Generalized differential quadrature finite element method for vibration analysis of arbitrarily shaped membranes

International Journal of Mechanical Sciences

Volumn 79, Issue , 2014, Pages 216-251

  Fantuzzi, Nicholas ^a   Tornabene, Francesco ^a   Viola, Erasmo ^a  

^a UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

Arbitrarily shaped membranes; Free vibrations; Generalized Differential Quadrature; Generalized Differential Quadrature Finite Element Method; Non homogeneous membranes

Indexed keywords

COMPOSITE MEMBRANES; DIFFERENTIAL EQUATIONS; GEOMETRY; MEMBRANES; VIBRATION ANALYSIS;

ARBITRARILY SHAPED MEMBRANES; CO-ORDINATE TRANSFORMATION; COMPATIBILITY CONDITIONS; FREE VIBRATION; GENERALIZED DIFFERENTIAL QUADRATURE; GOVERNING DIFFERENTIAL EQUATIONS; NON-HOMOGENEOUS; NUMERICAL IMPLEMENTATION;

FINITE ELEMENT METHOD;

EID: 84893055481     PISSN: 00207403     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijmecsci.2013.12.008     Document Type: Article

References (119)

1. Ö. Civalek, and M. Gurses Free vibration of curvilinear membranes by eight-node discrete singular convolution Int J Sci Technol 3 2008 165 171
2. B.C. Bhadra Symmetrical vibration of a composite circular membrane fastened at the circumference Indian J Pure Appl Math 10 1979 223 229
3. G.R. Buchanan Jr., and J. Peddieson Vibration of circular and annular membranes with variable density J Sound Vib 226 1999 379 382 (Pubitemid 129533329)
4. G.R. Buchanan Vibration of circular membranes with linearly varying density along a diameter J Sound Vib 280 2005 407 414
5. P.A.A. Laura, R.E. Rossi, and R.H. Gutierrez The fundamental frequency of non-homogeneous rectangular membranes J Sound Vib 204 1997 373 376 (Pubitemid 127402365)
6. P.A.A. Laura, D.V. Bambill, and R.H. Gutierrez A note on transverse vibrations of circular, annular, composite membranes J Sound Vib 205 1997 692 697 (Pubitemid 127401221)
7. Ö. Civalek Eigenvalues of membranes having skew and rhombic geometry using discrete singular convolution algorithm Commun Nonlinear Sci Numer Simul 14 2009 4003 4009
8. H. Ersoy, Ö. Civalek, and L. Özpolat Free vibration analysis of rectangular membranes with variable density using the discrete singular convolution approach Asian J Civ Eng 11 2010 83 94
9. H. Ersoy, L. Özpolat, and Ö. Civalek Free vibration of circular and annular membranes with varying density by the method of discrete singular convolution Struct Eng Mech 32 2009 621 634
10. J.T. Chen, J.H. Lin, S.R. Kuo, and S.W. Chyuan Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain Proc R Soc A 457 2001 2521 2546
11. S.Y. Reutskiy The methods of external excitation for analysis of arbitrary-shaped hollow conducting waveguides Prog Electromagn Res 82 2008 203 226
12. P. Amore, and D. Chowell Collocation approach to the Helmholtz eigenvalue problem on multiply connected domains J Sound Vib 329 2010 1362 1375
13. C. Shu, and H. Xue Solution of Helmholtz by differential quadrature method Comput Methods Appl Mech Eng 175 1999 203 212 (Pubitemid 29373440)
14. C. Shu, and Y.T. Chew Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems Commun Numer Methods Eng 13 1997 643 653 (Pubitemid 127548006)
15. C. Shu, and Y.T. Chew Application of multi-domain GDQ method to analysis of waveguides with rectangular boundaries Prog Electromagn Res 21 1999 1 19
16. G. Chen, and J. Zhou Boundary element methods 1992 Academic Press London
17. Z. Ding Vibration of arbitrarily shaped membranes with elastic supports at points Appl Math Mech 10 1989 1171 1177
18. S.W. Kang, J.M. Lee, and Y.J. Kang Vibration analysis of arbitrarily shaped membranes using non-dimensional dynamic influence function J Sound Vib 221 1999 117 132 (Pubitemid 129602578)
19. S.W. Kang, and J.M. Lee Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point-matching J Acoust Soc Am 107 2000 1153 1160 (Pubitemid 30141659)
