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Frontiers of Structural and Civil Engineering

, Volume 13, Issue 6, pp 1289–1300 | Cite as

Global sensitivity analysis of certain and uncertain factors for a circular tunnel under seismic action

  • Nazim Abdul NarimanEmail author
  • Raja Rizwan Hussain
  • Ilham Ibrahim Mohammad
  • Peyman Karampour
Research Article
  • 20 Downloads

Abstract

There are many certain and uncertain design factors which have unrevealed rational effects on the generation of tensile damage and the stability of the circular tunnels during seismic actions. In this research paper, we have dedicated three certain and four uncertain design factors to quantify their rational effects using numerical simulations and the Sobol’s sensitivity indices. Main effects and interaction effects between the design factors have been determined supporting on variance-based global sensitivity analysis. The results detected that the concrete modulus of elasticity for the tunnel lining has the greatest effect on the tensile damage generation in the tunnel lining during the seismic action. In the other direction, the interactions between the concrete density and both of concrete modulus of elasticity and tunnel diameter have appreciable effects on the tensile damage. Furthermore, the tunnel diameter has the deciding effect on the stability of the tunnel structure. While the interaction between the tunnel diameter and concrete density has appreciable effect on the stability process. It is worthy to mention that Sobol’s sensitivity indices manifested strong efficiency in detecting the roles of each design factor in cooperation with the numerical simulations explaining the responses of the circular tunnel during seismic actions.

Keywords

shear waves Sobol’s sensitivity indices maximum principal stress maximum overall displacement tensile damage 

