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An extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) for free vibration analysis of cracked Reissner-Mindlin shells

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Abstract

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) was recently proposed and proven to be robust for free vibration analyses of Reissner-Mindlin shell. The method improves significantly the accuracy of the solution due to softening effect of the cell-based strain smoothing technique. In addition, due to using only three-node triangular elements generated automatically, the CS-FEM-DSG3 can be applied flexibly for arbitrary complicated geometric domains. However so far, the CS-FEM-DSG3 has been only developed for analyzing intact structures without possessing internal cracks. The paper hence tries to extend the CS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells by integrating the original CS-FEM-DSG3 with discontinuous and crack–tip singular enrichment functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3). The accuracy and reliability of the novel XCS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells are investigated through solving three numerical examples and comparing with commercial software ANSYS.

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References

  1. Kwon Y W. Development of finite element shape functions with derivative singularity. Computers & Structures, 1988, 30(5): 1159–1163

    Article  MATH  Google Scholar 

  2. Krawczuk M. Rectangular shell finite element with an open crack. Finite Elements in Analysis and Design, 1994, 15(3): 233–253

    Article  MATH  Google Scholar 

  3. Liu R, Zhang T, Wu X, Wang C. Determination of stress intensity factors for a cracked shell under bending with improved shell theories. Journal of Aerospace Engineering, 2006, 19(1): 21–28

    Article  Google Scholar 

  4. Vaziri A, Estekanchi H E. Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression. Thinwalled Structures, 2006, 44(2): 141–151

    Google Scholar 

  5. Fu J, To C W S. Bulging factors and geometrically nonlinear responses of cracked shell structures under internal pressure. Engineering Structures, 2012, 41: 456–463

    Article  Google Scholar 

  6. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 1999, 45(5): 601–620

    Article  MATH  MathSciNet  Google Scholar 

  7. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46: 131–150

    Article  MATH  Google Scholar 

  8. Stolarska M, Chopp D L, Moës N, Belytschko T. Modelling crack growth by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering, 2001, 51(8): 943–960

    Article  MATH  Google Scholar 

  9. Bachene M, Tiberkak R, Rechak S. Vibration analysis of cracked plates using the extended finite element method. Archive of Applied Mechanics, 2009, 79(3): 249–262

    Article  MATH  Google Scholar 

  10. Natarajan S, Baiz P M, Bordas S, Rabczuk T, Kerfriden P. Natural frequencies of cracked functionally graded material plates by the extended finite element method. Composite Structures, 2011, 93(11): 3082–3092

    Article  Google Scholar 

  11. 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: 115–130

    Google Scholar 

  12. Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548

    Article  MATH  MathSciNet  Google Scholar 

  13. Zhuang X, Augarde C E, Mathisen K M. Fracture modeling using meshless methods and level sets in 3D: Framework and modeling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998

    Article  MathSciNet  Google Scholar 

  14. 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: 242–256

    Article  Google Scholar 

  15. Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146

    Article  Google Scholar 

  16. 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 Kirchhoff–Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

    Article  MathSciNet  Google Scholar 

  17. Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122

    Article  MathSciNet  Google Scholar 

  18. Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947

    Article  MATH  Google Scholar 

  19. Areias P, Rabczuk T, Dias-da- Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

    Article  Google Scholar 

  20. Areias P, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63

    Article  Google Scholar 

  21. Liu G R, Nguyen-Thoi T. Smoothed Finite Element Methods. NewYork: Taylor and Francis Group, 2010

    Book  Google Scholar 

  22. Liu G R, Nguyen-Thoi T, Nguyen-Xuan H, Dai K Y, Lam K Y. On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering, 2009, 77(13): 1863–1869

    Article  MATH  MathSciNet  Google Scholar 

  23. Nguyen T T, Liu G R, Dai K Y, Lam K Y. Selective smoothed finite element method. Tsinghua Science and Technology, 2007, 12(5): 497–508

    Article  MathSciNet  Google Scholar 

  24. Liu G R, Dai K Y, Nguyen T T. A smoothed finite element method for mechanics problems. Computational Mechanics, 2007, 39(6): 859–877

    Article  MATH  Google Scholar 

  25. Nguyen-Thoi T, Liu G R, Nguyen-Xuan H. Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems. International Journal of Computational Methods, 2009, 06(04): 633–666

    Article  MathSciNet  Google Scholar 

  26. Nguyen-Thoi T, Liu G R, Nguyen-Xuan H, Nguyen-Tran C. Adaptive analysis using the node-based smoothed finite element method (NS-FEM). International Journal for Numerical Methods in Biomedical Engineering, 2011, 27(2): 198–218

