Abstract

Metamodel technology provides an efficient method to approximate complex engineering design problems. However, the approximation for high-dimensional problems usually requires a large number of samples for most traditional metamodeling methods, which leads to the difficulty of “curse of dimensionality.” To address the aforementioned issue, this paper presents the Net-high dimension model representation (HDMR) method based on the Cut-HDMR framework. Compared with traditional HDMR modeling, the Net-HDMR method incorporates two novel modeling approaches that improve the modeling efficiency of high-dimensional problems. The first approach enhances the modeling accuracy of HDMR by using the net function interpolation method to decompose the component functions into a series of one-dimensional net functions. The second approach adopts the CV-Voronoi sequence sampling method to effectively represent one-dimensional net functions with limited samples. Overall, the proposed method transforms complex high-dimensional problems into fitting finite one-dimensional splines, thereby increasing the modeling efficiency while ensuring approximate accuracy. Six numerical benchmark examples with different dimensions are examined to demonstrate the accuracy and efficiency of the proposed Net-HDMR. An engineering problem of thermal stress and deformation analysis for a jet engine turbine blade was introduced to verify the engineering feasibility of the proposed Net-HDMR.

References

1.
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
.
2.
Jiang
,
P.
,
Zhou
,
Q.
, and
Shao
,
X.
,
2020
,
Surrogate Model-Based Engineering Design and Optimization
,
Springer
,
Singapore
, pp.
135
236
.
3.
Jensen
,
W. A.
,
2017
, “
Response Surface Methodology: Process and Product Optimization Using Designed Experiments
,”
J. Qual. Technol.
,
49
(
2
), p.
186
.
4.
Liu
,
X.
,
Liu
,
X.
,
Zhou
,
Z.
, and
Hu
,
L.
,
2021
, “
An Efficient Multi-Objective Optimization Method Based on the Adaptive Approximation Model of the Radial Basis Function
,”
Struct. Multidiscip. Optim.
,
63
(
3
), pp.
1385
1403
.
5.
Fuhg
,
J. N.
,
Fau
,
A.
, and
Nackenhorst
,
U.
,
2021
, “
State-of-the-Art and Comparative Review of Adaptive Sampling Methods for Kriging
,”
Arch. Comput. Meth. Eng.
,
28
(
4
), pp.
2689
2747
.
6.
Kabasi
,
S.
,
Roy
,
A.
, and
Chakraborty
,
S.
,
2021
, “
A Generalized Moving Least Square-Based Response Surface Method for Efficient Reliability Analysis of Structure
,”
Struct. Multidiscip. Optim.
,
63
(
3
), pp.
1085
1097
.
7.
Cheng
,
K.
, and
Lu
,
Z.
,
2021
, “
Adaptive Bayesian Support Vector Regression Model for Structural Reliability Analysis
,”
Reliab. Eng. Syst. Saf.
,
206
, p.
107286
.
8.
Xu
,
Z.
,
Zhang
,
X.
,
Wang
,
S.
, and
He
,
G.
,
2022
, “
Artificial Neural Network Based Response Surface for Data-Driven Dimensional Analysis
,”
J. Comput. Phys.
,
459
, p.
111145
.
9.
Rabitz
,
H.
, and
Aliş
,
ÖF
,
1999
, “
General Foundations of High-Dimensional Model Representations
,”
J. Math. Chem.
,
25
(
2–3
), pp.
197
233
.
10.
Rabitz
,
H.
,
Aliş
,
ÖF
,
Shorter
,
J.
, and
Shim
,
K.
,
1999
, “
Efficient Input-Output Model Representations
,”
Comput. Phys. Commun.
,
117
(
1–2
), pp.
11
20
.
11.
Alış
,
ÖF
, and
Rabitz
,
H.
,
2001
, “
Efficient Implementation of High Dimensional Model Representations
,”
J. Math. Chem.
,
29
(
2
), pp.
127
142
.
12.
Shan
,
S.
, and
Wang
,
G. G.
,
2010
, “
Metamodeling for High Dimensional Simulation-Based Design Problems
,”
ASME J. Mech. Des.
,
132
(
5
), p.
051009
.
13.
Boussaidi
,
M. A.
,
Ren
,
O.
,
Voytsekhovsky
,
D.
, and
Manzhos
,
S.
,
2020
, “
Random Sampling High Dimensional Model Representation Gaussian Process Regression (RS-HDMR-GPR) for Multivariate Function Representation: Application to Molecular Potential Energy Surfaces
,”
J. Phys. Chem. A
,
124
(
37
), pp.
7598
7607
.
14.
Yue
,
X.
,
Zhang
,
J.
,
Gong
,
W.
,
Luo
,
M.
, and
Duan
,
L.
,
2021
, “
An Adaptive PCE-HDMR Metamodeling Approach for High-Dimensional Problems
,”
Struct. Multidiscip. Optim.