20. S.W. Kang, and J.M. Lee Application of free vibration analysis of membranes using the non-dimensional dynamic influence function J Sound Vib 234 2000 455 470
21. S.W. Kang, and J.M. Lee Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type functions J Sound Vib 242 2002 9 26
22. S.W. Kang, and J.M. Lee Free vibration analysis of an unsymmetric trapezoidal membrane J Sound Vib 272 2004 450 460
23. S.W. Kang, and J.M. Lee Free vibration analysis of composite rectangular membranes with an oblique interface J Sound Vib 251 2002 505 517 (Pubitemid 35262874)
24. S.W. Kang, and S.H. Kim Free vibration analysis of free plates with smoothly varying boundary shapes using non-dimensional dynamic influence functions J Sound Vib 130 2008 1 8
25. S.W. Kang, and S.N. Atluri Free vibration analysis of arbitrarily shaped polygonal plates with simply supported edges using a sub-domain method J Sound Vib 327 2009 271 284
26. W.X. Wu, C. Shu, and C.M. Wang Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method J Sound Vib 306 2007 252 270 (Pubitemid 47069471)
27. M.G. Milsted, and J.R. Hutchinson Use of trigonometric terms in the finite element with application to vibrating membranes J Sound Vib 32 1974 327 346
28. H.D. Conway, and K.A. Farnham The free flexural vibrations of triangular, rhombic and parallelogram plates and some analogies Int J Mech Sci 7 1965 811 816
29. S. Durvasula Natural frequencies and modes of skew membranes J Acoust Soc Am 44 1968 1636 1646
30. L. Bauer, and E.L. Reiss Free vibrations of rhombic plates and membranes J Acoust Soc Am 54 1973 1373 1375
31. T. Irie, G. Yamada, and K. Umesato Free vibration of regular polygonal plates with simply supported edges J Acoust Soc Am 69 1981 1330 1336 (Pubitemid 11417960)
32. T. Irie, G. Yamada, and M. Tsujino Natural frequencies of concavely shaped polygonal plates with simply supported edges J Acoust Soc Am 69 1981 1507 1509
33. A.W. Leissa Vibration of plates 1993 Acoustical Society San Francisco
34. C.Y. Wang, and C.M. Wang Exact solutions for vibrating rectangular membranes placed in a vertical plane Int J Appl Mech 3 2011 625 631
35. J.A. Cottrell, A. Reali, Y. Bazilevs, and T.J.R. Hughes Isogeometric analysis of structural vibrations Comput Methods Appl Mech Eng 195 2006 5257 5296 (Pubitemid 44172039)
36. T.J.R. Hughes, A. Reali, and G. Sangalli Efficient quadrature for NURBS-based isogeometric analysis Comput Methods Appl Mech Eng 199 2010 301 313
37. A.J. Burton, and G.F. Miller The application of integral equation methods to the numerical solution of some exterior boundary-value problems Proc R Soc A 323 1971 201 210
38. C. Shu Differential quadrature and its application in engineering 2000 Springer London
39. C. Bert, and M. Malik Differential quadrature method in computational mechanics Appl Mech Rev 49 1996 1 27 (Pubitemid 126614778)
40. K.Y. Lam, and L. Hua Influence of initial pressure on frequency characteristics of a rotating truncated circular conical shell Int J Mech Sci 42 2000 213 236 (Pubitemid 30511249)
41. K.M. Liew, T.M. Teo, and J.B. Han Three-dimensional static solutions of rectangular plates by variant differential quadrature method Int J Mech Sci 43 2001 1611 1628 (Pubitemid 32396306)
42. T.Y. Ng, H. Li, and K.Y. Lam Generalized differential quadrature method for the free vibration of rotating composite laminated conical shell with various boundary conditions Int J Mech Sci 45 2003 567 587
43. E. Viola, and F. Tornabene Vibration analysis of damaged circular arches with varying cross-section Struct Integr Durab (SID-SDHM) 1 2005 155 169 (Pubitemid 43024470)
44. L. Zhang, Y. Xiang, and G.W. Wei Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions Int J Mech Sci 48 2006 1126 1138 (Pubitemid 44142516)
45. X. Wang, X. Wang, and X. Shi Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method Int J Mech Sci 49 2007 447 453 (Pubitemid 46356369)
46. E. Viola, and F. Tornabene Vibration analysis of conical shell structures using GDQ method Far East J Appl Math 25 2006 23 39
