NDSolve`FEM`
NDSolve`FEM`

# 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 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)

Create a mesh with one triangle element:

## Scope(1)

A boundary mesh with triangle elements:

Visualize the wireframe:

## Generalizations & Extensions(4)

The base coordinates of the linear element:

The base incidents of the linear element:

A mesh with a linear unit element:

Visualization of the linear unit element:

The base coordinates of the quadratic element:

The base incidents of the quadratic element:

The base face incidents of the linear element:

The base face incidents of the quadratic element:

## Applications(3)

Linear triangle elements in a mesh:

Visualize the mesh with the elements' vertices:

Quadratic triangle elements in a mesh:

Visualize the mesh with the elements' vertices:

A triangle element mesh with markers:

Visualize the mesh with the elements' markers:

## Possible Issues(6)

The incidents must be of the appropriate length:

The incidents order cannot be mixed:

The incidents must be lists of integers:

The number of markers must match the number of incidents:

Markers must be a vector of integers:

When possible, noninteger markers will be converted to integers:

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.

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2024_triangleelement, author="Wolfram Research", title="{TriangleElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html}", note=[Accessed: 22-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_triangleelement, organization={Wolfram Research}, title={TriangleElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/TriangleElement.html}, note=[Accessed: 22-May-2024 ]}