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Multi-fidelity surrogate model-assisted fatigue analysis of welded joints

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Abstract

In this study, Kriging based multi-fidelity (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 high-fidelity (HF) model, while the finite element model that is not calibrated with the experiment is considered as the low-fidelity (LF) model, aiming to capture the trend of the HF model. The Leave-one-out (LOO) verification method is used to evaluate the benefits of the three types of Kriging-based MF surrogate models comparing to the single-fidelity 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 weighting-based technique for order preference by similarity to ideal solution (EW-TOPSIS). 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 Hyundai-MOBIS, and the National Natural Science Foundation of China (NSFC) under Grant Nos. 51,775,203/51721092.

Author information

Authors and Affiliations

  1. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science &Technology, Wuhan, 430074, Hubei, China

    Lili Zhang & Ping Jiang

  2. G.W.W. School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Lili Zhang, Seung-Kyum Choi & Tingli Xie

  3. School of Aerospace Engineering, Huazhong University of Science & Technology, Wuhan, 430074, Hubei, China

    Jiexiang Hu

  4. CAE Team, Hyundai MOBIS Co., Ltd, Ansan-si, Gyeonggi-Do, 16891, South Korea

    Jasuk Koo

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  1. Lili Zhang
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Contributions

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 co-corresponding authors of this paper.

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Correspondence to Seung-Kyum Choi.

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The authors declare that they have no conflict of interest.

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 EW-TOPSIS method is shown in Sections 4. To help readers understand better, the code for the EW-TOPSIS 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 EW-TOPSIS method for determining the most suitable MF surrogate models. The expressions of eight numerical examples are as follows, where the yh denotes the HF surrogate model and yl denotes the LF surrogate model.

  • Example 1 (Ex 1)

$$ {\displaystyle \begin{array}{l}{y}_l=-\sin (x)-{e}^{\frac{x}{100}}+10.3+0.03\times {\left(x-3\right)}^2\\ {}{y}_h=-\sin (x)-{e}^{\frac{x}{100}}+10\kern3.499999em x\in \left[0,10\right]\kern3.099999em \end{array}} $$
(A1)

  • Example 2 (Ex 2)

$$ {\displaystyle \begin{array}{l}{y}_l=0.5\times \sin \left(12x-4\right)\times {\left(6x-2\right)}^2+10\times \left(x-0.5\right)+5\\ {}{y}_h=\sin \left(12x-4\right)\times {\left(6x-2\right)}^2\kern4.099998em x\in \left[0,1\right]\kern3.099999em \end{array}} $$
(A2)

  • Example 3 (Ex 3)

$$ {\displaystyle \begin{array}{l}{y}_l=4\times {\left(0.7{x}_1\right)}^2-2.1\times {\left(0.7{x}_1\right)}^4+{\left(0.7{x}_1\right)}^6/3+\left(0.7{x}_1\times 0.7{x}_2\right)-4\times {\left(0.7{x}_2\right)}^2\\ {}\kern2em +4\times {\left(0.7{x}_2\right)}^4\\ {}{y}_h=4{x}_1^2-2.1{x}_1^4+{x}_1^6/3+{x}_1{x}_2-4{x}_2^2+4\kern0.1em {x}_2^4\kern1.5em {x}_1,{x}_2\in \left[-2,2\right]\end{array}} $$
(A3)

  • Example 4 (Ex 4)

$$ {\displaystyle \begin{array}{l}{y}_l={\left({\left(0.5{x}_1\right)}^2+0.8{x}_2-11\right)}^2+{\left({\left(0.8{x}_2\right)}^2+0.5{x}_1-7\right)}^2+{x}_2^3-{\left({x}_2+1\right)}^2\\ {}{y}_h={\left({x}_1^2+{x}_2-11\right)}^2+{\left({x}_2^2+{x}_1-7\right)}^2\kern0.5em {x}_1,{x}_2\in \left[-3,3\right]\end{array}} $$
(A4)

  • Example 5 (Ex 5)

$$ {\displaystyle \begin{array}{l}{y}_l={\left({x}_1-1\right)}^2+2\times {\left(2{x}_2^2-0.75{x}_1\right)}^2+3\times {\left(3{x}_3^2-0.75{x}_2\right)}^2+4\\ {}\kern2.5em \times {\left(4{x}_4^2-0.75{x}_3\right)}^2\\ {}\begin{array}{l}{y}_h={\left({x}_1-1\right)}^2+2\times {\left(2{x}_2^2-{x}_1\right)}^2+3\times {\left(3{x}_3^2-{x}_2\right)}^2+4\times {\left(4{x}_4^2-{x}_3\right)}^2\\ {}{x}_1,{x}_2,{x}_3,{x}_4\in \left[-10,10\right]\end{array}\end{array}} $$
(A5)

