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Dune Core Modules (2.5.0)

Reference Elements

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...
 

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 \mathbf{R}^w .

The basic building block for these elements is given by a recursion formula which assigns to each set E \subset \mathbf{R}^d either a prism element E^\vert\subset \mathbf{R}^{d+1} or a pyramid element E^\circ\subset \mathbf{R}^{d+1} with E^\vert = \lbrace (x,\bar{x}) \mid x \in E, \bar{x} \in [0,1] \rbrace and E^\circ = \lbrace ((1-\bar{x})x,\bar{x}) \mid x \in E, \bar{x} \in [0,1] \rbrace . The recursion starts with a single point P_0=0\in\mathbf{R}^0 .

For d=1,2,3 this leads to the following elements

  • d=1 : L_1 = P_0^\vert = P_0^\circ = [0,1] is a line.
  • d=2 : Q_2 = L_1^\vert is a cube and S_2 = L_1^\circ is a simplex.
  • d=3 : Q_3 = Q_2^\vert is a cube, S_3 = S_2^\circ is a simplex, \mathrm{pyramid}_3 = Q_2^\circ is a pyramid, and \mathrm{prism}_3 = S_2^\vert is a prism.

In general if Q_d is a cube then Q_d^\vert is also a cube and if S_d is a simplex then S_d^\circ 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 E^\vert or E^\circ based on a numbering of E . For the subentities of codimension c we use the numbering

  • E^\vert : the first numbers are assigned to the entities parallel to the x_d-axis in the same order as the subentites of the same codimension in E ; then to the subentities of codimension c-1 in the bottom followed by those in the top.
  • E^\circ : in this case we first number the subentities of codimension c-1 in the bottom, followed by each subentity based on a subentity of codimension c in E .

For the subentity of codimension cc in a codimension c subentity E' we use the numbering induced by the numbering the reference element corresponding to E' .

Here is a graphical representation of the reference elements:

  1. One-dimensional reference element. For d=1 the simplex and cube are identical
  2. Two-dimensional reference simplex (a.k.a. triangle)
  3. Three-dimensional reference simplex (a.k.a. tetrahedron)
    Face Numbering
    Edge Numbering
  4. Two-dimensional reference cube (a.k.a. quadrilateral)
  5. Three-dimensional reference cube (a.k.a. hexahedron)
    Face Numbering
    Edge Numbering
  6. Prism reference element
    Face Numbering
    Edge Numbering
  7. Pyramid reference element
    Face Numbering
    Edge Numbering

In addition to the numbering and the corner coordinates of a reference element \hat{E} we also define the barycenters b(\hat{E}) , the volume |\hat{E}| and the normals n_i(\hat{E}) to all codimension one subelements.

The recursion formula is also used to define mappings from reference elements E to general polytop given by a set of coordinates for the corner points - together with the mapping \Phi , the transpose of the Jacobian D\Phi^T(x)\in\mathbf{R}^{d,w} is also defined where d is the dimension of the reference element and w the dimension of the coordinates. This suffices to define other necessary parts of a Dune geometry by LQ-decomposing D\Phi^T : let D\Phi^T(x) = LQ be given with a lower diagonal matrix L\in \mathbf{R}^{d,d} and a matrix Q\in\mathbf{R}^{d,w} which satisfies Q Q^T = I :

  • Jacobian inverse transpose

    D\Phi(x)^{-T}:=Q^T L^{-1}.

  • Integration element

    \sqrt{\mathrm{det}(D\Phi(x)^T D\Phi(x))} = \sqrt{\mathrm{det}(LL^T)} = \Pi_{i=1}^d l_{ii}.

  • Volume

    \int_{\hat{E}} \Pi_{i=1}^d l_{ii}(x) d\;x = |\hat{E}| \Pi_{i=1}^d l_{ii}(b(\hat{E})).

    (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 \mathcal{T} of reference topologies by the following rules:

  • \mathcal{T} contains an element \mathbf{p} that we call the point topology.
  • For T \in \mathcal{T}, \mathcal{T} contains an element T^\vert that we call the prism over T.
  • For T \in \mathcal{T}, \mathcal{T} contains an element T^\circ that we call the pyramid over T.

