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Keywords
(6)
Finite Element
Mixed Finite Element
Mixed Method
Scalar Field
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Related Publications
(16)
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Mixed finite elements for elasticity
Mixed finite elements for elasticity,10.1007/s002110100348,Numerische Mathematik,Douglas N. Arnold,Ragnar Winther
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Mixed finite elements for elasticity
(
Citations: 89
)
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Douglas N. Arnold
,
Ragnar Winther
Summary. There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stressdisplacement formulation of the plane elasticity system. This requires the development of a compatible pair of
finite element
spaces, one to discretize the space of symmetric tensors in which the
stress field
is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of wellknown
mixed finite element
pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the
stress field
is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of
finite element
spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such
finite element
spaces, which account for the paucity of elements available.
Journal:
Numerische Mathematik  NUMER MATH
, vol. 92, no. 3, pp. 401419, 2002
DOI:
10.1007/s002110100348
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Citation Context
(38)
...Not until the year 2002, was the first family of stable mixed finite elements [
6
] proposed for HellingerReissner formulation, using polynomial shape functions with respect to a single arbitrary triangular mesh for both the stress and the displacement spaces...
...As shown in [
6
], it is unavoidable to use vertex degrees of freedom for symmetric stress spaces and for conforming finite elements with continuous shape functions on an arbitrary triangulation...
...The approach of the analysis comes from [2,
6
]. In Sect...
...The elasticity complex in two dimensions [
6
] takes the form:...
...An analogous exact sequence with less smoothness [
6
]i s:...
...To this end, we invoke [
6
] the fact that for each v ∈ L 2 (Ω, R 2 ) there exists a τ ∈ H 1 (Ω, S) such that...
ShaoChun Chen
,
et al.
Conforming Rectangular Mixed Finite Elements for Elasticity
...The following two conditions are sufficient for (A1) and (A2) [
6
]: (A1) � divhΣh ⊂ Vh. (A2) � There exists a linear operator Πh : H 1 (Ω, S) ∩ H0,N(div, Ω, S) → Σh, bounded in L(H 1 , L 2 )...
...In [
6
], Arnold and Winther proposed a family of stable conforming triangular elements satisfying the stability conditions (A1) � and (A2) � , and derived optimal order approximation...
...We adopt some plane elasticity mixed finite element spaces, including Arnold and Winther’s conforming and nonconforming families [
6
, 7], and the nonconforming elements we propose in this contribution, for the approximation of the stress and velocity, whereas continuous piecewise polynomials for the approximation of the pressure...
...T, S). There hold the following properties [
6
]...
...Thus, by following a discussion similar to that in [
6
, 7], i.e., using the matrix Piola transformation and a standard scaling argument, we can show (A3) holds for Πh =Π (1) with m =1 , 2,...
...They are: ArnoldWinther’s conforming element family [
6
], ArnoldWinther’s nonconforming elements [7], and the nonconforming elements proposed in Section 3...
Xiaoping Xie
,
et al.
New mixed finite elements for plane elasticity and Stokes equations
...We consider the nite element pairs ( Sh;Uh) of symmetric tensors and vectors constructed by Arnold and Winther [
7
,1] which satisfy the infsup condition...
...In [
7
,1] an interpolation operator h : S\H1( )d d! Sh is constructed for which...
Jason S. Howell
,
et al.
Inf–sup conditions for twofold saddle point problems
...Conversely, the complementary energy principle (equilibrium method) that uses stresses as variables will give an upper bound on the global strain energy [11,
12
]...
Z. C. Xuan
,
et al.
On computing upper and lower bounds on the outputs of linear elasticit...
...This difficulty, if not circumvented, is severely restrictive (see [21] and [
22
, 23] for the analysis of admissible elements in linear elasticity)...
M. Cervera
,
et al.
Mixed stabilized finite element methods in nonlinear solid mechanics
References
(16)
Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity
(
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PEERS: A new mixed finite element for plane elasticity
(
Citations: 93
)
Douglas N. Arnold
,
Franco Brezzi
,
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Journal:
Japan Journal of Industrial and Applied Mathematics  JPN J IND APPL MATH
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A family of higher order mixed finite element methods for plane elasticity
(
Citations: 70
)
Douglas N. Arnold
,
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,
Chaitan P. Gupta
Journal:
Numerische Mathematik  NUMER MATH
, vol. 45, no. 1, pp. 122, 1984
Sort by:
Citations
(89)
Conforming Rectangular Mixed Finite Elements for Elasticity
(
Citations: 1
)
ShaoChun Chen
,
YaNa Wang
Journal:
Journal of Scientific Computing
, vol. 47, no. 1, pp. 93108, 2011
New mixed finite elements for plane elasticity and Stokes equations
(
Citations: 1
)
Xiaoping Xie
,
Jinchao Xu
Journal:
Science Chinamathematics  SCI CHINAMATH
, vol. 54, no. 7, pp. 14991519, 2011
A residualbased a posteriori error estimator for a fullymixed formulation of the Stokes–Darcy coupled problem
Gabriel N. Gatica
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Computer Methods in Applied Mechanics and Engineering  COMPUT METHOD APPL MECH ENG
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Characterizing the Antibacterial Mechanism of Three Novel HistoneDerived Antimicrobial Peptides
Sara A. Spinella
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Donald E. Elmore
Journal:
Biophysical Journal  BIOPHYS J
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UvVisible Spectroscopy of Glycoproteins Interacting with Lipid Vesicles
Andrew K. Fraser
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Kwame K. E. Newon
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Bruce D. Ray
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Horia I. Petrache
Journal:
Biophysical Journal  BIOPHYS J
, vol. 100, no. 3, pp. 508a508a, 2011