4,033 research outputs found
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
Finite element differential forms on cubical meshes
We develop a family of finite element spaces of differential forms defined on
cubical meshes in any number of dimensions. The family contains elements of all
polynomial degrees and all form degrees. In two dimensions, these include the
serendipity finite elements and the rectangular BDM elements. In three
dimensions they include a recent generalization of the serendipity spaces, and
new H(curl) and H(div) finite element spaces. Spaces in the family can be
combined to give finite element subcomplexes of the de Rham complex which
satisfy the basic hypotheses of the finite element exterior calculus, and hence
can be used for stable discretization of a variety of problems. The
construction and properties of the spaces are established in a uniform manner
using finite element exterior calculus.Comment: v2: as accepted by Mathematics of Computation after minor revisions;
v3: this version corresponds to the final version for Math. Comp., after
copyediting and galley proof
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
Computational Serendipity and Tensor Product Finite Element Differential Forms
Many conforming finite elements on squares and cubes are elegantly classified
into families by the language of finite element exterior calculus and presented
in the Periodic Table of the Finite Elements. Use of these elements varies,
based principally on the ease or difficulty in finding a "computational basis"
of shape functions for element families. The tensor product family,
, is most commonly used because computational basis functions
are easy to state and implement. The trimmed and non-trimmed serendipity
families, and respectively, are used less
frequently because they are newer to the community and, until now, lacked a
straightforward technique for computational basis construction. This represents
a missed opportunity for computational efficiency as the serendipity elements
in general have fewer degrees of freedom than elements of equivalent accuracy
from the tensor product family. Accordingly, in pursuit of easy adoption of the
serendipity families, we present complete lists of computational bases for both
serendipity families, for any order and for any differential form
order , for problems in dimension or . The bases are
defined via shared subspace structures, allowing easy comparison of elements
across families. We use and include code in SageMath to find, list, and verify
these computational basis functions.Comment: 19 page manuscript; 8 page appendix. Code available at
http://math.arizona.edu/~agillette/research/computationalBases
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
Trimmed Serendipity Finite Element Differential Forms
We introduce the family of trimmed serendipity finite element differential
form spaces, defined on cubical meshes in any number of dimensions, for any
polynomial degree, and for any form order. The relation between the trimmed
serendipity family and the (non-trimmed) serendipity family developed by Arnold
and Awanou [Math. Comp. 83(288) 2014] is analogous to the relation between the
trimmed and (non-trimmed) polynomial finite element differential form families
on simplicial meshes from finite element exterior calculus. We provide degrees
of freedom in the general setting and prove that they are unisolvent for the
trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a
fixed polynomial order r provides an explicit example of a system described by
Christiansen and Gillette [ESAIM:M2AN 50(3) 2016], namely, a minimal compatible
finite element system on squares or cubes containing order r-1 polynomial
differential forms.Comment: Improved results, detailed comparison to prior and contemporary work,
and further explanation of computational benefits have been added since the
original version. This version has been accepted for publication in
Mathematics of Computatio
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
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