Reference cells

The reference cells are used to i) define grid cells, ii) define shape functions, and iii) define quadrature rules. The numbering of vertices, edges, faces are visualized below. See also FerriteViz.elementinfo.

`AbstractRefShape` subtypes

Ferrite.AbstractRefShape — Type

AbstractRefShape{refdim}

Supertype for all reference shapes, with reference dimension refdim. Reference shapes are used to define grid cells, shape functions, and quadrature rules. Currently existing reference shapes are: RefLine, RefTriangle, RefQuadrilateral, RefTetrahedron, RefHexahedron, RefPrism.

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Ferrite.RefLine — Type

RefLine <: AbstractRefShape{1}

Reference line/interval, reference dimension 1.

----------------+--------------------
Vertex numbers: | Vertex coordinates:
  1-------2     | v1: 𝛏 = (-1.0,)
    --> ξ₁      | v2: 𝛏 = ( 1.0,)
----------------+--------------------
Face numbers:   | Face identifiers:
  1-------2     | f1: (v1,)
                | f2: (v2,)
----------------+--------------------

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Ferrite.RefTriangle — Type

RefTriangle <: AbstractRefShape{2}

Reference triangle, reference dimension 2.

----------------+--------------------
Vertex numbers: | Vertex coordinates:
    2           |
    | \         | v1: 𝛏 = (1.0, 0.0)
    |   \       | v2: 𝛏 = (0.0, 1.0)
ξ₂^ |     \     | v3: 𝛏 = (0.0, 0.0)
  | 3-------1   |
  +--> ξ₁       |
----------------+--------------------
Face numbers:   | Face identifiers:
    +           |
    | \         | f1: (v1, v2)
    2   1       | f2: (v2, v3)
    |     \     | f3: (v3, v1)
    +---3---+   |
----------------+--------------------

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Ferrite.RefQuadrilateral — Type

RefQuadrilateral <: AbstractRefShape{2}

Reference quadrilateral, reference dimension 2.

----------------+---------------------
Vertex numbers: | Vertex coordinates:
    4-------3   |
    |       |   | v1: 𝛏 = (-1.0, -1.0)
    |       |   | v2: 𝛏 = ( 1.0, -1.0)
ξ₂^ |       |   | v3: 𝛏 = ( 1.0,  1.0)
  | 1-------2   | v4: 𝛏 = (-1.0,  1.0)
  +--> ξ₁       |
----------------+---------------------
Face numbers:   | Face identifiers:
    +---3---+   | f1: (v1, v2)
    |       |   | f2: (v2, v3)
    4       2   | f3: (v3, v4)
    |       |   | f4: (v4, v1)
    +---1---+   |
----------------+---------------------

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Ferrite.RefTetrahedron — Type

RefTetrahedron <: AbstractRefShape{3}

Reference tetrahedron, reference dimension 3.

---------------------------------------+-------------------------
Vertex numbers:                        | Vertex coordinates:
             4                4        |
  ^ ξ₃      /  \             /| \      |  v1: 𝛏 = (0.0, 0.0, 0.0)
  |        /     \          / |   \    |  v2: 𝛏 = (1.0, 0.0, 0.0)
  +-> ξ₂  /        \       /  1___  \  |  v3: 𝛏 = (0.0, 1.0, 0.0)
 /       /      __--3     / /    __‾-3 |  v4: 𝛏 = (0.0, 0.0, 1.0)
ξ₁      2 __--‾‾         2/__--‾‾      |
---------------------------------------+-------------------------
Edge numbers:                          | Edge identifiers:
             +                +        | e1: (v1, v2)
            /  \             /| \      | e2: (v2, v3)
         5 /     \ 6      5 / |4  \ 6  | e3: (v3, v1)
          /        \       /  +__3  \  | e4: (v1, v4)
         /      __--+     / /1   __‾-+ | e5: (v2, v4)
        + __--‾‾2        +/__--‾‾2     | e6: (v3, v4)
---------------------------------------+-------------------------
Face numbers:                          | Face identifiers:
             +                +        |
            /  \             /| \      | f1: (v1, v3, v2)
           /     \          / | 4 \    | f2: (v1, v2, v4)
          /   3    \       /2 +___  \  | f3: (v2, v3, v4)
         /      __--+     / /  1 __‾-+ | f4: (v1, v4, v3)
        + __--‾‾         +/__--‾‾      |
---------------------------------------+-------------------------

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Ferrite.RefHexahedron — Type

RefHexahedron <: AbstractRefShape{3}

Reference hexahedron, reference dimension 3.

