dune-geometry
2.4
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Classes | |
struct | Dune::ReferenceElements< ctype, dim > |
Class providing access to the singletons of the reference elements. More... | |
class | Dune::ReferenceElement< ctype, dim > |
This class provides access to geometric and topological properties of a reference element. More... | |
In the following we will give a definition of reference elements and subelement numbering. This is used to define geometries by prescibing a set of points in the space .
The basic building block for these elements is given by a recursion formula which assignes to each set either a prism element
or a pyramid element
with
and
. The recursion starts with a single point
.
For this leads to the following elements
In general if is a cube then
is also a cube and if
is a simplex then
is also a simplex.
Based on the recursion formula we can also define a numbering of the subentities and also of the sub-subentities of or
based on a numbering of
. For the subentities of codimension
we use the numbering
For the subentity of codimension in a codimension
subentity
we use the numbering induced by the numbering the reference element corresponding to
.
Here is a graphical representation of the reference elements:
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Face Numbering | ![]()
Edge Numbering |
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Face Numbering | ![]()
Edge Numbering |
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Face Numbering | ![]()
Edge Numbering |
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Face Numbering | ![]()
Edge Numbering |
In addition to the numbering and the corner coordinates of a reference element we also define the barycenters
, the volume
and the normals
to all codimension one subelements.
The recursion formula is also used to define mappings from reference elements to general polytop given by a set of coordinates for the corner points - together with the mapping
, the transpose of the jacobian
is also defined where
is the dimension of the reference element and
the dimension of the coordinates. This sufficies to define other necessary parts of a Dune geometry by LQ-decomposing
: let
be given with a lower diagonal matrix
and a matrix
which satisfies
:
The next sections describe the details of the construction.
We define the set of reference topologies by the following rules:
For each reference topology we define the following values:
Notice that the number of vertices (i.e., subtopologies of codimension ) of a topology
does not uniquely identify the topology. To see this, consider the topologies
and
. For these topologies we have
.
For each reference topology we assosiate the set of corners
defined through
The convex hall of the set of points defines the reference domain
for the reference topology
; it follows that
A pair of a topology
and a map
with
is called an element.
The reference element is the pair .
For a given set of points we define a mapping
through
for all
. This mapping can be expressed using the recurive definition of the reference topologies through:
Given a reference topology , a codimension
and a subtopology
we define a subset of the corner set
given by the subsequence
of
:
,
, and for
we define
through the recursion
Given these subsets we define subreference elements of
given by the following mapping
.
Furthermore we define a numbering of the subreference elements of each subreference element in . This is the number
for
,
, and
,
for which