Page 189 - Using ANSYS for Finite Element Analysis Dynamic, Probabilistic, Design and Heat Transfer Analysis

176 • using ansys for finite eLement anaLysis
• The first-order method is based on design sensitivities and is more
suitable for problems that require high accuracy.

For both the subproblem approximation and first-order methods, the
program performs a series of analysis-evaluation-modification cycles.
That is, an analysis of the initial design is performed, the results are evalu-
ated against specified design criteria, and the design is modified as neces-
sary. The process is repeated until all specified criteria are met.

5.2.2.1 Subproblem Approximation Method

The subproblem approximation method can be described as an advanced
zero-order method in that it requires only the values of the dependent vari-
ables, and not their derivatives. There are two concepts that play a key
role in the subproblem approximation method: the use of approximations
for the objective function and state variables, and the conversion of the
constrained optimization problem to an unconstrained problem.
Approximations: For this method, the program establishes the relation-
ship between the objective function and the DVs by curve fitting. This is done
by calculating the objective function for several sets of DV values (i.e., for
several designs) and performing a least squares fit between the data points.
The resulting curve (or surface) is called an approximation. Each optimization
loop generates a new data point, and the objective function approximation is
updated. It is this approximation that is minimized instead of the actual objec-
tive function. State variables are handled in the same manner. An approxima-
tion is generated for each state variable and updated at the end of each loop.
Conversion to an Unconstrained Problem: State variables and limits
on design variables are used to constrain the design and make the optimiza-
tion problem a constrained one. The ANSYS program converts this problem
to an unconstrained optimization problem because minimization techniques
for the latter are more efficient. The conversion is done by adding penal-
ties to the objective function approximation to account for the imposed
constraints.
Convergence Checking: At the end of each loop, a check for convergence
(or termination) is made. The problem is said to be converged if the current, pre-
vious, or best design is feasible and any of the following conditions are satisfied:

• The change in objective function from the best feasible design to
the current design is less than the objective function tolerance.
• The change in objective function between the last two designs is
less than the objective function tolerance.