The weak form quadrature element method (QEM) combines the generality of the finite element method (FEM) with the accuracy of spectral techniques and thus has been projected by its proponents as a potential alternative to the conventional finite element method. The progression on the QEM and its applications is clear from past research, but this has been scattered over many papers. This paper presents a state-of-the-art review of the QEM employed to analyze a variety of problems in science and engineering, which should be of general interest to the community of the computational mechanics. The difference between the weak form quadrature element method (WQEM) and the time domain spectral element method (SEM) is clarified. The review is carried out with an emphasis to present static, buckling, free vibration, and dynamic analysis of structural members and structures by the QEM. A subroutine to compute abscissas and weights in Gauss–Lobatto–Legendre (GLL) quadrature is provided in the Appendix.
Issue Section:
Review Article
Keywords:
Computational mechanics
References
1.
Bert
, C. W.
, and Malik
, M.
, 1996
, “Differential Quadrature Method in Computational Mechanics: A Review
,” ASME Appl. Mech. Rev.
, 49
(1
), pp. 1
–28
.2.
Striz
, A. G.
, Chen
, W. L.
, and Bert
, C. W.
, 1994
, “Static Analysis of Structures by the Quadrature Element Method (QEM)
,” Int. J. Solids Struct.
, 31
(20
), pp. 2807
–2818
.3.
Zhong
, H.
, and He
, Y.
, 1998
, “Solution of Poisson and Laplace Equations by Quadrilateral Quadrature Element
,” Int. J. Solids Struct.
, 35
(21
), pp. 2805
–2819
.4.
Wang
, X.
, and Gu
, H. Z.
, 1997
, “Static Analysis of Frame Structures by the Differential Quadrature Element Method
,” Int. J. Numer. Methods Eng.
, 40
(4
), pp. 759
–772
.5.
Wang
, X.
, 2015
, Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications
, Butterworth-Heinemann
, Oxford, UK
.6.
Tornabene
, F.
, Fantuzzi
, N.
, Ubertini
, F.
, and Viola
, E.
, 2015
, “Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey
,” ASME Appl. Mech. Rev.
, 67
(2
), p. 020801
.7.
Striz
, A.
, Chen
, W.
, and Bert
, C.
, 1995
, “High-Accuracy Plane Stress and Plate Elements in the Quadrature Element Method
,” AIAA
Paper No. 95-1267.8.
Striz
, A. G.
, Chen
, W. L.
, and Bert
, C.
, 1997
, “Free Vibration of Plates by the High Accuracy Quadrature Element Method
,” J. Sound Vib.
, 202
(5
), pp. 689
–702
.9.
Chen
, W. L.
, Striz
, A. G.
, and Bert
, C. W.
, 2000
, “High-Accuracy Plane Stress and Plate Elements in the Quadrature Element Method
,” Int. J. Solids Struct.
, 37
(4
), pp. 627
–647
.10.
Zhong
, H.
, and Yu
, T.
, 2007
, “Flexural Vibration Analysis of an Eccentric Annular Mindlin Plate
,” Arch. Appl. Mech.
, 77
(4
), pp. 185
–195
.11.
Xing
, Y.
, and Liu
, B.
, 2009
, “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
(13
), pp. 1718
–1742
.12.
Zhong
, H.
, and Yue
, Z. G.
, 2012
, “Analysis of Thin Plates by the Weak Form Quadrature Element Method
,” Sci. China Phys. Mech.
, 55
(5
), pp. 861
–871
.13.
Jin
, C.
, Wang
, X.
, and Ge
, L.
, 2014
, “Novel Weak Form Quadrature Element Method With Expanded Chebyshev Nodes
,” Appl. Math. Lett.
, 34
, pp. 51
–59
.14.
Jin
, C.
, and Wang
, X.
, 2015
, “Accurate Free Vibration Analysis of Euler Functionally Graded Beams by the Weak Form Quadrature Element Method
,” Compos. Struct.
, 125
, pp. 41
–50
.15.
Franciosi
, C.
, and Tomasiello
, S.
, 2004
, “A Modified Quadrature Element Method to Perform Static Analysis of Structures
,” Int. J. Mech. Sci.