-----------------------------------------+----------------------------
Vertex numbers:                          | Vertex coordinates:
            5--------8        5--------8 | v1: 𝛏 = (-1.0, -1.0, -1.0)
           /        /|       /|        | | v2: 𝛏 = ( 1.0, -1.0, -1.0)
          /        / |      / |        | | v3: 𝛏 = ( 1.0,  1.0, -1.0)
  ^ ξ₃   6--------7  |     6  |        | | v4: 𝛏 = (-1.0,  1.0, -1.0)
  |      |        |  4     |  1--------4 | v5: 𝛏 = (-1.0, -1.0,  1.0)
  +-> ξ₂ |        | /      | /        /  | v6: 𝛏 = ( 1.0, -1.0,  1.0)
 /       |        |/       |/        /   | v7: 𝛏 = ( 1.0,  1.0,  1.0)
ξ₁       2--------3        2--------3    | v8: 𝛏 = (-1.0,  1.0,  1.0)
-----------------------------------------+-----------------------------
Edge numbers:                            | Edge identifiers:
            +----8---+        +----8---+ |
          5/        /|      5/|        | |  e1: (v1, v2),  e2: (v2, v3)
          /       7/ |12    / |9     12| |  e3: (v3, v4),  e4: (v4, v1)
         +----6---+  |     +  |        | |  e5: (v5, v6),  e6: (v6, v7)
         |        |  +     |  +---4----+ |  e7: (v7, v8),  e8: (v8, v5)
       10|      11| /    10| /1       /  |  e9: (v1, v5), e10: (v2, v6)
         |        |/3      |/        /3  | e11: (v3, v7), e12: (v4, v8)
         +---2----+        +---2----+    |
-----------------------------------------+-----------------------------
Face numbers:                            | Face identifiers:
            +--------+        +--------+ |
           /   6    /|       /|        | |  f1: (v1, v4, v3, v2)
          /        / |      / |   5    | |  f2: (v1, v2, v6, v5)
         +--------+ 4|     +  |        | |  f3: (v2, v3, v7, v6)
         |        |  +     |2 +--------+ |  f4: (v3, v4, v8, v7)
         |    3   | /      | /        /  |  f5: (v1, v5, v8, v4)
         |        |/       |/    1   /   |  f6: (v5, v6, v7, v8)
         +--------+        +--------+    |
-----------------------------------------+-----------------------------

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Ferrite.RefPrism — Type

RefPrism <: AbstractRefShape{3}

Reference prism, reference dimension 3.

-----------------------------------------+----------------------------
Vertex numbers:                          | Vertex coordinates:
            4-------/6       4--------6  |
           /     /   |      /|        |  |  v1: 𝛏 = (0.0, 0.0, 0.0)
          /   /      |     / |        |  |  v2: 𝛏 = (1.0, 0.0, 0.0)
  ^ ξ₃   5 /         |    5  |        |  |  v3: 𝛏 = (0.0, 1.0, 0.0)
  |      |          /3    |  1-------/3  |  v4: 𝛏 = (0.0, 0.0, 1.0)
  +-> ξ₂ |       /        | /     /      |  v5: 𝛏 = (1.0, 0.0, 1.0)
 /       |    /           |/   /         |  v6: 𝛏 = (0.0, 1.0, 1.0)
ξ₁       2 /              2 /            |
-----------------------------------------+----------------------------
Edge numbers:                            | Edge identifiers:
            +---8---/+       +---8----+  |
          7/     /   |     7/|        |  | e1: (v2, v1),  e2: (v1, v3)
          /   / 9    |6    / |3       |6 | e3: (v1, v4),  e4: (v3, v2)
         + /         |    +  |        |  | e5: (v2, v5),  e6: (v3, v6)
         |          /+    |  +--2----/+  | e7: (v4, v5),  e8: (v4, v6)
        5|       /       5| /1    /      | e9: (v6, v5)
         |    / 4         |/   / 4       |
         + /              + /            |
-----------------------------------------+----------------------------
Face numbers:                            | Face identifiers:
            +-------/+       +--------+  |
           /  5  /   |      /|        |  | f1: (v1, v3, v2)
          /   /      |     / |    3   |  | f2: (v1, v2, v5, v4)
         + /         |    +  |        |  | f3: (v3, v1, v4, v6)
         |     4    /+    |2 +-------/+  | f4: (v2, v3, v6, v5)
         |       /        | /  1  /      | f5: (v4, v5, v6)
         |    /           |/   /         |
         + /              + /            |
-----------------------------------------+----------------------------

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Required methods to implement for all subtypes of `AbstractRefShape` to define a new reference shape

Ferrite.reference_vertices — Method

reference_vertices(::Type{<:AbstractRefShape})
reference_vertices(::AbstractCell)

Returns a tuple of integers containing the local node indices corresponding to the vertices (i.e. corners or endpoints) of the cell.

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Ferrite.reference_edges — Method

reference_edges(::Type{<:AbstractRefShape})
reference_edges(::AbstractCell)

Returns a tuple of 2-tuples containing the ordered local node indices (corresponding to the vertices) that define an edge.

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Ferrite.reference_faces — Method

reference_faces(::Type{<:AbstractRefShape})
reference_faces(::AbstractCell)

Returns a tuple of n-tuples containing the ordered local node indices (corresponding to the vertices) that define a face.

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which automatically defines

Ferrite.reference_facets — Method

Ferrite.reference_facets(::Type{<:AbstractRefShape})
Ferrite.reference_facets(::AbstractCell)

Returns a tuple of n-tuples containing the ordered local node indices (corresponding to the vertices) that define a facet.

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Applicable methods to `AbstractRefShape`s

Ferrite.getrefdim — Method

Ferrite.getrefdim(RefShape::Type{<:AbstractRefShape})

Get the dimension of the reference shape

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Reference cells

AbstractRefShape subtypes

Required methods to implement for all subtypes of AbstractRefShape to define a new reference shape

Applicable methods to AbstractRefShapes

`AbstractRefShape` subtypes

Required methods to implement for all subtypes of `AbstractRefShape` to define a new reference shape

Applicable methods to `AbstractRefShape`s