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


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12


  1. Cheng GH, Younis A, Haji Hajikolaei K, Gary Wang G (2015) Trust region based mode pursuing sampling method for global optimization of high dimensional design problems. J Mech Des 137(2):021407

    Article  Google Scholar 

  2. Clarke SM, Griebsch JH, Simpson TW (2003) Analysis of support vector regression for approximation of complex engineering analyses. ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference 37009:535–543

  3. de Baar J, Roberts S, Dwight R, Mallol B (2015) Uncertainty quantification for a sailing yacht hull, using multi-fidelity kriging. Comput Fluids 123:185–201

    MathSciNet  MATH  Article  Google Scholar 

  4. Efron B (1992) Bootstrap methods: another look at the jackknife. In: Breakthroughs in statistics. Springer, New York, NY, pp 569–593

  5. Ertas AH, Vardar O, Sonmez FO, Solim Z (2009) Measurement and assessment of fatigue life of spot-weld joints. J Eng Mater Technol 131(1):011011

    Article  Google Scholar 

  6. Fernández-Godino MG, Park C, Kim N-H, Haftka RT (2016) Review of multi-fidelity models. arXiv preprint arXiv:160907196

  7. Fernández-Godino MG, Park C, Kim N-H, Haftka RT (2019a) Issues in deciding whether to use multifidelity surrogates. AIAA J 57(5):2039–2054

  8. Fernández-Godino MG, Dubreuil S, Bartoli N, Gogu C, Balachandar S, Haftka RT (2019b) Linear regression-based multifidelity surrogate for disturbance amplification in multiphase explosion. Struct Multidiscip Optim 60(6):2205–2220

    Article  Google Scholar 

  9. Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79

    Article  Google Scholar 

  10. Goyal R, Bogdanov S, El-zein M, Glinka G (2018) Fracture mechanics based estimation of fatigue lives of laser welded joints. Eng Fail Anal 93:340–355

    Article  Google Scholar 

  11. Han Z, Zimmerman R, Görtz S (2012) Alternative cokriging method for variable-fidelity surrogate modeling. AIAA J 50(5):1205–1210

    Article  Google Scholar 

  12. Han Z-H, Görtz S, Zimmermann R (2013) Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function. Aerosp Sci Technol 25(1):177–189

    Article  Google Scholar 

  13. Hu J, Zhou Q, Jiang P, Shao X, Xie T (2017) An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging. Eng Optim 50(1):145–163

    Article  Google Scholar 

  14. Hu J, Yang Y, Zhou Q, Jiang P, Shao X, Shu L, Zhang Y (2018) Comparative studies of error metrics in variable fidelity model uncertainty quantification. J Eng Des 29(8–9):512–538

    Article  Google Scholar 

  15. Huang D, Allen TT, Notz WI, Miller RA (2006) Sequential kriging optimization using multiple-fidelity evaluations. Struct Multidiscip Optim 32(5):369–382

    Article  Google Scholar 

  16. Jiang P, Zhang Y, Zhou Q, Shao X, Hu J, Shu L (2018) An adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS. Appl Intell 48(6):1644–1656

    Article  Google Scholar 

  17. Kennedy MC, O'Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13

    MathSciNet  MATH  Article  Google Scholar 

  18. Kohavi R (1995) A study of cross-validation and bootstrap for accuracy estimation and model selection. Ijcai 14:1137–1145

  19. Koziel S, Bandler JW (2010) Recent advances in space-mapping-based modeling of microwave devices. Int J Numer Model Electron Netw Devices Fields 23(6):425–446

    MATH  Article  Google Scholar 

  20. Lataniotis C, Marelli S, Sudret B (2017) The Gaussian process modelling module in UQLab. arXiv preprint arXiv 1709:09382

  21. Le Gratiet L (2013) Bayesian analysis of hierarchical multifidelity codes. SIAM/ASA J Uncertain Quantif 1(1):244–269

    MathSciNet  MATH  Article  Google Scholar 

  22. Leusink D, Alfano D, Cinnella P (2015) Multi-fidelity optimization strategy for the industrial aerodynamic design of helicopter rotor blades. Aerosp Sci Technol 42:136–147