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References

  1. 1.
    Akhlaghi T, Nikkar A. Effect of vertically propagating shear waves on seismic behavior of circular tunnels. Hindawi the Scientific World Journal, 2014, 2014, 1–10Google Scholar
  2. 2.
    Hashash Y M A, Hook J J, Schmidt B, I-Chiang Yao J. Seismic design and analysis of underground structure. Tunnelling and Underground Space Technology, 2001, 16(4): 247–293Google Scholar
  3. 3.
    Nariman N A, Hussain R R, Msekh M A, Karampour P. Prediction meta-models for the responses of a circular tunnel during earthquakes. Underground Space, 2019, 4(1): 31–47Google Scholar
  4. 4.
    Nariman N A, Ramazan A, Mohammad I I. Application of coupled XFEM-BCQO in the structural optimization of a circular tunnel lining subjected to a ground motion. Frontiers of Structural and Civil Engineering, 2019 (in press)Google Scholar
  5. 5.
    Mollon G, Dias D, Soubra A H. Probabilistic analysis and design of circular tunnels against face stability. International Journal of Geomechanics, 2009, 9(6): 237–249Google Scholar
  6. 6.
    Kalab Z, Stemon P. Influence of seismic events on shallow geotechnical structures. Acta Montanistica Slovaca, 2017, 22(4): 412–421Google Scholar
  7. 7.
    Ghasemi H, Park H S, Rabczuk T. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258MathSciNetGoogle Scholar
  8. 8.
    Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. Optimum fiber content and distribution in fiber-reinforced solids using a reliability and NURBS based sequential optimization approach. Structural and Multidisciplinary Optimization, 2015, 51(1): 99–112MathSciNetGoogle Scholar
  9. 9.
    Zhang C, Nanthakumar S S, Lahmer T, Rabczuk T. Multiple cracks identification for piezoelectric structures. International Journal of Fracture, 2017, 206(2): 151–169Google Scholar
  10. 10.
    Nanthakumar S, Zhuang X, Park H, Rabczuk T. Topology optimization of flexoelectric structures. Journal of the Mechanics and Physics of Solids, 2017, 105: 217–234MathSciNetGoogle Scholar
  11. 11.
    Nanthakumar S, Lahmer T, Zhuang X, Park H S, Rabczuk T. Topology optimization of piezoelectric nanostructures. Journal of the Mechanics and Physics of Solids, 2016, 94: 316–335MathSciNetGoogle Scholar
  12. 12.
    Nanthakumar S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176MathSciNetGoogle Scholar
  13. 13.
    Nanthakumar S, Valizadeh N, Park H, Rabczuk T. Surface effects on shape and topology optimization of nanostructures. Computational Mechanics, 2015, 56(1): 97–112MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nanthakumar S S, Lahmer T, Rabczuk T. Detection of flaws in piezoelectric structures using extended FEM. International Journal for Numerical Methods in Engineering, 2013, 96(6): 373–389MathSciNetzbMATHGoogle Scholar
  15. 15.
    Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31Google Scholar
  16. 16.
    Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535Google Scholar
  17. 17.
    Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464Google Scholar
  18. 18.
    Rabczuk T, Akkermann J, Eibl J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5-6): 1327–1354zbMATHGoogle Scholar
  19. 19.
    Bazant Z P. Why continuum damage is nonlocal: Micromechanics arguments. Journal of Engineering Mechanics, 1991, 117(5): 1070–1087Google Scholar
  20. 20.
    Thai T Q, Rabczuk T, Bazilevs Y, Meschke G. A higher-order stress-based gradient-enhanced damage model based on isogeo-metric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604MathSciNetzbMATHGoogle Scholar
  21. 21.
    Fleck N A, Hutchinson J W. A phenomenological theory for strain gradient effects in plasticity. Journal of the Mechanics and Physics of Solids, 1993, 41(12): 1825–1857MathSciNetzbMATHGoogle Scholar
  22. 22.
    Rabczuk T, Eibl J. Simulation of high velocity concrete fragmentation using SPH/MLSPH. International Journal for Numerical Methods in Engineering, 2003, 56(10): 1421–1444zbMATHGoogle Scholar
  23. 23.
    Rabczuk T, Eibl J, Stempniewski L. Numerical analysis of high speed concrete fragmentation using a meshfree Lagrangian method. Engineering Fracture Mechanics, 2004, 71(4-6): 547–556Google Scholar
  24. 24.
    Rabczuk T, Xiao S P, Sauer M. Coupling of meshfree methods with nite elements: Basic concepts and test results. Communications in Numerical Methods in Engineering, 2006, 22(10): 1031–1065MathSciNetzbMATHGoogle Scholar
  25. 25.
    Rabczuk T, Eibl J. Modelling dynamic failure of concrete with meshfree methods. International Journal of Impact Engineering, 2006, 32(11): 1878–1897Google Scholar
  26. 26.
    Etse G, Willam K. Failure analysis of elastoviscoplastic material models. Journal of Engineering Mechanics, 1999, 125(1): 60–69Google Scholar
  27. 27.
    Miehe C, Hofacker M, Welschinger F. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45-48): 2765–2778MathSciNetzbMATHGoogle Scholar
  28. 28.
