35,053 research outputs found
Nonconforming tetrahedral mixed finite elements for elasticity
This paper presents a nonconforming finite element approximation of the space
of symmetric tensors with square integrable divergence, on tetrahedral meshes.
Used for stress approximation together with the full space of piecewise linear
vector fields for displacement, this gives a stable mixed finite element method
which is shown to be linearly convergent for both the stress and displacement,
and which is significantly simpler than any stable conforming mixed finite
element method. The method may be viewed as the three-dimensional analogue of a
previously developed element in two dimensions. As in that case, a variant of
the method is proposed as well, in which the displacement approximation is
reduced to piecewise rigid motions and the stress space is reduced accordingly,
but the linear convergence is retained.Comment: 13 pages, 2 figure
Stabilized mixed finite element methods for linear elasticity on simplicial grids in
In this paper, we design two classes of stabilized mixed finite element
methods for linear elasticity on simplicial grids. In the first class of
elements, we use - and
- to approximate the stress
and displacement spaces, respectively, for , and employ a
stabilization technique in terms of the jump of the discrete displacement over
the faces of the triangulation under consideration; in the second class of
elements, we use - to
approximate the displacement space for , and adopt the
stabilization technique suggested by Brezzi, Fortin, and Marini. We establish
the discrete inf-sup conditions, and consequently present the a priori error
analysis for them. The main ingredient for the analysis is two special
interpolation operators, which can be constructed using a crucial
bubble function space of polynomials on each
element. The feature of these methods is the low number of global degrees of
freedom in the lowest order case. We present some numerical results to
demonstrate the theoretical estimates.Comment: 16 pages, 1 figur
Mixed finite element methods for linear elasticity with weakly imposed symmetry
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.Comment: to appear in Mathematics of Computatio
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