using Response Surface-Based Variable Screening and Optimization
,”
Eng. Comput.
0177-0667,
22
(
1
), pp.
38
61
.
2.
Wang
,
X.
,
Liu
,
Y.
, and
Antonsson
,
E.
, 1999, “
Fitting Functions to Data in High Dimensional Design Space
,” in
Proceedings of DETC 99, 1999 ASME Design Engineering Technical Conferences
, DETC99∕DAC-8622.
3.
Sobieski
,
J.
,
Kodiyalam
,
S.
, and
Yang
,
R. J.
, 2001, “
Optimization of Car Body under Constraints of Noise, Vibration, and Harshness (NVH), and Crash
,”
Struct. Multidiscip. Optim.
1615-147X,
22
, pp.
295
306
.
4.
Jin
,
R.
,
Chen
,
W.
, and
Simpson
,
T. W.
, 2000, “
Comparative Studies of Metamodeling Techniques under Multiple Modeling Criteria
,” AIAA-2000-4801.
5.
Yang
,
R. J.
,
Gu
,
L.
,
Liaw
,
L.
,
Gearhart
,
C.
,
Tho
,
C. H.
,
Liu
,
X.
, and
Wang
,
B. P.
, 2000, “
Approximations for Safety Optimization of Large Systems
,”
ASME Design Automation Conference
, Paper No. DETC2000∕DAC-14245.
6.
Lu
,
Stephen C-Y.
, and
Cheng
,
D. K.
, 1991, “
Building Layered Models to Support Engineering Decision Making: A Machine Learning Approach
,”
J. Eng. Ind.
0022-0817, ASME Transactions,
13
(
1
), pp.
1
9
.
7.
Ge
,
P.
, and
Wang
,
N.
, 2000, “
DoE Study in Building Surrogate Models for Complex Systems
,” in
Proceedings of DETC’2000
, Baltimore.
8.
Simpson
,
T. W.
,
Booker
,
A. J.
,
Ghosh
,
D.
,
Giunta
,
A. A.
,
Koch
,
P. N.
, and
Yang
,
R. J.
, 2004, “
Approximation Methods in Multidisciplinary Analysis and Optimization: A Panel Discussion
,”
Struct. Multidiscip. Optim.
1615-147X,
27
, pp.
302
313
.
9.
Simpson
,
T. W.
,
Clarke
,
S. M.
, and
Griebsch
,
J. H.
, 2005, “
Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses
,”
ASME J. Mech. Des.
1050-0472,
MD-03-1319.
10.
Efroymson
,
M.
, 1960, “
Multiple Regression Analysis
,”
Mathematical Methods for Digital Computers
, edited by
A.
Ralston
and
H.
Wilf
,
Wiley
, New York, pp.
191
203
.
11.
Lancaster
,
P.
, and
Salkauskas
,
K.
, 1981, “
Surfaces Generated by Moving Least Squares Methods
,”
Math. Comput.
0025-5718,
37
, pp.
141
158
.
12.
Hawkins
,
D. M.
, and
Cressie
,
N.
, 1984, “
Robust Kriging—a Proposal
,”
Journal of the International Association for Mathematical Geology
,
16
, pp.
3
18
.
13.
Oliver
,
M. A.
, and
Webster
,
R.
, 1990, “
Kriging: A Method of Interpolation for Geographical Information Systems
,”
Int. J. Geograph. Inf. Sci.
1365-8816,
4
, pp.
313
332
.
14.
Cressie
,
N. C. A.
, 1993,
Statistics for Spatial Data
,
Wiley
, New York.
15.
Ripley
,
B. D.
, 1987,
Stochastic Simulation
,
Wiley
, New York.
16.
Kodiyalam
,
S.
,
Yang
,
R. J.
,
Gu
,
L.
, and
Tho
,
C-H.
, 2004, “
Multidisciplinary Design Optimization of a Vehicle System in a Scalable, High Performance Computing Environment
,”
Struct. Multidiscip. Optim.
1615-147X,
26
, pp.
235
248
.
17.
Hardy
,
R. L.
, 1971, “
Multiquadric Equations Of Topography and Other Irregular Surfaces
,”
J. Geophys. Res.
0148-0227,
76
, pp.
1905
1915
.
18.
Hardy
,
R. L.
, 1990, “
Theory and Application of the Multiquadric-Biharmonic Method
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
19
, pp.
163
208
.
19.
Wang
,
B. P.
, 2004, “
Parameter Optimization in Multiquadric Response Surface Approximation
,”
Struct. Multidiscip. Optim.
1615-147X,
26
, pp.
219
223
.
20.
RADIOSS User’s Manual, Version 4.1, MECALOG, France, 1998.
21.
MADYMO User’s Manual, Version 5.3, TNO, Netherlands, 1998.
You do not currently have access to this content.