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

and 2.95) and are given as
…
Mu ˆ
…
K ˆ
B T
I0 ÿ 2
3 mmT
ÿ
N T u N u d C u ˆ
…
B d f ˆ
…
N T u …r…uNu†† d
N T …
u g d ‡ N
ÿ
T u t d dÿ
…3:50†
where g is ‰g 1 g 2 g 3Š T and t d is the traction corresponding to the deviatoric stress
components. The matrix K is also de®ned at several places in Volume 1 (for instance
A in Chapter 12).
In Eq. (3.49) K u and f s come from the terms introduced by the discretization along
the characteristics. After integration by parts, the expressions for K u and f s are
and
Ku ˆÿ1 …
2
fs ˆÿ1 …
2
…r T …uN u†† T …r T …uN u†† d …3:51†
…r T …uN u†† T g d …3:52†
The weak form of the density±pressure equation is
…
N k p d
…
ˆ N k 1
p
c2 …
ˆÿ t
p d
@
U n i ‡ 1 Ui ÿ 1 t @pn ‡ !
2
d
N k p
@xi ˆ
… k
@Np t U
@xi n i ‡ 1 Ui ÿ t @pn ‡ " ! #
2
d
@xi ÿ
…
t 1 N k p U n i ‡ Ui ÿ t @pn ‡ !
2
ni dÿ …3:53†
ÿ
In the above, the pressure and U i terms are integrated by parts. Further we shall
discretize directly only in problems of compressible gas ¯ows and therefore below
we retain p as the main variable. Spatial discretization of the above equation gives
Step 2
…M p ‡ t 2 1 2H† ~p ˆ t‰G ~U n ‡ 1G ~U ÿ t 1H~p n ÿ f pŠ …3:54†
which can be solved for ~p.
The new matrices arising here are
…
H ˆ …rNp† T …
rNp d Mp ˆ
…
G ˆ
Characteristic-based split (CBS) algorithm 73
N T p
1
c 2
@x i
n
N p d
…rNp† T Nu d fp ˆ t
…
N
ÿ
T p n T ‰ ~U n ‡ 1… ~U ÿ trp n ‡ 2 †Š dÿ …3:55†
In the above f p contains boundary conditions as shown above as indicated. We
shall discuss these forcing terms fully in a later section as this form is vital to the
@x i