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^(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)

Load the package:

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: 05-November-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: 05-November-2024 ]}