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NDSolve`FEM`
NDSolve`FEM`

HexahedronElement

HexahedronElement[{{i11,…,i18},…,{in1,…,in8}}]

represents n linear hexahedron elements ek with incidents {ik1,…ik8}.

HexahedronElement[{{i11,…,i120},…,{in1,…,in20}}]

represents n quadratic hexahedron elements ek with incidents {ik1,…,ik20}.

HexahedronElement[{e1,…,en},{m1,…,mn}]

represents n hexahedron elements ek and n integer markers mk.

Details and Options

  • HexahedronElement is used to represent hexahedron mesh elements in ElementMesh.
  • HexahedronElement can be used as an input to ToElementMesh or ToBoundaryMesh.
  • Incidents ik,j are integers that index an array of spatial coordinates. The coordinates referenced by ek={ik1,…} are the nodes of the k^(th) triangle.
  • The first three incidents ik1 until ik8 are always vertices.
  • For quadratic triangle elements, the next 12 incidents are mid-side nodes of possibly curved edges.
  • Linear elements are order 1 elements and quadratic elements are order 2 elements.
  • In HexahedronElement[{e1,…,en}], all elements ek need to be of the same order.
  • The hexahedra in HexahedronElement[{e1,…,en}] will share common nodes, edges, and faces, but cannot intersect with each other or with themselves.
  • The nodes for a linear and a quadratic hexahedron are shown:
  • For a HexahedronElement, the incidents on the faces must be counterclockwise when viewed from inside the element. An element {i1,…,i8} has face incidents {i1,i2,i3,i4}, {i8,i7,i6,i5}, {i1,i5,i6,i2}, {i2,i6,i7,i3}, {i3,i7,i8,i4}, and {i4,i8,i5,i1} for the six faces.
  • The hexahedron element is known in the finite element method as a Serendipity element.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Load the package:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-h9vte9

Create a mesh with one quad element:

In[2]:=2
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-jmidlk
Out[2]=2

Generalizations & Extensions  (4)Generalized and extended use cases

The base coordinates of the linear element:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-bdlhqr
Out[1]=1

The base incidents of the linear element:

In[2]:=2
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-e986sw
Out[2]=2

A mesh with a linear unit element:

In[3]:=3
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-yi7d7y
Out[3]=3

Visualization of the linear unit element:

In[4]:=4
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-gr7kca
Out[4]=4

The base coordinates of the quadratic element:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-xkd20r
Out[1]=1

The base incidents of the quadratic element:

In[2]:=2
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-4b03mm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-12438n
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-79tbf8
Out[4]=4

The base face incidents of the linear element:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-gf6ti
Out[1]=1

The base face incidents of the quadratic element:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-j7cpks
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

A linear hexahedron element mesh with markers:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-vflsr7

Visualizing the index of the coordinates at their respective positions:

In[2]:=2
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-vf52b6
Out[2]=2

Create the mesh:

In[3]:=3
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-4ufzk
Out[3]=3

Visualize the mesh with the element markers:

In[4]:=4
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-fr0s8d
Out[4]=4

Possible Issues  (6)Common pitfalls and unexpected behavior

The incidents must be of the appropriate length:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-lvcvm4
Out[1]=1

The incidents order cannot be mixed:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-29oxt4
Out[1]=1

The incidents must be lists of integers:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-cm1i73
Out[1]=1

The number of markers must match the number of incidents:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-3ab5ie
Out[1]=1

Markers must be a vector of integers:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-neqwpm
Out[1]=1

When possible, noninteger markers will be converted to integers:

In[1]:=1
✖
https://wolfram.com/xid/0cm68ddr75ate6h0w7i78wo23547ki-yp4gm6
Out[1]=1
Wolfram Research (2014), HexahedronElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html.
✖
Wolfram Research (2014), HexahedronElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html.

Text

Wolfram Research (2014), HexahedronElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html.

✖
Wolfram Research (2014), HexahedronElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html.

CMS

Wolfram Language. 2014. "HexahedronElement." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html.

✖
Wolfram Language. 2014. "HexahedronElement." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html.

APA

Wolfram Language. (2014). HexahedronElement. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html

✖
Wolfram Language. (2014). HexahedronElement. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html

BibTeX

@misc{reference.wolfram_2025_hexahedronelement, author="Wolfram Research", title="{HexahedronElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html}", note=[Accessed: 26-March-2025 ]}

✖
@misc{reference.wolfram_2025_hexahedronelement, author="Wolfram Research", title="{HexahedronElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hexahedronelement, organization={Wolfram Research}, title={HexahedronElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_hexahedronelement, organization={Wolfram Research}, title={HexahedronElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html}, note=[Accessed: 26-March-2025 ]}