    Amiri F, Millan D, Arroyo M, Silani M, Rabczuk T. Fourth order phase-field model for local max-ent approximants applied to crack propagation. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 254–275MathSciNetGoogle Scholar
  29. 29.
    Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 322–350MathSciNetGoogle Scholar
  30. 30.
    Msekh M A, Silani M, Jamshidian M, Areias P, Zhuang X, Zi G, He P, Rabczuk T. Predictions of J integral and tensile strength of clay/epoxy nanocomposites material using phase-field model. Composites. Part B, Engineering, 2016, 93: 97–114Google Scholar
  31. 31.
    Hamdia K, Msekh M A, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T. Uncertainty quantication of the fracture properties of polymeric nanocomposites based on phase-field modeling. Composite Structures, 2015, 133: 1177–1190Google Scholar
  32. 32.
    Msekh M A, Sargado M, Jamshidian M, Areias P, Rabczuk T. ABAQUS implementation of phase-field model for brittle fracture. Computational Materials Science, 2015, 96: 472–484Google Scholar
  33. 33.
    Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109Google Scholar
  34. 34.
    Hamdia K M, Zhuang X, He P, Rabczuk T. Fracture toughness of polymeric particle nanocomposites: Evaluation of Models performance using Bayesian method. Composites Science and Technology, 2016, 126: 122–129Google Scholar
  35. 35.
    Rabczuk T, Belytschko T, Xiao S P. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14): 1035–1063MathSciNetzbMATHGoogle Scholar
  36. 36.
    Rabczuk T, Belytschko T. Adaptivity for structured meshfree particle methods in 2D and 3D. International Journal for Numerical Methods in Engineering, 2005, 63(11): 1559–1582MathSciNetzbMATHGoogle Scholar
  37. 37.
    Nguyen V P, Rabczuk T, Bordas S, Duflot M. Meshless methods: A review and computer implementation aspects. Mathematics and Computers in Simulation, 2008, 79(3): 763–813MathSciNetzbMATHGoogle Scholar
  38. 38.
    Zhuang X, Cai Y, Augarde C. A meshless sub-region radial point interpolation method for accurate calculation of crack tip elds. Theoretical and Applied Fracture Mechanics, 2014, 69: 118–125Google Scholar
  39. 39.
    Zhuang X, Zhu H, Augarde C. An improved meshless Shepard and least square method possessing the delta property and requiring no singular weight function. Computational Mechanics, 2014, 53(2): 343–357MathSciNetzbMATHGoogle Scholar
  40. 40.
    Zhuang X, Augarde C, Mathisen K. Fracture modelling using meshless methods and level sets in 3D: Framework and modelling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998MathSciNetzbMATHGoogle Scholar
  41. 41.
    Chen L, Rabczuk T, Bordas S, Liu GR, Zeng KY, Kerfriden P. Extended finite element method with edge-based strain smoothing (Esm-XFEM) for linear elastic crack growth. Computer Methods in Applied Mechanics and Engineering, 2012, 209-212(4): 250–265MathSciNetzbMATHGoogle Scholar
  42. 42.
    Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 1999, 45(5): 601–620zbMATHGoogle Scholar
  43. 43.
    Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46(1): 131–150MathSciNetzbMATHGoogle Scholar
  44. 44.
    Vu-Bac N, Nguyen-Xuan H, Chen L, Lee C K, Zi G, Zhuang X, Liu G R, Rabczuk T. A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. Journal of Applied Mathematics, 2013, 2013, 978026MathSciNetzbMATHGoogle Scholar
  45. 45.
    Bordas S P A, Natarajan S, Kerfriden P, Augarde C E, Mahapatra D R, Rabczuk T, Pont S D. On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/ GFEM/PUFEM). International Journal for Numerical Methods in Engineering, 2011, 86(4-5): 637–666zbMATHGoogle Scholar
  46. 46.
    Bordas S P A, Rabczuk T, Hung N X, Nguyen V P, Natarajan S, Bog T, Quan D M, Hiep N V. Strain Smoothing in FEM and XFEM. Computers & Structures, 2010, 88(23-24): 1419–1443Google Scholar
  47. 47.
    Rabczuk T, Zi G, Gerstenberger A, Wall W A. A new crack tip element for the phantom node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599zbMATHGoogle Scholar
  48. 48.
    Chau-Dinh T, Zi G, Lee P S, Rabczuk T, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92-93: 242–256Google Scholar
  49. 49.
    Song J H, Areias P M A, Belytschko T. A method for dynamic crack and shear band propagation with phantom nodes. International Journal for Numerical Methods in Engineering, 2006, 67(6): 868–893zbMATHGoogle Scholar
  50. 50.
    Areias P M A, Song J H, Belytschko T. Analysis of fracture in thin shells by overlapping paired elements. Computer Methods in Applied Mechanics and Engineering, 2006, 195(41-43): 5343–5360zbMATHGoogle Scholar
  51. 51.
    Rabczuk T, Areias P M A. A meshfree thin shell for arbitrary evolving cracks based on an external enrichment. CMES-Computer Modeling in Engineering and Sciences, 2006, 16(2): 115–130Google Scholar
  52. 52.
    Zi G, Rabczuk T, Wall W A. Extended meshfree methods without branch enrichment for cohesive cracks. Computational Mechanics, 2007, 40(2): 367–382zbMATHGoogle Scholar
