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5. Finite Element Methods for Solving Hydrodynamie Dispersion Equations
3
FIGURE 5.4. A quadratie triangle.
6
5
~----------~4----------~2
2
FIGURE 5.5. A standard quadratie triangle in a loeal eoordinate system.
values on six points are given in eaeh element, the eonditions will be satisfied.
Therefore, we ean designate six nodes in eaeh tri angle element, whieh are
respeetively the three vertiees and three middle points on the three sides, as
shown in Figure 5.4.
Through the transformation
{
X = Xl ~ + X21J + x 3 (1 - ~ - 1J),
(5.1.42)
Y = Yl ~ + Y21J + Y3(1 - ~ - 1J),
the triangle (.1) in the xy plane as shown in Figure 5.4 beeomes a standard
triangle (.1') in the ~1J plane in a loeal eoordinate system, as shown in Figure
5.5.
In the standard element (.1'), the eomplete quadratie funetion of ~, 1J has
the following form.
r/J(~, 1J) = a + b~ + C1J + d~2 + e~1J + /1J2.
(5.1.43)
Ifthe value of quadratie basis funetion r/Ji(~' 1J) assoeiated with node i within
element (.1') is required to be 1 at node i and zero at the other five points, we
then have six eonditions. With the six eonditions, the six eoeffieients of r/Ji(~' 1J)
ean be determined, as listed in Table 5.1.