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.

References

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
.
You do not currently have access to this content.