    Article  MATH  MathSciNet  Google Scholar 

  27. Liu G R, Nguyen-Thoi T, Nguyen-Xuan H, Lam K Y. A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Computers & Structures, 2009, 87(1–2): 14–26

    Article  Google Scholar 

  28. Nguyen-Thoi T, Liu G R, Nguyen-Xuan H, Nguyen-Tran C. An nsided polygonal edge-based smoothed finite element method (nESFEM) for solid mechanics. International Journal for Numerical Methods in Biomedical Engineering, 2011, 27: 1446–1472

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu G R, Nguyen-Thoi T, Lam K Y. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. Journal of Sound and Vibration, 2009, 320(4–5): 1100–1130

    Article  Google Scholar 

  30. Nguyen-Thoi T, Liu G R, Lam K Y, Zhang G Y. A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements. International Journal for Numerical Methods in Engineering, 2009, 78(3): 324–353

    Article  MATH  MathSciNet  Google Scholar 

  31. Liu G R, Nguyen-Xuan H, Nguyen-Thoi T, Xu X. A novel Galerkin-like weakform and a superconvergent alpha finite element method (S-alpha FEM) for mechanics problems using triangular meshes. Journal of Computational Physics, 2009, 228(11): 4055–4087

    Article  MATH  MathSciNet  Google Scholar 

  32. Liu G R, Nguyen-Thoi T, Lam K Y. A novel alpha finite element method (aFEM) for exact solution to mechanics problems using triangular and tetrahedral elements. Computer Methods in Applied Mechanics and Engineering, 2008, 197(45–48): 3883–3897

    Article  MATH  MathSciNet  Google Scholar 

  33. Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. A variationally consistent alpha FEM (VC alpha FEM) for solution bounds and nearly exact solution to solid mechanics problems using quadrilateral elements. International Journal for Numerical Methods in Engineering, 2011, 85(4): 461–497

    Article  MATH  MathSciNet  Google Scholar 

  34. Liu G R, Nguyen-Thoi T, Lam K Y. A novel FEM by scaling the gradient of strains with factor alpha (alpha FEM). Computational Mechanics, 2009, 43(3): 369–391

    Article  MATH  MathSciNet  Google Scholar 

  35. Liu G R, Nguyen T T, Dai K Y, Lam K Y. Theoretical aspects of the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering, 2007, 71(8): 902–930

    Article  MATH  MathSciNet  Google Scholar 

  36. Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates. International Journal for Numerical Methods in Engineering, 2010, 84(10): 1222–1256

    Article  MATH  MathSciNet  Google Scholar 

  37. Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie J F. A smoothed finite element method for plate analysis. Computer Methods in Applied Mechanics and Engineering, 2008, 197(13–16): 1184–1203

    Article  MATH  Google Scholar 

  38. Nguyen-Xuan H, Liu G R, Thai-Hoang C, Nguyen-Thoi T. An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 2010, 199(9–12): 471–489

    Article  MATH  MathSciNet  Google Scholar 

  39. Nguyen-Xuan H, Rabczuk T, Nguyen-Thanh N, Nguyen-Thoi T, Bordas S. A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates. Computational Mechanics, 2010, 46(5): 679–701

    Article  MATH  MathSciNet  Google Scholar 

  40. Nguyen-Xuan H, Tran L V, Nguyen-Thoi T, Vu-Do H C. Analysis of functionally graded plates using an edge-based smoothed finite element method. Composite Structures, 2011, 93(11): 3019–3039

    Article  Google Scholar 

  41. Nguyen-Xuan H, Tran L V, Thai C H, Nguyen-Thoi T. Analysis of functionally graded plates by an efficient finite element method with node-based strain smoothing. Thin-walled Structures, 2012, 54: 1–18

    Article  Google Scholar 

  42. Thai C H, Tran L V, Tran D T, Nguyen-Thoi T, Nguyen-Xuan H. Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method. Applied Mathematical Modelling, 2012, 36(11): 5657–5677

    Article  MATH  MathSciNet  Google Scholar 

  43. Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Xuan H, Ngo- Thanh P. Static, free vibration and buckling analyses of stiffened plates by CS-FEM-DSG3 using triangular elements. Computers & Structures, 2013, 125: 100–113

    Article  Google Scholar 

  44. Nguyen-Thoi T, Phung-Van P, Luong-Van H, Nguyen-Van H, Nguyen-Xuan H. A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates. Computational Mechanics, 2013, 51(1): 65–81

