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TECHNICAL PAPERS

A Stabilized Mixed Finite Element Method for Nearly Incompressible Elasticity

[+] Author and Article Information
Arif Masud1

Department of Civil and Materials Engineering,  The University of Illinois at Chicago, Chicago, IL 60607-7023amasud@uic.edu

Kaiming Xia2

Department of Civil and Materials Engineering,  The University of Illinois at Chicago, Chicago, IL 60607-7023

1

Author to whom correspondence should be addressed.

2

Formerly Graduate Research Assistant.

J. Appl. Mech. 72(5), 711-720 (Jan 11, 2005) (10 pages) doi:10.1115/1.1985433 History: Received August 09, 2004; Revised January 11, 2005

We present a new multiscale/stabilized finite element method for compressible and incompressible elasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting stabilized-mixed form consistently represents the fine computational scales in the solution and thus possesses higher coarse mesh accuracy. The ensuing finite element formulation allows arbitrary combinations of interpolation functions for the displacement and stress fields. Specifically, equal order interpolations that are easy to implement but violate the celebrated Babushka-Brezzi inf-sup condition, become stable and convergent. Since the proposed framework is based on sound variational foundations, it provides a basis for a priori error analysis of the system. Numerical simulations pass various element patch tests and confirm optimal convergence in the norms considered.

Copyright © 2005 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A family of 2D linear and quadratic elements

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Figure 2

Bubble functions employed for quadrilaterals and triangles

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Figure 3

Patch test with regular elements

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Figure 4

Patch test with distorted elements

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Figure 5

Sensitivity to mesh distortion. 9-node quadrilaterals.

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Sensitivity to mesh distortion. 6-node triangles.

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Cantilever beam with edge shear

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Figure 8

Convergence rates for the L2 norm of the displacement field (linear elements)

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Figure 9

Convergence rates for the energy norm (linear elements)

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Convergence rates for the L2 norm of the pressure field (linear elements)

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Convergence rates for the L2 norm of the displacement field (quadratic elements)

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Figure 12

Convergence rates for the energy norm (quadratic elements)

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Convergence rates for the L2 norm of the pressure field (quadratic elements)

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Contours of the pressure field for the 3-node element mesh

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Contours of the pressure field for the 4-node element mesh

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Contours of the pressure field for the composite mesh

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Tip deflection convergence for 3-node and 4-node elements

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Figure 18

Stress convergence for 3-node and 4-node elements

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Tip deflection convergence for 6-node and 9-node elements

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Stress convergence for 6-node and 9-node elements

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Cook’s membrane

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Figure 22

Tip deflection convergence

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Tip deflection convergence as a function of Poisson’s ratio

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Figure 24

Contours of the pressure field for the 4-node element mesh

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Figure 25

Contours of the pressure field for the composite mesh

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