5.3. Finite Element Methods for Three-Dimensional Problems
141
Using Eq. (5.3.37), the value of the second integral on the right-hand side
of Eq. (5.3.38) can be calculated for element (g). After some elementary integration, we have
ff r Va a
ac dR = t f?r~:'
J~O)
x"
,=1
(5.3.40)
where (ße) is the part of the exclusive subdomain of node P in (g). In the
above equation
For other Q, (r = 2,3, ... ,6), we have similar expressions.
Using tlie approximate expression of ac/at in Eq. (5.3.37), the value of the
integral on the left-hand side of Eq. (5.3.38) can be obtained for element (g). It
also has the form of a summation of weighted nodal concentrations, that is,
(5.3.41)
(L\z) 11
(L\z) 7
where 111 = 12· 36L\, 112 = 12· 36L\, and so on.
Now, let us consider the calculation of each term of Eq. (5.3.38) for the
upper part of the exclusive subdomain of anode. Assurne that (e) is an
element with node P as its vertex and located above P. The part of the
exclusive subdomain of node P in this element is marked as (Re) and the
relevant boundary surface is (se) as in Figure 5.24. Node P in element (e)
corresponds to node 1. By the same method as the one mentioned above, it
can be calculated as fi
ac
f -_ D"p-a n"dS = L. T,C;,
(SO)
Xp
,=1
FIGURE 5.24. The part (Re) of the exclusive
subdomain of node P in element (e) above P
(node P is just node 1 in this element).
P
(5.3.42)