Abstract
In this study, Kriging based multifidelity (MF) surrogate models are constructed to accelerate the fatigue analysis of welded joints. The influence of leg length, leg height, the width of the specimen, and load in the fatigue test are taken into consideration. In the construction of the MF surrogate model, the finite element model that is calibrated with the experiment is chosen as the highfidelity (HF) model, while the finite element model that is not calibrated with the experiment is considered as the lowfidelity (LF) model, aiming to capture the trend of the HF model. The Leaveoneout (LOO) verification method is used to evaluate the benefits of the three types of Krigingbased MF surrogate models comparing to the singlefidelity one. The results show that the accuracy improvement of MF surrogate models compared with the HF Kriging surrogate model is between 49.17 and 79.92%, while it is between 53.4 and 87.99% compared with the LF Kriging surrogate model. To determine the most suitable MF surrogate models for different responses of the welded single lap joint, three different MF uncertainty quantification (UQ) metrics are used to evaluate the prediction errors of the MF surrogate models. Based on the results of the UQ, a comprehensive ranking for the MF surrogate models is provided by introducing the entropy weightingbased technique for order preference by similarity to ideal solution (EWTOPSIS). The developed methods can also be generalized to the selection of the MF surrogate model for other engineering applications.
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Acknowledgments
The authors also would like to thank the anonymous referees for their valuable comments.
Funding
This research has been partially supported by HyundaiMOBIS, and the National Natural Science Foundation of China (NSFC) under Grant Nos. 51,775,203/51721092.
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L. Zhang wrote the manuscript with support from T. Xie and J. Hu. J. Koo generated the experimental results. SK. Choi and P. Jiang (jiangping@hust.edu.cn) supervised the project, and they are cocorresponding authors of this paper.
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Replication of results
The main step for generating the HF simulation and the LF simulation has been presented in Section 3. The process of the proposed EWTOPSIS method is shown in Sections 4. To help readers understand better, the code for the EWTOPSIS method and the corresponding modeling data can be downloaded from the website: https://pan.baidu.com/s/10Nnx5jp8irqrwwVbdYSw9Q by using the code wt6s.
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Appendix 1
Appendix 1
In this Appendix, eight numerical examples are used to test the effectiveness of the EWTOPSIS method for determining the most suitable MF surrogate models. The expressions of eight numerical examples are as follows, where the y_{h} denotes the HF surrogate model and y_{l} denotes the LF surrogate model.

Example 1 (Ex 1)

Example 2 (Ex 2)

Example 3 (Ex 3)

Example 4 (Ex 4)

Example 5 (Ex 5)

Example 6 (Ex 6)

Example 7 (Ex 7)

Example 8 (Ex 8)
The implementation steps are as follows,

Step 1: LHS method is used to generate the initial sample points. The number of LF and HF sample points are set to be 25 times and 5 times the number of design variables, respectively.

Step 2: Construct three types of MF surrogate models (ASFMF model, ASFMFρ model, and Cokriging model).

Step 3: Quantify the uncertainties of three MF surrogate models using three error metrics, including LOO, Bootstrap, and MSE. Considering the randomness, each numerical case is repeated 100 times and the mean value of the 100 results is recorded as the final error.

Step 4: The EWTOPSIS method is used to determine the most suitable MF surrogate models.

Step 5: Compare the ranking results of MF surrogate models from the EWTOPSIS method and that according to the true error. The results are listed in Appendix Table 7. In Appendix Table 7, the numbers in brackets after each MF surrogate model denotes the ranks.
As can be seen from Appendix Table 7, in these eight numerical examples, the ranking results of EWTOPSIS method are always the same as those of the true error, indicating the effectiveness of the EWTOPSIS method for selecting the most suitable MF surrogate models.
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Zhang, L., Choi, SK., Xie, T. et al. Multifidelity surrogate modelassisted fatigue analysis of welded joints. Struct Multidisc Optim (2021). https://doi.org/10.1007/s00158020028409
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Keywords
 Multifidelity surrogate model
 Fatigue life
 Stress distribution
 Finite element model
 Kriging model