53,104 research outputs found
Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection
Polygonal finite elements generally do not pass the patch test as a result of
quadrature error in the evaluation of weak form integrals. In this work, we
examine the consequences of lack of polynomial consistency and show that it can
lead to a deterioration of convergence of the finite element solutions. We
propose a general remedy, inspired by techniques in the recent literature of
mimetic finite differences, for restoring consistency and thereby ensuring the
satisfaction of the patch test and recovering optimal rates of convergence. The
proposed approach, based on polynomial projections of the basis functions,
allows for the use of moderate number of integration points and brings the
computational cost of polygonal finite elements closer to that of the commonly
used linear triangles and bilinear quadrilaterals. Numerical studies of a
two-dimensional scalar diffusion problem accompany the theoretical
considerations
A volume-averaged nodal projection method for the Reissner-Mindlin plate model
We introduce a novel meshfree Galerkin method for the solution of
Reissner-Mindlin plate problems that is written in terms of the primitive
variables only (i.e., rotations and transverse displacement) and is devoid of
shear-locking. The proposed approach uses linear maximum-entropy approximations
and is built variationally on a two-field potential energy functional wherein
the shear strain, written in terms of the primitive variables, is computed via
a volume-averaged nodal projection operator that is constructed from the
Kirchhoff constraint of the three-field mixed weak form. The stability of the
method is rendered by adding bubble-like enrichment to the rotation degrees of
freedom. Some benchmark problems are presented to demonstrate the accuracy and
performance of the proposed method for a wide range of plate thicknesses
Energy-corrected FEM and explicit time-stepping for parabolic problems
The presence of corners in the computational domain, in general, reduces the
regularity of solutions of parabolic problems and diminishes the convergence
properties of the finite element approximation introducing a so-called
"pollution effect". Standard remedies based on mesh refinement around the
singular corner result in very restrictive stability requirements on the
time-step size when explicit time integration is applied. In this article, we
introduce and analyse the energy-corrected finite element method for parabolic
problems, which works on quasi-uniform meshes, and, based on it, create fast
explicit time discretisation. We illustrate these results with extensive
numerical investigations not only confirming the theoretical results but also
showing the flexibility of the method, which can be applied in the presence of
multiple singular corners and a three-dimensional setting. We also propose a
fast explicit time-stepping scheme based on a piecewise cubic energy-corrected
discretisation in space completed with mass-lumping techniques and numerically
verify its efficiency
Numerical analysis for the pure Neumann control problem using the gradient discretisation method
The article discusses the gradient discretisation method (GDM) for
distributed optimal control problems governed by diffusion equation with pure
Neumann boundary condition. Using the GDM framework enables to develop an
analysis that directly applies to a wide range of numerical schemes, from
conforming and non-conforming finite elements, to mixed finite elements, to
finite volumes and mimetic finite differences methods. Optimal order error
estimates for state, adjoint and control variables for low order schemes are
derived under standard regularity assumptions. A novel projection relation
between the optimal control and the adjoint variable allows the proof of a
super-convergence result for post-processed control. Numerical experiments
performed using a modified active set strategy algorithm for conforming,
nonconforming and mimetic finite difference methods confirm the theoretical
rates of convergence
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