Frontiers of Structural and Civil Engineering

, Volume 13, Issue 6, pp 1495–1509 | Cite as

Application of coupled XFEM-BCQO in the structural optimization of a circular tunnel lining subjected to a ground motion

  • Nazim Abdul NarimanEmail author
  • Ayad Mohammad Ramadan
  • Ilham Ibrahim Mohammad
Research Article


A new structural optimization method of coupled extended finite element method and bound constrained quadratic optimization method (XFEM-BCQO) is adopted to quantify the optimum values of four design parameters for a circular tunnel lining when it is subjected to earthquakes. The parameters are: tunnel lining thickness, tunnel diameter, tunnel lining concrete modulus of elasticity and tunnel lining concrete density. Monte-Carlo sampling method is dedicated to construct the meta models so that to be used for the BCQO method using matlab codes. Numerical simulations of the tensile damage in the tunnel lining due to a real earthquake in the literature are created for three design cases. XFEM approach is used to show the cracks for the mentioned design cases. The results of the BCQO method for the maximum design case for the tunnel tensile damage was matching the results obtained from XFEM approach to a fair extent. The new coupled approach manifested a significant capability to predict the cracks and spalling of the tunnel lining concrete under the effects of dynamic earthquakes.


ovaling deformation monte carlo sampling XFEM-BCQO maximum principal stress 


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  1. 1.
    Pescara M, Gaspari G M, Repetto L. Design of underground structures under seismic conditions: A long deep tunnel and a metro tunnel. In: Colloquium on Seismic Design of Tunnels. Torino: Geodata Engineering SpA, 2011Google Scholar
  2. 2.
    Hashash Y M A, Hook J J, Schmidt B, I-Chiang Yao J. Seismic design and analysis of underground structure. Journal of Tunneling and Underground Space Technology, 2001, 16(4): 247–293Google Scholar
  3. 3.
    Hashash Y M A, Park D, Yao J I. Ovaling deformations of circular tunnels under seismic loading: An update on seismic design and analysis of underground structures. Journal of Tunneling and Underground Space Technology, 2005, 20(5): 435–441Google Scholar
  4. 4.
    St John C M, Zahrah T F. Aseismic design of underground structures. Tunnelling and Underground Space Technology, 1987, 2(2): 165–197Google Scholar
  5. 5.
    Kawashima K. Seismic design of underground structures in soft ground: A review. In: Kusakabe, Fujita, Miyazaki, eds. Geotechnical Aspects of Underground Construction in Soft Ground. Rotterdam, 1999Google Scholar
  6. 6.
    Fabozzi S. Behaviour of segmental tunnel lining under static and dynamic loads. Dissertation for the Doctoral Degree. Naples: University of Naples Federico II, 2017Google Scholar
  7. 7.
    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
  8. 8.
    Moller S C, Vermeer P A. On numerical simulation of tunnel installation. Tunnelling and Underground Space Technology (Oxford, England), 2008, 23(4): 461–475Google Scholar
  9. 9.
    Lekhnitskii S G. Anisotropic plates. London: Foreign Technology Div Wright-Patterson Afb Oh, 1968Google Scholar
  10. 10.
    Lu A Z, Zhang L Q, Zhang N. Analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(3): 514–519Google Scholar
  11. 11.
    Yasuda N, Tsukada K, Asakura T. Elastic solutions for circular tunnel with void behind lining. Tunnelling and Underground Space Technology (Oxford, England), 2017, 70: 274–285Google Scholar
  12. 12.
    Han X, Xia Y. Analytic solutions of the forces and displacements for multicentre circular arc tunnels. Hindawi Mathematical Problems in Engineering, 2018, 2018: 8409129MathSciNetGoogle Scholar
  13. 13.
    Schmid H. Static problems of tunnels and pressure tunnels construction and their mutual relationships. Berlin: Springer, 1926Google Scholar
  14. 14.
    Morgan H. A contribution to the analysis of stress in a circular tunnel. Geotechnique, 1961, 11(1): 37–46Google Scholar
  15. 15.
    Windels R. Kreisring im elastischen continuum. Bauingenieur, 1967, 42: 429–439Google Scholar
  16. 16.
    Duddeck H, Erdmann J. On structural design models for tunnels in soft soil. Underground Space (United States), 1985, 9(5–6): 246–259Google Scholar
  17. 17.
    Do N A, Dias D, Oreste P, Djeran-Maigre I. A new numerical approach to the hyperstatic reaction method for segmental tunnel linings. International Journal for Numerical and Analytical Methods in Geomechanics, 2014, 38(15): 1617–1632Google Scholar
  18. 18.
    Vu Minh N, Broere W, Bosch J W. Structural analysis for shallow tunnels in soft soils. International Journal of Geomechanics, 2017, 17(8): 04017038Google Scholar
  19. 19.
    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
  20. 20.
    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
  21. 21.
    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
  22. 22.
    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
  23. 23.
    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
  24. 24.
    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
  25. 25.
    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
  26. 26.
    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
  27. 27.
    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
  28. 28.
    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
