Table Of ContentEngineering Applications of Computational Methods 12
Qi Zhou
Min Zhao
Jiexiang Hu
Mengying Ma
Multi-fidelity
Surrogates
Modeling, Optimization and
Applications
Engineering Applications of Computational
Methods
Volume 12
Series Editors
Liang Gao, State Key Laboratory of Digital Manufacturing Equipment
and Technology, Huazhong University of Science and Technology,
Wuhan, Hubei, China
Akhil Garg, School of Mechanical Science and Engineering, Huazhong University
of Science and Technology, Wuhan, Hubei, China
The book series Engineering Applications of Computational Methods addresses the
numerous applications of mathematical theory and latest computational or
numerical methods in various fields of engineering. It emphasizes the practical
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developing computational methods using big data, machine learning and AI are
discussed in this book series, and could be applied to engineering fields, such as
manufacturing, industrial engineering, control engineering, civil engineering,
energy engineering and material engineering.
The book series Engineering Applications of Computational Methods aims to
introduce important computational methods adopted in different engineering
projects to researchers and engineers. The individual book volumes in the series
are thematic. The goal of each volume is to give readers a comprehensive overview
of how the computational methods in a certain engineering area can be used. As a
collection, the series provides valuable resources to a wide audience in academia,
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· · ·
Qi Zhou Min Zhao Jiexiang Hu Mengying Ma
Multi-fidelity Surrogates
Modeling, Optimization and Applications
Qi Zhou Min Zhao
School of Aerospace Engineering China Academy of Launch Vehicle
Huazhong University of Science Technology
and Technology Beijing, China
Wuhan, Hubei, China
Mengying Ma
Jiexiang Hu China Academy of Launch Vehicle
School of Aerospace Engineering Technology
Huazhong University of Science Beijing, China
and Technology
Wuhan, Hubei, China
This work was supported by [National Natural Science Foundation of China]
([52105254,52175231])
ISSN 2662-3366 ISSN 2662-3374 (electronic)
Engineering Applications of Computational Methods
ISBN 978-981-19-7209-6 ISBN 978-981-19-7210-2 (eBook)
https://doi.org/10.1007/978-981-19-7210-2
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Singapore Pte Ltd. 2023
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Contents
1 Introduction .................................................. 1
1.1 Concept of Multi-fidelity Surrogates ........................ 1
1.2 Review of Multi-fidelity Surrogates in Engineering Design ..... 3
1.2.1 Classification of Multi-fidelity Surrogate Modeling
Methods .......................................... 3
1.2.2 Research Progress on Multi-fidelity Surrogate
Modeling Methods ................................. 4
1.3 Sampling Method for Multi-fidelity Surrogates ............... 11
1.3.1 One-Shot Sampling Method ......................... 11
1.3.2 Sequential Sampling Method ........................ 13
1.4 Optimization Design Based on MF Surrogates ................ 16
1.4.1 MF Surrogate Management ......................... 17
1.4.2 Heuristic Optimization Algorithms Based on MF
Surrogates ........................................ 24
1.4.3 Uncertainty Optimization Design Based on MF
Surrogates ........................................ 26
1.5 Application of MF Surrogates in the Optimization Design
of Complex Equipment .................................... 28
References .................................................... 30
2 Hierarchical Multi-fidelity Surrogate Modeling .................. 39
2.1 Difference Mapping Based on Ensembles of Surrogates
for Multi-fidelity Surrogate Modeling ....................... 40
2.1.1 Tuning the LF Surrogate ............................ 40
2.1.2 Difference Mapping Based on Ensembles
of Surrogates ...................................... 41
2.1.3 Process Flow of the Proposed DMF-EM .............. 44
2.1.4 Examples and Results .............................. 46
2.2 Bumpiness of the Scaling-Function Reduction Method
for Multi-fidelity Surrogate Modeling ....................... 53
2.2.1 Introduction of the BR-SF Model .................... 53
v
vi Contents
2.2.2 Numerical Tests ................................... 57
2.3 Space-Mapping Method for Multi-fidelity Surrogate
Modeling ............................................... 61
2.3.1 Introduction of the SM-MF .......................... 61
2.3.2 Introduction of the SM-RBF ......................... 67
2.3.3 Numerical Tests ................................... 71
2.4 Generalized Hierarchical Cokriging Model
