Civil Engineering Reference
In-Depth Information
and the sensitivity analysis, are not based on the same discretization scheme. This is rather unavoidable
when the sensitivity analysis is carried out independently, for instance, when the source code of the direct
problem is not available. On the other hand, these approaches make it possible to extend the mathematical
developments, so gaining the favors for a more fundamental approach. In the frame of fluid mechanics,
they are rather popular and their extension to metal forming problems has been studied [36].
Generally, the objective function can be written as an integral of a function of various state variables.
For example, the forming energy is such a function:
t
t
end
:
˙
d
wd
(5.6)
t
t
0
(see Section 5.3 for the problem notations). The differentiation of
with respect to the shape parameters
p
gives:
t
t
end
t
t
end
d
:
˙
˙
v
s
:
d
wd
:
n
sdd
(5.7)
d
p
p
p
t
t
0
t
0
v
s
where
, due to the parameter variations.
When a flow formulation is used to solve the viscoplastic forging problem, all the state variables can
be expressed as functions of the velocity field,
v
. This way, all the shape derivatives of Eq. (5.7) can be
easily calculated once
is the displacement velocity of the boundary
∂
p
is known.
In order to achieve this, the problem equations are differentiated with respect to
p
: the equilibrium
and constitutive equations, and the corresponding boundary conditions. This leads to a linear system of
equations, which is solved using a variational approach and a finite element discretization [Pir84].
However, when it is applied to a non-steady-state forming problem, this formulation exhibits boundary
integrals on
∂
v
/
∂
which comprise gradients of the stress tensor [37]. This poses a difficult problem because
these gradients are not straight results of the finite element resolution: they are either unknown or
calculated with poor accuracy, which results in a significant loss of accuracy in the sensitivity analysis.
In order to overpass these problems, a slightly different method has been developed [5], [6]. It is based
on a total lagrangian formulation, rather than on an updated eulerian formulation. The total deformation
gradient
F
is the main variable, rather than the velocity field
v
. The equilibrium equation is expressed in
the reference configuration
∂
0
, using the Piola Kirchoff stress measure:
T
PK
()
det
F
F
(5.8)
0
,
PK
x
t
[
0,
t
end
]
,
div
(
)
0
(5.9)
This local equilibrium equation is differentiated with respect to the shape parameters
p
, and the weak
variational form of the problem is written, still in the reference configuration:
d
PK
u
,
div
u
d
w
0
0
(5.10)
d
p
0
where
u
are the test functions for the sensitivity problem.
After integration by parts:
d
PK
u
d
PK
n
0
--------
d
w
0
u
d
s
0
(5.11)
d
p
x
0
d
p
0
0
As the reference configuration does not depend on the parameter values, the right hand term is easily
written onto the current configuration. Then, the constitutive equation is differentiated so that
d
PK
/
dp
can be expressed as a function of the derivatives of the deformation gradient
dF/dp
. Thus, Eq. (5.11) is
