A mixed finite element family in plane elasticity

Mignot, Alain; Surry, C.

doi:10.1016/s0307-904x(81)80076-5

Cited by 8 publications

(5 citation statements)

References 5 publications

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“…Because of the lack of suitable mixed elasticity elements, several authors have resorted to the use of Lagrangian functionals which are modifications of the Hellinger-Reissner functional given above [1,3,5,[15][16][17][18], in which the symmetry of the stress tensor is enforced only weakly or abandoned altogether. Another modification is analyzed in [14], where the solution space is altered such that Σ h is only required to be a subspace of L 2 (Ω, S).…”

Section: Introductionmentioning

confidence: 99%

Mixed finite elements for elasticity

Arnold

Winther

2002

Numerische Mathematik

367

373

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There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement 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 well-known 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.

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Section: Introductionmentioning

confidence: 99%

Mixed finite elements for elasticity

Arnold

Winther

2002

Numerische Mathematik

367

373

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“…Other researchers have abandoned the symmetry condition, used a different variational formulation, or weakened the regularity conditions on the stress fields in various ways [34][35][36][37][38][39]. Stable mixed finite elements with weakly imposed symmetry have been introduced in [17,[40][41][42][43][44], but the simplest ones have only been recently discovered [13,[45][46][47].…”

Section: Discussionmentioning

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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“…Owing to the lack of simple stable elements, some authors also resorted to the use of Lagrangian functionals based on modified variational principles, in which the symmetry of the stress tensor is enforced only weakly or abandoned altogether [1,3,5,[27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

New mixed finite elements for plane elasticity and Stokes equations

Xie¹,

2011

Sci. China Math.

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We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified Hellinger-Reissner formulation of elasticity in which the stress and displacement fields are the primary unknowns, we derive two new nonconforming mixed finite elements of triangle type. Both elements use piecewise rigid motions to approximate the displacement and piecewise polynomial functions to approximate the stress, where no vertex degrees of freedom are involved. The two stress finite element spaces consist respectively of piecewise quadratic polynomials and piecewise cubic polynomials such that the divergence of each space restricted to a single simplex is contained in the corresponding displacement approximation space. We derive stability and optimal order approximation for the elements. We also give some numerical results to verify the theoretical results.For the Stokes equation, introducing the symmetric part of the gradient tensor of the velocity as a stress variable, we present a stress-velocity-pressure field Stokes system. We use some plane elasticity mixed finite elements, including the two elements we proposed, to approximate the stress and velocity fields, and use continuous piecewise polynomial functions to approximate the pressure with the gradient of the pressure approximation being in the corresponding velocity finite element spaces. We derive stability and convergence for these methods.

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A mixed finite element family in plane elasticity

Cited by 8 publications

References 5 publications

Mixed finite elements for elasticity

Mixed finite elements for elasticity

Symmetric Matrix Fields in the Finite Element Method

New mixed finite elements for plane elasticity and Stokes equations

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