, 46
(6
), pp. 945
–959
.16.
Hu
, Y. C.
, Sze
, K. Y.
, and Zhou
, Y. X.
, 2015
, “Stabilized Plane and Axisymmetric Lobatto Finite Element Models
,” Comput. Mech.
, 56
(5
), pp. 879
–903
.17.
Brutman
, L.
, 1978
, “On the Lebesgue Function for Polynomial Interpolation
,” SIAM J. Numer. Anal.
, 15
(4
), pp. 694
–704
.18.
Boyd
, J. P.
, 1999
, “A Numerical Comparison of Seven Grids for Polynomial Interpolation on the Interval
,” Comput. Math. Appl.
, 38
(3–4), pp. 35
–50
.19.
Ibrahimoglu
, B. A.
, 2016
, “Lebesgue Functions and Lebesgue Constants in Polynomial Interpolation
,” J. Inequalities Appl.
, 93
, pp. 1
–15
.20.
Davis
, P. J.
, and Robinowitz
, P.
, 1975
, Methods of Numerical Integration
, Academic Press
, New York
.21.
Striz
, A. G.
, Wang
, X.
, and Bert
, C. W.
, 1995
, “Harmonic Differential Quadrature Method and Applications to Structural Components
,” Acta Mech.
, 111
(1
), pp. 85
–94
.22.
Wang
, X.
, and He
, B.
, 1995
, “An Explicit Formulation for Weighting Coefficients of Harmonic Differential Quadrature
,” J. Nanjing Univ. Aeronaut. Astronaut.
, 27
(4
), pp. 496
–501
(in Chinese).23.
Wang
, Y. L.
, 2001
, “Differential Quadrature Method and Differential Quadrature Element Method-Theory and Application
,” Ph.D. thesis
, Nanjing University of Aeronautics and Astronautics, Nanjing, China (in Chinese).24.
Yang
, T. Y.
, 1986
, Finite Element Structural Analysis
, Prentice-Hall
, Englewood Cliffs, NJ
.25.
Jin
, C.
, and Wang
, X.
, 2015
, “Weak Form Quadrature Element Method for Accurate Free Vibration Analysis of Thin Skew Plates
,” Comput. Math. Appl.
, 70
(8
), pp. 2074
–2086
.26.
Liu
, F.
, and Wang
, X.
, 2011
, “Modeling and Simulation of Lamb Wave Propagation in Composite Panels Based on the Spectral Element Method
,” Acta Mater. Compositae Sin.
, 28
(5
), pp. 174
–180
(in Chinese).27.
Xu
, C.
, and Wang
, X.
, 2011
, “Modeling of Lamb Wave Propagations in Composite Plates by the Spectral Element Method
,” Chin. Q. Mech.
, 32
(1
), pp. 10
–18
(in Chinese).28.
Xu
, C.
, and Wang
, X.
, 2012
, “Efficient Numerical Method for Dynamic Analysis of Flexible Rod Hit by Rigid Ball
,” Trans. Nanjing Univ. Aeronaut. Astronaut.
, 29
(4
), pp. 338
–344
.29.
Xu
, C.
, and Wang
, X.
, 2012
, “Efficient Modeling and Simulations of Lamb Wave Propagation in Thin Plates by Using a New Spectral Plate Element
,” J. Vibroengineering
, 14
(3
), pp. 1187
–1199
.30.
Wang
, X.
, Wang
, F.
, Xu
, C.
, and Ge
, L.
, 2012
, “New Spectral Plate Element for Simulating Lamb Wave Propagations in Plate Structures
,” J. Nanjing Univ. Aeronaut. Astronaut.
, 44
(5
), pp. 645
–651(in Chinese)
.31.
Wang
, F.
, Wang
, X.
, and Feng
, Z.
, 2013
, “Simulation of Wave Propagation in Plate Structures by Using New Spectral Element With Piezoelectric Coupling
,” J. Vibroengineering
, 15
(1
), pp. 214
–222
.32.
Feng
, Z.
, and Wang
, X.
