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NDSolve`FEM`
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TriangleElement

TriangleElement[{{i11,i12,i13},…,{in1,in2,in3}}]

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

TriangleElement[{{i11,…,i16},…,{in1,…,in6}}]

represents n quadratic triangle elements ek with incidents {ik1,…,ik6}.

TriangleElement[{e1,…,en},{m1,…,mn}]

represents n triangle elements ek and n integer markers mk.

Details and Options

  • TriangleElement is used to represent triangle mesh elements in ElementMesh.
  • TriangleElement 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, ik2, and ik3 are always vertices.
  • For quadratic triangle elements, the next three incidents are mid-side nodes of possibly curved edges.
  • Linear elements are order 1 elements and quadratic elements are order 2 elements.
  • In TriangleElement[{e1,…,en}], all elements ek need to be of the same order.
  • The triangles in TriangleElement[{e1,…,en}] will share common nodes and edges but cannot intersect with each other, or for second order triangles, with themselves.
  • The nodes for a linear and a quadratic triangle are shown:
  • The incidents {i1,i2,i3} must be ordered so that going from the coordinates referenced by i1 to i2 to i3 is in the counterclockwise direction.
  • Typically, TriangleElement is used for two-dimensional regions, but may be embedded in three dimensions, for example, as a part of a boundary mesh.
  • The triangle 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
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https://wolfram.com/xid/0md07wcqj294skxml1ma9u-h9vte9

Create a mesh with one triangle element:

In[2]:=2
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-jmidlk
Out[2]=2

Scope  (1)Survey of the scope of standard use cases

A boundary mesh with triangle elements:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-tuj67q
Out[1]=1

Visualize the wireframe:

In[2]:=2
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-685ujy
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/0md07wcqj294skxml1ma9u-bdlhqr
Out[1]=1

The base incidents of the linear element:

In[2]:=2
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-e986sw
Out[2]=2

A mesh with a linear unit element:

In[3]:=3
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https://wolfram.com/xid/0md07wcqj294skxml1ma9u-yi7d7y
Out[3]=3

Visualization of the linear unit element:

In[4]:=4
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-gr7kca
Out[4]=4

The base coordinates of the quadratic element:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-xkd20r
Out[1]=1

The base incidents of the quadratic element:

In[2]:=2
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-4b03mm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-12438n
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-79tbf8
Out[4]=4

The base face incidents of the linear element:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-gf6ti
Out[1]=1

The base face incidents of the quadratic element:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-j7cpks
Out[1]=1

Applications  (3)Sample problems that can be solved with this function

Linear triangle elements in a mesh:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-iqxiai
Out[1]=1

Visualize the mesh with the elements' vertices:

In[2]:=2
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-810cbo
Out[2]=2

Quadratic triangle elements in a mesh:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-uctnw6
In[2]:=2
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-dh1uqk
Out[2]=2

Visualize the mesh with the elements' vertices:

In[3]:=3
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-ihw9kv
Out[3]=3

A triangle element mesh with markers:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-y7m4zj
In[2]:=2
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-5ksxcf
Out[2]=2

Visualize the mesh with the elements' markers:

In[3]:=3
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-fr0s8d
Out[3]=3

Possible Issues  (6)Common pitfalls and unexpected behavior

The incidents must be of the appropriate length:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-lvcvm4
Out[1]=1

The incidents order cannot be mixed:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-29oxt4
Out[1]=1

The incidents must be lists of integers:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-cm1i73
Out[1]=1

The number of markers must match the number of incidents:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-3ab5ie
Out[1]=1

Markers must be a vector of integers:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-neqwpm
Out[1]=1

When possible, noninteger markers will be converted to integers:

In[1]:=1
✖
https://wolfram.com/xid/0md07wcqj294skxml1ma9u-yp4gm6
Out[1]=1
Wolfram Research (2014), TriangleElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html.
✖
Wolfram Research (2014), TriangleElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_triangleelement, author="Wolfram Research", title="{TriangleElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html}", note=[Accessed: 26-March-2025 ]}

✖
@misc{reference.wolfram_2025_triangleelement, author="Wolfram Research", title="{TriangleElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_triangleelement, organization={Wolfram Research}, title={TriangleElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_triangleelement, organization={Wolfram Research}, title={TriangleElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html}, note=[Accessed: 26-March-2025 ]}