Reference Element

Reference elements play a crucial role in finite element methods. In finite element space domain is discretized by using simple elements such as line, triangle, quadrangle, tetrahedron, hexahedron, prism, pyramid, etc. The collection of these elements is called the mesh. The elements of mesh are called physical elements. The physical elements can have different size and orientation. Therefore, we usually do not construct basis functions on these elements. Instead, we map these elements to "nicer" element. These nicer looking elements will be called the reference elements or master elements.

Reference Line element

The reference element for line element can be:

• unit line
• bi-unit line

note

The unit line is given by $[0, 1]$, and the biunit line is given by $[-1, 1]$. The reference line has two end points.

Reference Triangle element

The reference element for triangle is given by a right triangle (i.e., a triangle in which one angle is a right angle).

The reference triangle has

• Three nodes (vertices)
• Three edges

The reference triangle can be unit triangle or biunit triangle.

The coordinates of a unit triangle are given by:

vertex	x	y	z
1	0	0	0
2	1	0	0
3	0	1	0

The area of a unit triangle is 0.5.

The coordinates of a biunit triangle are given by:

vertex	x	y	z
1	-1	-1	0
2	1	-1	0
3	-1	1	0

The area of a biunit triangle is 2.0.

Reference Quadrangle element

Reference Tetrahedron element

Reference Hexahedron element

Reference Prism element

Reference Pyramid element

The reference element

Line

Triangle

Quadrangle

Hexahedron

Tetrahedron

Prism

Pyramid