A Mixed Finite Element Method for Elasticity in Three Dimensions

Adams, Scot; Cockburn, Bernardo

doi:10.1007/s10915-004-4807-3

Cited by 98 publications

(93 citation statements)

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“…Mixed method finite elements for linear elasticity generally fall into two categories, those that enforce symmetry of the stress tensor exactly [2,3,6], and the ones that enforce it weakly [4,5,18,12,8]. Here, we consider the latter methods.…”

Section: Mixed Methods For Linear Elasticitymentioning

confidence: 99%

A new elasticity element made for enforcing weak stress symmetry

2010

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Abstract. We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer k, the row-wise divergence of the element space spans the set of polynomials of total degree k. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress. This is achieved using certain "bubble matrices", which are special divergence-free matrix-valued polynomials. We prove that the approximation error is of order k + 1 in both the displacement and the stress, and that a postprocessed displacement approximation converging at order k + 2 can be computed element by element. We also show that the globally coupled degrees of freedom can be reduced by hybridization to those of a displacement approximation on the element boundaries.

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Section: Mixed Methods For Linear Elasticitymentioning

confidence: 99%

A new elasticity element made for enforcing weak stress symmetry

2010

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“…An analogous element was obtained in [22], where the last set of degrees of freedom was replaced by ∫ e T t · t dx for each of the six edges. The other elements of the family can be described as…”

Section: The Value Of the Moments ∫mentioning

confidence: 99%

Symmetric Matrix Fields in the Finite Element Method

Awanou

2010

Symmetry

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The theory of elasticity is used to predict the response of a material body subject to applied forces. In the linear theory, where the displacement is small, the stress tensor which measures the internal forces is the variable of primal importance. However the symmetry of the stress tensor which expresses the conservation of angular momentum had been a challenge for finite element computations. We review in this paper approaches based on mixed finite element methods.

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“…Note that, in contrast toΠ 1 h , in the definition ofΠ 1 0h , we have set the troublesome edge degrees of freedom to zero. LetΠ 1 0 : H 1 Λ 1 (T ) → P + r+1 Λ 1 (T ) be defined analogously on the reference element.…”

Section: Stable Mixed Finite Elements For Elasticitymentioning

confidence: 99%

Untitled

2007

Math. Comp.

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Abstract. In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

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A Mixed Finite Element Method for Elasticity in Three Dimensions

Abstract: We describe a stable mixed finite element method for linear elasticity in three dimensions.

Cited by 98 publications

References 1 publication

A new elasticity element made for enforcing weak stress symmetry

A new elasticity element made for enforcing weak stress symmetry

Symmetric Matrix Fields in the Finite Element Method

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