  • Example 6 (Ex 6)

$$ {\displaystyle \begin{array}{l}f\left({x}_1,\cdots, {x}_6\right)=-\sum \limits_{i=1}^4{c}_i\,\exp \left[-\sum \limits_{j=1}^6{a}_{ij}{\left({x}_j-{p}_{ij}\right)}^2\right]\kern0.5em {x}_j\in \left[0,1\right]\\ {}\begin{array}{cc}\left[{c}_i\right]={\left[1\ 1.2\ 3\ 3.2\right]}^T,& \left[{a}_{ij}\right]=\left[\begin{array}{ccccccc}10& 3& 17& 3.05& 1.7& 8& \\ {}0.05& 10& 17& 0.1& 8& 4& \\ {}& 3& 3.5& 1.7& 10& 17& 8\\ {}17& 8& 0.05& 10& 0.1& 14& \end{array}\right]\end{array}\\ {}\left[{p}_{ij}\right]=\left[\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ {}0.2329& 0.4139& 0.8307& 0.3736& 0.1004& 0.9991\\ {}0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ {}0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right]\\ {}\left[l{c}_i\right]=\begin{array}{cc}{\left[1.1\ 0.8\ 2.5\ 3\right]}^T,& \left[{l}_j\right]={\left[0.75\ 1\ 0.8\ 1.3\ 0.7\ 1.1\right]}^T\end{array}\\ {}{y}_l=-\sum \limits_{i=1}^4l{c}_i\,\exp \left[-\sum \limits_{j=1}^6{a}_{ij}{\left({l}_j{x}_j-{p}_{ij}\right)}^2\right]\\ {}\begin{array}{cc}{y}_h=f\left({x}_1,\cdots, {x}_6\right)& {x}_j\in \left[0,1\right]\kern0.1em \end{array}\end{array}} $$
(A6)

  • Example 7 (Ex 7)

$$ {\displaystyle \begin{array}{c}{f}_{borehole}=\frac{2\pi {x}_3\left({x}_4-{x}_6\right)}{\ln \left({x}_2/{x}_1\right)\left[1+2{x}_7{x}_4/\left(\ln \left({x}_2/{x}_1\right){x}_1^2{x}_8\right)+{x}_3/{x}_5\right]}\\ {}{y}_l=0.4{f}_{borehole}(x)+0.07{x}_1^2{x}_8+{x}_1{x}_7/{x}_3+{x}_1{x}_6/{x}_2+{x}_1^2{x}_4\\ {}{y}_h={f}_{borehole}(x)\kern2.1em \\ {}\kern0.4em {x}_1\in \left[\mathrm{0.05,0.15}\right]\kern0.3em ,\kern0.7em {x}_2\in \left[100,50000\right],\kern0.7em {x}_3\in \left[63070,115600\right],\\ {}\kern0.4em {x}_4\in \left[990,1110\right]\kern0.3em ,\kern0.7em {x}_5\in \left[\mathrm{63.1,116}\right],\kern0.7em {x}_6\in \left[\mathrm{700,820}\right]\kern0.3em ,\\ {}\kern0.4em {x}_7\in \left[1120,1680\right]\kern0.3em ,\kern0.7em {x}_8\in \left[9855,12045\right]\kern0.4em \end{array}} $$
(A7)

  • Example 8 (Ex 8)

$$ {\displaystyle \begin{array}{l}{y}_l=\sum \limits_{i=1}^{10}{x}_i^3+{\left(\sum \limits_{i=1}^{10}2i{x}_i\right)}^2+{\left(\sum \limits_{i=1}^{10}3i{x}_i\right)}^4\\ {}{y}_h=\sum \limits_{i=1}^{10}{x}_i^2+{\left(\sum \limits_{i=1}^{10}0.5i{x}_i\right)}^2+{\left(\sum \limits_{i=1}^{10}0.5i{x}_i\right)}^4\kern0.5em {x}_i\in \left[-5,10\right]\kern0.1em \end{array}} $$
(A8)

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 (ASF-MF model, ASF-MF-ρ model, and Co-kriging 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 EW-TOPSIS method is used to determine the most suitable MF surrogate models.

  • Step 5: Compare the ranking results of MF surrogate models from the EW-TOPSIS 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.

    Table 7 The ranking of MF surrogate models
    Full size table

As can be seen from Appendix Table 7, in these eight numerical examples, the ranking results of EW-TOPSIS method are always the same as those of the true error, indicating the effectiveness of the EW-TOPSIS method for selecting the most suitable MF surrogate models.

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Zhang, L., Choi, SK., Xie, T. et al. Multi-fidelity surrogate model-assisted fatigue analysis of welded joints. Struct Multidisc Optim 63, 2771–2787 (2021). https://doi.org/10.1007/s00158-020-02840-9

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