Nonlinear models based on enhanced Kriging interpolation for prediction of rock joint shear strength

Abstract

One of the most basic topics in rock mechanic is the shear strength criteria for rock joints. Thus, it is of high importance to accurately predict the shear strength of rock joints. In this study, the abilities for accuracy and agreement of Kriging model-based nonlinear interpolation strategy are investigated in terms of predicting the shear strength of rock joints. Totally 84 datasets were used to construct the Kriging models; the datasets were divided into two main parts: training and testing. The prepared database was applied to the training phase in the Kriging model; this way, several nonlinear basic functions were introduced to enhance the predictions of the Kriging model. The examined functions in this paper were linear, 2-order, 3-order, exponential, logarithmic, logistic, hyperbolic tangent, and hyperbolic sine. The sigmoid forms of the basic functions, including logistic and hyperbolic tangent, provide the superior predictions compared to other mathematical functions, while the 2-order and 3-order forms provide the worst performances than the linear, exponential, and logarithmic functions. According to the obtained results, the logistic-based model with coefficient of determination (R2) of 0.916 was found the optimal model that can be successfully applied to estimating the shear strength of rock joints.

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References

  1. 1.

    Babanouri N, Fattahi H (2020) An ANFIS–TLBO criterion for shear failure of rock joints. Soft Comput 24(7):4759–4773. https://doi.org/10.1007/s00500-019-04230-w

    Article  Google Scholar 

  2. 2.

    Patton FD (1966) Multiple modes of shear failure in rock. In: 1st ISRM congress, 1966. International Society for Rock Mechanics and Rock Engineering

  3. 3.

    Barton N (1973) Review of a new shear-strength criterion for rock joints. Eng Geol 7(4):287–332

    Article  Google Scholar 

  4. 4.

    Maksimović M (1992) New description of the shear strength for rock joints. Rock Mech Rock Eng 25(4):275–284

    Article  Google Scholar 

  5. 5.

    Tang Z-C, Liu Q-S, Huang J-H (2014) New criterion for rock joints based on three-dimensional roughness parameters. J Cent S Univ 21(12):4653–4659

    Article  Google Scholar 

  6. 6.

    Hossaini KA, Babanouri N, Nasab SK (2014) The influence of asperity deformability on the mechanical behavior of rock joints. Int J Rock Mech Min Sci 70:154–161

    Article  Google Scholar 

  7. 7.

    Wei Y, Fu W, Nie D (2015) Nonlinearity of the rock joint shear strength. Strength Mater 47(1):205–212

    Article  Google Scholar 

  8. 8.

    Zhang X, Jiang Q, Chen N, Wei W, Feng X (2016) Laboratory investigation on shear behavior of rock joints and a new peak shear strength criterion. Rock Mech Rock Eng 49(9):3495–3512

    Article  Google Scholar 

  9. 9.

    Sarfarazi V, Haeri H, Shemirani AB, Zhu Z (2017) Shear behavior of non-persistent joint under high normal load. Strength Mater 49(2):320–334

    Article  Google Scholar 

  10. 10.

    Gentier SS, Hopkins DL (1997) Mapping fracture aperture as a function of normal stress using a combination of casting, image analysis and modeling techniques. Int J Rock Mech Min Sci 34(3–4):132-e1

    Google Scholar 

  11. 11.

    Grasselli G, Egger P (2003) Constitutive law for the shear strength of rock joints based on three-dimensional surface parameters. Int J Rock Mech Min Sci 40(1):25–40

    Article  Google Scholar 

  12. 12.

    Li K-h, Cao P, Zhang K, Zhong Y-f (2015) Macro and meso characteristics evolution on shear behavior of rock joints. J Cent S Univ 22(8):3087–3096

    Article  Google Scholar 

  13. 13.

    Babanouri N, Nasab SK, Baghbanan A, Mohamadi HR (2011) Over-consolidation effect on shear behavior of rock joints. Int J Rock Mech Min Sci 48(8):1283–1291

    Article  Google Scholar 

  14. 14.

    Babanouri N, Nasab SK (2017) Proposing triangulation-based measures for rock fracture roughness. Rock Mech Rock Eng 50(4):1055–1061

    Article  Google Scholar 

  15. 15.

    Chen X, Fu J, Yao J, Gan J (2018) Prediction of shear strength for squat RC walls using a hybrid ANN–PSO model. Eng Comput 34(2):367–383

    Article  Google Scholar 

  16. 16.

    Sarkar K, Tiwary A, Singh TN (2010) Estimation of strength parameters of rock using artificial neural networks. Bull Eng Geol Environ 69(4):599–606. https://doi.org/10.1007/s10064-010-0301-3

    Article  Google Scholar 

  17. 17.

    Dantas Neto SA, Indraratna B, Oliveira DAF, de Assis AP (2017) Modelling the shear behaviour of clean rock discontinuities using artificial neural networks. Rock Mech Rock Eng 50(7):1817–1831. https://doi.org/10.1007/s00603-017-1197-z

    Article  Google Scholar 

  18. 18.

