Civil Engineering Reference
In-Depth Information
We will partly follow the modification of Chapelle and Stenberg [1998]. The ad-
vantage and the disadvantagge of the two models were elucidated by Pitkäranta
and Suri [2000].
Let
t,h <
1. We again start with the minimization of the functional (6.7), but
now combine a part of the shear term with the bending part:
t
−
2
2
1
2
a
p
(w, θ
;
w, θ)
+
2
dx
−
(u)
=
|∇
w
−
θ
|
fwdx,
(
6
.
19
)
where
+
t
2
1
a
p
(w, θ
;
v, φ)
:
=
a(θ, φ)
+
(
∇
w
−
θ)
·
(
∇
v
−
φ) dx,
h
2
(
6
.
20
)
h
2
1
t
2
=
1
1
t
2
or
t
2
=
t
2
+
t
2
+
+
t
2
.
h
2
h
2
Thus, we seek
(w, θ)
∈
X
=
H
0
()
×
H
0
()
2
such that
1
t
2
(
∇
w
−
θ,
∇
v
−
φ)
0
=
(f, w)
0
a
p
(w, θ
;
v, φ)
+
for all
(v, φ)
∈
X. (
6
.
21
)
By analogy with the derivation of (6.2) from (5.7), with the introduction of (mod-
ified) shear terms
γ
:
=
t
−
2
(
∇
w
−
θ)
, we now get the following mixed problem
with penalty term. Find
(w, θ)
∈
X
and
γ
∈
M
such that
a
p
(w, θ
;
v, φ)
+
(
∇
v
−
φ,γ)
0
=
(f, v)
0
for all
(v, φ)
∈
X,
(
6
.
22
)
(
∇
w
−
θ,η)
0
−
t
2
(γ, η)
0
=
for all
η
∈
M.
0
The essential difference compared to (6.2) is the coercivity of the enhanced form
a
p
.
6.7 Lemma.
There exists a constant c
:
=
c() >
0
such that
2
1
2
for all w
∈
H
0
(), θ
∈
H
0
()
2
.
6
.
23
)
a
p
(w, θ
;
w, θ)
≥
c(
w
+
θ
1
)
2
Proof.
By Korn's inequality,
a(φ, φ)
≥
c
1
φ
1
. In addition,
2
0
≤
(
∇
w
−
θ
0
+
θ
0
)
2
2
0
2
∇
w
≤
2
∇
w
−
θ
+
2
θ
1
.
Friedrichs' inequality now implies
2
1
2
1
2
0
2
w
≤
c
2
|
w
|
≤
2
c
2
(
∇
w
−
θ
+
θ
1
),
and so
2
c
2
)
∇
w
−
θ
1
2
1
2
1
2
0
2
w
+
θ
≤
(
1
+
+
θ
2
c
2
)
∇
c
−
1
a(θ, θ)
2
0
≤
(
1
+
w
−
θ
+
+
c
−
1
)a
p
(w, θ
;
w, θ).
This establishes the coercivity with the constant
c
:
≤
(
1
+
2
2
)(
1
2
c
2
)
−
1
(
1
+
c
−
1
)
−
1
.
=
(
1
+
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