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Review Article

Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey

[+] Author and Article Information
Francesco Tornabene

DICAM Department,
University of Bologna,
Viale del Risorgimento 2,
Bologna 40136, Italy
e-mail: francesco.tornabene@unibo.it

Nicholas Fantuzzi

DICAM Department,
University of Bologna,
Viale del Risorgimento 2,
Bologna 40136, Italy
e-mail: nicholas.fantuzzi@unibo.it

Francesco Ubertini

DICAM Department,
University of Bologna,
Viale del Risorgimento 2,
Bologna 40136, Italy
e-mail: francesco.ubertini@unibo.it

Erasmo Viola

DICAM Department,
University of Bologna,
Viale del Risorgimento 2,
Bologna 40136, Italy
e-mail: erasmo.viola@unibo.it

1Corresponding author. ASME 2012 5th Annual Dynamic Systems and Control Conference joint with the JSME 2012 11th Motion and Vibration Conference.

Manuscript received April 29, 2014; final manuscript received September 30, 2014; published online January 15, 2015. Editor: Harry Dankowicz.

Appl. Mech. Rev 67(2), 020801 (Mar 01, 2015) (55 pages) Paper No: AMR-14-1041; doi: 10.1115/1.4028859 History: Received April 29, 2014; Revised September 30, 2014; Online January 15, 2015

A survey of several methods under the heading of strong formulation finite element method (SFEM) is presented. These approaches are distinguished from classical one, termed weak formulation finite element method (WFEM). The main advantage of the SFEM is that it uses differential quadrature method (DQM) for the discretization of the equations and the mapping technique for the coordinate transformation from the Cartesian to the computational domain. Moreover, the element connectivity is performed by using kinematic and static conditions, so that displacements and stresses are continuous across the element boundaries. Numerical investigations integrate this survey by giving details on the subject.

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Figures

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Fig. 1

General engineering problem: (a) physical problem geometry, (b) discretization for numerical simulation, (c) two adjacent regular elements, (d) regular discretization of a rectangular domain, (e) general discretization of an arbitrarily shaped domain, and (f) a single element computational scheme

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Fig. 2

Mapping of a linear 4 node element: (a) physical domain, (b) computational domain. Mapping of a quadratic 8 node element: (c) physical domain, (d) computational domain. Mapping of a cubic 12 node element: (e) physical domain and (f) computational domain.

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Fig. 3

(a) Internal and external boundary conditions for element edges and corners. (b) Outward unit normal vectors definition for a generic subdivision.

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Fig. 4

Global stiffness (a) and mass (b) matrices for a sample mesh made of two adjacent elements

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Fig. 5

Static analysis for single element structural components varying the number of points N. Effect of the basis function choice using Che–Gau–Lob distribution (except for Jac DQ basis and SM): (a) a CC rod, (b) a SS EB beam without δ-points, (c) a SS EB beam with δ-points (1/2), and (d) a CC Tim beam. Effect of the grid distribution choice using PDQ basis functions: (e) a CC rod, (f) a SS EB beam without δ-points, (g) a SS EB beam with δ-points (1/2), and (h) a CC Tim beam.

Grahic Jump Location
Fig. 6

Static analysis for structural components varying the number of elements ne with N = 7 for each ones. Effect of the basis function choice using Che–Gau–Lob distribution (except for Jac DQ basis and SM): (a) a CC rod, (b) a SS EB beam without δ-points, (c) a SS EB beam with δ-points (10-5), (d) a CC Tim beam. Effect of the grid distribution choice using PDQ basis functions: (e) a CC rod, (f) a SS EB beam without δ-points, (g) a SS EB beam with δ-points (1/2), and (h) a CC Tim beam.

Grahic Jump Location
Fig. 7

Static analysis for structural components varying the number of elements ne and using PDQ basis functions and Leg–Gau distribution. Effect of the grid point number N inside each element: (a) a CC rod, (b) a SS EB beam without δ-points, (c) a SS EB beam with δ-points (1/2), and (d) a CC Tim beam.

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Fig. 8

Static analysis for a SS EB beam. Effect of the grid distribution choice: (a) a single element beam varying the grid point number N using GDQR method, (b) a beam structure varying the number of elements ne with N = 7 for each ones using GDQR method, (c) a single element beam varying the grid point number N using κ technique, (d) a beam structure varying the number of elements ne with N = 7 for each ones using κ technique. Effect of the basis function choice using Che–Gau–Lob distribution (except for Jac DQ basis and SM) and κ technique: (e) a single element beam varying the grid point number N, and (f) a beam structure varying the number of elements ne with N=7 for each ones.