    Khandelwal M, Armaghani DJ (2016) Prediction of drillability of rocks with strength properties using a hybrid GA-ANN technique. Geotech Geol Eng 34(2):605–620. https://doi.org/10.1007/s10706-015-9970-9

    Article  Google Scholar 

  19. 19.

    Murlidhar BR, Ahmed M, Mavaluru D, Siddiqi AF, Mohamad ET (2019) Prediction of rock interlocking by developing two hybrid models based on GA and fuzzy system. Eng Comput 35(4):1419–1430. https://doi.org/10.1007/s00366-018-0672-9

    Article  Google Scholar 

  20. 20.

    Xia C, Huang M, Qian X, Hong C, Luo Z, Du S (2019) Novel intelligent approach for peak shear strength assessment of rock joints on the basis of the relevance vector machine. Math Probl Eng 2019:3182736. https://doi.org/10.1155/2019/3182736

    Article  Google Scholar 

  21. 21.

    Zhou J, Li E, Wei H, Li C, Qiao Q, Armaghani DJ (2019) Random forests and cubist algorithms for predicting shear strengths of rockfill materials. Appl Sci 9(8):1621

    Article  Google Scholar 

  22. 22.

    Krige DG (1952) A statistical approach to some basic mine valuation problems on the Witwatersrand. J South Afr Inst Min Metall 52(9):201–203

    Google Scholar 

  23. 23.

    Matheron G (1963) Principles of geostatistics. Econ Geol 58(8):1246–1266

    Article  Google Scholar 

  24. 24.

    Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423. https://doi.org/10.1214/ss/1177012413

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Heddam S, Keshtegar B, Kisi O (2019) Predicting total dissolved gas concentration on a daily scale using Kriging interpolation, response surface method and artificial neural network: case study of Columbia River Basin Dams, USA. Natl Resour Res. https://doi.org/10.1007/s11053-019-09524-2

    Article  Google Scholar 

  26. 26.

    Keshtegar B, Mert C, Kisi O (2018) Comparison of four heuristic regression techniques in solar radiation modeling: Kriging method vs RSM, MARS and M5 model tree. Renew Sustain Energy Rev 81:330–341

    Article  Google Scholar 

  27. 27.

    Sakata S, Ashida F, Zako M (2003) Structural optimization using Kriging approximation. Comput Methods Appl Mech Eng 192(7):923–939. https://doi.org/10.1016/S0045-7825(02)00617-5

    Article  MATH  Google Scholar 

  28. 28.

    Huang D, Allen TT, Notz WI, Miller RA (2006) Sequential Kriging optimization using multiple-fidelity evaluations. Struct Multidiscip Optim 32(5):369–382. https://doi.org/10.1007/s00158-005-0587-0

    Article  Google Scholar 

  29. 29.

    Zhang J, Xiao M, Gao L, Qiu H, Yang Z (2018) An improved two-stage framework of evidence-based design optimization. Struct Multidiscip Optim 58(4):1673–1693

    MathSciNet  Article  Google Scholar 

  30. 30.

    Xiao M, Zhang J, Gao L (2020) A system active learning Kriging method for system reliability-based design optimization with a multiple response model. Reliab Eng Syst Saf 199:106935. https://doi.org/10.1016/j.ress.2020.106935

    Article  Google Scholar 

  31. 31.

    Zhang J, Xiao M, Gao L, Chu S (2019) A combined projection-outline-based active learning Kriging and adaptive importance sampling method for hybrid reliability analysis with small failure probabilities. Comput Methods Appl Mech Eng 344:13–33

    MathSciNet  Article  Google Scholar 

  32. 32.

    Xiao N-C, Yuan K, Zhou C (2020) Adaptive Kriging-based efficient reliability method for structural systems with multiple failure modes and mixed variables. Comput Methods Appl Mech Eng 359:112649

    MathSciNet  Article  Google Scholar 

  33. 33.

    Li H, Liu T, Wang M, Zhao D, Qiao A, Wang X, Gu J, Li Z, Zhu B (2017) Design optimization of stent and its dilatation balloon using Kriging surrogate model. BioMed Eng OnLine 16(1):13. https://doi.org/10.1186/s12938-016-0307-6

    Article  Google Scholar 

  34. 34.

    Simpson TW, Mauery TM, Korte JJ, Mistree F (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241. https://doi.org/10.2514/2.1234

    Article  Google Scholar 

  35. 35.

    Lu C, Feng Y-W, Liem RP, Fei C-W (2018) Improved Kriging with extremum response surface method for structural dynamic reliability and sensitivity analyses. Aerosp Sci Technol 76:164–175. https://doi.org/10.1016/j.ast.2018.02.012

    Article  Google Scholar 

  36. 36.

    Keshtegar B, Meng D, Ben Seghier MEA, Xiao M, Trung N-T, Bui DT (2020) A hybrid sufficient performance measure approach to improve robustness and efficiency of reliability-based design optimization. Eng Comput. https://doi.org/10.1007/s00366-019-00907-w

    Article  Google Scholar 

  37. 37.

