Let P be a point inside the tetrahedral element as shown in Fig. <4.7>. It divides the tetrahedron in four sub-tetrahedrons
The barycentric coordinates of the point P are given by
(4.66) |
(4.67) |
(4.68) |
(4.69) |
where is the area of the triangular face of the tetrahedron opposite to the vertex
The vector is normal to its according face, has the length and points outwards. The position vector is written as
(4.70) |
and is the position of the vertex .
Analogously to the two-dimensional case it can be shown that the barycentric coordinates are equal to the linear element shape functions in (4.65). Thus the same notation is used.
The gradient of the barycentric coordinates is a constant vector
Analogously to the two-dimensional case it is valid
(4.75) |
The barycentric coordinate is constant on a plane parallel to the element facet opposite to the -th node and it is zero on this opposite facet, which ensures the inter-element continuity of the element interpolation function (4.58).
Only three of the four linear element form functions are independent
For points inside the element and on the element facets
(4.77) |
is satisfied. Similar to the two-dimensional case some of the barycentric coordinates of a point outside the element can be negative or greater than 1.
Otherwise, the barycentric coordinates can be used to represent the coordinates of each point inside the tetrahedral element
Using (4.76), equation (4.78) leads to
which gives the coordinate transformation as shown in Fig. <4.8>.