  53. 53.
    Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495zbMATHGoogle Scholar
  54. 54.
    Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760zbMATHGoogle Scholar
  55. 55.
    Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548MathSciNetzbMATHGoogle Scholar
  56. 56.
    Bordas S, Rabczuk T, Zi G. Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Engineering Fracture Mechanics, 2008, 75(5): 943–960Google Scholar
  57. 57.
    Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically nonlinear three dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758Google Scholar
  58. 58.
    Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71MathSciNetzbMATHGoogle Scholar
  59. 59.
    Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23-24): 1391–1411Google Scholar
  60. 60.
    Amiri F, Anitescu C, Arroyo M, Bordas S, Rabczuk T. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57MathSciNetzbMATHGoogle Scholar
  61. 61.
    Talebi H, Samaniego C, Samaniego E, Rabczuk T. On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods. International Journal for Numerical Methods in Engineering, 2012, 89(8): 1009–1027MathSciNetzbMATHGoogle Scholar
  62. 62.
    Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T. Isogeometric analysis of large-deformation thin shells using RHTsplines for multiple-patch coupling. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 1157–1178MathSciNetGoogle Scholar
  63. 63.
    Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchho-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291MathSciNetzbMATHGoogle Scholar
  64. 64.
    Jia Y, Anitescu C, Ghorashi S, Rabczuk T. Extended isogeometric analysis for material interface problems. IMA Journal of Applied Mathematics, 2015, 80(3): 608–633MathSciNetzbMATHGoogle Scholar
  65. 65.
    Ghorashi S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture nalysis of orthotropic media. Computers & Structures, 2015, 147: 138–146Google Scholar
  66. 66.
    Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343zbMATHGoogle Scholar
  67. 67.
    Rabczuk T, Areias P M A. A new approach for modelling slip lines in geological materials with cohesive models. International Journal for Numerical and Analytical Methods in Engineering, 2006, 30(11): 1159–1172zbMATHGoogle Scholar
  68. 68.
    Rabczuk T, Belytschko T. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1-4): 19–49zbMATHGoogle Scholar
  69. 69.
    Rabczuk T, Belytschko T. A three dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29-30): 2777–2799MathSciNetzbMATHGoogle Scholar
  70. 70.
    Rabczuk T, Areias P M A, Belytschko T. A simplied meshfree method for shear bands with cohesive surfaces. International Journal for Numerical Methods in Engineering, 2007, 69(5): 993–1021zbMATHGoogle Scholar
  71. 71.
    Rabczuk T, Samaniego E. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6-8): 641–658MathSciNetzbMATHGoogle Scholar
  72. 72.
    Rabczuk T, Song J H, Belytschko T. Simulations of instability in dynamic fracture by the cracking particles method. Engineering Fracture Mechanics, 2009, 76(6): 730–741Google Scholar
  73. 73.
    Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37-40): 2437–2455zbMATHGoogle Scholar
  74. 74.
    Cai Y, Zhuang X, Zhu H. A generalized and ecient method for nite cover generation in the numerical manifold method. International Journal of Computational Methods, 2013, 10(5): 1350028MathSciNetzbMATHGoogle Scholar
  75. 75.
    Liu G, Zhuang X, Cui Z. Three-dimensional slope stability analysis using independent cover based numerical manifold and vector method. Engineering Geology, 2017, 225: 83–95Google Scholar
  76. 76.
    Nguyen B H, Zhuang X, Wriggers P, Rabczuk T, Mear M E, Tran H D. Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems. Computer Methods in Applied Mechanics and Engineering, 2017, 323: 132–150MathSciNetGoogle Scholar
  77. 77.
    Nguyen B H, Tran H D, Anitescu C, Zhuang X, Rabczuk T. An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems. Computer Methods in Applied Mechanics and Engineering, 2016, 306: 252–275MathSciNetGoogle Scholar
  78. 78.
    Zhu H, Wu W, Chen J, Ma G, Liu X, Zhuang X. Integration of three dimensional discontinuous deformation analysis (DDA) with binocular photogrammetry for stability analysis of tunnels in blocky rock mass. Tunnelling and Underground Space Technology, 2016, 51: 30–40Google Scholar
  79. 79.
    Wu W, Zhu H, Zhuang X, Ma G, Cai Y. A multi-shell cover algorithm for contact detection in the three dimensional discontinuous deformation analysis. Theoretical and Applied Fracture Mechanics, 2014, 72: 136–149Google Scholar
  80. 80.
    Cai Y, Zhu H, Zhuang X. A continuous/discontinuous deformation analysis (CDDA) method based on deformable blocks for fracture modelling. Frontiers of Structural and Civil Engineering, 2013, 7(4): 369–378Google Scholar
