5.2. The Multiple Cell Balance Method
117
tion of each point can be represented approximately by linear interpolation
of its nodal values, i.e.,
C(x, y, t) = (Mx, y)C;(t) + (x,y) E (e),
where interpolation functions
1
e
(5.2.4)
(5.2.5)
are just the basis functions of linear FEM (see Eq. (5.1.32)). C;(t), Cj(t), Ck(t)
in Eq. (5.2.4) are the solute concentrations of nodes i, j, k at time t, respectively. Coefficients a" b" and c, in the above equation are given by Eq.
(5.1.33), ~e is the area of element (e).
From Eq. (5.2.4) we can directly obtain the following expressions for element (e):
OC
1
ox = 2~e (biC; + bjCj + bkCk),
(5.2.6)
oC
1
a = 2~ (CiCi + CjCj + CkCk),
Y
e
(5.2.7)
oC = ot
I ot
J ot
ot
(5.2.8)
Saturated thickness m, head hand velocity components V x , V, can also be
represented by the same linear interpolation functions defined in Eq. (5.2.5).
Now let us consider all the elements with node i as one of their vertices.
Link up the center and the midpoints of the sides of each triangle to form a
subdomain surrounding node i (see Figure 5.10). This area is called the
exclusive subdomain of node i and is taken as the domain (D) in Eq. (5.2.3).
The part of element (e) in domain (D), i.e., the quadrilateral iACB in Figure
5.10, is denoted by (e;). Substituting Eqs. (5.2.6) and (5.2.7) into the first line
FIGURE 5.10. The exclusive subdomain for node i.