,
64
(
1
), pp.
141
162
.
15.
Luo
,
X.
,
Lu
,
Z.
, and
Xu
,
X.
,
2014
, “
Reproducing Kernel Technique for High Dimensional Model Representations (HDMR)
,”
Comput. Phys. Commun.
,
185
(
12
), pp.
3099
3108
.
16.
Hajikolaei
,
K. H.
, and
Gary Wang
,
G.
,
2014
, “
High Dimensional Model Representation With Principal Component Analysis
,”
ASME J. Mech. Des.
,
136
(
1
), p.
011003
.
17.
Jiang
,
L.
, and
Li
,
X.
,
2015
, “
Multi-Element Least Square HDMR Methods and Their Applications for Stochastic Multiscale Model Reduction
,”
J. Comput. Phys.
,
294
, pp.
439
461
.
18.
Cheng
,
K.
,
Lu
,
Z.
, and
Chaozhang
,
K.
,
2019
, “
Gradient-Enhanced High Dimensional Model Representation Via Bayesian Inference
,”
Knowl. Based Syst.
,
184
, p.
104903
.
19.
Zhou
,
Y.
, and
Lu
,
Z.
,
2020
, “
An Enhanced Kriging Surrogate Modeling Technique for High-Dimensional Problems
,”
Mech. Syst. Signal Process.
,
140
, p.
106687
.
20.
Chen
,
L.
,
Qiu
,
H.
,
Gao
,
L.
,
Jiang
,
C.
, and
Yang
,
Z.
,
2019
, “
A Screening-Based Gradient-Enhanced Kriging Modeling Method for High-Dimensional Problems
,”
Appl. Math. Model.
,
69
, pp.
15
31
.
21.
Chen
,
L.
,
Qiu
,
H.
,
Gao
,
L.
,
Yang
,
Z.
, and
Xu
,
D.
,
2022
, “
Exploiting Active Subspaces of Hyperparameters for Efficient High-Dimensional Kriging Modeling
,”
Mech. Syst. Signal Process.
,
169
, p.
108643
.
22.
Garud
,
S. S.
,
Karimi
,
I. A.
, and
Kraft
,
M.
,
2017
, “
Smart Sampling Algorithm for Surrogate Model Development
,”
Comput. Chem. Eng.
,
96
, pp.
103
114
.
23.
Garud
,
S. S.
,
Karimi
,
I. A.
,
Brownbridge
,
G. P. E.
, and
Kraft
,
M.
,
2018
, “
Evaluating Smart Sampling for Constructing Multidimensional Surrogate Models
,”
Comput. Chem. Eng.
,
108
, pp.
276
288
.
24.
Li
,
G.
,
Aute
,
V.
, and
Azarm
,
S.
,
2010
, “
An Accumulative Error Based Adaptive Design of Experiments for Offline Metamodeling
,”
Struct. Multidiscip. Optim.
,
40
(
1–6
), pp.
137
155
.
25.
Eason
,
J.
, and
Cremaschi
,
S.
,
2014
, “
Adaptive Sequential Sampling for Surrogate Model Generation With Artificial Neural Networks
,”
Comput. Chem. Eng.
,
68
, pp.
220
232
.
26.
Ajdari
,
A.
, and
Mahlooji
,
H.
,
2014
, “
An Adaptive Exploration-Exploitation Algorithm for Constructing Metamodels in Random Simulation Using a Novel Sequential Experimental Design
,”
Commun. Stat. Simul. Comput.
,
43
(
5
), pp.
947
968
.
27.
Van den Bos
,
L. M. M.
,
Sanderse
,
B.
, and
Bierbooms
,
W.
,
2020
, “
Adaptive Sampling-Based Quadrature Rules for Efficient Bayesian Prediction
,”
J. Comput. Phys.
,
417
, p.
109537
.
28.
Crombecq
,
K.
,
De Tommasi
,
L.
,
Gorissen
,
D.
, and
Dhaene
,
T.
,
2009
, “
A Novel Sequential Design Strategy for Global Surrogate Modeling
,”
Proceedings of the 2009 Winter Simulation Conference (WSC)
,
Austin, TX
,
Dec. 13–16
, IEEE, pp.
731
742
.
29.
Crombecq
,
K.
,
Gorissen
,
D.
,
Deschrijver
,
D.
, and
Dhaene
,
T.
,
2011
, “
A Novel Hybrid Sequential Design Strategy for Global Surrogate Modeling of Computer Experiments
,”
SIAM J. Sci. Comput.
,
33
(
4
), pp.
1948
1974
.
30.
Wang
,
H.
,
Tang
,
L.
, and
Li
,
G. Y.
,
2011
, “
Adaptive MLS-HDMR Metamodeling Techniques for High Dimensional Problems
,”
Expert Syst. Appl.
,
38
(
11
), pp.
14117
14126
.
31.
Li
,
E.
