
Free vibration analysis of cracked thin plates by quasi-convex coupled isogeometric-meshfree method
Hanjie ZHANG, Junzhao WU, Dongdong WANG
Front. Struct. Civ. Eng. ›› 2015, Vol. 9 ›› Issue (4) : 405-419.
Free vibration analysis of cracked thin plates by quasi-convex coupled isogeometric-meshfree method
The free vibration analysis of cracked thin plates via a quasi-convex coupled isogeometric-meshfree method is presented. This formulation employs the consistently coupled isogeometric-meshfree strategy where a mixed basis vector of the convex B-splines is used to impose the consistency conditions throughout the whole problem domain. Meanwhile, the rigid body modes related to the mixed basis vector and reproducing conditions are also discussed. The mixed basis vector simultaneously offers the consistent isogeometric-meshfree coupling in the coupled region and the quasi-convex property for the meshfree shape functions in the meshfree region, which is particularly attractive for the vibration analysis. The quasi-convex meshfree shape functions mimic the isogeometric basis function as well as offer the meshfree nodal arrangement flexibility. Subsequently, this approach is exploited to study the free vibration analysis of cracked plates, in which the plate geometry is exactly represented by the isogeometric basis functions, while the cracks are discretized by meshfree nodes and highly smoothing approximation is invoked in the rest of the problem domain. The efficacy of the present method is illustrated through several numerical examples.
meshfree method / isogeometric analysis / quasi-convex isogeometric-meshfree method / free vibration / cracked thin plate
[1] |
Lynn P P, Kumbasar N. Free vibration of thin rectangular plates having narrow cracks with simply supported edges. Developments in Mechanics, 1967, 4: 911–928
|
[2] |
Stahl B, Keer L M. Vibration and stability of cracked rectangular plates. International Journal of Solids and Structures, 1972, 8(1): 69–91
|
[3] |
Nezu K. Free vibration of a simply-supported rectangular plate with a straight through-notch. Bulletin of the Japan Society of Mechanical Engineers, 1982, 25(199): 16–23
|
[4] |
Solecki R. Bending vibration of a simply supported rectangular plate with a crack parallel to one edge. Engineering Fracture Mechanics, 1983, 18(6): 1111–1118
|
[5] |
Hirano Y, Okazaki K. Vibration of cracked rectangular plates. Bulletin of the Japan Society of Mechanical Engineers, 1980, 23(179): 732–740
|
[6] |
Leissa A W, McGee O G, Huang C S. Vibration of circular plates having V-notches or sharp radial cracks. Journal of Sound and Vibration, 1993, 161(2): 227–239
|
[7] |
Liew K M, Hung K C, Lim M K. A solution method for analysis of cracked plates under vibration. Engineering Fracture Mechanics, 1994, 48(3): 393–404
|
[8] |
Huang C S, Leissa A W. Vibration analysis of rectangular plates with side cracks via the Ritz method. Journal of Sound and Vibration, 2009, 323(3−5): 974–988
|
[9] |
Zienkiewicz O C, Taylor R L. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, 2005
|
[10] |
Belytschko T, Lu Y Y, Gu L. Element-free Gakerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256
|
[11] |
Liu W K, Jun S, Zhang Y F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 1995, 20(8−9): 1081–1106
|
[12] |
Sukumar N. Construction of polygonal interpolants: a maximum entropy approach. International Journal for Numerical Methods in Engineering, 2004, 61(12): 2159–2181
|
[13] |
Arroyo M, Ortiz M. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerical Methods in Engineering, 2006, 65(13): 2167–2202
|
[14] |
Rabczuk T, Samaniego E. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6): 641–658
|
[15] |
Chen J S, Chi S W, Hu H Y. Recent developments in stabilized Galerkin and collocation meshfree methods. Computer Assisted Mechanics and Engineering Sciences, 2011, 18: 3–21
|
[16] |
Wang D, Chen P. Quasi-convex reproducing kernel meshfree method. Computational Mechanics, 2014, 54(3): 689–709
|
[17] |
Kwok O L A, Guan P C, Cheng W P, Sun C T. Semi-Lagrangian reproducing kernel particle method for slope stability analysis and post-failure simulation. KSCE Journal of Civil Engineering, 2015, 19(1): 107–115
|
[18] |
Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39−41): 4135–4195
|
[19] |
Cottrell J A, Hughes T J R, Reali A. Studies of refinement and continuity in isogeometric structural analysis. Computer Methods in Applied Mechanics and Engineering, 2007, 196(41): 4160–4183
|
[20] |
De Luycker E, Benson D J, Belytschko T, Bazilevs Y, Hsu M C. X‐FEM in isogeometric analysis for linear fracture mechanics. International Journal for Numerical Methods in Engineering, 2011, 87(6): 541–565
|
[21] |