for Multi-fidelity Surrogate Modeling ....................... 83
2.4.1 Introduction of the GCK Model ...................... 83
2.4.2 Numerical Tests ................................... 90
References .................................................... 98
3 Nonhierarchical Multi-fidelity Surrogate Modeling ............... 101
3.1 Variance-Weighted-Sum Method for Multi-fidelity
Surrogate Modeling ...................................... 101
3.1.1 Introduction of the VWS-NMF Model ................ 101
3.1.2 Numerical Tests of VWS-MFS ....................... 110
3.2 Derivative of the Scaling Function Reduction Method
for Multi-fidelity Surrogate Modeling ....................... 123
3.2.1 Introduction of the DR-SF Surrogate .................. 123
3.2.2 Numerical Tests of NHLF-Cokriging Surrogate ........ 131
3.3 Multi-Output Gaussian Process Model for Multi-fidelity
Surrogate Modeling ...................................... 144
3.3.1 Introduction of the NH-MOMF Surrogate ............. 144
3.3.2 Numerical Tests of the NH-MOMF Surrogate .......... 148
References .................................................... 155
4 Sequential Multi-fidelity Surrogate Modeling .................... 157
4.1 Difference Mapping Method for Multi-fidelity Surrogate
Modeling ............................................... 158
4.1.1 Introduction of the DM-EM Model ................... 158
4.1.2 Numerical Tests ................................... 165
4.2 Weighted Cumulative-Error-Based Sequential
Multi-fidelity Surrogate Modeling .......................... 174
4.2.1 Introduction of the AL-SMF Method ................. 174
4.2.2 Numerical Tests of the AL-VFM Method .............. 180
4.3 Predicted-Improvement-Level-Based Sequential
Multi-fidelity Surrogate Modeling .......................... 186
4.3.1 Introduction of the SMF Method ..................... 186
4.3.2 Numerical Tests of the SMF Method .................. 192
4.4 Bootstrap-Estimator-Based Sequential Multi-fidelity
Surrogate Modeling ...................................... 195
4.4.1 Introduction of the BB-SMF Method ................. 195
4.4.2 Numerical Tests ................................... 203
References .................................................... 209
Contents vii
5 Multi-fidelity Surrogate Assisted Efficient Global
Optimization ................................................. 213
5.1 Lower Confidence Bounding Method for Efficient
Multi-fidelity Global Optimization .......................... 214
5.1.1 Introduction of the LCB-MFO Method ................ 214
5.1.2 Numerical Tests ................................... 219
5.2 Probability of Improvement Method for Efficient
Multi-fidelity Global Optimization .......................... 221
5.2.1 Introduction of the MF-PI Method .................... 221
5.2.2 Numerical Tests ................................... 233
5.3 Space Preselection Method for Efficient Multi-fidelity
Global Optimization ...................................... 237
5.3.1 Introduction of the SP-MFO Method .................. 237
5.3.2 Numerical Tests ................................... 243
References .................................................... 246
6 Multi-fidelity Surrogate Assisted Reliability Design
Optimization ................................................. 249
6.1 Augmented Expected Feasibility Function-Based Method
for Multi-fidelity Surrogate Assisted Reliability Design
Optimization ............................................ 250
6.1.1 Introduction of the EGRA-MF Method ................ 250
6.1.2 Numerical Tests of EGRA-MF ....................... 253
6.2 Contour Prediction Method for Multi-fidelity Surrogate
Assisted Reliability Design Optimization .................... 260
6.2.1 Introduction of the EEI Method ...................... 260
6.2.2 Numerical Tests ................................... 264
References .................................................... 276
7 Multi-fidelity Surrogate Assisted Robust Design Optimization ..... 279
7.1 Multi-fidelity Surrogate Assisted Six-Sigma Robust
Optimization ............................................ 280
7.1.1 Introduction of the MF-RO Method ................... 280
7.1.2 Numerical Tests of the MF-RO Method ............... 286
7.2 Multi-fidelity Surrogate Assisted Sequential Robust
Optimization ............................................ 297
7.2.1 Introduction of the MF-SRO Method ................. 297
7.2.2 Numerical Tests of the MF-SRO Method .............. 307
7.3 Conservative Multi-fidelity Surrogate Assisted Robust
Optimization ............................................ 316
7.3.1 Introduction of the CMF-RO Method ................. 316
7.3.2 Numerical Tests of the CMF-RO Method .............. 322