    Article  MATH  MathSciNet  Google Scholar 

  45. Nguyen-Thoi T, Phung-Van P, Thai-Hoang C, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements for static and free vibration analyses of shell structures. International Journal of Mechanical Sciences, 2013, 74: 32–45

    Article  Google Scholar 

  46. Phan-Dao H H, Nguyen-Xuan H, Thai-Hoang C, Nguyen-Thoi T, Rabczuk T. An edge-based smoothed finite element method for analysis of laminated composite plates. International Journal of Computational Methods, 2013, 10(01): 1340005

    Article  MathSciNet  Google Scholar 

  47. Phung-Van P, Nguyen-Thoi T, Tran L V, Nguyen-Xuan H. A cellbased smoothed discrete shear gap method (CS-DSG3) based on the C0-type higher-order shear deformation theory for static and free vibration analyses of functionally graded plates. Computational Materials Science, 2013, 79: 857–872

    Article  Google Scholar 

  48. Luong-Van H, Nguyen-Thoi T, Liu G R, Phung-Van P. A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation. Engineering Analysis with Boundary Elements, 2014, 42: 8–19

    Article  MATH  MathSciNet  Google Scholar 

  49. Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Hoang S, Nguyen-Xuan H. An edge-based smoothed three-node mindlin plate element (ES-MIN3) for static and free vibration analyses of plates. KSCE Journal of Civil Engineering, 2014, 18(4): 1072–1082

    Article  Google Scholar 

  50. Phung-Van P, Nguyen-Thoi T, Luong-Van H, Lieu-Xuan Q. Geometrically nonlinear analysis of functionally graded plates using a cell-based smoothed three-node plate element (CS-MIN3) based on the C0-HSDT. Computer Methods in Applied Mechanics and Engineering, 2014, 270: 15–36

    Article  MATH  MathSciNet  Google Scholar 

  51. Phung-Van P, Nguyen-Thoi T, Le-Dinh T, Nguyen-Xuan H. Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3). Smart Materials and Structures, 2013, 22(9): 17

    Article  Google Scholar 

  52. Nguyen-Xuan H, Liu G R, Nguyen-Thoi T, Nguyen-Tran C. An edge-based smoothed finite element method for analysis of twodimensional piezoelectric structures. Smart Materials and Structures, 2009, 18(6): 1–12

    Article  Google Scholar 

  53. Liu G R, Chen L, Nguyen-Thoi T, Zeng K Y, Zhang G Y. A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems. International Journal for Numerical Methods in Engineering, 2010, 83(11): 1466–1497

    Article  MATH  MathSciNet  Google Scholar 

  54. Nguyen-Thoi T, Liu G R, Vu-Do H C, Nguyen-Xuan H. A facebased smoothed finite element method (FS-FEM) for viscoelastoplastic analyses of 3D solids using tetrahedral mesh. Computer Methods in Applied Mechanics and Engineering, 2009, 198(41–44): 3479–3498

    Article  MATH  Google Scholar 

  55. Nguyen-Thoi T, Vu-Do H C, Rabczuk T, Nguyen-Xuan H. A nodebased smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45–48): 3005–3027

    Article  MATH  MathSciNet  Google Scholar 

  56. Nguyen-Thoi T, Liu G R, Vu-Do H C, Nguyen-Xuan H. An edgebased smoothed finite element method for visco-elastoplastic analyses of 2D solids using triangular mesh. Computational Mechanics, 2009, 45(1): 23–44

    Article  MATH  MathSciNet  Google Scholar 

  57. Nguyen-Xuan H, Rabczuk T, Nguyen-Thoi T, Tran T N, Nguyen- Thanh N. Computation of limit and shakedown loads using a nodebased smoothed finite element method. International Journal for Numerical Methods in Engineering, 2012, 90(3): 287–310

    Article  MATH  MathSciNet  Google Scholar 

  58. Tran T N, Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. An edgebased smoothed finite element method for primal–dual shakedown analysis of structures. International Journal for Numerical Methods in Engineering, 2010, 82: 917–938

    MATH  MathSciNet  Google Scholar 

  59. Nguyen-Thoi T, Phung-Van P, Rabczuk T, Nguyen-Xuan H, Le- Van C. An application of the ES-FEM in solid domain for dynamic analysis of 2d fluid-solid interaction problems. International Journal of Computational Methods, 2013, 10

