Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

of the equations obtained in steady-state conditions. For simplicity we shall consider
here only the Stokes form of the governing equations in which the convective terms
disappear. Further we shall take the ¯uid as incompressible and thus uncoupled
from the energy equations. Now the three steps of Eqs. (3.49), (3.54) and (3.57) are
written as
~U ˆÿ tM ÿ1
u ‰K ~u n ÿ fŠ
~p ˆ 1
H
t 1 2
ÿ1 ‰G ~U n ‡ 1G ~U ÿ t 1H~p n ÿ fpŠ ~U ˆ ~U ÿ tM ÿ1
u G T …~p n ‡ 2 ~p†
…3:72†
In steady state we have ~p ˆ ~U ˆ 0 and eliminating ~U we can write (dropping
now the superscript n)
from the ®rst and third of Eqs. (3.72) and
K ~u ‡ G T ~p ˆ f …3:73†
G ~U ‡ 1 tGM ÿ1
u G T up~p ÿ t 1H~p ÿ f p ˆ 0 …3:74†
from the second and third of Eqs. (3.72)
We ®nally have a system which can be written in the form
K = G T
G t 1‰GM ÿ1
u G T " # ( )
~U
ˆ
ÿ HŠ ~p
f ( )
1
f2 …3:75†
here f 1 and f 2 arise from the forcing terms.
The system is now always positive de®nite and therefore leads to a non-singular
solution for any interpolation functions N u, N p chosen. In most of the examples
discussed in this book and elsewhere equal interpolation is chosen for both the U i
and p variables, i.e. N u = N p. We must however stress that any other interpolation
can be used without violating the stability. This is an important reason for the
preferred use of the Split A form.
It can be easily veri®ed that if the pressure gradient term is retained as in Eq. (3.27),
i.e. if we use Split B the lower diagonal term of Eq. (3.75) is identically zero and the BB
conditions in the full scheme cannot be avoided. Now we show this below. From
Eqs. (3.63), (3.65) and (3.66), for incompressible Stokes ¯ow we have
~U i ˆÿM ÿ1
u
~p ˆ 1
H
t 1 2
ÿ1
t K ~u ‡ G T ~p ÿ f† n
~U ˆ ~ U ÿ M ÿ1
u t 2G T ~p
t‰G ~U n ‡ 1G ~U ÿ f pŠ n
At steady state ~p ˆ ~U ˆ 0, which gives the following two equations:
and
`Circumventing' the BabusÆka±Brezzi (BB) restrictions 79
…3:76†
K ~u ‡ G T ~p ˆ f …3:77†
G ~U ˆ f b
…3:78†