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6.8
Hybrid Methods
For the displacement and force methods, it is sometimes difficult to find trial solutions
that satisfy the required conditions. This is particularly true for the force method. This
shortcoming can be corrected with the use of extended variational theorems, in which
those conditions causing difficulties are attached to the variational work expressions using
Lagrange multipliers. Recall that the Lagrange multipliers can be interpreted as physical
variables. These extended variational principles may be used as the basis of finite element
approximations. In the literature, many of these formulations are referred to as hybrid or
mixed methods [Bathe, et al., 1977; Wunderlich, 1983; Wunderlich, 1972].
The hybrid method has features of the displacement and force methods. Typically, the
extended variational principle is discretized using two sets of trial solutions, one for the
interior of the element and the other especially for the boundaries. There is literature of
considerable size on the hybrid method (see, for example, Pian (1964) and Pian and Tong
(1969)).
We choose to introduce the hybrid method by treating it as an extended force method.
This is one of several available hybrid techniques. Begin with the principle of stationary
complementary energy or the principle of complementary virtual work. Suppose the clas-
sical principle [Chapter 2, Eq. (2.78)]
W
∗
=
V
δ
σ
T
dV
p
T
u
dS
−
δ
−
S
u
δ
=
0
(6.149)
is to be extende
d
with the addition of the global (integral) form of the static boundary
conditions (
p
−
p
=
0
on
S
p
), i.e.,
u
T
δ
(
−
)
=
p
p
dS
0
S
p
This means that these static boundary conditions are to supplement Eq. (6.149) with the aid
of the boundary displacements
u
as Lagrange multipliers. Then
V
δ
W
∗
=
σ
T
dV
p
T
u
dS
u
T
−
δ
−
S
u
δ
−
δ
(
p
−
p
)
dS
S
p
V
δ
σ
T
E
−
1
σ
dV
S
p
δ
S
p
(δ
p
T
u
dS
u
T
p
dS
u
T
p
p
T
u
=
−
S
u
δ
+
−
+
δ
)
dS
=
0
(6.150)
The stresses still must satisfy the conditions of equilibrium in
V
, i.e.,
D
T
σ
0
. The
extended principle of Eq. (6.150) corresponds to the hybrid functional of Chapter 2, Eq.
(2.104). It should be observed that both displacements and stresses on the boundaries are
unknowns, for which trial functions have to be chosen.
For a hybrid method based on an extended displacement formulation, the principle of
virtual work expression [Chapter 2, Eq. (2.5
4
)] is supplemented with an integral form of
the displacement boundary condition (
u
+
p
V
=
−
=
u
0
on
S
u
), with the boundary forces
p
as
Lagrange multipliers.
The hybrid method utilizing the extended principle complementary virtual work func-
tional of Eq. (6.150) can be implemented by selecting stress trial functions for the interior
of the element and on the
S
p
boundary, e.g.,
N
σ
σ
=
σ
in
V
with
A
T
σ
on
=
p
S
p
(6.151)
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