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Review Article

Weak Form Quadrature Element Method and Its Applications in Science and Engineering: A State-of-the-Art Review

[+] Author and Article Information
Xinwei Wang

State Key Laboratory of Mechanics and
Control of Mechanical Structures,
Nanjing University of
Aeronautics and Astronautics,
Nanjing 210016, China

Zhangxian Yuan

School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0150

Chunhua Jin

State Key Laboratory of Mechanics and
Control of Mechanical Structures,
Nanjing University of
Aeronautics and Astronautics,
Nanjing 210016, China;
School of Architecture Engineering,
Nantong University,
Nantong 224019, China

1Corresponding author.

Manuscript received January 5, 2017; final manuscript received April 24, 2017; published online May 16, 2017. Assoc. Editor: Martin Schanz.

Appl. Mech. Rev 69(3), 030801 (May 16, 2017) (19 pages) Paper No: AMR-17-1002; doi: 10.1115/1.4036634 History: Received January 05, 2017; Revised April 24, 2017

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.

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Figures

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Fig. 1

Sketch of an N-node bar element (N = 5)

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Fig. 2

Sketch of an N-node beam element (N = 5)

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Fig. 3

Sketch of an N × N-node rectangular plate element in bending (N = 7)

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Fig. 7

Sketch of a sandwich beam

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Fig. 5

Lagrange interpolation functions (N = 7): (a) equidistant nodes, (b) CGL nodes, (c) optimal (EC) node-1, (d) grid V nodes, (e) optimal node-2, and (f) GLL nodes

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Fig. 6

Hermite interpolation functions (N = 7): (a) equidistant nodes, (b) CGL nodes, (c) optimal (EC) node-1, (d) grid V nodes, (e) optimal node-2, and (f) GLL nodes

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Fig. 4

Sketch of an N × N-node curved plate element in bending (N = 7)

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Fig. 8

Sketch of an N-node sandwich beam element (N = 5)

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