247 research outputs found
Nodal bases for the serendipity family of finite elements
Using the notion of multivariate lower set interpolation, we construct nodal
basis functions for the serendipity family of finite elements, of any order and
any dimension. For the purpose of computation, we also show how to express
these functions as linear combinations of tensor-product polynomials.Comment: Pre-print of version that will appear in Foundations of Computational
Mathematic
Serendipity Nodal VEM spaces
We introduce a new variant of Nodal Virtual Element spaces that mimics the
"Serendipity Finite Element Methods" (whose most popular example is the 8-node
quadrilateral) and allows to reduce (often in a significant way) the number of
internal degrees of freedom. When applied to the faces of a three-dimensional
decomposition, this allows a reduction in the number of face degrees of
freedom: an improvement that cannot be achieved by a simple static
condensation. On triangular and tetrahedral decompositions the new elements
(contrary to the original VEMs) reduce exactly to the classical Lagrange FEM.
On quadrilaterals and hexahedra the new elements are quite similar (and have
the same amount of degrees of freedom) to the Serendipity Finite Elements, but
are much more robust with respect to element distortions. On more general
polytopes the Serendipity VEMs are the natural (and simple) generalization of
the simplicial case
Serendipity and Tensor Product Affine Pyramid Finite Elements
Using the language of finite element exterior calculus, we define two
families of -conforming finite element spaces over pyramids with a
parallelogram base. The first family has matching polynomial traces with tensor
product elements on the base while the second has matching polynomial traces
with serendipity elements on the base. The second family is new to the
literature and provides a robust approach for linking between Lagrange elements
on tetrahedra and serendipity elements on affinely-mapped cubes while
preserving continuity and approximation properties. We define shape functions
and degrees of freedom for each family and prove unisolvence and polynomial
reproduction results.Comment: Accepted to SMAI Journal of Computational Mathematic
Nonstandard finite element de Rham complexes on cubical meshes
We propose two general operations on finite element differential complexes on
cubical meshes that can be used to construct and analyze sequences of
"nonstandard" convergent methods. The first operation, called DoF-transfer,
moves edge degrees of freedom to vertices in a way that reduces global degrees
of freedom while increasing continuity order at vertices. The second operation,
called serendipity, eliminates interior bubble functions and degrees of freedom
locally on each element without affecting edge degrees of freedom. These
operations can be used independently or in tandem to create nonstandard
complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show
that the resulting elements provide convergent, non-conforming methods for
problems requiring stronger regularity and satisfy a discrete Korn inequality.
We discuss potential benefits of applying these elements to Stokes, biharmonic
and elasticity problems.Comment: 31 page
Efficient finite element analysis using graph-theoretical force method; rectangular plane stress and plane strain serendipity family elements
Formation of a suitable null basis for equilibrium matrix is the main part of finite elements analysis via force method. Foran optimal analysis, the selected null basis matrices should be sparse and banded corresponding to produce sparse, banded and well-conditioned flexibility matrices. In this paper, an efficient method is developed for the formation of null bases of finite element models (FEMs) consisting of rectangular plane stress and plane strain serendipity family elements, corresponding to highly sparse and banded flexibility matrices. This is achieved by associating special graphs with the FEM and selecting appropriate subgraphs and forming the self-equilibrating systems (SESs) on these subgraphs. The efficiency of the present method is illustrated through three examples
General complete Lagrange family for the cube in finite element interpolations
In this paper, we have first derived the interpolation polynomials for the General Serendipity elements which allow arbitrarily placed nodes along the edges. We have then presented a method to determine the interpolation functions for the General Complete Lagrange elements which allow arbitrarily placed nodes. Explicit expressions for interpolation functions of the Serendipity and Complete Lagrange family elements which allow uniform spacing of nodes over the element domain are derived for elements of orders 4–10. We have also modified the Shape functions of Complete Lagrange family so that they can correctly interpolate the complete polynomial in the global space for angular distortions
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
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