    Zhang J, Xiao M, Gao L, Fu J (2018) A novel projection outline based active learning method and its combination with Kriging metamodel for hybrid reliability analysis with random and interval variables. Comput Methods Appl Mech Eng 341:32–52

    MathSciNet  Article  Google Scholar 

  38. 38.

    Zhang Y, Gao L, Xiao M (2020) Maximizing natural frequencies of inhomogeneous cellular structures by Kriging-assisted multiscale topology optimization. Comput Struct 230:106197

    Article  Google Scholar 

  39. 39.

    Sun Z, Wang J, Li R, Tong C (2017) LIF: a new Kriging based learning function and its application to structural reliability analysis. Reliab Eng Syst Saf 157:152–165. https://doi.org/10.1016/j.ress.2016.09.003

    Article  Google Scholar 

  40. 40.

    Xiao M, Zhang J, Gao L, Lee S, Eshghi AT (2019) An efficient Kriging-based subset simulation method for hybrid reliability analysis under random and interval variables with small failure probability. Struct Multidiscip Optim 59(6):2077–2092

    MathSciNet  Article  Google Scholar 

  41. 41.

    Keshtegar B, MeAB Seghier (2018) Modified response surface method basis harmony search to predict the burst pressure of corroded pipelines. Eng Fail Anal 89:177–199

    Article  Google Scholar 

  42. 42.

    Keshtegar B, Heddam S (2018) Modeling daily dissolved oxygen concentration using modified response surface method and artificial neural network: a comparative study. Neural Comput Appl 30(10):2995–3006

    Article  Google Scholar 

  43. 43.

    Keshtegar B, Kisi O (2017) Modified response-surface method: new approach for modeling pan evaporation. J Hydrol Eng 22(10):04017045

    Article  Google Scholar 

  44. 44.

    Coleman JN (2004) Method and apparatus for determining the approximate valve of a logarithmic function. Google Patents

  45. 45.

    Jordan MI (1995) Why the logistic function? A tutorial discussion on probabilities and neural networks. Computational cognitive science technical report

  46. 46.

    Mathias AC, Rech PC (2012) Hopfield neural network: the hyperbolic tangent and the piecewise-linear activation functions. Neural Netw 34:42–45

    Article  Google Scholar 

  47. 47.

    Karlik B, Olgac AV (2011) Performance analysis of various activation functions in generalized MLP architectures of neural networks. Int J Artif Intell Expert Syst 1(4):111–122

    Google Scholar 

  48. 48.

    Keshtegar B, Ozbakkaloglu T, Gholampour A (2017) Modeling the behavior of FRP-confined concrete using dynamic harmony search algorithm. Eng Comput 33(3):415–430

    Article  Google Scholar 

  49. 49.

    Gao L, Xiao M, Shao X, Jiang P, Nie L, Qiu H (2012) Analysis of gene expression programming for approximation in engineering design. Struct Multidiscip Optim 46(3):399–413. https://doi.org/10.1007/s00158-012-0767-7

    Article  Google Scholar 

  50. 50.

    Keshtegar B, Bagheri M, Yaseen ZM (2019) Shear strength of steel fiber-unconfined reinforced concrete beam simulation: application of novel intelligent model. Compos Struct 212:230–242

    Article  Google Scholar 

  51. 51.

    Chiu SL (1994) Fuzzy model identification based on cluster estimation. J Intell Fuzzy Syst 2(3):267–278

    Article  Google Scholar 

  52. 52.

    Bezdek JC, Ehrlich R, Full W (1984) FCM: the fuzzy c-means clustering algorithm. Comput Geosci 10(2):191–203. https://doi.org/10.1016/0098-3004(84)90020-7

    Article  Google Scholar 

  53. 53.

    Qiu B-Z, Li X-L, Shen J-Y (2007) Grid-based clustering algorithm based on intersecting partition and density estimation. In: Washio T, Zhou ZH, Huang JZ, Hu XT, Li J, Xie C, He J, Zou D, Li KC, Freire MM (eds) Emerging technologies in knowledge discovery and data mining. Springer, Berlin, pp 368–377

    Chapter  Google Scholar 

  54. 54.

    Kowsar R, Keshtegar B, Marey MA, Miyamoto A (2017) An autoregressive logistic model to predict the reciprocal effects of oviductal fluid components on in vitro spermophagy by neutrophils in cattle. Sci Rep 7(1):4482

    Article  Google Scholar 

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Correspondence to Behrooz Keshtegar.

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Hasanipanah, M., Meng, D., Keshtegar, B. et al. Nonlinear models based on enhanced Kriging interpolation for prediction of rock joint shear strength. Neural Comput & Applic 33, 4205–4215 (2021). https://doi.org/10.1007/s00521-020-05252-4

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Keywords

  • Shear strength of rock joints
  • Kriging
  • Logistic function
  • Predicted models