104
5. Finite Element Methods for Solving Hydrodynamie Dispersion Equations
0
y
FIGURE 5.2. A rectangular element
and its nodes.
j
I
J
k t
~
(-\,1)
0
(-\,-\)
(e)
(1,1)
e
(1,-1)
m
x
FIGURE 5.3. A square element and its nodes in
a loeal eoordinate system.
The next step is the same as that for solving the problems of groundwater
flow, that is, to form the global matrices [A], [H] and vector F through
assembling their components of each element. Thus, the system of Eq. (5.1.12)
is built up. From Eq. (5.1.35), it is c1ear that [A] is an asymmetric matrix
because of the existence of advection terms.
Rectangular Elements and Bilinear Basis Functions
In this case, domain (R) is divided into a net of rectangles, and all grid points
are taken as nodes as they were in the finite difTerence method. Figure 5.2
shows an arbitrary rectangular element (e) with length a and width b. Its four
nodes are numbered as i,i, k, and m, respectively.
Using the transformation
j
e = 2 x ~ x o ,
y- Yo
'1 = 2 - b -,
(5.1.38)
where (xo,Yo) are the coordinates of the central point of the rectangular
element, the element can be transformed into a standard square element in
the local coordinate system, as shown in Figure 5.3.