For the following explanation of edge elements exclusively barycentric
coordinates will be used, because they offer advantages in understanding the following
calculations. A point
inside the triangle
(Fig. <4.1>) divides this
triangle into three sub triangles, namely
,
, and
, with the corresponding
areas of these sub triangles
,
and
. The barycentric
coordinate
is the ratio of the area
of the sub-face opposite
to the i-th node to the whole area
. Thus the barycentric coordinates are given
by the following definitions (see Fig. <4.1>)
where the following expressions have been used
![]() |
(4.33) |
![]() |
(4.34) |
![]() |
(4.35) |
,
and
are the areas of the triangles
,
and
,
respectively
![]() |
(4.36) |
This definition is very helpful for understanding of some of the following properties of the barycentric coordinates or linear element form functions, respectively.
The reason for the introduction of the barycentric coordinates at this place
is, that they are identical to the linear triangular element shape functions.
For instance, the expression for the barycentric coordinate
is given by
![]() |
(4.37) |
which complies with the first linear triangular element form function in
(4.29). Thus the linear triangular barycentric
coordinates and element shape functions use the same notation
.
The barycentric coordinate
is constant along a line parallel to the element
edge opposite to the
-th node and is zero on the opposite edge.
Two barycentric coordinates are sufficient to
determine the position of the point
inside the triangle (see Fig. <4.1>).
However, a third barycentric coordinate is introduced such that
Only two of the three linear element form functions are independent.
For the points inside the element and on the element edges it holds
![]() |
(4.39) |
For points outside the element the barycentric coordinates can take
arbitrary values. Of coarse (4.38) must be
fulfilled. At least one and at most two barycentric coordinates
must have negative values. This is illustrated in the example (Fig. <4.2>),
where
and
are negative and
is
greater than one.
For instance, this relation can be used to determine, if a point
lies inside or outside a given element.
If there is at least one negative barycentric
coordinate, the point lies outside. If one or two barycentric
coordinates are zero, the point lies on the corresponding
one or two edges of the element. Otherwise, if all are positive
the point is inside.
Points located at a vertex of the element satisfy
In the calculation of the finite elements also the gradients are needed. In barycentric coordinates the gradients are constant and expressed as
![]() |
(4.42) |
The coordinates of each point in the element are determined by
where
,
and
are the barycentric coordinates
of the point inside the triangle element and
,
and
are the coordinates of the element vertices. Using (4.38) the
expression (4.43) can be written as
Equations (4.44) transform
the local coordinates
and
to the
global
coordinate system. Reversely each global element
is mapped to the unit triangle in the
plane (Fig. <4.3>).
With the properties mentioned above the barycentric coordinates guarantee the
continuity of the element interpolation function
(4.29) across the element sides. For instance,
on the Side
the barycentric coordinates are restricted by
![]() |
Thus the element interpolation function
(4.29) on the Side
is given by
![]() |
(4.45) |
depends only on the position
of
on the Edge
. Thus for the two
neighbor elements, which share the Edge
, the corresponding
element interpolation functions will be equal for the
same position on Edge
.