  29. 29.
    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
  30. 30.
    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
  31. 31.
    Bažant Z P. Why continuum damage is nonlocal: Micromechanics arguments. Journal of Engineering Mechanics, 1991, 117(5): 1070–1087Google Scholar
  32. 32.
    Thai T Q, Rabczuk T, Bazilevs Y, Meschke G. A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604MathSciNetzbMATHGoogle Scholar
  33. 33.
    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
  34. 34.
    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
  35. 35.
    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
  36. 36.
    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
  37. 37.
    Rabczuk T, Eibl J. Modelling dynamic failure of concrete with meshfree methods. International Journal of Impact Engineering, 2006, 32(11): 1878–1897Google Scholar
  38. 38.
    Etse G, Willam K. Failure analysis of elastoviscoplastic material models. Journal of Engineering Mechanics, 1999, 125(1): 60–69Google Scholar
  39. 39.
    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
  40. 40.
    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
  41. 41.
    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
  42. 42.
    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-eld model. Composites. Part B, Engineering, 2016, 93: 97–114Google Scholar
  43. 43.
    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
  44. 44.
    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
  45. 45.
    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
  46. 46.
    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
  47. 47.
    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
  48. 48.
    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
  49. 49.
    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
  50. 50.
    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
  51. 51.
    Zhuang X, Zhu H, Augarde C. An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function. Computational Mechanics, 2014, 53(2): 343–357MathSciNetzbMATHGoogle Scholar
  52. 52.
    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
  53. 53.
    Chen L, Rabczuk T, Bordas S, Liu G R, Zeng K Y, 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
  54. 54.
    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
  55. 55.
    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
  56. 56.
    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
  57. 57.
    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
  58. 58.
    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
  59. 59.
    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
  60. 60.
    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
  61. 61.
    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
  62. 62.
    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
  63. 63.
    Zamani R, Motahari M R. The effect of soil stiffness variations on Tunnel Lining Internal Forces under seismic loading and Case comparison with existing analytical methods. Ciência e Natura, Santa Maria, 2015, 37(1): 476–487Google Scholar
  64. 64.
    Möller S C. Tunnel induced settlements and structural forces in linings. Dissertation for the Doctoral Degree. Stuttgart: University of Stuttgart, 2006Google Scholar
  65. 65.
    Lu Q, Chen S, Chan Y, He C. Comparison between numerical and analytical analysis of the dynamic behavior of circular tunnels. Earth Sciences Research Journal, 2018, 22(2): 119–128Google Scholar
  66. 66.
    Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D. Global Sensitivity Analysis: The Primer. Hoboken: John Wiley & Sons Ltd., 2008zbMATHGoogle Scholar
  67. 67.
    Burhenne S, Jacob D, Henze G P. Sampling based on Sobol sequences for monte carlo techniques applied to building simulations. In: The 12th Conference of International Building Performance Simulation Association. Sydney, 2011Google Scholar
  68. 68.
    Myers R H, Montgomery D C. Response Surface Methodology: Product and Process Op-timization Using Designed Experiments. 2nd ed. New York: John Wiley & Sons, 2002zbMATHGoogle Scholar
  69. 69.
    Zhao J, Tiede C. Using a variance-based sensitivity analysis for analyzing the relation between measurements and unknown parameters of a physical model. Nonlinear Processes in Geophysics, 2011, 18(3): 269–276Google Scholar
  70. 70.
    Khuril A I, Mukhopadhyay S. Response surface methodology. WIREs Computational Statistics, 2010, 2(2): 128–149Google Scholar
  71. 71.
    Luenberger D G, Ye Y. Linear and Non-linear programming. In: International Series in Operations Research & Management Science. Palo Alto, CA: Stanford University, 2015Google Scholar
  72. 72.
    Box M J. A new method of constrained optimization and a comparison with other methods. Computer Journal, 1965, 8(1): 42–52MathSciNetzbMATHGoogle Scholar
  73. 73.
    Hunt B R, Lipsman R L, Rosenberg J M. A Guide to MATLAB for Beginner and Experienced Users. Cambridge: Cambridge University Press, 2006zbMATHGoogle 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
  • Ayad Mohammad Ramadan
    • 2
  • Ilham Ibrahim Mohammad
    • 1
  1. 1.Department of Civil EngineeringTishk International University-SulaimaniSulaimaniyaIraq
  2. 2.Mathematics Department-College of ScienceSulaimani UniversitySulaimaniyaIraq

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