    Google Scholar 

  60. Nguyen-Thoi T, Phung-Van P, Rabczuk T, Nguyen-Xuan H, Le-Van C. Free and forced vibration analysis using the n-sided polygonal Cell-Based Smoothed Finite Element Method (NCSFEM). International Journal of Computational Methods, 2013, 10 (01): 1340008

    Article  MathSciNet  Google Scholar 

  61. Bletzinger K U, Bischoff M, Ramm E. A unified approach for shearlocking-free triangular and rectangular shell finite elements. Computers & Structures, 2000, 75(3): 321–334

    Article  Google Scholar 

  62. Phung-Van P, Nguyen-Thoi T, Tran L V, Nguyen-Xuan H. A cellbased smoothed discrete shear gap method (CS-DSG3) based on the C-0-type higher-order shear deformation theory for static and free vibration analyses of functionally graded plates. Computational Materials Science, 2013, 79: 857–872

    Article  Google Scholar 

  63. Phung-Van P, Nguyen-Thoi T, Dang-Trung H, Nguyen-Minh N. A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise theory based on the C0-type higher-order shear deformation for static and free vibration analyses of sandwich and composite plates. Composite Structures, 2014, 111: 553–565

    Article  Google Scholar 

  64. Phung-Van P, Luong-Van H, Nguyen-Thoi T, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-DSG3) based on the higher-order shear deformation theory for dynamic responses of Mindlin plates on the viscoelastic foundation subjected to a moving sprung vehicle. International Journal for Numerical Methods in Engineering, 2014, 98(13): 988–1014

    Article  MathSciNet  Google Scholar 

  65. Phung-Van P, Nguyen-Thoi T, Luong-Van H, Thai-Hoang C, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise deformation theory for dynamic response of composite plates resting on viscoelastic foundation. Computer Methods in Applied Mechanics and Engineering, 2014, 272: 138–159

    Article  MATH  MathSciNet  Google Scholar 

  66. Bischoff M, Bletzinger K U. Stabilized DSG plate and shell elements. Trends in Computational structural mechanics. CIMNE. Barcelona, Spain, 2001

    Google Scholar 

  67. Lyly M, Stenberg R, Vihinen T. A stable bilinear element for the Reissner-Mindlin plate model. Computer Methods in Applied Mechanics and Engineering, 1993, 110(3–4): 343–357

    Article  MATH  MathSciNet  Google Scholar 

  68. Babuška I, Caloz G, Osborn J. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM Journal on Numerical Analysis, 1994, 31(4): 945–981

    Article  MATH  MathSciNet  Google Scholar 

  69. Melenk JM. On Generalized Finite Element Methods. University of Maryland, 1995

    Google Scholar 

  70. Babuška I, Melenk J. The partition of unity finite element method. International Journal for Numerical Methods in Engineering, 1997, 40(4): 727–758

    Article  MATH  MathSciNet  Google Scholar 

  71. Simone A, Duarte C A, Van der Giessen E. A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries. International Journal for Numerical Methods in Engineering, 2006, 67(8): 1122–1145

    Article  MATH  MathSciNet  Google Scholar 

  72. Babuška I, Nistor V, Tarfulea N. Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions. Journal of Computational and Applied Mathematics, 2008, 218(1): 175–183

    Article  MATH  MathSciNet  Google Scholar 

  73. Dolbow J, Moës N, Belytschko T. Modeling fracture in Mindlin–Reissner plates with the extended finite element method. International Journal of Solids and Structures, 2000, 37(48–50): 7161–7183

    Article  MATH  Google Scholar 

  74. Ventura G. On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method. International Journal for Numerical Methods in Engineering, 2006, 66(5): 761–795

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Division of Computational Mathematics and Engineering (CME), Institute for Computational Science (INCOS), Ton Duc Thang University, Hochiminh city, Vietnam

    M. H. Nguyen-Thoi, L. Le-Anh, V. Ho-Huu, H. Dang-Trung & T. Nguyen-Thoi

  2. Faculty of Civil Engineering, Ton Duc Thang University, Hochiminh city, Vietnam

    M. H. Nguyen-Thoi, L. Le-Anh, V. Ho-Huu, H. Dang-Trung & T. Nguyen-Thoi

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  1. M. H. Nguyen-Thoi
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Nguyen-Thoi, M.H., Le-Anh, L., Ho-Huu, V. et al. An extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) for free vibration analysis of cracked Reissner-Mindlin shells. Front. Struct. Civ. Eng. 9, 341–358 (2015). https://doi.org/10.1007/s11709-015-0302-1

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