Civil Engineering Reference
In-Depth Information
The Mixed Method of Hellinger and Reissner
The two-field formulation with the displacements and the stresses is usually de-
noted as the
Hellinger-Reissner principle
, but also as the
Hellinger-Prange-
Reissner principle
. The basic idea is contained in Hellinger [1914]; a first proof for
the traction problem is due to Prange [1916]
18
and for mixed boundary conditions
due to Reissner [1950]; see Gurtin [1972], p. 124 and Orava and McLean [1966].
The method has many similarities to the mixed formulation of the Poisson
equation in Ch. III, §5. The variational formulation according to (3.12) is
(
C
−
1
σ, τ)
0
−
(τ,
∇
(s)
u)
0
=
for all
τ
∈
L
2
(),
0
g
·
vdx
for all
v
∈
H
1
(),
(
3
.
21
)
which corresponds to the standard displacement formulation. Since
ν<
2
,
(s)
v)
0
−
(σ,
∇
=−
(f, v)
0
+
1
is
positive definite and the bilinear form
(
C
−
1
σ, τ)
0
is
L
2
-elliptic. The following
lemma shows that the inf-sup condition follows from Korn's inequality.
C
3.6 Lemma.
Suppose the hypotheses for Korn's second inequality are satisfied.
Then for all v
∈
H
1
(),
(s)
v)
0
τ
0
(τ,
∇
≥
c
v
1
,
sup
3
)
τ
∈
L
2
(,
S
where c
is the constant in (3.18).
Proof.
Given
v
∈
H
1
()
,
τ
:
(s)
v
is a symmetric
L
2
-tensor. Moreover, by
=∇
(s)
v
0
≥
c
v
1
. It suffices to consider the case
v
=
0:
(3.18),
τ
0
=∇
2
0
(s)
v)
0
τ
0
(s)
v
(τ,
∇
=
∇
(s)
v
0
≥
c
v
1
,
∇
which establishes the inf-sup condition.
The formulation with the spaces as in (3.21) is almost equivalent to the dis-
placement formulation. Specifically, it can be understood as a displacement for-
mulation combined with a softening of the energy. It is suitable for the method of
enhanced assumed strains
by Simo and Rifai [1990]; see Ch. III, §5. As in the dis-
cretization of the Poisson equation using the Raviart-Thomas element, generally
18
Prange's "Habilitationsschrift" (a thesis for an academic degree at a level above the
doctorate, which is usually a prerequisite for a professorship in Germany) was unpublished
due to the first world war. It was only edited with an introduction by K. Knothe in 1999.
Search WWH ::
Custom Search