, 2014
, “Spectral Element With Piezoelectric Patches and Its Applications in Simulation of Lamb Wave Propagation in Plates
,” China Mech. Eng.
, 25
(3
), pp. 377
–382
(in Chinese).33.
Ge
, L.
, Wang
, X.
, and Wang
, F.
, 2014
, “Accurate Modeling of PZT-Induced Lamb Wave Propagation in Structures by Using a Novel Spectral Finite Element Method
,” Smart Mater. Struct.
, 23
(9
), p. 095018
.34.
Ge
, L.
, Wang
, X.
, and Jin
, C.
, 2014
, “Numerical Modeling of PZT-Induced Lamb Wave-Based Crack Detection in Plate-Like Structures
,” Wave Motion
, 51
(6
), pp. 867
–885
.35.
Patera
, A. T.
, 1984
, “A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion
,” J. Comput. Phys.
, 54
(3
), pp. 468
–488
.36.
Seriani
, G.
, and Priolo
, E.
, 1994
, “Spectral Element Method for Acoustic Wave Simulation in Heterogeneous Media
,” Finite Elem. Anal. Des.
, 16
(3–4
), pp. 337
–348
.37.
Padovani
, E.
, Priolo
, E.
, and Seriani
, G.
, 1994
, “Low- and High-Order Finite Element Method: Experience in Seismic Modeling
,” J. Comput. Acoust.
, 2
(04
), pp. 371
–422
.38.
Seriani
, G.
, and Oliveira
, S. P.
, 2008
, “Dispersion Analysis of Spectral Element Methods for Elastic Wave Propagation
,” Wave Motion
, 45
(6
), pp. 729
–744
.39.
Dauksher
, W.
, and Emery
, A. F.
, 1997
, “Accuracy in Modeling the Acoustic Wave Equation With Chebyshev Spectral Finite Elements
,” Finite Elem. Anal. Des.
, 26
(1
), pp. 115
–128
.40.
van de Vosse
, F. N.
, and Minev
, P. D.
, 1996
, Spectral Element Methods: Theory and Applications
(EUT Report W, Department of Mechanical Engineering; Vol. 96-W-001), Eindhoven University of Technology, Eindhoven
, The Netherlands
.41.
Choi
, J.
, and Inman
, D. J.
, 2014
, “Spectral Element Method for Cable Harnessed Structure
,” Topics in Modal Analysis
(Conference Proceedings of the Society for Experimental Mechanics Series), Vol. 7
, Springer
, New York
, pp. 377
–387
.42.
Brito
, K. D.
, and Sprague
, M. A.
, 2012
, “Reissner–Mindlin Legendre Spectral Finite Elements With Mixed Reduced Quadrature
,” Finite Elem. Anal. Des.
, 58
, pp. 74
–83
.43.
Komatitsch
, D.
, Vilotte
, J. P.
, Vai
, R.
, Castillo-Covarrubias
, J. M.
, and Sanchez-Sesma
, F. J.
, 1999
, “The Spectral Element Method for Elastic Wave Equations-Application to 2-D and 3-D Seismic Problems
,” Int. J. Numer. Methods Eng.
, 45
(9
), pp. 1139
–1164
.44.
Komatitsch
, D.
, Martin
, R.
, Tromp
, J.
, Taylor
, M. A.
, and Wingate
, B. A.
, 2001
, “Wave Propagation in 2-D Elastic Media Using a Spectral Element Method With Triangles and Quadrangles
,” J. Comput. Acoust.
, 9
(2
), pp. 703
–718
.45.
Kudela
, P.
, Krawczuk
, M.
, and Ostachowicz
, W.
, 2007
, “Wave Propagation Modeling in 1D Structure Using Spectral Finite Elements
,” J. Sound Vib.
, 300
, pp. 88
–100
.46.
Kudela
, P.
, Zak
, A.
, Krawczuk
, M.
, and Ostachowicz
, W.
, 2007
, “Modelling of Wave Propagation in Composite Plates Using the Time Domain Spectral Element Method
,” J. Sound Vib.
, 302
, pp. 728
–745
.47.
Kim
, Y.
, Ha
, S.
