35 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 Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations
In this work we develop new finite element discretisations of the
shear-deformable Reissner--Mindlin plate problem based on the
Hellinger-Reissner principle of symmetric stresses. Specifically, we use
conforming Hu-Zhang elements to discretise the bending moments in the space of
symmetric square integrable fields with a square integrable divergence
. The
latter results in highly accurate approximations of the bending moments
and in the rotation field being in the discontinuous Lebesgue
space , such that the Kirchhoff-Love constraint
can be satisfied for . In order to preserve optimal convergence rates
across all variables for the case , we present an extension of the
formulation using Raviart-Thomas elements for the shear stress .
We prove existence and uniqueness in the continuous setting and rely on exact
complexes for inheritance of well-posedness in the discrete setting.
This work introduces an efficient construction of the Hu-Zhang base functions
on the reference element via the polytopal template methodology and Legendre
polynomials, making it applicable to hp-FEM. The base functions on the
reference element are then mapped to the physical element using novel polytopal
transformations, which are suitable also for curved geometries.
The robustness of the formulations and the construction of the Hu-Zhang
element are tested for shear-locking, curved geometries and an L-shaped domain
with a singularity in the bending moments . Further, we compare
the performance of the novel formulations with the primal-, MITC- and recently
introduced TDNNS methods.Comment: Additional implementation material in:
https://github.com/Askys/NGSolve_HuZhang_Elemen
Novel -conforming finite elements for the relaxed micromorphic sequence
In this work we construct novel -conforming
finite elements for the recently introduced relaxed micromorphic sequence,
which can be considered as the completion of the -sequence with respect to the -space. The elements respect -regularity and
their lowest order versions converge optimally for -fields. This work introduces a
detailed construction, proofs of linear independence and conformity of the
basis, and numerical examples. Further, we demonstrate an application to the
computation of metamaterials with the relaxed micromorphic model
Mixed finite elements for Kirchhoff-Love plate bending
We present a mixed finite element method with parallelogram meshes for the
Kirchhoff-Love plate bending model. Critical ingredient is the construction of
appropriate basis functions that are conforming in terms of a sufficiently
large tensor space and allow for any kind of physically relevant Dirichlet and
Neumann boundary conditions. For Dirichlet boundary conditions, and polygonal
convex or non-convex plates that can be discretized by parallelogram meshes, we
prove quasi-optimal convergence of the mixed scheme. Numerical results for
regular and singular examples with different boundary conditions illustrate our
findings.Comment: corrected versio