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Fig. 9

Static analysis for structural components using weak formulation (WFEM). Effect of the basis function choice using Leg–Gau–Lob distribution with N = 7 for each element: (a) a CC rod, (b) a CC Tim beam. Effect of the grid distribution choice using PDQ basis functions with N = 7 for each element: (c) a CC rod, (d) a CC Tim beam. Effect of the grid point number N inside each element using PDQ basis functions and Leg–Gau–Lob distribution: (e) a CC rod, (f) a CC Tim beam. Absolute error of a SS EB beam: (g) effect of the grid point distribution on a single element using PDQ basis functions and δ-point technique (1/2), and (h) comparison between SFEM and WFEM using PDQ basis functions and Leg–Gau–Lob distribution.

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Fig. 10

Dynamic discrete spectra for single element structural components with N = 151. Effect of the basis function choice using Che–Gau–Lob distribution (except for Jac DQ basis and SM): (a) a CC rod, (b) a SS EB beam without δ-points, (c) a SS EB beam with δ-points (1/2), (d) a SS Tim beam. Effect of the grid distribution choice using PDQ basis functions: (e) a CC rod, (f) a SS EB beam without δ-points, (g) a SS EB beam with δ-points (1/2), and (h) a SS Tim beam.

Grahic Jump Location
Fig. 11

Dynamic discrete spectra for various structural components with N = 7 and ne = 100. Effect of the basis function choice using Leg–Gau distribution (except for Jac DQ basis and SM): (a) a CC rod, (b) a SS EB beam without δ-points, (c) a SS EB beam with δ-points (1/2), (d) a SS Tim beam. Effect of the grid distribution choice using PDQ basis functions: (e) a CC rod, (f) a SS EB beam without δ-points, g) a SS EB beam with δ-points (1/2), and (h) a SS Tim beam.

Grahic Jump Location
Fig. 12

Relative error of the first frequency for various structural components with N = 7. Effect of the grid distribution choice using PDQ basis functions: (a) a CC rod, (b) a SS EB beam without δ-points, (c) a SS EB beam with δ-points (1/2), (d) a SS Tim beam. Effect of the basis function choice using Leg–Gau distribution (except for Jac DQ basis and SM): (e) a CC rod, (f) a SS EB beam without δ-points, (g) a SS EB beam with δ-points (1/2), and (h) a SS Tim beam.

Grahic Jump Location
Fig. 13

Relative error of the first three frequencies for several structural components varying the number of grid points N and the number of elements ne and using PDQ basis functions and Leg–Gau distribution: (a) a CC rod, (b) a SS EB beam without δ-points, (c) a SS EB beam with δ-points (1/2), and (d) a SS Tim beam

Grahic Jump Location
Fig. 14

Dynamic discrete spectra for a SS EB beam. GDQR method: (a) effect of the grid distribution choice with N = 151 and ne = 1, (b) effect of the grid distribution choice with N = 7 and ne = 100, (c) effect of the grid point number N considering ne = 100 and Leg–Gau distribution, (d) effect of the number of element ne considering N = 13 and Leg–Gau distribution. κ technique: (e) effect of the basis function choice using Leg–Gau distribution with N = 151 and ne = 1, (f) effect of the grid distribution choice using PDQ basis functions with N = 151 and ne = 1, (g) effect of the basis function choice using Leg–Gau distribution with N = 7 and ne = 100, and (h) effect of the grid distribution choice using PDQ basis functions with N = 7 and ne = 100.

Grahic Jump Location
Fig. 15

Dynamic discrete spectra for several structural components using weak formulation (WFEM). Effect of the basis function choice using Leg–Gau–Lob distribution with N = 7 and ne = 100: (a) a CC rod, (b) a SS Tim beam. Effect of the grid distribution choice using PDQ basis functions with N = 7 and ne = 100: (c) a CC rod, (d) a SS Tim beam. (e) Effect of the basis function choice for a SS EB beam single element with N = 151 and Leg–Gau–Lob distribution without δ-points. Effect of the grid distribution choice for a SS EB beam single element with N = 151: (f) PDQ basis functions without δ-points, (g) GDQR method. (h) Comparison between SFEM and WFEM using PDQ basis functions with N = 11 and ne = 20 for a SS EB beam.

Grahic Jump Location
Fig. 16

Comparison between SFEM with Leg–Gau distribution and WFEM with Leg–Gau–Lob distribution using PDQ basis functions. Dynamic discrete spectra using N = 7 and ne = 100: (a) a CC rod, (b) a SS Tim beam. Effect of the grid point number N using ne = 100: (c) a CC rod, (d) a SS Tim beam. Effect of the number of elements ne with N = 13 for each ones: (e) a CC rod, (f) a SS Tim beam. Relative error of the first three frequencies varying the number of elements ne with N = 7 for each ones: (g) a CC rod, and (h) a SS Tim beam.