47. F. Tornabene, and E. Viola Vibration analysis of spherical structural elements using the GDQ method Comput Math Appl 53 2007 1538 1560 (Pubitemid 46783384)
48. E. Viola, M. Dilena, and F. Tornabene Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches J Sound Vib 299 2007 143 163 (Pubitemid 44485703)
49. P. Malekzadeh, and G. Karami A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations Appl Math Model 32 2008 1381 1394 (Pubitemid 351384329)
50. A. Marzani, F. Tornabene, and E. Viola Nonconservative stability problems via generalized differential quadrature method J Sound Vib 315 2008 176 196
51. F. Tornabene, and E. Viola 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method Eur J Mech - A/Solids 27 2008 1001 1025
52. F. Tornabene Vibration analysis of functionally graded conical, cylindrical and annular shell structures with a four-parameter power-law distribution Comput Methods Appl Mech Eng 198 2009 2911 2935
53. F. Tornabene, and E. Viola Free vibrations of four-parameter functionally graded parabolic panels and shell of revolution Eur J Mech - A/Solids 28 2009 991 1013
54. F. Tornabene, and E. Viola Free vibration analysis of functionally graded panels and shells of revolution Meccanica 44 2009 255 281
55. F. Tornabene, E. Viola, and D.J. Inman 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical and annular shell structures J Sound Vib 328 2009 259 290
56. E. Viola, and F. Tornabene Free vibrations of three parameter functionally graded parabolic panels of revolution Mech Res Commun 36 2009 587 594
57. A. Alibeigloo, and A.M. Kani 3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method Appl Math Model 34 2010 4123 4137
58. S.h. Hosseini-Hashemi, M. Fadaee, and M. Es'haghi A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates Int J Mech Sci 52 2010 1025 1035
59. B. Sobhani Aragh, and M.H. Yas Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panels Int J Mech Sci 52 2010 1047 1063
60. F. Tornabene, A. Marzani, E. Viola, and I. Elishakoff Critical flow speeds of pipes conveying fluid by the generalized differential quadrature method Adv Theor Appl Mech 3 2010 121 138
61. B.S. Aragh, and M.H. Yas Effect of continuously grading fiber orientation face sheets on vibration of sandwich panels with FGM core Int J Mech Sci 53 2011 628 638
62. M. Es'haghi, Sh. Hosseini-Hashemi, and M. Fadaee Vibration analysis of piezoelectric FGM sensors using an accurate method Int J Mech Sci 53 2011 585 594
63. Sh. Hosseimi-Hashemi, M. Fadaee, and S.R. Atashipour A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates Int J Mech Sci 53 2011 11 22
64. F. Tornabene Free Vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method Comput Methods Appl Mech Eng 200 2011 931 952
65. F. Tornabene 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution Compos Struct 93 2011 1854 1876
66. F. Tornabene Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler-Pasternak elastic foundations Compos Struct 94 2011 186 206
67. F. Tornabene, A. Liverani, and G. Caligiana FGM and laminated doubly-curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations Int J Mech Sci 53 2011 446 470
68. V.V. Zozulya, and Ch. Zhang A high order theory for functionally graded axisymmetric cylindrical shells Int J Mech Sci 60 2012 12 22
69. Sh. Hosseini-Hashemi, K. Khorshidi, M. Es'haghi, M. Fadaee, and M. Karimi On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates Appl Math Model 36 2012 1132 1147
70. C.H. Thai, L.V. Tran, D.T. Tran, T. Nguyen-Thoi, and H. Nguyen-Xuan Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method Appl Math Model 36 2012 5657 5677
71. F. Tornabene, A. Liverani, and G. Caligiana Laminated composite rectangular and annular plates: a GDQ solution for static analysis with a posteriori shear and normal stress recovery Compos Part B Eng 43 2012 1847 1872
72. F. Tornabene, A. Liverani, and G. Caligiana Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method Int J Mech Sci 61 2012 71 87
