Although the internal optimizer of Ansys Polyflow accepts only one objective
function 
for minimization, it is possible to define multiple objective
functions noted 
, and each of these objective functions can be either minimized or
maximized. When the goal is to maximize the objective function 
, Ansys Polyflow will redefine it such that 
is minimized.
Three schemes are implemented in Ansys Polyflow, which allow for single and multiple objective functions:
single scheme
When you define only one objective function for optimization (
), the following equation is used: 
(36–36)
where
is a factor that depends upon the goal of the objective
function (1 for minimization and -1 for maximization), and 
is the initial value of the design variable. For die shape
optimization, this value is set by default to the amplitude of displacement
you specified during the mesh deformation setup. multi-compromise scheme
You can use the following equation when you have multiple objective functions to optimize, so that
is obtained by weighting the different objective
functions: 
(36–37)
In the previous equation,
is the number of objective functions, 
is a factor that depends upon the goal of objective
function 
(1 for minimization and -1 for maximization), and
is a weight factor associated with the objective function
. The weight factor transforms 
into a dimensionless quantity. 
can be a very small or very large value, as it can be used
not only to control the relative weight of an objective function with
respect to the other objective functions, but also to ensure that the
objective functions are of the same order of magnitude numerically. multi-priority scheme
Another approach for handling multiple objective functions involves defining a priority
for each objective function 
(where 
varies from 1 to the total number of objective functions).
The objective functions are then optimized separately using Equation 36–36, in the order of
decreasing priority. Consider the following example, in which
is the highest priority and 
is the next highest priority. The first step is to
optimize the die shape while only considering the objective function
, in order to obtain the minimum 
. Then 
is removed from the list of objective functions, and is
converted into the equality constraint 
. Next, the die shape is optimized while only considering
the objective function 
, in order to obtain the minimum 
. Again, this objective function is removed from the list
of objective functions, and is converted into the equality constraint
. This process continues until all of the objective
functions have been optimized. During the optimization process, the number
of constraints increases and the number of objective functions to be
satisfied decreases accordingly. Important: When the multi-priority scheme is used, it is a good practice to make sure that the different objective functions have the same order of magnitude numerically, in the manner described at the end of Constraints.
Since the multi-priority scheme converts the objective functions into constraints, see Constraints for recommendations to consider when defining the form of each of the objective functions.
The parameters for an objective function include the following:
name
Short names are recommended for objective functions, for the sake of readability in the output files.
priority, ranging between 0 (lowest) and 20 (highest)
This parameter must be provided when the multi-priority scheme for multiple objective functions is used for optimization. When a single objective function is defined or the multi-compromise scheme is used, the priority is not taken into account.
goal of the optimization: to minimize or maximize the function
form of the function to minimize or maximize
This parameter allows you to use predefined functions to specify the form (noted as
in the following set of equations) of the objective
function, which will then be minimized or maximized for the set of design
variables 
. These predefined functions involve parameters
, 
, 
, 
, and 
, as well as one or two extracted values that are noted in
the following list of equations as 
and 
(these latter two variables represent 
from Equation 36–25). Note that the objective function must be
defined in such a way so as to have a minimum. The predefined functions
include the following: Which predefined function you select will depend upon the behavior of the extracted values
and 
: Consider the case where
exhibits a minimum, such as when it represents the
extracted value of the flow balance. Its value is always greater
than 0, where 0 corresponds to a perfect die. In this situation, it
is recommended that you use Equation 36–38. Alternatively, you could use the linear
function expressed in Equation 36–39 if you want to shift and/or normalize the
objective function.On the other hand, consider the case where the field
does not exhibit a minimum, such as when you want
to minimize the difference between the temperature (that is,
is the extracted value of 
) and a given value (for example, 150). In this
situation, it is recommended that you use the square of the linear
function expressed in Equation 36–40
, 
, and 
. With such parameters, the function will reach a
minimum when 
equals the ideal temperature of 150.
extracted value
extracted value
, if required parameters
, 
, 
, 
, and 
, if required Important: You must choose values for parameters
, 
, 
, 
, and 
such that 
is always a positive value, otherwise the minimization
of Equation 36–36 or Equation 36–37 will not necessarily yield correct
results.weight factor
This parameter must be provided when the multi-compromise scheme for multiple objective functions is used for optimization. When a single objective function is defined or the multi-priority scheme is used, the weight factor is not taken into account.
