Civil Engineering Reference
In-Depth Information
Locking
The concept of
locking effect
is used frequently by engineers to describe the case
where a finite element computation produces significantly smaller displacements
than it should. In addition to
volume locking
, we also know
shear locking
,
mem-
brane locking
, and
thickness locking
as well as others with no special names. The
essential point is that because of a small parameter
t
, as in (3.28) the quotient
C/α
grows, and the convergence of the finite element solution to the true solution is
not
uniform in t
as
h
→
0. The papers of Arnold [1981], Babuska and Suri [1992], and
Suri, Babuska, and Schwab [1995] have made fundamental contributions to the
understanding of locking effects. The following general framework covers nearly
incompressible material and applies to the treatment of the Mindlin-Reissner plate
in §6.
Let
X
be a Hilbert space,
a
0
:
X
×
X
→ R
a continuous, symmetric, coercive
2
, and
B
:
X
→
L
2
()
a continuous linear
mapping. Generally,
B
has a nontrivial kernel and dim ker
B
=∞
. In addition,
let
t
be a parameter 0
<t
≤
bilinear form with
a
0
(v, v)
≥
α
0
v
1
.
Given
∈
X
, we seek a solution
u
:
=
u
t
∈
X
of
the equation
1
t
2
(Bu
t
,Bv)
0
,
=
a
0
(u
t
,v)
+
, v
for all
v
∈
X.
(
3
.
42
)
The existence and uniqueness are guaranteed by the coercivity of
a(u, v)
:
=
a
0
(u, v)
+
t
−
2
(Bu, Bv)
0
,
.
Suppose there exists
u
0
∈
X
with
Bu
0
=
0
,d
:
=
, u
0
>
0
.
(
3
.
43
)
After multiplying
u
0
by a suitable factor, we can assume that
a
0
(u
0
,u
0
)
≤
, u
0
.
In particular, the energy of the minimal solution satisfies
2
a
0
(u
0
,u
0
)
+
0
,
−
, u
0
≤−
1
1
t
2
Bu
0
1
2
d
2
(u
t
)
≤
(u
0
)
=
1
with a bound that is independent of
t
. Thus,
, u
t
≥−
(u
t
)
≥
2
d
, and so
u
t
≥
−
1
1
2
d
for all
t>
0
(
3
.
44
)
is bounded below, where
X
.
We now consider the solution of the variational problem in the finite element
space
X
h
⊂
X
. Looking at the results of Babuska and Suri [1992] or Braess [1998],
we recognize that the locking effect occurs when
X
h
∩
:
=
and
19
ker
B
={
0
}
Bv
h
0
,
≥
C(h)
v
h
X
for all
v
h
∈
X
h
.
(
3
.
45
)
19
To be precise, we have to exclude functions from
X
h
which are polynomials in
or belong to the low-dimensional kernel. This is why on the right-hand side of (3.45) we
should replace
v
h
X
by a norm of a quotient space as in (3.47).
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