501 research outputs found
The TDNNS method for Reissner-Mindlin plates
A new family of locking-free finite elements for shear deformable
Reissner-Mindlin plates is presented. The elements are based on the
"tangential-displacement normal-normal-stress" formulation of elasticity. In
this formulation, the bending moments are treated as separate unknowns. The
degrees of freedom for the plate element are the nodal values of the
deflection, tangential components of the rotations and normal-normal components
of the bending strain. Contrary to other plate bending elements, no special
treatment for the shear term such as reduced integration is necessary. The
elements attain an optimal order of convergence
A class of nonparametric DSSY nonconforming quadrilateral elements
A new class of nonparametric nonconforming quadrilateral finite elements is
introduced which has the midpoint continuity and the mean value continuity at
the interfaces of elements simultaneously as the rectangular DSSY element
[J.Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming {G}alerkin
methods based on quadrilateral elements for second order elliptic problems.
ESAIM--Math. Model. Numer. Anal., 33(4):747--770, 1999]. The parametric DSSY
element for general quadrilaterals requires five degrees of freedom to have an
optimal order of convergence [Z. Cai, J. Douglas, Jr., J. E. Santos, D. Sheen,
and X. Ye. Nonconforming quadrilateral finite elements: A correction. Calcolo,
37(4):253--254, 2000], while the new nonparametric DSSY elements require only
four degrees of freedom. The design of new elements is based on the
decomposition of a bilinear transform into a simple bilinear map followed by a
suitable affine map. Numerical results are presented to compare the new
elements with the parametric DSSY element.Comment: 20 page
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
A superconvergent hybridisable discontinuous Galerkin method for linear elasticity
The first superconvergent hybridisable discontinuous Galerkin method for linear elastic problems capable of using the same degree of approximation for both the primal and mixed variables is presented. The key feature of the method is the strong imposition of the symmetry of the stress tensor by means of the well known and extensively used Voigt notation, circumventing the use of complex mathematical concepts to enforce the symmetry of the stress tensor either weakly or strongly. A novel procedure to construct element by element a superconvergent postprocessed displacement is proposed. Contrary to other hybridisable discontinuous Galerkin formulations, the methodology proposed here is able to produce a superconvergent displacement field for low-order approximations. The resulting method is robust and locking-free in the nearly incompressible limit. An extensive set of numerical examples is utilised to provide evidence of the optimality of the method and its superconvergent properties in two and three dimensions and for different element type
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