, and Chang
, F. K.
, 2008
, “Time-Domain Spectral Element Method for Built-In Piezoelectric-Actuator-Induced Lamb Wave Propagation Analysis
,” AIAA J.
, 46
(3
), pp. 591
–600
.48.
Ha
, S.
, and Chang
, F. K.
, 2010
, “Optimizing a Spectral Element for Modeling PZT-Induced Lamb Wave Propagation in Thin Plates
,” Smart Mater. Struct.
, 19
(1
), p. 015015
.49.
Witkowski
, W.
, Rucka
, M.
, ChróŚcielewski
, J.
, and Wilde
, K.
, 2012
, “On Some Properties of 2D Spectral Finite Elements in Problems of Wave Propagation
,” Finite Elem. Anal. Des.
, 55
, pp. 31
–41
.50.
Li
, Y.
, and Li
, X.
, 2016
, “The Chebyshev Spectral Element Approximation With Exact Quadratures
,” J. Comput. Appl. Math.
, 296
, pp. 320
–333
.51.
Dauksher
, W.
, and Emery
, A. F.
, 2000
, “The Solution of Elastostatic and Elastodynamic Problems With Chebyshev Spectral Finite Elements
,” Comput. Methods Appl. Mech. Eng.
, 188
(1–3), pp. 217
–233
.52.
Zak
, A.
, 2009
, “A Novel Formulation of a Spectral Plate Element for Wave Propagation in Isotropic Structures
,” Finite Elem. Anal. Des.
, 45
(10
), pp. 650
–658
.53.
Phan
, C. N.
, Frostig
, Y.
, and Kardomateas
, G. A.
, 2012
, “Analysis of Sandwich Beams With a Compliant Core and With In-Plane Rigidity—Extended High-Order Sandwich Panel Theory Versus Elasticity
,” ASME J. Appl. Mech.
, 79
(4
), p. 041001
.54.
Wang
, Y.
, and Wang
, X.
, 2014
, “Static Analysis of Higher Order Sandwich Beams by Weak Form Quadrature Element Method
,” Compos. Struct.
, 116
, pp. 841
–848
.55.
Yuan
, Z.
, Kardomateas
, G. A.
, and Frostig
, Y.
, 2015
, “Finite Element Formulation Based on the Extended High-Order Sandwich Panel Theory
,” AIAA J.
, 53
(10
), pp. 3006
–3015
.56.
Yuan
, Z.
, Kardomateas
, G. A.
, and Frostig
, Y.
, 2016
, “Geometric Nonlinearity Effects in the Response of Sandwich Wide Panels
,” ASME J. Appl. Mech.
, 83
(9
), p. 091008
.57.
Jones
, R. M.
, 1975
, Mechanics of Composite Materials
, Hemisphere Publishing
, New York
.58.
Ashton
, J. E.
, 1970
, “Anisotropic Plate Analysis—Boundary Conditions
,” J. Compos. Mater.
, 4
(2
), pp. 162
–171
.59.
Whitney
, J. M.
, 1987
, Structural Analysis of Laminated Plates
, Technomic Publishing
, Lancaster, PA
.60.
Leissa
, A. W.
, 1973
, “The Free Vibration of Rectangular Plates
,” J. Sound Vib.
, 31
(3
), pp. 257
–293
.61.
Bogner
, F. K.
, Fox
, R. L.
, and Schmit
, L. A.
, 1965
, “The Generation of Inter-Element-Compatible Stiffness and Mass Matrices by the Use of Interpolation Formulas
,” Wright-Patterson Air Force Base
, Dayton, OH, pp. 397
–443
, Report No. AFFDL TR-66-80
.62.
Zhong
, H.
, and Gao
, M.
, 2007
, “Transverse Vibration Analysis of an Arbitrarily-Shaped Membrane by the Weak-Form Quadrature Element Method
,” Computational Mechanics: ISCM
, Z. Yao and M. Yuan, eds., Tsinghua University Press and Springer, Beijing, China, July 30–Aug. 1, pp. 1009
–1018
.63.
Mo
, Y.
, Ou
, L.
