Civil Engineering Reference
In-Depth Information
FIGURE 5.11
The different strategies of the simplex algorithm for the selection of a new point.
. Formulated, this problem
is similar to the optimization problems met in structural mechanics. Several optimization algorithm can
be used to solve it.
The numerical simulation is used to compute the different values of
and
C
Simplex Optimization Algorithm
This algorithm is based on a preliminary definition of a set of possible optimal points. At each iteration,
a new point is added while the worst point is removed [39]. In a first step, the objective function is
evaluated for a prescribed number of points, belonging to the space of the parameters. The number of
points must be greater than the number of parameters. During the iterations of the algorithm, the point
which provides the larger value of the objective function is removed. It is replaced by a new point, derived
from its value, whose position is computed using the following: first reflection with respect to the gravity
center of the set of points, then expansion or contraction according to the value of the objective function
This method has been applied to the shape optimization of an extrusion die profile in steady-state
forming, in order to minimize the extrusion force [49]. Although it requires a large number of function
evaluations, in this specific two-dimensional steady state example, the algorithm shows as efficient as a
gradient based method.
Genetic Optimization Algorithm
Here also, the method is based on a set of possible points, which is used to generate new solutions. However,
the algorithm is quite different, as it is derived from the biological evolution theory, where the best
specimens of a population are used to generate a new and better generation of points. Here, better refers
to the value of the objective function. Reproduction, crossing, and mutations rules govern the generation
of the new set of points [24], [33]. This kind of probabilistic algorithm makes it possible to obtain a local
minimum of the objective function, even when the function is not continuous nor differentiable. This is
its greater power. On the other hand, they require a large number of function evaluations. For instance,
some version of the algorithm using a small population needs about 200 function calculations to find an
optimum [39], [48], which is too many for complex forging problems.
It has been applied to the design of an industrial four steps forging sequence, in order to minimize a
damage criterion [21]. The process parameters are the angles of the different preforming dies, the
radius/length ratio of the initial billet, and the number of forging operations. Although this last parameter
takes integer values and so is discontinuous, the genetic algorithm can handle it. For the next more
efficient methods, the continuities of the
,
C
functions, and
p
are the basic requirements.
Analytical Approximation of the Objective Function
Another type of optimization algorithm is based on a preliminary approximation of the objective func-
tion, which can then be easily minimized using any efficient algorithms.
As for the simplex and genetic algorithms, an initial set of parameters values (
is first defined.
This set must be large enough in order to represent all the possible variations of the parameters. It can
be designed using the plane of experiments methods. The objective and constraint functions,
p
)
k
k
1,Nbset
(
p
) and
k
C(
p
), are evaluated for all these parameter values. If Nbset, the number of
p
points, is sufficiently larger
k
k