,
Wang
,
H.
, and
Li
,
G.
,
2012
, “
High Dimensional Model Representation (HDMR) Coupled Intelligent Sampling Strategy for Nonlinear Problems
,”
Comput. Phys. Commun.
,
183
(
9
), pp.
1947
1955
.
32.
Liu
,
Y.
,
Hussaini
,
M. Y.
, and
Ökten
,
G.
,
2016
, “
Accurate Construction of High Dimensional Model Representation With Applications to Uncertainty Quantification
,”
Reliab. Eng. Syst. Saf.
,
152
, pp.
281
295
.
33.
Zhang
,
X.
,
Zhang
,
H.
,
Zhu
,
J.
, and
Li
,
Z.
,
2022
, “
Revealing the Structure of Prediction Models Through Feature Interaction Detection
,”
Knowl. Based Syst.
,
236
, p.
107737
.
34.
Liu
,
J.
,
Cai
,
H.
,
Jiang
,
C.
,
Han
,
X.
, and
Zhang
,
Z.
,
2018
, “
An Interval Inverse Method Based on High Dimensional Model Representation and Affine Arithmetic
,”
Appl. Math. Model.
,
63
, pp.
732
743
.
35.
Rao
,
B. K.
, and
Balu
,
A. S.
,
2019
, “
Assessment of Cohesive Parameters Using High Dimensional Model Representation for Mixed Mode Cohesive Zone Model
,”
Structures
,
19
, pp.
156
160
.
36.
Huang
,
H.
,
Wang
,
H.
,
Gu
,
J.
, and
Wu
,
Y.
,
2019
, “
High-Dimensional Model Representation-Based Global Sensitivity Analysis and the Design of a Novel Thermal Management System for Lithium-Ion Batteries
,”
Energy Convers. Manag.
,
190
, pp.
54
72
.
37.
Vessia
,
G.
,
Kozubal
,
J.
, and
Puła
,
W.
,
2017
, “
High Dimensional Model Representation for Reliability Analyses of Complex Rock-Soil Slope Stability
,”
Arch. Civ. Mech. Eng.
,
17
(
4
), pp.
954
963
.
38.
Sahu
,
D.
,
Nishanth
,
M.
,
Dhir
,
P. K.
,
Sarkar
,
P.
,
Davis
,
R.
, and
Mangalathu
,
S.
,
2019
, “
Stochastic Response of Reinforced Concrete Buildings Using High Dimensional Model Representation
,”
Eng. Struct.
,
179
, pp.
412
422
.
39.
Cheng
,
K.
, and
Lu
,
Z.
,
2019
, “
Time-Variant Reliability Analysis Based on High Dimensional Model Representation
,”
Reliab. Eng. Syst. Saf.
,
188
, pp.
310
319
.
40.
Balu
,
A. S.
, and
Rao
,
B. N.
,
2012
, “
Inverse Structural Reliability Analysis Under Mixed Uncertainties Using High Dimensional Model Representation and Fast Fourier Transform
,”
Eng. Struct.
,
37
, pp.
224
234
.
41.
Wang
,
Z.
, and
Cai
,
J.
,
2018
, “
Inversion of Radiation Field on Nuclear Facilities: A Method Based on Net Function Interpolation
,”
Radiat. Phys. Chem.
,
153
, pp.
27
34
.
42.
Wang
,
Z.
, and
Cai
,
J.
,
2020
, “
Reconstruction of the Neutron Radiation Field on Nuclear Facilities Near the Shield Using Bayesian Inference
,”
Prog. Nucl. Energy
,
118
, p.
103070
.
43.
Qiu
,
P. Z.
, and
Chen
,
Q. H.
,
2007
,
Theory and Application of net Function Interpolation
,
Shanghai Science and Technology Press
,
Shanghai, China
.
44.
Xu
,
S.
,
Liu
,
H.
,
Wang
,
X.
, and
Jiang
,
X.
,
2014
, “
A Robust Error-Pursuing Sequential Sampling Approach for Global Metamodeling Based on Voronoi Diagram and Cross Validation
,”
ASME J. Mech. Des.
,
136
(
7
), p.
071009
.
45.
De Boor
,
C.
,
1978
,
A Practical Guide to Splines
,
Springer-Verlag
,
New York
.
46.
Aurenhammer
,
F.
,
1991
, “
Voronoi Diagrams-A Survey of a Fundamental Geometric Data Structure
,”
ACM Comput. Surv.
,
23
(
3
), pp.
345
405
.
47.
Hong
,
L.
,
Li
,
H.
,
Fu
,
J.
,
Li
,
J.
, and
Peng
,
K.
,
2022
, “
Hybrid Active Learning Method for Non-Probabilistic Reliability Analysis With Multi-Super-Ellipsoidal Model
,”
Reliab. Eng. Syst. Saf.
,
222
, p.
108414
.
You do not currently have access to this content.