    Article  Google Scholar 

  23. Lophaven SN, Nielsen HB, Søndergaard J (2002a) Aspects of the matlab toolbox DACE. IMM-TR2002–13. Technical University of Denmark, Lyngby, Denmark

  24. Lophaven SN, Nielsen HB, Søndergaard J (2002b) DACE: a Matlab kriging toolbox. Citeseer 2

  25. Morris MD (2004) The design and analysis of computer experiments. J Am Stat Assoc 99(468):1203–1204

    Article  Google Scholar 

  26. Pan N, Sheppard S (2002) Spot welds fatigue life prediction with cyclic strain range. Int J Fatigue 24(5):519–528

    Article  Google Scholar 

  27. Park C, Haftka RT, Kim NH (2017) Remarks on multi-fidelity surrogates. Struct Multidiscip Optim 55(3):1029–1050

    MathSciNet  Article  Google Scholar 

  28. Park C, Haftka RT, Kim NH (2018) Low-fidelity scale factor improves Bayesian multi-fidelity prediction by reducing bumpiness of discrepancy function. Struct Multidiscip Optim 58(2):399–414

    Article  Google Scholar 

  29. Peherstorfer B, Willcox K, Gunzburger M (2018) Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Rev 60(3):550–591

    MathSciNet  MATH  Article  Google Scholar 

  30. Perdikaris P, Raissi M, Damianou A, Lawrence N, Karniadakis GE (2017) Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling. Proc R Soc A Math Phys Eng Sci 473(2198):20160751

    MATH  Google Scholar 

  31. Qian PZG, Wu CFJ (2008) Bayesian hierarchical modeling for integrating low-accuracy and high-accuracy experiments. Technometrics 50(2):192–204

    MathSciNet  Article  Google Scholar 

  32. Romero DA, Marin VE, Amon CH (2015) Error metrics and the sequential refinement of Kriging Metamodels. J Mech Des 137(1):011402

    Article  Google Scholar 

  33. Shan S, Wang G (2009) Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct Multidiscip Optim 41(2):219–241

    MathSciNet  MATH  Article  Google Scholar 

  34. Şimşek B, İç YT, Şimşek EHJC, Systems IL (2013) A TOPSIS-based Taguchi optimization to determine optimal mixture proportions of the high strength self-compacting concrete. Chemom Intell Lab Syst 125:18–32

    Article  Google Scholar 

  35. Song X, Lv L, Sun W, Zhang J (2019) A radial basis function-based multi-fidelity surrogate model: exploring correlation between high-fidelity and low-fidelity models. Struct Multidiscip Optim 60(3):965–981

    Article  Google Scholar 

  36. Sun G, Li G, Stone M, Li Q (2010) A two-stage multi-fidelity optimization procedure for honeycomb-type cellular materials. Comput Mater Sci 49(3):500–511

    Article  Google Scholar 

  37. Sun G, Li G, Li Q (2012) Variable fidelity design based surrogate and artificial bee colony algorithm for sheet metal forming process. Finite Elem Anal Des 59:76–90

    Article  Google Scholar 

  38. Swanson SR (1974) Handbook of Fatigue Testing, ASTM International. PA No, West Conshohocken, p 566

  39. Tao J, Sun G (2019) Application of deep learning based multi-fidelity surrogate model to robust aerodynamic design optimization. Aerosp Sci Technol 92:722–737

    Article  Google Scholar 

  40. Tovo R, Livieri P (2011) A numerical approach to fatigue assessment of spot weld joints. Fatigue Fract Eng Mater Struct 34(1):32–45

    Article  Google Scholar 

  41. Ulaganathan S, Couckuyt I, Ferranti F, Laermans E, Dhaene T (2014) Performance study of multi-fidelity gradient enhanced kriging. Struct Multidiscip Optim 51(5):1017–1033

    Article  Google Scholar 

  42. Variyar A, Economon TD, Alonso JJ (2016) Multifidelity conceptual design and optimization of strut-braced wing aircraft using physics based methods. In 54th AIAA Aerospace Sciences Meeting (p. 2000).