  81. 81.
    Nguyen-Xuan H, Liu G R, Bordas S, Natarajan S, Rabczuk T. An adaptive singular ES-FEM for mechanics problems with singular eld of arbitrary order. Computer Methods in Applied Mechanics and Engineering, 2013, 253: 252–273MathSciNetzbMATHGoogle Scholar
  82. 82.
    Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41Google Scholar
  83. 83.
    Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315MathSciNetzbMATHGoogle Scholar
  84. 84.
    Areias P M A, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63Google Scholar
  85. 85.
    Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137Google Scholar
  86. 86.
    Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947zbMATHGoogle Scholar
  87. 87.
    Areias P, Rabczuk T. Finite strain fracture of plates and shells with congurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122MathSciNetzbMATHGoogle Scholar
  88. 88.
    Silani M, Talebi H, Hamouda A S, Rabczuk T. Nonlocal damage modeling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23Google Scholar
  89. 89.
    Talebi H, Silani M, Rabczuk T. Concurrent multiscale modelling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92Google Scholar
  90. 90.
    Silani M, Talebi H, Ziaei-Rad S, Hamouda A M S, Zi G, Rabczuk T. A three dimensional extended Arlequin method for dynamic fracture. Computational Materials Science, 2015, 96: 425–431Google Scholar
  91. 91.
    Silani M, Ziaei-Rad S, Talebi H, Rabczuk T. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38Google Scholar
  92. 92.
    Talebi H, Silani M, Bordas S, Kerfriden P, Rabczuk T. A computational library for multiscale modelling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071MathSciNetGoogle Scholar
  93. 93.
    Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541Google Scholar
  94. 94.
    Yang S W, Budarapu P R, Mahapatra D R, Bordas S P A, Zi G, Rabczuk T. A meshless adaptive multiscale method for fracture. Computational Materials Science, 2015, 96: 382–395Google Scholar
  95. 95.
    Budarapu P, Gracie R, Bordas S, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148Google Scholar
  96. 96.
    Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Ecient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143Google Scholar
  97. 97.
    Zhuang X, Wang Q, Zhu H. Multiscale modelling of hydro-mechanical couplings in quasi-brittle materials. International Journal of Fracture, 2017, 204(1): 1–27Google Scholar
  98. 98.
    Zhu H, Wang Q, Zhuang X. A nonlinear semi-concurrent multiscale method for fractures. International Journal of Impact Engineering, 2016, 87: 65–82Google Scholar
  99. 99.
    Zhuang X, Wang Q, Zhu H. A 3D computational homogenization model for porous material and parameters identification. Computational Materials Science, 2015, 96: 536–548Google Scholar
  100. 100.
    Kouznetsova V, Geers M G D, Brekelmans W A M. Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering, 2002, 54(8): 1235–1260zbMATHGoogle Scholar
  101. 101.
    Rabczuk T, Ren H. A peridynamics formulation for quasi-static fracture and contact in rock. Engineering Geology, 2017, 225: 42–48Google Scholar
  102. 102.
    Amani J, Oterkus E, Areias P, Zi G, Nguyen-Thoi T, Rabczuk T. A non-ordinary state-based peridynamics formulation for thermoplastic fracture. International Journal of Impact Engineering, 2016, 87: 83–94Google Scholar
  103. 103.
    Ren H, Zhuang X, Rabczuk T. A new Peridynamic formulation with shear deformation for elastic solid. Journal of Micromecha-nics and Molecular Physics, 2016, 1(2): 1650009Google Scholar
  104. 104.
    Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476MathSciNetGoogle Scholar
  105. 105.
    Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782MathSciNetGoogle Scholar
  106. 106.
    Glen G, Isaacs K. Estimating sobol sensitivity indices using correlations. Journal of Environmental Modelling and Software, 2012, 37: 157–166Google Scholar
  107. 107.
    Nossent J, Elsen P, Bauwens W. Sobol sensitivity analysis of a complex environmental model. Journal of Environmental Modelling and Software, 2011, 26(12): 1515–1525Google Scholar
  108. 108.
    Zhang X Y, Trame M N, Lesko L J, Schmidt S. Sobol sensitivity analysis: A tool to guide the development and evaluation of systems pharmacology models. CPT: Pharmacometrics & Systems Pharmacology, 2015, 4(2): 69–79Google Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Nazim Abdul Nariman
    • 1
    Email author
  • Raja Rizwan Hussain
    • 2
  • Ilham Ibrahim Mohammad
    • 1
  • Peyman Karampour
    • 3
  1. 1.Department of Civil EngineeringTishk International University SulaimaniSulaimaniyaIraq
  2. 2.Civil Engineering Department, College of EngineeringKing Saud UniversityRiyadhSaudi Arabia
  3. 3.Department of Mechanical Engineering, Faculty of EngineeringArak UniversityArakIran

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