Civil Engineering Reference
In-Depth Information
There is a qualitative difference between a poor approximation due to using
too coarse a grid in the discretization and a poor approximation due to locking. It
can be understood in terms of the associated eigenvalue problem; cf. Braess [1988].
As illustrated in Problem I.4.8, the lower eigenvalues are normally approximated
well. This observation was also an important argument in the multigrid theory.
On the other hand, in the discretization even the lower eigenvalues will clearly be
shifted, once we have a tendency toward locking.
3.12 Remark.
There are several approaches to reducing the effects of locking:
1. Convert the variational problem (3.42) to a saddle point problem with a
penalty term (as explained above for nearly incompressible materials). With
p
=
t
−
2
Bu
,weget
a
0
(u, v)
+
(Bv, p)
0
,
=
, v
for all
v
∈
X,
(
3
.
48
)
(Bu, q)
0
,
−
t
2
(p, q)
0
,
=
0
for all
q
∈
L
2
().
2. Use
selective reduced integration
. Then in setting up the system matrix, the
term
t
−
2
(Bu
h
,Bv
h
)
0
,
will be relaxed as in (3.36) so that the constant in (3.45) is also reduced. This
makes sense if the process can be understood as one where the approximating
functions
v
h
are replaced in a neighborhood by others for which
Bv
is small.
Strict mathematical proofs usually involve working with equivalent mixed
formulations; see §6.
3. Simo and Rifai [1990] expand the space in the spirit of Remark III.5.7 by
including nonconforming approximating functions so that sufficiently many
functions are available in order to make
Bv
0
,h
small (if an appropriate
discrete norm is chosen).
The first two methods listed in 3.12 are partially equivalent; cf. Remark 3.9.
The finite element approximation of (3.48),
a
0
(u
h
,v)
+
(Bv, p
h
)
0
,
=
, v
for all
v
∈
X
h
,
(
3
.
49
)
(Bu
h
,q)
0
,
−
t
2
(p, q)
0
,
=
0
for all
q
∈
M
h
,
is equivalent to the minimization problem
1
2
a
0
(u
h
,u
h
)
+
1
2
t
−
2
R
h
Bu
h
0
−
, u
h
−→
min
u
h
∈
X
h
!
(
3
.
50
)
Here
R
h
:
L
2
()
→
M
h
is the orthogonal
L
2
-projector. In practice other projectors
are used along with so-called
selected reduced integration.
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