  43. Wang G, Shan S (2006) Review of metamodeling techniques in support of engineering design optimization, vol 4255. ASME 2006 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp 415–426

  44. Wang R, Shang D (2009) Low-cycle fatigue life prediction of spot welds based on hardness distribution and finite element analysis. Int J Fatigue 31(3):508–514

    MathSciNet  Article  Google Scholar 

  45. Wang YM, Réthoré PE, van der Laan MP, Leon JPM, Liu YQ, Li L (2016) Multi-fidelity wake modelling based on co-Kriging method. J Phys Conf Ser 753(3):032065

    Article  Google Scholar 

  46. Wang Y, Zhao W, Zhou G, Gao Q, Wang C (2017) Optimization of an auxetic jounce bumper based on Gaussian process metamodel and series hybrid GA-SQP algorithm. Struct Multidiscip Optim 57(6):2515–2525

    Article  Google Scholar 

  47. Williams CK, Rasmussen CE (2006) Gaussian processes for machine learning. MIT Press, Cambridge

    Google Scholar 

  48. Yang Y, Gao Z, Cao L (2018) Identifying optimal process parameters in deep penetration laser welding by adopting hierarchical-Kriging model. Infrared Phys Technol 92:443–453

    Article  Google Scholar 

  49. Yu J (2018) State of health prediction of lithium-ion batteries: multiscale logic regression and Gaussian process regression ensemble. Reliab Eng Syst Saf 174:82–95

    Article  Google Scholar 

  50. Zhang Y, Kim NH, Park C, Haftka RTJAJ (2018) Multifidelity surrogate based on single linear regression. AIAA J 56(12):4944–4952

    Article  Google Scholar 

  51. Zheng J, Shao X, Gao L, Jiang P, Qiu H (2014) A prior-knowledge input LSSVR metamodeling method with tuning based on cellular particle swarm optimization for engineering design. Expert Syst Appl 41(5):2111–2125

    Article  Google Scholar 

  52. Zhou Q, Shao X, Jiang P, Gao Z, Wang C, Shu L (2016) An active learning metamodeling approach by sequentially exploiting difference information from variable-fidelity models. Adv Eng Inform 30(3):283–297

    Article  Google Scholar 

  53. Zhou Q, Jiang P, Shao X, Hu J, Cao L, Wan L (2017a) A variable fidelity information fusion method based on radial basis function. Adv Eng Inform 32:26–39

    Article  Google Scholar 

  54. Zhou Q, Wang Y, Choi S-K, Jiang P, Shao X, Hu J (2017b) A sequential multi-fidelity metamodeling approach for data regression. Knowl-Based Syst 134:199–212

    Article  Google Scholar 

  55. Zhou Q et al (2017c) A multi-fidelity information fusion metamodeling assisted laser beam welding process parameter optimization approach. Adv Eng Softw 110:85–97

    Article  Google Scholar 

Download references


The authors also would like to thank the anonymous referees for their valuable comments.


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




L. Zhang wrote the manuscript with support from T. Xie and J. Hu. J. Koo generated the experimental results. SK. Choi and P. Jiang ( supervised the project, and they are co-corresponding authors of this paper.

Corresponding author

Correspondence to Seung-Kyum Choi.

Ethics declarations

Conflict of interest

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: by using the code wt6s.

Additional information

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Responsible Editor: Erdem Acar

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)

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

$$ {\displaystyle \begin{array}{l}{y}_l=0.5\times \sin \left(12x-4\right)\times {\left(6x-2\right)}^2+10\times \left(x-0.5\right)+5\\ {}{y}_h=\sin \left(12x-4\right)\times {\left(6x-2\right)}^2\kern4.099998em x\in \left[0,1\right]\kern3.099999em \end{array}} $$
  • Example 3 (Ex 3)

