FEMDocumentation`
FEMDocumentation`

# 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

• 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: