18 research outputs found
A short note on a Bernstein-Bezier basis for the pyramid
We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to
the face reduces to the Bernstein-Bezier basis on the triangle or
quadrilateral. The basis satisfies the standard positivity and partition of
unity properties common to Bernstein polynomials, and spans the same space as
non-polynomial pyramid bases in the literature.Comment: Submitte
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
Low-order continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes
This article deals with solving partial differential equations with the
finite element method on hybrid non-conforming hexahedral-tetrahedral meshes.
By non-conforming, we mean that a quadrangular face of a hexahedron can be
connected to two triangular faces of tetrahedra. We introduce a set of
low-order continuous (C0) finite element spaces defined on these meshes. They
are built from standard tri-linear and quadratic Lagrange finite elements with
an extra set of constraints at non-conforming hexahedra-tetrahedra junctions to
recover continuity. We consider both the continuity of the geometry and the
continuity of the function basis as follows: the continuity of the geometry is
achieved by using quadratic mappings for tetrahedra connected to tri-affine
hexahedra and the continuity of interpolating functions is enforced in a
similar manner by using quadratic Lagrange basis on tetrahedra with constraints
at non-conforming junctions to match tri-linear hexahedra. The so-defined
function spaces are validated numerically on simple Poisson and linear
elasticity problems for which an analytical solution is known. We observe that
using a hybrid mesh with the proposed function spaces results in an accuracy
significantly better than when using linear tetrahedra and slightly worse than
when solely using tri-linear hexahedra. As a consequence, the proposed function
spaces may be a promising alternative for complex geometries that are out of
reach of existing full hexahedral meshing methods
Numerical integration for high order pyramidal finite elements
We examine the effect of numerical integration on the convergence of high
order pyramidal finite element methods. Rational functions are indispensable to
the construction of pyramidal interpolants so the conventional treatment of
numerical integration, which requires that the finite element approximation
space is piecewise polynomial, cannot be applied. We develop an analysis that
allows the finite element approximation space to include rational functions and
show that despite this complication, conventional rules of thumb can still be
used to select appropriate quadrature methods on pyramids. Along the way, we
present a new family of high order pyramidal finite elements for each of the
spaces of the de Rham complex.Comment: 28 page
A Comparison of High Order Interpolation Nodes for the Pyramid
The use of pyramid elements is crucial to the construction of efficient
hex-dominant meshes. For conforming nodal finite element methods with mixed
element types, it is advantageous for nodal distributions on the faces of the
pyramid to match those on the faces and edges of hexahedra and tetrahedra. We
adapt existing procedures for constructing optimized tetrahedral nodal sets for
high order interpolation to the pyramid with constrained face nodes, including
two generalizations of the explicit Warp and Blend construction of nodes on the
tetrahedron.Comment: Submitted to SIAM:SIS
High-order finite elements on pyramids: approximation spaces, unisolvency and exactness
We present a family of high-order finite element approximation spaces on a
pyramid, and associated unisolvent degrees of freedom. These spaces consist of
rational basis functions. We establish conforming, exactness and polynomial
approximation properties.Comment: 37 pages, 3 figures. This work was originally in one paper, then
split into two; it has now been recombined into one paper, with substantial
changes from both of its previous form
GPU-accelerated discontinuous Galerkin methods on hybrid meshes
We present a time-explicit discontinuous Galerkin (DG) solver for the
time-domain acoustic wave equation on hybrid meshes containing vertex-mapped
hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable
formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto
(Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions
for hybrid meshes are derived by bounding the spectral radius of the DG
operator using order-dependent constants in trace and Markov inequalities.
Computational efficiency is achieved under a combination of element-specific
kernels (including new quadrature-free operators for the pyramid), multi-rate
timestepping, and acceleration using Graphics Processing Units.Comment: Submitted to CMAM