Grahic Jump Location
Fig. 17

Dynamic discrete spectra for several structural components. (a) Comparison between SFEM and WFEM using PDQ basis functions with N = 7 and ne = 100: a CC rod with Leg–Gau–Lob, a SS EB beam using Quad distribution with δ-points (1/2) for SFEM and using Leg–Gau–Lob distribution with δ-points (10-5) for WFEM, a SS Tim beam with Leg–Gau–Lob. (b) LGDQ method with PDQ basis functions with Unif grid distribution, N = 1000 and ne = 1 using Ni = 15: a CC rod, a SS EB beam without δ-points, with δ-points (1/2) and with κ technique, a SS Tim beam. (c) Gaussian DQ (or RBF) method with N = 7 and ne = 100: a CC rod using Leg–Gau–Lob distribution, a SS EB beam using Leg–Gau with δ-points (1/2), a SS Tim beam using Leg–Gau distribution with SFEM and using Leg–Gau–Lob with WFEM.

Grahic Jump Location
Fig. 18

Dynamic discrete spectra for several structural components. Effect of the stretching parameter α using PDQ basis functions, Unif grid point discretization, N = 7 and ne = 100: (a) a CC rod, (b) a SS EB beam with δ-delta (1/2), (c) a SS EB beam with GDQR, and (d) a SS Tim beam.

Grahic Jump Location
Fig. 19

Comparison among SFEM with Leg–Gau distribution, WFEM and SEM with Leg–Gau–Lob using PDQ basis functions and FEM with linear and quadratic shape functions. (a) Effect of the grid point number inside each element for the static analysis of a CC rod. (b) Relative error of the first frequency varying the number of elements ne for a CC rod. (c) Dynamic discrete spectra with ne = 100 for a CC rod.

Grahic Jump Location
Fig. 20

Static analysis for single element structural components varying the number of points N. Effect of the basis function choice using Che–Gau–Lob distribution (except for Jac DQ basis and SM): (a) a membrane, (b) a KL plate without δ-points, (c) a KL plate with δ-points (10-5), (d) a RM plate. Effect of the grid distribution choice using PDQ basis functions: (e) a membrane, (f) a KL plate without δ-points, (g) a KL plate with δ-points (10-5), and (h) a RM plate.

Grahic Jump Location
Fig. 21

Static analysis for structural components varying the number of elements ne. Effect of the basis function choice using Leg–Gau distribution (except for Jac DQ basis and SM) with N = 7: (a) a membrane, (b) a RM plate. Effect of the grid distribution choice using PDQ basis functions with N = 7: (c) a membrane, (d) a RM plate. Effect of the grid point number N inside each element using PDQ basis functions and Leg–Gau distribution: (e) a membrane, and (f) a RM plate.

Grahic Jump Location
Fig. 22

Dynamic discrete spectra for single element structural components with N = 31. Effect of the basis function choice using Che–Gau–Lob distribution (except for Jac DQ basis and SM): (a) a membrane, (b) a KL plate without δ-points, (c) a KL plate with δ-points (10-5), (d) a RM plate. Effect of the grid distribution choice using PDQ basis functions: (e) a membrane, (f) a KL plate without δ-points, (g) a KL plate with δ-points (10-5), and (h) a RM plate.

Grahic Jump Location
Fig. 23

Dynamic discrete spectra for various structural components with N = 7 and ne = 64. Effect of the basis function choice using Leg–Gau distribution (except for Jac DQ basis and SM): (a) a membrane, (b) a RM plate. Effect of the grid distribution choice using PDQ basis functions: (c) a membrane, (d) a RM plate. Relative error of the first frequency for various structural components with N = 7. Effect of the basis function choice using Leg–Gau distribution (except for Jac DQ basis and SM): (e) a membrane, (f) a RM plate. Effect of the grid distribution choice using PDQ basis functions: (g) a membrane, and (h) a RM plate.

Grahic Jump Location
Fig. 24

Relative error of the first four frequencies for several structural components varying the number of grid points N and the number of elements ne and using PDQ basis functions and Leg–Gau distribution: (a) a membrane and (b) a RM plate

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Fig. 25

Dynamic discrete spectra for several structural components. Effect of the stretching parameter α using PDQ basis functions and Unif grid distribution. A single element with N = 31: (a) a membrane, (b) a RM plate. Discretized domains with N = 7 and ne = 64: (c) a membrane, and (d) a RM plate.

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Fig. 26

Comparison between structured and distorted meshes using PDQ basis function and Leg–Gau distribution. Study on the number of grid points per element: (a) static analysis of a membrane, (b) first three frequencies of a membrane, (c) static analysis of a RM plate, (d) first three frequencies of a RM plate. Dynamic discrete spectra using different meshes: (e) a membrane, and (f) a RM plate.

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Fig. 27

Distorted meshes used in the computations: (a) single element, (b) two element mesh, (c) two element distorted (skew) mesh, (d) four element mesh, (e) four element distorted (skew) mesh, and (f) nine element mesh

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Fig. 28

Dynamic discrete spectra for a single element structural component using κ technique applied to KL plate with N = 31. (a) Effect of the basis function choice using Che-Gau-Lob distribution (except for Jac DQ basis and SM). (b) Effect of the grid distribution choice using PDQ basis functions.

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