, and Zhong
, H.
, 2009
, “Vibration Analysis of Timoshenko Beams on a Nonlinear Elastic Foundation
,” Tsinghua Sci. Technol.
, 14
(3
), pp. 322
–326
.64.
Zhong
, H.
, and Yu
, T.
, 2009
, “A Weak Form Quadrature Element Method for Plane Elasticity Problems
,” Appl. Math. Modell.
, 33
(10
), pp. 3801
–3814
.65.
Xing
, Y.
, Liu
, B.
, and Liu
, G.
, 2010
, “A Differential Quadrature Finite Element Method
,” Int. J. Appl. Mech.
, 2
(1
), pp. 1
–20
.66.
Zhong
, H.
, and Gao
, M.
, 2010
, “Quadrature Element Analysis of Planar Frameworks
,” Arch. Appl. Mech.
, 80
(12
), pp. 1391
–1405
.67.
Gautschi
, W.
, 1991
, “Gauss-Radau and Gauss-Lobatto Quadratures With Double End Points
,” J. Comput. Appl. Math.
, 34
(3
), pp. 343
–360
.68.
Zhong
, H.
, and Wang
, Y.
, 2010
, “Weak Form Quadrature Element Analysis of Bickford Beams
,” Eur. J. Mech. A: Solids
, 29
(5
), pp. 851
–858
.69.
Xia
, F.
, Wu
, X.
, and Li
, H.
, 2010
, “Elastoplastic Analysis of Timoshenko Beam Based on Weakform Quadrature Element Method
,” J. Xinyang Norm. Univ. (Nat. Sci.)
, 23
(4
), pp. 527
–536
(in Chinese).70.
He
, R.
, Zhang
, R.
, and Zhong
, H.
, 2011
, “An Efficient Quadrature Beam Model to Simulate Inelastic Seismic Behavior of Steel Frames
,” III ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering
, M.
Papadrakakis
, M.
Fragiadakis
, and V.
Plevris
, eds., Corfu, Greece, May 25–28, Paper No. 166.71.
Zhong
, H.
, Pan
, C.
, and Yu
, H.
, 2011
, “Buckling Analysis of Shear Deformable Plates Using the Quadrature Element Method
,” Appl. Math. Modell.
, 35
(10
), pp. 5059
–5074
.72.
Zhong
, H.
, Zhang
, R.
, and Yu
, H.
, 2011
, “Buckling Analysis of Planar Frameworks Using the Quadrature Element Method
,” Int. J. Struct. Stab. Dyn.
, 11
(2
), pp. 363
–378
.73.
Wang
, X.
, Feng
, T.
, and Li
, D.
, 1983
, “Finite Element Analysis of Laminated Composite Plates
,” J. Nanjing Aeronaut. Inst.
, 15
(3
), pp. 66
–83
(in Chinese).74.
Adini
, A.
, and Clough
, R. W.
, 1960
, “Analysis of Plate Bending by the Finite Element Method
,” Report Submitted to National Science Foundation Grant G-7337, University of California, Berkeley, CA.75.
Shen
, Z.
, and Zhong
, H.
, 2012
, “Static and Vibrational Analysis of Partially Composite Beams Using the Weak-Form Quadrature Element Method
,” Math. Probl. Eng.
, 2012
, p. 974023
.76.
Xiao
, N.
, and Zhong
, H.
, 2012
, “Non-Linear Quadrature Element Analysis of Planar Frames Based on Geometrically Exact Beam Theory
,” Int. J. Non-Linear Mech.
, 47
(5
), pp. 481
–488
.77.
He
, R.
, and Zhong
, H.
, 2012
, “Large Deflection Elasto-Plastic Analysis of Frames Using the Weak-Form Quadrature Element Method
,” Finite Elem. Anal. Des.
, 50
, pp. 125
–133
.78.
Shen
, Z.
, and Zhong
, H.
, 2013
, “Long-Term Quadrature Element Analysis of Steel-Concrete Composite Beams With Partial Interactions and Shear-Lag Effects
,” J. Tsinghua Univ. (Sci. Technol.)
, 53
(4
), pp. 493
–498
(in Chinese).79.
