Refer to the tetrahedron from Fig. <4.7>.
The vector field in the tetrahedral element is interpolated by
means of the Whitney
-form basis function
associated to the element edge
[100].
For example along Edge
between Node
and Node
this function is given by
![]() |
This approach was introduced first by Whitney [101].
The functions
are the barycentric coordinates from Subsection
4.2.2. If only one
subscript index is used for the vector basis functions (for example
),
this index means an edge number (for example Edge 1). If two
subscript indexes are used (for example
), these indexes
comply with the corresponding node numbers (for example Edge 1 belongs
to the node pair Node
and Node
). The edge indexes are associated to node
pairs in Table 5.1.
Furthermore the following notation is used
![]() |
![]() |
Since
vanishes on the facet defined by
and
is perpendicular to that facet,
has no tangential component on this facet.
Analogously it can be stated that
has no
tangential component on the facet defined by
.
Thus the edge function
has no tangential
component along the edges different from the Edge
and it contributes only to their normal components.
This is also valid for the Whitney vector basis
functions
on the remaining element edges.
For the divergence and for the rotor of the Whitney function
it is written
![]() |
![]() |
![]() |
In direction parallel to Edge
and
are constant. On Edge
and
are zero.
With (4.76)
are also
constant parallel to the edge
. On Edge
. Consequently parallel to Edge
![]() |
(5.43) |
![]() |
(5.44) |
Thus, the continuity of the tangential component along the element edge across
the elements is guaranteed. Otherwise the normal component must not be continuous.
Exactly this behavior is required for the fields
and
(if no surface current is available) and for the magnetic vector potential
.
Thus, the element vector basis functions for the field associated with the edges
are given by the very well suited
-form Whitney functions
![]() |
(5.46) |
![]() |
(5.47) |
![]() |
(5.48) |
![]() |
(5.49) |
The performed multiplication with
delivers the normalized and dimensionless edge functions
.
The edge functions
and
are visualized in Fig. <5.5>
and Fig. <5.6>, respectively.
If the field in the tetrahedral element (for instance the magnetic field
) is given by
![]() |
(5.51) |
the coefficient
corresponds to the tangential component
of the field
on the Edge
. An example of the field in
the element is depicted in Fig. <5.7> where all
coefficients are set to
.