This class provides access to geometric and topological properties of a reference element. More...
Detailed Description
Generic Geometries
Introduction
In the following we will give a definition of reference elements and subelement numbering. This is used to define geometries by prescribing a set of points in the space .
The basic building block for these elements is given by a recursion formula which assigns 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
: is a line.
: is a cube and is a simplex.
: is a cube, is a simplex, is a pyramid, and is a prism.
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
: the first numbers are assigned to the entities parallel to the axis in the same order as the subentites of the same codimension in ; then to the subentities of codimension in the bottom followed by those in the top.
: in this case we first number the subentities of codimension in the bottom, followed by each subentity based on a subentity of codimension in .
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:
One-dimensional reference element. For d=1 the simplex and cube are identical
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 suffices 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 :
Jacobian inverse transpose
Integration element
Volume
(Here some assumptions on the degree of the integration element is used.)
The next sections describe the details of the construction.
Reference Topology
We define the set of reference topologies by the following rules:
contains an element that we call the point topology.
For , contains an element that we call the prism over .
For , contains an element that we call the pyramid over .
For each reference topology we define the following values:
Dimension: The point topology has dimension zero and the dimension of a prism or a pyramid topology over has dimension .
Size: For with we define the number through
.
If then and for we have .
If then and for we have .
Subtopology: Given a reference topology of dimension and a codimension we now define the subtopology for :
and
For and we define using the abbreviations
For and we define using the abbreviations
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 .
Reference Domains
For each reference topology we assosiate the set of corners defined through
:
: for , with .
: for and with
The convex hall of the set of points defines the reference domain for the reference topology ; it follows that
,
,
.
Reference Elements and Mappings
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 recursive definition of the reference topologies through:
,
with and .
with and .
Numbering of Subelements
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
:
For we define with .
For we define with .
For we define with .
:
For we define with .
For we define with .
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