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

74 A general algorithm for compressible and incompressible ¯ows
success of the solution process. The weak form of the correction step from Eq. (3.25) is
…
N k u
n ‡ 1
Ui d
…
ˆ N k u Ui d ÿ
…
t N k u
@p n
@ p
‡ 2
@xi @xi d
ÿ t2
…
2
@
…u
@x
jN
j
k u † @pn
d
@xi …3:56†
n ‡ 1
The ®nal stage of the computation of the mass ¯ow vector Ui is completed by
following matrix form
Step 3
where
~U ˆ ~U ÿ M ÿ1
u t G T …~p n ‡ 2 ~p†‡ t
2 P~pn
…3:57†
…
P ˆ …r…uNu†† T rNp d …3:58†
At the completion of this stage the values of ~U n ‡ 1 and ~p n ‡ 1 are fully determined
but the computation of the energy … E† n ‡ 1 is needed so that new values of c n ‡ 1 ,
the speed of sound, can be determined.
Once again the energy equation (3.6) is identical in form to that of the scalar
problem of convection±di€usion if we observe that p, Ui, etc. are known. The
weak form of the energy equation is written using the characteristic±Galerkin
approximation of Eq. (2.91) as
…
N k E … E† n ‡ 1 d
…
ˆ t ÿ N k … k
@
@NE E …u
@x
i… E ‡ p†† d ÿ
i
@x
ijuj ‡ k
i
@T
" # n
d
@xi ‡ t2
…
@
…u
2 @x
jN
j
k E† @
n
… ÿu
@x
i… E ‡ p† † d
i
…
‡ t N
ÿ
k E ijuj ‡ k @T
n
ni dÿ
…3:59†
@xi With
we have
E ˆ N E ~E T ˆ N T ~T …3:60†
Step 4
~E ˆÿM ÿ1
E t C ~E E ‡ Cp~p ‡ K ~T T ‡ K E~u ‡ fe ÿ t…K ~E uE ‡ Kup~p ‡ fes† n
…3:61†
where ~E contains the nodal values of E and again the matrices are similar to those
previously obtained (assuming that E and T can be suitably scaled in the conduction
term).