The Neumann boundary integral (5.29) becomes
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(C.1) |
The assembly of the matrix
with entries
![]() |
(C.2) |
is performed element by element, whereby only
the elements lying on the Neumann boundary are
considered. The elements are tetrahedra and in each element
is represented by the vector edge functions (5.45) to
(5.50).
The integral domain transformation for an arbitrary surface in the three-dimensional space is the same as in Appendix A and is derived in a similar manner. The integral is represented as a sum
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(C.3) |
The surface
is subdivided into pieces
with areas
.
is
point inside
. The transformation is given by
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(C.4) |
An area
is calculated as
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(C.5) |
which leads again to (A.8) and (A.7).
Now the element matrix
can be computed. As there are four outer triangular
faces on each tetrahedral element, there will be four different element matrices for
each face which lies on the Neumann boundary
![]() |
(C.6) |
and using (5.52)
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(C.7) |
is the face opposite to the node
.
is
a constant vector with the characteristic length 1, perpendicular to
and points outwards.
is assumed to be constant
in each element. Only the three
functions
for the edges in the
plane are not perpendicular to
. The remaining three vector functions are perpendicular to
.
Consequently these vectors are parallel to
and the
corresponding vector product
becomes zero.
For the element face
the element matrix is given as follows
![]() |
(C.8) |
![]() |
(C.9) |
The integral is computed using the integral domain transformation discussed above.
![]() |
(C.10) |
![]() |
(C.11) |
Thus the final solution for the fourth edge (
) is given by the expression
![]() |
(C.12) |
The same procedure is used for remaining edges (with index
):
![]() |
(C.13) |
![]() |
(C.14) |
Analogously it is proceeded for the remaining faces.
For the face
:
![]() |
(C.15) |
![]() |
(C.16) |
![]() |
(C.17) |
![]() |
(C.18) |
For the face
:
![]() |
(C.19) |
![]() |
(C.20) |
![]() |
(C.21) |
![]() |
(C.22) |
For the face
:
![]() |
(C.23) |
![]() |
(C.24) |
![]() |
(C.25) |
![]() |
(C.26) |