References .................................................... 332
viii Contents
8 Multi-fidelity Surrogate Assisted Evolutional Optimization ........ 335
8.1 Multi-fidelity Surrogate Assisted Multi-objective Genetic
Algorithm ............................................... 336
8.1.1 Introduction of the AMFS-MOGA Method ............ 336
8.1.2 Numerical Tests of AMFS-MOGA ................... 345
8.2 Multilevel Multi-fidelity Surrogate Assisted
Multi-objective Genetic Algorithm .......................... 351
8.2.1 Introduction of the TSMA-MOGA Method ............ 351
8.2.2 Numerical Tests of TSMA-MOGA ................... 360
8.3 Online Multi-fidelity Surrogate Assisted Multi-objective
Genetic Algorithm ........................................ 366
8.3.1 Introduction of the OLMF-MOGA Method ............ 366
8.3.2 Numerical Tests of OLMF-MOGA ................... 370
References .................................................... 384
9 Engineering Applications ...................................... 387
9.1 Prediction of Angular Distortion in Laser Welding ............ 387
9.1.1 Laser Welding Experiment .......................... 387
9.1.2 Finite Element Simulation ........................... 389
9.1.3 Results and Discussion ............................. 394
9.2 Optimization of Laser Beam Welding Parameters ............. 401
9.2.1 Two Models with Different Fidelity Levels ............ 401
9.2.2 Proposed Approach ................................ 407
9.2.3 Results and Discussion ............................. 412
9.3 Optimization of Metamaterial Vibration Isolator Design ....... 420
9.3.1 Establishment of the FEM of the Honeycomb
Structure Vibration Isolator .......................... 420
9.3.2 Experimentation and Validation of the FE Model ....... 423
9.3.3 Optimization Design of the MI300-Type
Honeycomb Structure Vibration Isolator .............. 429
9.4 Optimization Design of a Stiffened Cylindrical Shell
with Variable Ribs ........................................ 436
References .................................................... 442
10 Concluding Remarks .......................................... 445
10.1 Conclusions ............................................. 445
10.2 Remaining Challenges .................................... 455
Chapter 1
Introduction
1.1 Concept of Multi-fidelity Surrogates
Physics-based simulation models from different disciplines are becoming indispens-
able in modern product design. In the preliminary design phase, these simulation
models can help predict the performance of products to expedite the design space
exploration and search for the optimal design. Notably, to obtain accurate predictions,
high-fidelity (HF) but computationally expensive simulations must be performed.
Therefore, despite the present processing power of computers and parallel computing
techniques, it remains impractical to rely only on simulations to clarify the compre-
hensive relationships between the design variables (inputs) and product performance
values (outputs).
Surrogates have been widely used in engineering design and optimization to
replace expensive simulations to relieve the computational burden [1, 2]. To obtain
sufficient data, designers must manually select a design of experimental and simu-
lation models with appropriate fidelities to obtain the quantity of interests (QoIs) at
reasonable cost levels. Generally, HF simulation models can provide more reliable
and accurate simulation results than low-fidelity (LF) models. However, the use of
only HF models to obtain QoIs for building surrogates may be time-and resource-
intensive. Although LF models are less computationally demanding, the obtained
QoIs may result in inaccurate or incorrect surrogates due to noise and stochastic
effects.
A promising approach to balance the prediction accuracy and computational cost
is to integrate the information from HF and LF simulations by constructing multi-
fidelity (MF) surrogates [3, 4]. The core concept of MF surrogates is that a large
number of LF samples are evaluated to clarify the general trend of the system
behavior, and a small number of HF samples are used to enhance the prediction
accuracy in important regions. To ensure the effectiveness of MF surrogates, the LF
model must capture the overall trend of the responses from the HF model. In MF
surrogates, the HF and LF models are relative concepts describing the same complex
engineering product at different complexity levels. LF models are typically derived
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 1
Q. Zhou et al., Multi-fidelity Surrogates, Engineering Applications of Computational
Methods 12, https://doi.org/10.1007/978-981-19-7210-2_1
