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NDSolve`FEM`
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TetrahedronElement

TetrahedronElement[{{i11,i12,i13,i14},…,{in1,in2,in3,in4}}]

represents n linear tetrahedron elements ek with incidents {ik1,ik2,ik3,ik4}.

TetrahedronElement[{{i11,…,i110},…,{in1,…,in10}}]

represents n quadratic tetrahedron elements ek with incidents {ik1,…,ik10}.

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

represents n tetrahedron elements ek and n integer markers mk.

Details and Options

  • TetrahedronElement is used to represent tetrahedron mesh elements in ElementMesh.
  • TetrahedronElement can be used as an input to ToElementMesh.
  • 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) tetrahedron.
  • The first four incidents ik1, ik2, ik3, and ik4 are always vertices.
  • For quadratic tetrahedron elements, the next six incidents are mid-side nodes of possibly curved edges.
  • Linear elements are order 1 elements and quadratic elements are order 2 elements.
  • In TetrahedronElement[{e1,…,en}], all elements ek need to be of the same order.
  • The tetrahedra in TetrahedronElement[{e1,…,en}] will share common nodes, edges, and faces but cannot intersect with each other, or for second-order tetrahedra, with themselves.
  • The nodes for a linear and a quadratic tetrahedra are shown:
  • For a TetrahedronElement, the face incidents opposite a vertex ij must be counterclockwise. An element {i1,i2,i3,i4} has the face incidents {i4,i3,i2}, {i4,i1,i3}, {i4,i2,i1}, and {i1,i2,i3} for the four faces.
  • The tetrahedron 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/0eg8uvsh6he8azj042i2piux1fjc5ea-h9vte9

Create a mesh with one tetrahedron element:

In[2]:=2
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-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/0eg8uvsh6he8azj042i2piux1fjc5ea-bdlhqr
Out[1]=1

The base incidents of the linear element:

In[2]:=2
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-e986sw
Out[2]=2

A mesh with a linear unit element:

In[3]:=3
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-yi7d7y
Out[3]=3

Visualization of the linear unit element:

In[4]:=4
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-gr7kca
Out[4]=4

The base coordinates of the quadratic element:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-xkd20r
Out[1]=1

The base incidents of the quadratic element:

In[2]:=2
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-4b03mm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-12438n
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-79tbf8
Out[4]=4

The base face incidents of the linear element:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-gf6ti
Out[1]=1

The base face incidents of the quadratic element:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-j7cpks
Out[1]=1

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

A set of linear tetrahedron elements mesh with markers:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-y7m4zj

Visualizing the index of the coordinates at their respective positions:

In[2]:=2
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-vf52b6
Out[2]=2

Create the mesh:

In[3]:=3
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-5ksxcf
Out[3]=3

Visualize the mesh with the elements' markers:

In[4]:=4
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-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/0eg8uvsh6he8azj042i2piux1fjc5ea-lvcvm4
Out[1]=1

The incident order cannot be mixed:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-29oxt4
Out[1]=1

The incidents must be lists of integers:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-cm1i73
Out[1]=1

The number of markers must match the number of incidents:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-3ab5ie
Out[1]=1

Markers must be a vector of integers:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-neqwpm
Out[1]=1

When possible, noninteger markers will be converted to integers:

In[1]:=1
✖
https://wolfram.com/xid/0eg8uvsh6he8azj042i2piux1fjc5ea-yp4gm6
Out[1]=1
Wolfram Research (2014), TetrahedronElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/TetrahedronElement.html.
✖
Wolfram Research (2014), TetrahedronElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/TetrahedronElement.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_tetrahedronelement, author="Wolfram Research", title="{TetrahedronElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/TetrahedronElement.html}", note=[Accessed: 24-May-2025 ]}

✖
@misc{reference.wolfram_2025_tetrahedronelement, author="Wolfram Research", title="{TetrahedronElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/TetrahedronElement.html}", note=[Accessed: 24-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_tetrahedronelement, organization={Wolfram Research}, title={TetrahedronElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/TetrahedronElement.html}, note=[Accessed: 24-May-2025 ]}

✖
@online{reference.wolfram_2025_tetrahedronelement, organization={Wolfram Research}, title={TetrahedronElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/TetrahedronElement.html}, note=[Accessed: 24-May-2025 ]}