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

116 Incompressible laminar ¯ow
(b) the shear modulus G by the viscosity and
(c) the mean stress by negative pressure.
We have discussed such equations in Chapters 1 and 3.
4.8.2 Mixed and penalty discretization
The discretization can be started from the mixed form with independent approximations
of u and p, i.e.
u ˆ N u~u p ˆ N p~p …4:39†
or by a penalty form in which Eq. (4.36) is augmented by p= where is a large
penalty parameter
m T Su ‡ p ˆ 0 …4:40†
allowing p to be eliminated from the computation. Such penalty forms are only
applicable with reduced integration and their general equivalence with the mixed
form in which p is discretized by a discontinuous choice of N p between elements
has been demonstrated. 86 (See Chapter 12, Volume 1 for details.)
As computationally it is advantageous to use the mixed form and introduce the
penalty parameter only to eliminate the p values at the element levels, we shall
presume such penalization to be done after the mixed discretization.
The use of penalty forms in ¯uid mechanics was introduced early in the 1970s 87ÿ89
and is fully discussed elsewhere. 90ÿ92
The discretized equations will always be of the form
K ÿG
ÿG T ÿh 2 I=
~u
~p
ˆ f
0
where h is a typical element size, I an identity matrix,
…
K ˆ B T I0B d where B SNu …
G ˆ …rNu† T Np d
…
f ˆ
N T u g d ‡
…
N
ÿt T u t dÿ
…4:41†
…4:42†
and the penalty number, , is introduced purely as a numerical convenience. This is
taken generally as 90;92
ˆ…10 7 -10 8 †
There is little more to be said about the solution procedures for creeping incompressible
¯ow with constant viscosity. The range of applicability is of course limited
to low velocities of ¯ow or high viscosity ¯uids such as oil, blood in biomechanics
applications, etc. It is, however, important to recall here that the mixed form
allows only certain combinations of N u and N p interpolations to be used without
violating the convergence conditions. This is discussed in detail in Chapter 12 of
Volume 1, but for completeness Fig. 4.18 lists some of the available elements together