Yu
, H.
, 2013
, “Application of Quadrature Element Method to Fire Analysis of Planar Steel Frames
,” Eng. Mech.
, 30
(6
), pp. 191
–196
(in Chinese).80.
Shen
, Z.
, and Zhong
, H.
, 2013
, “Geometrically Nonlinear Quadrature Element Analysis of Composite Beams With Partial Interaction
,” Eng. Mech.
, 30
(3
), pp. 270
–275
(in Chinese).81.
Zhang
, R.
, and Zhong
, H.
, 2013
, “Weak Form Quadrature Element Analysis of Planar Slender Beams Based on Geometrically Exact Beam Theory
,” Arch. Appl. Mech.
, 83
(9
), pp. 1309
–1325
.82.
Bathe
, K. J.
, 1982
, Finite Element Procedures in Engineering Analysis
, Prentice-Hall
, Englewood Cliffs, NJ
.83.
Jin
, C.
, and Wang
, X.
, 2014
, “Dynamic Analysis of Functionally Graded Material Bars by Using Novel Weak Form Quadrature Element Method
,” J. Vibroengineering
, 16
(6
), pp. 2790
–2799
.84.
Liu
, B.
, and Xing
, Y.
, 2014
, “Thickness-Shear Vibration Analysis of Rectangular Quartz Plates by a Differential Quadrature Finite Element Method
,” AIP Conf. Proc.
, 1618
(1
), pp. 41
–44
.85.
Zhang
, R.
, and Zhong
, H.
, 2014
, “Weak Form Quadrature Element Analysis of Spatial Geometrically Exact Shear-Rigid Beams
,” Finite Elem. Anal. Des.
, 87
, pp. 22
–31
.86.
Zhong
, H.
, Zhang
, R.
, and Xiao
, N.
, 2014
, “A Quaternion-Based Weak Form Quadrature Element Formulation for Spatial Geometrically Exact Beams
,” Arch. Appl. Mech.
, 84
(12
), pp. 1825
–1840
.87.
Yue
, Z.
, and Zhong
, H.
, 2014
, “Geometrical Nonlinear Quadrature Element Analysis on Thin Plate
,” Comput.-Aided Eng.
, 23
(2
), pp. 37
–40
(in Chinese).88.
Li
, Y.
, and Zhong
, H.
, 2014
, “Seismic Analysis of Steel-Concrete Bridges With Weak Form Quadrature Beam Element
,” Earthquake Eng. Eng. Dyn.
, 34
(Suppl
.), pp. 302
–306
(in Chinese).89.
Yuan
, S.
, and Zhong
, H.
, 2014
, “Consolidation Analysis of Non-Homogeneous Soil by the Weak Form Quadrature Element Method
,” Comput. Geotech.
, 62
, pp. 1
–10
.90.
Yuan
, S.
, and Zhong
, H.
, 2015
, “Seepage Analysis Using the Weak Form Quadrature Element Method
,” Chin. J. Geotech. Eng.
, 37
(2
), pp. 257
–262
(in Chinese).91.
Guan
, Y.
, and Zhong
, H.
, 2015
, “Weak Form Quadrature Element Analysis of Elastostatic and Free Vibration Problems of Toroidal Shells
,” Mech. Eng.
, 37
(3
), pp. 338
–343
.92.
Zhang
, R.
, and Zhong
, H.
, 2015
, “Weak Form Quadrature Element Analysis of Geometrically Exact Shells
,” Int. J. Non-Linear Mech.
, 71
, pp. 63
–71
.93.
Liao
, M.
, Tang
, A.
, and Hu
, Y. G.
, 2015
, “Calculation of Mode III Stress Intensity Factors by the Weak-Form Quadrature Element Method
,” Arch. Appl. Mech.
, 85
(11
), pp. 1595
–1605
.94.
Liao
, M.
, Tang
, A.
, Hu
, Y.-G.
, and Guo
, Z.
, 2015
, “Computation of Coefficients of Crack-Tip Asymptotic Fields Using the Weak Form Quadrature Element Method
,” J. Eng. Mech.
