The variables ξ and η are local variables; for an individual element they are
8
x = ∑ Ni (ξ, η) xi
(91)
i =1
8
∑ Ni (ξ, η) yi .
y=
(92)
i =1
The derivatives N/ ξ and N/ η can be found directly; however, these deriva-
tives must be related to ξ and η in order to integrate eq 74 through 87. This is done
using the chain rule of differentiation and the Jacobian matrix J; the following rela-
tionship is obtained (Huebner and Thornton 1982):
Ni
Ni
x
ξ
= [ J ] -1, i = 1, 2, ... 8.
(93)
Ni
Ni
y
η
Using the relationship dxdy = det J d ξdη, the above area integrals can all be writ-
ten in terms of ξ and η and integrated from 1 to 1 using Gaussian quadrature.
same element as in Figure 9 is used; however, the sides are assumed to be straight
and the element is defined only by nodes 1, 3, 5, and 7. The interpolation func-
tions are
1
(1 + ξξ i ) (1 + ηη i )
p
Ni =
(94)
4
where ξ and η take on their nodal values (Fig. 9 and eq 8890). The evaluation of
the derivatives and integrals follows the same procedure as above.
The surface integral, eq 86, must also be expressed in terms of the parametric
order to simplify programming it is assumed here that the boundary s is made up
of at least one full side of an element; thus from Figure 9, side 1 is described by
nodes 1, 2, and 3; side two is nodes 3, 4, 5; side three is nodes 5, 6, 7; and side 4 is
nodes 7, 8, and 1. In this development no other combinations are allowed; how-
ever, more than one side per element can be specified as a boundary segment. For
each side either ξ and η will be a constant and ds is
1
1
ds =
Ledη or ds = Le dξ
(95)
2
2
where Le is the length of the side. The integral is now evaluated from 1 to 1, using
Gaussian quadrature. The integration of eq 87 is carried out similarly, except that
the term Ni Nj is a two-dimensional matrix.
Solution procedure
The computer model FECOME (Finite Element COMbined Equations, Rich-
mond 1995) solves eq 6871 simultaneously for u, v, T, and p and uses either direct
substitution or the Newton-Raphson iteration procedure. The solution procedure
requires the use of a previous solution (the old solution) or an initial estimate, which
is then used to obtain a new solution. Between iterations, both the direct substitu-
17
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