Civil Engineering Reference
In-Depth Information
Optimization Problem
So, as mentioned in Section 5.2, the optimization problem is written:
find
p
such that:
()
p
MIN
()
p
C
1
p
()
0
(5.54)
p
,
C
2
p
()
0
can be one of the functions previously described. However, it is important to check that the problem
is well posed, which might not be the case with all the proposed functions. If this property is not satisfied,
the optimization method, whichever it is, will not work. Several techniques can be used to obtain a well
posed problem, either by introducing more constraints, or by adding a regularization function:
• energy method: among the possible solutions, the selected solution minimizes the forming energy;
as already mentioned, often this is also an actual objective of the design
reg
p
()
(
1
)
()
ene
p
p
()
(5.55)
where
is a regularization parameter
• Tikhonov method [62]: the selected solution is the nearest solution from the initial set of param-
eters
N
par
2
p
i
p
i
reg
p
)
()
()
(
1
p
(
)
(5.56)
i
1
where
N
par
is the number of the parameter values,
p
i
()
,
i
p
i
p
i
are the initial values of .
Often, the design objectives are not unique and several of them have to be considered in the optimization
procedure. A first approach is to decompose the problem into several sub-problems. This is possible if
the objectives are not too strongly coupled, or if it is possible to define a clear hierarchy between them.
The method can be first applied to a small number of important objectives, and then, starting again
from the optimized design, the other objective functions are progressively introduced. During the second
step, the previous functions can be considered as constraints, in order to prevent too large variations of
their values. For instance, it is possible to first optimize the shape of the preforming dies in order to
obtain the right final part without any defects, and then to optimize the process parameters such as the
initial temperature of the billet, the temperature of the dies, and the forging velocity, in order to
optimize the thermo dependent metallurgical properties of the part, for instance by minimizing the grain
size. However, when the functions are strongly coupled or when it is expensive to carry out to many
optimizations, a multi-objective function can be defined:
()
()
i
N
obj
tot
p
()
i
i
p
()
(5.57)
i
1
where
N
obj
is the number of different objective functions,
i
,
i
are the weighting coefficients, taking into account both the physical dimensions of
i
, its interval
of variations, and its priority with respect to the other functions.
Obviously, the choice of the
i
coefficients is delicate and will influence the final solution.
