NDSolve`FEM`
NDSolve`FEM`

QuadElement

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

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

QuadElement[{{i11,…,i18},…,{in1,…,in8}}]

represents n quadratic quad elements ek with incidents {ik1,…,ik8}.

QuadElement[{e1,…,en},{m1,…,mn}]

represents n quad elements ek and n integer markers mk.

Details and Options

  • QuadElement is used to represent quad mesh elements in ElementMesh.
  • QuadElement 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) quad.
  • The first four incidents ik1, ik2, ik3, and ik4 are always vertices.
  • For quadratic quad elements, the next four incidents are mid-side nodes of possibly curved edges.
  • Linear elements are order 1 elements and quadratic elements are order 2 elements.
  • In QuadElement[{e1,…,en}], all elements ek need to be of the same order.
  • The quads in QuadElement[{e1,…,en}] will share common nodes and edges but cannot intersect with each other or with themselves.
  • The nodes for a linear and a quadratic quad are shown:
  • The incidents {i1,i2,i3,i4} must be ordered so that going from the coordinates referenced by i1 to i2 to i3 to i4 is in the counterclockwise direction.
  • Typically, QuadElement is used for two-dimensional regions, but may be embedded in three dimensions, for example, as a part of a boundary mesh.
  • The quad 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 quad element:

Scope  (1)

A boundary mesh with quad 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 quad elements in a mesh:

Visualize the mesh with the elements vertices.

Quadratic quad elements in a mesh:

Visualize the mesh with the elements' vertices:

A mixed quad and 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), QuadElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/QuadElement.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_quadelement, author="Wolfram Research", title="{QuadElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/QuadElement.html}", note=[Accessed: 03-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_quadelement, organization={Wolfram Research}, title={QuadElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/QuadElement.html}, note=[Accessed: 03-December-2024 ]}