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

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

$$ {\displaystyle \begin{array}{l}{y}_l={\left({x}_1-1\right)}^2+2\times {\left(2{x}_2^2-0.75{x}_1\right)}^2+3\times {\left(3{x}_3^2-0.75{x}_2\right)}^2+4\\ {}\kern2.5em \times {\left(4{x}_4^2-0.75{x}_3\right)}^2\\ {}\begin{array}{l}{y}_h={\left({x}_1-1\right)}^2+2\times {\left(2{x}_2^2-{x}_1\right)}^2+3\times {\left(3{x}_3^2-{x}_2\right)}^2+4\times {\left(4{x}_4^2-{x}_3\right)}^2\\ {}{x}_1,{x}_2,{x}_3,{x}_4\in \left[-10,10\right]\end{array}\end{array}} $$
  • Example 6 (Ex 6)

$$ {\displaystyle \begin{array}{l}f\left({x}_1,\cdots, {x}_6\right)=-\sum \limits_{i=1}^4{c}_i\,\exp \left[-\sum \limits_{j=1}^6{a}_{ij}{\left({x}_j-{p}_{ij}\right)}^2\right]\kern0.5em {x}_j\in \left[0,1\right]\\ {}\begin{array}{cc}\left[{c}_i\right]={\left[1\ 1.2\ 3\ 3.2\right]}^T,& \left[{a}_{ij}\right]=\left[\begin{array}{ccccccc}10& 3& 17& 3.05& 1.7& 8& \\ {}0.05& 10& 17& 0.1& 8& 4& \\ {}& 3& 3.5& 1.7& 10& 17& 8\\ {}17& 8& 0.05& 10& 0.1& 14& \end{array}\right]\end{array}\\ {}\left[{p}_{ij}\right]=\left[\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ {}0.2329& 0.4139& 0.8307& 0.3736& 0.1004& 0.9991\\ {}0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ {}0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right]\\ {}\left[l{c}_i\right]=\begin{array}{cc}{\left[1.1\ 0.8\ 2.5\ 3\right]}^T,& \left[{l}_j\right]={\left[0.75\ 1\ 0.8\ 1.3\ 0.7\ 1.1\right]}^T\end{array}\\ {}{y}_l=-\sum \limits_{i=1}^4l{c}_i\,\exp \left[-\sum \limits_{j=1}^6{a}_{ij}{\left({l}_j{x}_j-{p}_{ij}\right)}^2\right]\\ {}\begin{array}{cc}{y}_h=f\left({x}_1,\cdots, {x}_6\right)& {x}_j\in \left[0,1\right]\kern0.1em \end{array}\end{array}} $$
  • Example 7 (Ex 7)

$$ {\displaystyle \begin{array}{c}{f}_{borehole}=\frac{2\pi {x}_3\left({x}_4-{x}_6\right)}{\ln \left({x}_2/{x}_1\right)\left[1+2{x}_7{x}_4/\left(\ln \left({x}_2/{x}_1\right){x}_1^2{x}_8\right)+{x}_3/{x}_5\right]}\\ {}{y}_l=0.4{f}_{borehole}(x)+0.07{x}_1^2{x}_8+{x}_1{x}_7/{x}_3+{x}_1{x}_6/{x}_2+{x}_1^2{x}_4\\ {}{y}_h={f}_{borehole}(x)\kern2.1em \\ {}\kern0.4em {x}_1\in \left[\mathrm{0.05,0.15}\right]\kern0.3em ,\kern0.7em {x}_2\in \left[100,50000\right],\kern0.7em {x}_3\in \left[63070,115600\right],\\ {}\kern0.4em {x}_4\in \left[990,1110\right]\kern0.3em ,\kern0.7em {x}_5\in \left[\mathrm{63.1,116}\right],\kern0.7em {x}_6\in \left[\mathrm{700,820}\right]\kern0.3em ,\\ {}\kern0.4em {x}_7\in \left[1120,1680\right]\kern0.3em ,\kern0.7em {x}_8\in \left[9855,12045\right]\kern0.4em \end{array}} $$
  • Example 8 (Ex 8)

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

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

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Choi, SK., Xie, T. et al. Multi-fidelity surrogate model-assisted fatigue analysis of welded joints. Struct Multidisc Optim (2021).

Download citation


  • Multi-fidelity surrogate model
  • Fatigue life
  • Stress distribution
  • Finite element model
  • Kriging model