73. F. Tornabene, A. Liverani, and G. Caligiana General anisotropic doubly-curved shell theory: a Differential Quadrature Solution for free vibrations of shells and panels of revolution with a free-form meridian J Sound Vib 331 2012 4848 4869
74. E. Viola, L. Rossetti, and N. Fantuzzi Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery Compos Struct 94 2012 3736 3758
75. U. Yücel, and K. Boubaker Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm-Liouville problems Appl Math Model 36 2012 158 167
76. F. Tornabene Meccanica delle strutture a guscio in materiale composito Il metodo generalizzato di quadratura differenziale 2012 Esculapio Bologna 978-88-7488-527-5
77. A. Najafov, and A. Sofiyev The non-linear dynamics of FGM truncated conical shells surrounded by an elastic medium Int J Mech Sci 66 2013 33 44
78. S. Hosseini-Hashemi, M. Derakhshani, and M. Fadaee An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates Appl Math Model 37 2013 4147 4164
79. S. Tomasiello Stability and accuracy of DQ-based step-by-step integration methods for structural dynamics Appl Math Model 37 2013 3426 3435
80. E. Viola, F. Tornabene, and N. Fantuzzi General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels Compos Struct 95 2013 639 666
81. E. Viola, F. Tornabene, and N. Fantuzzi Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories Compos Struct 101 2013 59 93
82. F. Tornabene, and E. Viola Static analysis of functionally graded doubly-curved shells and panels of revolution Meccanica 48 2013 901 930
83. F. Tornabene, and A. Ceruti Free-form laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations: a 2-D GDQ solution for static and free vibration analysis World J Mech 3 2013 1 25
84. F. Tornabene, E. Viola, and N. Fantuzzi General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels Compos Struct 104 2013 94 117
85. F. Tornabene, and A. Ceruti Mixed static and dynamic optimization of four-parameter functionally graded completely doubly-curved and degenerate shells and panels using GDQ method Math Probl Eng 2013 2013 1 33 (Article ID 867089)
86. F. Tornabene, N. Fantuzzi, E. Viola, and A.J.M. Ferreira Radial basis function method applied to doubly-curved laminated composite shells and panels with a general higher-order equivalent single layer theory Compos Part B Eng 55 2013 642 659
87. A.J.M. Ferreira, E. Viola, F. Tornabene, N. Fantuzzi, and A.M. Zenkour Analysis of sandwich plates by generalized differential quadrature method Math Probl Eng 2013 2013 1 12 (Article ID 964367)
88. F. Tornabene, and J.N. Reddy FGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foundation: a GDQ solution for static analysis with a Posteriori Stress and strain recovery J Indian Inst Sci 93 4 2014 635 688
89. F. Tornabene, N. Fantuzzi, E. Viola, and E. Carrera Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method Compos Struct 107 1 2014 675 697
90. F. Tornabene, N. Fantuzzi, E. Viola, and J.N. Reddy Winkler-Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels Compos Part B Eng 57 2014 269 296
91. A.J.M. Ferreira, E. Carrera, M. Cinefra, E. Viola, F. Tornabene, N. Fantuzzi, and A.M. Zenkour et al. Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a unified formulation Compos Part B Eng 58 2014 544 552
92. C. Shu, Y.T. Chew, B.C. Khoo, and K.S. Yeo Application of GDQ scheme to simulate incompressible viscous flows around complex geometries Mech Res Commun 95 1995 319 325
93. K.M. Kiew, and J.-B. Han A four-node differential quadrature method for straight-sided quadrilateral Reissner/Mindlin plates Commun Numer Methods Eng 13 1997 73 81 (Pubitemid 127565949)
94. J.-B. Han, and K.M. Liew An eight-node curvilinear differential quadrature formulation for Reissner/Mindlin plates Comput Methods Appl Mech Eng 141 1997 265 280 (Pubitemid 127411043)