, 141
(8
), p. 04015018
.95.
Liao
, M.
, and Zhong
, H.
, 2016
, “Application of a Weak Form Quadrature Element Method to Nonlinear Free Vibrations of Thin Rectangular Plates
,” Int. J. Struct. Stab. Dyn.
, 16
(1
), p. 1640001
.96.
Liu
, X.
, Shi
, Z.
, Mo
, Y. L.
, and Cheng
, Z.
, 2016
, “Effect of Initial Stress on Attenuation Zones of Layered Periodic Foundations
,” Eng. Struct.
, 121
, pp. 75
–84
.97.
Yuan
, S.
, and Zhong
, H.
, 2016
, “Analysis of Unbounded Domain Problems by the Weak Form Quadrature Element Method
,” Rock Soil Mech.
, 37
(4
), pp. 1187
–1194
(in Chinese).98.
Yuan
, S.
, and Zhong
, H.
, 2016
, “A Weak Form Quadrature Element Formulation for Coupled Analysis of Unsaturated Soils
,” Comput. Geotech.
, 76
, pp. 1
–11
.99.
Yuan
, S.
, and Zhong
, H.
, 2016
, “Three Dimensional Analysis of Unconfined Seepage in Earth Dams by the Weak Form Quadrature Element Method
,” J. Hydrol.
, 533
, pp. 403
–411
.100.
Simo
, J. C.
, Tarnow
, N.
, and Doblare
, M.
, 1995
, “Non-Linear Dynamics of Three-Dimensional Rods: Exact Energy and Momentum Conserving Algorithms
,” Int. J. Numer. Methods Eng.
, 38
(9
), pp. 1431
–1473
.101.
Zhang
, R.
, and Zhong
, H.
, 2016
, “A Quadrature Element Formulation of an Energy–Momentum Conserving Algorithm for Dynamic Analysis of Geometrically Exact Beams
,” Comput. Struct.
, 165
, pp. 96
–106
.102.
Liu
, B.
, Ferreira
, A. J. M.
, Xing
, Y. F.
, and Neves
, A. M. A.
, 2016
, “Analysis of Composite Plates Using a Layerwise Theory and a Differential Quadrature Finite Element Method
,” Compos. Struct.
, 156
, pp. 393
–398
.103.
Liu
, B.
, Ferreira
, A. J. M.
, Xing
, Y. F.
, and Neves
, A. M. A.
, 2016
, “Analysis of Functionally Graded Sandwich and Laminated Shells Using a Layerwise Theory and a Differential Quadrature Finite Element Method
,” Compos. Struct.
, 136
, pp. 546
–553
.104.
Liao
, M.
, and Zhong
, H.
, 2016
, “A Weak Form Quadrature Element Method for Nonlinear Free Vibrations of Timoshenko Beams
,” Eng. Comput.
, 33
(1
), pp. 274
–287
.105.
Liu
, X.
, Shi
, Z.
, and Mo
, Y. L.
, 2016
, “Effect of Initial Stress on Periodic Timoshenko Beams Resting on an Elastic Foundation
,” J. Vib. Control
, epub.106.
Wang
, Y.
, and Wang
, X.
, 2016
, “Free Vibration Analysis of Soft-Core Sandwich Beams by the Novel Weak Form Quadrature Element Method
,” J. Sandwich Struct. Mater.
, 18
(3
), pp. 294
–320
.107.
Wang
, X.
, and Wang
, Y.
, 2016
, “Static Analysis of Sandwich Panels With Non-Homogeneous Soft-Cores by Novel Weak Form Quadrature Element Method
,” Compos. Struct.
, 146
, pp. 207
–220
.108.
Tornabene
, F.
, Liverani
, A.
, and Caligiana
, G.
, 2012
, “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
(4
), pp. 1847
–1872
.109.
Liang
, X.
, Wang
, X.
, and Wang
, Y.
, 2016
, “Dynamic Response of Soft Core Sandwich Beams Under a Moving Point Load
,” J. Nanjing Univ. Aeronaut. Astronaut.
, 48
(4
), pp. 544
–550
(in Chinese).110.
