Page 181 - Using ANSYS for Finite Element Analysis Dynamic, Probabilistic, Design and Heat Transfer Analysis

168 • using ansys for finite eLement anaLysis
torque. In many other situations minimization of a single function may not
be the only goal, and attention must also be directed to the satisfaction of
predefined constraints placed on the design (e.g., limits on stress, geome-
try, displacement, heat flow).
The independent variables in an optimization analysis are the design
variables. The vector of design variables is indicated by:

X = [ X X X  X ] (5.1)
1 2 3 n
Design variables are subject to n constraints with upper and lower
limits, that is,

i (
X ≤ X ≤ X i = 12 3, ……, n) (5.2)
,,
i
i
Where: n = number of design variables.
The design variable constraints are often referred to as side con-
straints and define what is commonly called feasible design space.

f x
Now, minimize: f = () (5.3)
Subject to:

gx () ≤ ( ,, ……, m ) (5.4)
gi = 12 3,
i
i
1
hx i (
hx () ≤ () = 12 3, ……, m ) (5.5)
,,
i
i
2
W ≤ W x () ≤ ( ,, ……, m ) (5.6)
W i = 12 3,
i
i
i
3
Where:
f = objective function
g , h , w = state variables containing the design, with under bar and
i
i
i
over bars representing lower and upper bounds respectively.
m + m + m = number of state variables constraints with various
3
2
1
upper and lower limit values. The state variables can also be referred
to as dependent variables in that they vary with the vector x of design
variables.
(Equation 5.3) through (Equation 5.6) represent a constrained min-
imization problem whose aim is the minimization of the objective func-
tion f under the constraints imposed by (Equations 5.2, 5.4, 5.5, and 5.6).
Design configurations that satisfy all constraints are referred to as feasible
designs. Design configurations with one or more violations are termed