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

70 A general algorithm for compressible and incompressible ¯ows
It would appear that now U i is a better approximation of U n ‡ 1 . We can now write
the correction as
n ‡ 1
Ui ˆ Ui ÿ U n @ p
i ˆ Ui ÿ 2 t
@xi …3:28†
i.e. the correction to be applied is smaller than that assuming Split A, Eq. (3.24).
Further, if we use the fully explicit form with 2 = 0, no mass velocity (Ui) correction
is necessary. We proceed to calculate the pressure changes as in Split A as
ˆ 1
c 2 p ˆÿ t @Un i
@x i
" #
‡ 1
@ U i
@x i
The solution stages follow the same steps as in Split A.
ÿ t 1 2
3.2.3 Spatial discretization and solution procedure
@ 2 p
@x 2 i
…3:29†
Split A
In all of the equations given below the standard Galerkin procedure is used for spatial
discretization as this was fully justi®ed for the characteristic±Galerkin procedure in
Chapter 2. We now approximate spatially using standard ®nite element shape
functions as
Ui ˆ N ~
u Ui Ui ˆ N ~U u i Ui ˆ N ~U u i
…3:30†
ui ˆ Nu~u i p ˆ Np~p ˆ N ~q
In the above equation
~U i ˆ‰U 1 i U 2 i U k i U m i Š T
N ˆ‰N 1 N 2
N k
N m Š
…3:31†
where k is the node (or variable) identifying number (and varies between 1 and m).
Before introducing the above relations, we have the following weak form of
Eq. (3.23) for the standard Galerkin approximation (weighting functions are the
shape functions)
…
N k u Ui d
…
… k …
@
@Nu ˆ‡ t ÿ …ujUi† d ÿ
ij d ÿ N k " # n
u … gi† d
‡ t2
2
‡ t
…
…
ÿ
N k u
@xj @
@x l
N k u ijn j dÿ
@x j
…ulN k u † ÿ @
…u
@x
jUi†‡ gi d
j
n
n
…3:32†
It should be noted that in the above equations the weighting functions are the shape
functions as the standard Galerkin approximation is used. Also here, the viscous and
stabilizing terms are integrated by parts and the last term is the boundary integral