Wang
, X.
, Liang
, X.
, and Jin
, C.
, 2017
, “Accurate Dynamic Analysis of Functionally Graded Beams Under a Moving Point Load
,” Mech. Based Des. Struct.
, 45
(1
), pp. 76
–91
.111.
Liu
, C.
, Liu
, B.
, Zhao
, L.
, Xing
, Y.
, Ma
, C.
, and Li
, H.
, 2016
, “A Differential Quadrature Hierarchical Finite Element Method and Its Applications to Vibration and Bending of Mindlin Plates With Curvilinear Domains
,” Int. J. Numer. Methods Eng.
, 109
(2
), pp. 174
–197
.112.
Wang
, X.
, and Yuan
, Z.
, 2017
, “Weak Form Quadrature Element Analysis of Sandwich Panels With Functionally Graded Soft-Cores
,” Compos. Struct.
, 159
, pp. 157
–173
.113.
Jin
, C.
, and Wang
, X.
, 2017
, “Quadrature Element Method for Vibration Analysis of Functionally Graded Beams
,” Eng. Comput.
, 34
(2), epub.114.
Wang
, X.
, and Liang
, X.
, 2017
, “Free Vibration of Soft-Core Sandwich Panels With General Boundary Conditions by Harmonic Quadrature Element Method
,” Thin-Walled Struct.
, 113
, pp. 253
–261
.115.
Wang
, X.
, and Yuan
, Z.
, 2016
, “Harmonic Differential Quadrature Analysis of Soft-Core Sandwich Panels Under Locally Distributed Loads
,” Appl. Sci.
, 6
(11
), p. 361
.116.
Jin
, C.
, and Wang
, X.
, 2017
, “Accurate Free Vibration of Functionally Graded Skew Plates
,” Trans. Nanjing Univ. Aeronaut. Astronaut.
, 34
(2
), pp. 188
–194
.117.
Guo
, M.
, and Zhong
, H.
, 2016
, “Weak Form Quadrature Solution of 2mth-Order Fredholm Integro-Differential Equations
,” Int. J. Comput. Math.
, 93
(10
), pp. 1650
–1664
.118.
Xing
, Y.
, Liu
, B.
, and Xu
, T.
, 2013
, “Exact Solutions for Free Vibration of Circular Cylindrical Shells With Classical Boundary Conditions
,” Int. J. Mech. Sci.
, 75
, pp. 178
–188
.119.
Liu
, B.
, Xing
, Y.
, Wang
, W.
, and Yu
, W.
, 2015
, “Thickness-Shear Vibration Analysis of Circular Quartz Crystal Plates by a Differential Quadrature Hierarchical Finite Element Method
,” Compos. Struct.
, 131
, pp. 1073
–1080
.120.
Liu
, C.
, Liu
, B.
, Kang
, T.
, and Xing
, Y.
, 2016
, “Micro/Macro-Mechanical Analysis of the Interface of Composite Structures by a Differential Quadrature Hierarchical Finite Element Method
,” Compos. Struct.
, 154
, pp. 39
–48
.121.
Liu
, C.
, Liu
, B.
, Xing
, Y.
, Reddy
, J. N.
, Neves
, A. M. A.
, and Ferreira
, A. J. M.
, 2017
, “In-Plane Vibration Analysis of Plates in Curvilinear Domains by a Differential Quadrature Hierarchical Finite Element Method
,” Meccanica
, 52
(4
), pp. 1017
–1033
.122.
Liu
, B.
, Zhao
, L.
, Ferreira
, A. J. M.
, Xing
, Y.
, Neves
, A. M. A.
, and Wang
, J.
, 2017
, “Analysis of Viscoelastic Sandwich Laminates Using a Unified Formulation and a Differential Quadrature Hierarchical Finite Element Method
,” Compos. Part B: Eng.
, 110
, pp. 185
–192
.123.
Xing
, Y.
, Qin
, M.
, and Guo
, J.
, 2017
, “A Time Finite Element Method Based on the Differential Quadrature Rule and Hamilton's Variational Principle
,” Appl. Sci.
, 7
(2
), p. 138
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