Page 97 - Process Modelling and Simulation With Finite Element Methods

84 Process Modelling and Simulation with Finite Element Methods

Therefore we have

2
- +
i+j+l i+j+2 i+j+3

ij 2ij+i+ j (i+l)(j+l)
- +
i+j-1 i+j i+ j+l

(2.37)
Then we minimize the residual by tahng derivatives of R w.r.t ci. For
predetermined number N, this results in N algebraic equations that have to be
solved simultaneously.

(2.38)

For N = 2 there are only two unknowns; cl and c2. It produces two linear
equations.
0.122~~ +0.048~, = -0.120
+
0.048~~ 0.033~~ -0.063 (2.39a,b)
=
In matrix form

0.122 0.048 -0.120
(2.40)
c 0.048 0.033 [ :} = { - 0.0631

It resembles the general form

where K is the Jacobian (stiffness matrix), and x is the vector of unknowns. L is
the forcing vector (load vector).
Solution to (2.40) gives us
c1 = -0.554579

C, = -1.112560