95. H. Zhong, and Y. He Solution of Poisson and Laplace equations by quadrilateral quadrature element Int J Solids Struct 35 1998 2805 2819 (Pubitemid 128368524)
96. X.W. Wang, Y.-L. Wang, and R.-B. Chen Static and free vibrational analysis of rectangular plates by the differential quadrature element method Commun Numer Methods Eng 14 1998 1133 1141 (Pubitemid 128500380)
97. F.L. Liu, and K.M. Liew Differential quadrature element method: a new approach for free vibration of polar Mindlin plates having discontinuities Comput Methods Appl Mech Eng 179 1999 407 423 (Pubitemid 30504374)
98. C.-N. Chen The development of irregular elements for differential quadrature element method steady-state heat conduction analysis Comput Methods Appl Mech Eng 170 1999 1 14
99. C.-N. Chen The differential quadrature element method irregular element torsion analysis model Appl Math Model 23 1999 309 328
100. F.-L. Liu Differential quadrature element method for static analysis of shear de-formable cross-ply laminates Int J Numer Methods Eng 46 1999 1203 1219
101. C.N. Chen Efficient and reliable solutions of static and dynamic nonlinear structural mechanics problems by an integrated numerical approach using DQFEM and direct time integration with accelerated equilibrium iteration schemes Appl Math Model 24 2000 637 655
102. C.-N. Chen A generalized differential quadrature element method Comput Methods Appl Mech Eng 188 2000 553 566
103. F.-L. Liu Static analysis of thick rectangular laminated plates: three-dimensional elasticity solutions via differential quadrature element method Int J Solids Struct 37 2000 7671 7688
104. C.N. Chen Buckling equilibrium equations of arbitrarily loaded nonprismatic composite beams and the DQEM buckling analysis using EDQ Appl Math Model 27 2003 27 46
105. C.-N. Chen DQEM and DQFDM for the analysis of composite two-dimensional elasticity problems Compos Struct 59 2003 3 13 (Pubitemid 35369036)
106. H. Zhong, and Y. He A note on incorporation of domain decomposition into the differential quadrature method Commun Numer Methods Eng 19 2003 297 306 (Pubitemid 36447559)
107. C.N. Chen DQEM and DQFDM irregular elements for analyses of 2-D heat conduction in orthotropic media Appl Math Model 28 2004 617 638
108. Y. Wang, X. Wang, and Y. Zhou Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method Int J Numer Methods Eng 59 2004 1207 1226 (Pubitemid 38335649)
109. Z. Zong, K.Y. Lam, and Y.Y. Zhang A multi-domain differential quadrature approach to plane elastic problems with material discontinuity Math Comput Model 41 2005 539 553 (Pubitemid 41097587)
110. Y. Xing, and B. Liu High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain Int J Numer Methods Eng 80 2009 1718 1742
111. Z. Zong, and Y. Zhang Advanced differential quadrature methods 2009 CRC Press Boca Raton
112. Y. Xing, B. Liu, and G. Liu A differential quadrature finite element method Int J Appl Mech 2 2010 1 20
113. H. Zhong, C. Pan, and H. Yu Buckling analysis of shear deformable plates using the quadrature element method Appl Math Model 35 2011 5059 5074
114. Fantuzzi N. Generalized differential quadrature finite element method applied to advanced structural mechanics (PhD thesis). University of Bologna, DICAM Department; 2013.
115. E. Viola, F. Tornabene, and N. Fantuzzi Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape Compos Struct 106 1 2013 815 834
116. E. Viola, F. Tornabene, E. Ferretti, and N. Fantuzzi Plane, soft core state structures under static loads using GDQFEM and cell method Comput Model Eng Sci 94 4 2013 301 329
117. E. Viola, F. Tornabene, E. Ferretti, and N. Fantuzzi On static analysis of composite plane state structures via GDQFEM and cell method Comput Model Eng Sci 94 5 2013 419 456
118. E. Viola, F. Tornabene, E. Ferretti, and N. Fantuzzi GDQFEM numerical simulations of continuous media with cracks and discontinuities Comput Model Eng Sci 94 4 2013 331 368
119. R. Courant, and D. Hilbert Methods of mathematical physics 1989 John Wiley New York