Bazilevs Y, Hsu M C, Scott M A. Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Computer Methods in Applied Mechanics and Engineering, 2012, 249: 28–41
|
[22] |
Wang D, Liu W, Zhang H. Novel higher order mass matrices for isogeometric structural vibration analysis. Computer Methods in Applied Mechanics and Engineering, 2013, 260: 92–108
|
[23] |
Thai C H, Nguyen-Xuan H, Bordas S P A, Nguyen-Thanh N, Rabczuk T. Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory. Mechanics of Advanced Materials and Structures, 2014, 22(6): 451–469
|
[24] |
Elguedj T, Hughes T J R. Isogeometric analysis of nearly incompressible large strain plasticity. Computer Methods in Applied Mechanics and Engineering, 2014, 288: 388–416
|
[25] |
Zuo B Q, Huang Z D, Wang Y W, Wu Z J. Isogeometric analysis for CSG models. Computer Methods in Applied Mechanics and Engineering, 2015, 285: 102–124
|
[26] |
Wang D, Xuan J. An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37−40): 2425–2436
|
[27] |
Krysl P, Belytschko T. Analysis of thin plates by the element-free Galerkin method. Computational Mechanics, 1995, 16: 1–10
|
[28] |
Krysl P, Belytschko T. Analysis of thin plates by the element-free Galerkin method. Computational Mechanics, 1995, 16: 1–10
|
[29] |
Li S, Lu H, Han W, Liu W K, Simkins D C. Reproducing kernel element method, Part II. Global conforming
|
[30] |
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): 524–548
|
[31] |
Wang D, Peng H. A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates. Computational Mechanics, 2013, 51(6): 1013–1029
|
[32] |
Kiendl J, Bletzinger K U, Linhard J, Wüchner R. Isogeometric shell analysis with Kirchhoff−Love elements. Computer Methods in Applied Mechanics and Engineering, 2009, 198(49): 3902–3914
|
[33] |
Zhang H, Wang D, Xuan J. Non-uniform rational B spline-based isogeometric finite element analysis of thin beams and plates. Chinese Quarterly of Mechanics, 2010, 31: 469–477
|
[34] |
Benson D J, Bazilevs Y, Hsu M C, Hughes T J R. A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 2011, 200(13): 1367–1378
|
[35] |
Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger K U, Bazilevs Y, Rabczuk T. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47): 3410–3424
|
[36] |
Shojaee S, Izadpanah E, Valizadeh N, Kiendl J. Free vibration analysis of thin plates by using a NURBS-based isogeometric approach. Finite Elements in Analysis and Design, 2012, 61: 23–34
|
[37] |
Echter R, Oesterle B, Bischoff M. A hierarchic family of isogeometric shell finite elements. Computer Methods in Applied Mechanics and Engineering, 2013, 254: 170–180
|
[38] |
Wang D, Liu W, Zhang H. Superconvergent isogeometric free vibration analysis of Euler-Bernoulli beams and Kirchhoff plates with new higher order mass matrices. Computer Methods in Applied Mechanics and Engineering, 2015, 286: 230–267
|
[39] |
Organ D, Fleming M, Terry T, Belytschko T. Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Computational Mechanics, 1996, 18(3): 225–235
|
[40] |
Belytschko T, Fleming M. Smoothing, enrichment and contact in the element-free Galerkin method. Computers & Structures, 1999, 71(2): 173–195
|
[41] |
Rabczuk T, Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343
|
[42] |
De Luycker E, Benson D J, Belytschko T, Bazilevs Y, Hsu M C. X-FEM in isogeometric analysis for linear fracture mechanics. International Journal for Numerical Methods in Engineering, 2011, 87(6): 541–565
|
[43] |
Ghorashi S S, Valizadeh N, Mohammadi S. Extended isogeometric analysis for simulation of stationary and propagating cracks. International Journal for Numerical Methods in Engineering, 2012, 89(9): 1069–1101
|
[44] |
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
|
[45] |
Tran L V, Nguyan V P, Wahab M A, Nguyan-Xuan H. An extended isogeometric analysis for vibration of cracked FGM plates using higher-order shear deformation theory, arXiv preprint arXiv:1403.0306, 2014
|
[46] |
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
|
[47] |
Zhang H, Wang D. An isogeometric enriched quasi-convex meshfree formulation with application to material interface modeling. Engineering Analysis with Boundary Elements, 2015, 60: 37–50
|
[48] |
Wang D, Zhang H. A consistently coupled isogeometric-meshfree method. Computer Methods in Applied Mechanics and Engineering, 2014, 268: 843–870
|
[49] |
Zhang H, Wang D, Liu W. Isogeometric-meshfree coupled analysis of Kirchhoff plates. Advances in Structural Engineering, 2014, 17(8): 1159–1176
|
[50] |
Marsden M J. An identity for spline functions with applications to variation-diminishing spline approximation. Journal of Approximation Theory, 1970, 3(1): 7–49
|
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