Finite Element Techniques for Fluid Flow

38 WEIGHTED RESIDUAL AND VARIATIONAL METHODS

Let us now assume that the time variation of a within the At time

step is given by

a(i) = n

0

(l + ßt) (f)

<5a(i) = u

0

oßt

where u

0

is the initial value at the centre of the rod and ß is an

unknown parameter.

Hence (e) becomes

which gives

(•At

Jo

(l + ßt) + ~ß}tdt = 0 (g)

ï

+

'-r

+

è-

If we have for the sake of illustration, K = 1 and At = 001, then

ß — —246. u

0

will be taken as 100.

Hence for t = 0, we have u

c

(at the centre) equals 100; for t = 001,

u

c

= 100 x (1 + ßAt) = 100 x (1 - 0-0246) = 97-54; for t = 0-02,

u

c

= 97-54 x 0-9754 = 95-14; for t = 0-03, u

c

= 95-14 x 0-9754 =

92-80;

etc. In this