For each reference topology T we define the following values:

  • Dimension: The point topology has dimension zero and the dimension of a prism or a pyramid topology over t has dimension d(t)+1.
  • Size: For c=0,\dots,d with d=d(T) we define the number s_c(T) through
    • s_0(T)=1.
    • If T=t^\vert then s_d(T)=2s_{d-1}(t) and for c\in \{1,\dots,d-1\} we have s_c(T)=s_{c}(t)+2s_{c-1}(t).
    • If T=t^\circ then s_d(T)=s_{d-1}(t)+1 and for c\in \{1,\dots,d-1\} we have s_c(T)=s_{c-1}(t)+s_{c}(t).
  • Subtopology: Given a reference topology T of dimension d and a codimension c=0,\dots,d we now define the subtopology S_{c,i}(T)\in\mathcal{T} for i=1,\dots,s_c(T):
    • S_{0,1}(T)=T and S_{d,i}(T)=\mathbf{p}
    • For T=t^\vert and c\in \{1,\dots,d-1\} we define using the abbreviations p=s_{c}(t),q=s_{c-1}(t)

      S_{c,i}(T) = \left\{\begin{array}{ll} S_{c,i}(t)^\vert & \mbox{for}\; i=1,\dots,p, \\ S_{c-1,i-p}(t) & \mbox{for}\; i=p+1,\dots,p+q, \\ S_{c-1,i-p-q}(t) & \mbox{for}\; i=p+q+1,\dots,p+2q. \end{array}\right.

    • For T=t^\circ and c\in \{1,\dots,d-1\} we define using the abbreviations p=s_{c}(t),q=s_{c-1}(t)

      S_{c,i}(T) = \left\{\begin{array}{ll} S_{c-1,i}(t) & \mbox{for}\; i=1,\dots,q, \\ S_{c,i-q}(t)^\circ & \mbox{for}\; i=q+1,\dots,q+p. \end{array}\right.

Notice that the number of vertices (i.e., subtopologies of codimension d(T)) of a topology T does not uniquely identify the topology. To see this, consider the topologies T_1 = \mathbf{p}^{\vert\vert\vert\circ\circ} and T_2 = \mathbf{p}^{\circ\circ\circ\circ\vert}. For these topologies we have s_d( T_1 ) = s_d( T_2 ) = 10.

Reference Domains

For each reference topology T we assosiate the set of corners \mathcal{C}(T):=(p_i(T))_{1=1}^{s_d(T)}\subset\mathbf{R}^d(T) defined through

  • T=\mathbf{p}: p_0(T)=0\in\mathbf{R}^0
  • T=t^\vert: p_k(T)=(p_k(t),0),p_{d'+k}(T)=(p_k(t),1) for k=1,\dots,d', with d'=s_d(t).
  • T=t^\circ: p_k(T)=(p_k(t),0) for k=1,\dots,d-1 and p_d(T)=e_d with d=s_d(T)

The convex hall of the set of points \mathcal{C}(T) defines the reference domain \mathrm{domain}(T) for the reference topology T; it follows that

  • \mathrm{domain}( \mathbf{p} ) := \lbrace 0 \rbrace \subset \mathbf{R}^0,
  • \mathrm{domain}( T^\vert ) := \mathrm{domain}( T ) \times [0,1],
  • \mathrm{domain}( T^\circ ) := \lbrace ((1-\bar{x})x,\bar{x}) \mid x \in \mathrm{domain}( T ), \bar{x} \in [0,1] \rbrace.

Reference Elements and Mappings

A pair E=(T,\Phi) of a topology T and a map \Phi:\mathrm{domain}( T ) \to \mathbf{R}^w with w\geq d(T) is called an element.

The reference element is the pair E(T)=(T,{\rm id}).

