NDSolve`FEM`
NDSolve`FEM`

# HexahedronElement

HexahedronElement[{{i11,,i18},,{in1,,in8}}]

represents n linear hexahedron elements ek with incidents {ik1,ik8}.

HexahedronElement[{{i11,,i120},,{in1,,in20}}]

represents n quadratic hexahedron elements ek with incidents {ik1,,ik20}.

HexahedronElement[{e1,,en},{m1,,mn}]

represents n hexahedron elements ek and n integer markers mk.

# Details and Options

• HexahedronElement is used to represent hexahedron mesh elements in ElementMesh.
• HexahedronElement 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 until ik8 are always vertices.
• For quadratic triangle elements, the next 12 incidents are mid-side nodes of possibly curved edges.
• Linear elements are order 1 elements and quadratic elements are order 2 elements.
• In HexahedronElement[{e1,,en}], all elements ek need to be of the same order.
• The hexahedra in HexahedronElement[{e1,,en}] will share common nodes, edges, and faces, but cannot intersect with each other or with themselves.
• The nodes for a linear and a quadratic hexahedron are shown:
• For a HexahedronElement, the incidents on the faces must be counterclockwise when viewed from inside the element. An element {i1,,i8} has face incidents {i1,i2,i3,i4}, {i8,i7,i6,i5}, {i1,i5,i6,i2}, {i2,i6,i7,i3}, {i3,i7,i8,i4}, and {i4,i8,i5,i1} for the six faces.
• The hexahedron element is known in the finite element method as a Serendipity element.

# Examples

open allclose all

## Basic Examples(1)

Create a mesh with one quad element:

## 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(1)

A linear hexahedron element mesh with markers:

Visualizing the index of the coordinates at their respective positions:

Create the mesh:

Visualize the mesh with the element 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), HexahedronElement, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html.

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_hexahedronelement, author="Wolfram Research", title="{HexahedronElement}", year="2014", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html}", note=[Accessed: 07-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hexahedronelement, organization={Wolfram Research}, title={HexahedronElement}, year={2014}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/HexahedronElement.html}, note=[Accessed: 07-August-2024 ]}