Civil Engineering Reference
In-Depth Information
To define a simple member of the family of PEERS elements, we employ the
usual notation. (See also Fig. 59.) Let
T
be a triangulation of
, and let
k
M
:
={
v
∈
L
2
()
;
v
|
T
∈
P
k
for every
T
∈
T
}
,
0
:
k
∩
H
1
(),
0
,
0
:
k
∩
H
0
(),
M
=
M
M
=
M
k
+
1
)
2
RT
k
:
={
v
∈
(
M
∩
H(
div
,)
;
(
4
.
13
)
p
1
p
2
+
p
3
x
y
,p
1
,p
2
,p
3
∈
P
k
}
,
v
|
T
=
3
0
B
3
:
={
v
∈
M
;
v(x)
=
0 on every edge of the triangulation
}
.
The PEERS element is the one with the smallest number of local degrees of
freedom:
=
(RT
0
)
2
curl
(B
3
)
2
,
σ
h
∈
X
h
:
⊕
0
)
2
,
v
h
∈
W
h
:
=
(
M
1
0
.
Note that the divergence of the functions in curl
(B
3
)
2
vanishes, and that the di-
vergence of a piecewise differentiable function is an
L
2
function if and only if the
normal components are continuous on the inter-element boundaries.
By construction, div
τ
h
∈
γ
h
∈
h
:
=
M
W
h
. Thus, div
τ
h
=
0 follows immediately from
(
div
τ
h
,v
h
)
=
0 for all
v
h
∈
W
h
. Thus, the condition in III.4.7 is satisfied, and the
form
(
C
−
1
σ
h
,τ
h
)
is elliptic on the kernel. The inf-sup condition can be established
in a similar way as for the continuous problem. Since (4.11) must be replaced by
finite element approximations, however, here the details are more involved, see
Arnold, Brezzi, and Douglas [1984].
The implementation and postprocessing described by Arnold and Brezzi
[1985], see Ch. III, §5 is also advantageous for computations with the PEERS
element. The postprocessing was also used to estimate the local error by adaptive
grid refinement, see Braess, Klaas, Niekamp, Stein, and Wobschal [1995].
Problems
4.3
Show that we get the constitutive equations (4.3) for plane stress states by
restricting the relationship
ε
=
C
−
1
σ
to the components with
i
=
1
,
2.
What comparable assertion holds for plane strain states?
1
4.4
For a nearly incompressible material,
ν
is close to
2
. This causes difficulties
in the denominators
(
1
2
ν)
of (1.31) and (3.6), respectively. On the other hand,
in view of (4.3), for the plane stress states,
ν
=
−
1
2
does not cause any problem.
Give an explanation.
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