For a given set of points \mathcal{C}:=(p_i)_{1=1}^{s_d(T)}\subset\mathbf{R}^w we define a mapping \Phi_T(\mathcal{C},\cdot)\mathrm{domain}( T ) \to \mathbf{R}^w through \Phi_T(\mathcal{C};p_i(T))=p_i for all p_i(T)\in \mathcal{C}(T). This mapping can be expressed using the recursive definition of the reference topologies through:

  • \Phi_{\mathbf{p}}( (p_0) ; x ) = p_0,
  • \Phi_{T^\vert}( (p_1,\ldots,p_{s_d(T^\vert)} ; (x,\bar{x}) ) = (1 - \bar{x}) \Phi_T( p_1,\ldots,p_{s_d(T)} ; x ) + \bar{x} \Phi_T( p_{s_d(T)+1},\dots,p_{s_d(T^\vert)} ; x ) with x\in \mathrm{domain}( T ) and \bar{x}\in[0,1].
  • \Phi_{T^\circ}( p_1,\ldots,p_{s_d(T^\circ)}) ; (x,\bar{x}) ) = (1 - \bar{x}) \Phi_T( p_1,\ldots,p_{s_d(T)-1} ; x ) + \bar{x} p_{s_d(T^\circ)} with x\in \mathrm{domain}( T ) and \bar{x}\in[0,1].

Numbering of Subelements

Given a reference topology T, a codimension c\in\{0,\dots,d(T)\} and a subtopology S_{c,i}(T) we define a subset of the corner set \mathcal{C}(T) \mathcal{C}_{c,i}(T)= (p_{k_j}(T))_{j=1}^{s_{d(S_{c,i}(T)}(S_{c,i}(T))} given by the subsequence \mathcal{K}_{c,i}(T)=(k_j)_{j=1}^{s_{d(S_{c,i}(T)}(S_{c,i}(T))} of \{1,dots,s_{d(T)}(T)\} :

\mathcal{C}_{0,0}(T) = \mathcal{C}(T), \mathcal{C}_{d(T),i}(T) = (p_i(T)) , and for c\in\{1,\dots,d(T)-1\} we define \mathcal{K}_{c,i}(T)=(k_j)_{j=1}^{s_{d(S_{c,i}(T)}(S_{c,i}(T))} through the recursion

  • T=t^\vert:
    For i=1,\dots,s_c(T) we define \mathcal{K}_{c,i}(T)=(k_1,\dots,k_n, k_1+s_{d(t)}(t)\dots,k_n+s_{d(t)}(t)) with \mathcal{K}_{c,i}(T)= \{k_1,\dots,k_n\} .
    For i=s_(T)+1,\dots,s_c(T)+s_{c+1}(T) we define \mathcal{K}_{c,i}(T)=(k_1,\dots,k_n) with \mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\} .
    For i=s_c(T)+s_{c+1}(T)+1,\dots,s_c(T)+2s_{c+1}(T) we define \mathcal{K}_{c,i}(T)=(k_1+s_{d(t)}(t),\dots,k_n+s_{d(t)}(t)) with \mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\} .
  • T=t^\circ:
    For i=1,\dots,s_{c-1}(T) we define \mathcal{K}_{c,i}(T)=(k_1,\dots,k_n) with \mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\} .
    For i=s_{c-1}(T)+1,\dots,s_{c-1}(T)+s_c(T) we define \mathcal{K}_{c,i}(T)=(k_1,\dots,k_n,s_{d(T)}(T)) with \mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\} .

Given these subsets we define subreference elements E_{c,i}(T)=(S_{c,i}(T),\Phi_{c,i}(T)) of E(T) given by the following mapping \Phi_{c,i}(T,\cdot)=\Phi(S_{c,i}(T),\mathcal{C}_{c,i}(T),\cdot) .

Furthermore we define a numbering of the subreference elements of each subreference element in T. This is the number k=\mathrm{index}_{c,i,cc,ii}(T) for c\in\{0,\dots,d(T)\}, i\in\{1,\dots,s_c(T)\} , and cc\in\{0,\dots,d(S_{c,i})\}, ii\in\{1,\dots,s_{cc}(S_{c,i})\} for which

\Phi_{c,i}(S_{c,i}(T))\circ \Phi_{cc,ii}(S_{cc,ii}(S_{c,i}(T))) = \Phi_{c+cc,k}(T).

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