Page 183 - Using ANSYS for Finite Element Analysis Dynamic, Probabilistic, Design and Heat Transfer Analysis

170 • using ansys for finite eLement anaLysis
Design variables: A design variable is a specification that is control-
lable from the point of view of the designer. For instance, the thickness of
a structural member can be considered a design variable. Another might
be the material the member is made out of. Design variables can be con-
tinuous (such as a wing span), discrete (such as the number of ribs in a
wing), or boolean (such as whether to build a monoplane or a biplane).
Design problems with continuous variables are normally solved more eas-
ily. Design variables are often bounded, that is, they often have maximum
and minimum values. Depending on the solution method, these bounds
can be treated as constraints or separately.
Constraints: A constraint is a condition that must be satisfied in order
for the design to be feasible. An example of a constraint in aircraft design
is that the lift generated by a wing must be equal to the weight of the
aircraft. In addition to physical laws, constraints can reflect resource lim-
itations, user requirements, or bounds on the validity of the analysis mod-
els. Constraints can be used explicitly by the solution algorithm or can be
incorporated into the objective using Lagrange multipliers.
Objectives: An objective is a numerical value that is to be maximized
or minimized. For example, a designer may wish to maximize profit or
minimize weight. Many solution methods work only with single objec-
tives. When using these methods, the designer normally weighs the vari-
ous objectives and sums them to form a single objective. Other methods
allow multi-objective optimization.
Models: The designer must also choose models to relate the con-
straints and the objectives to the design variables. These models are depen-
dent on the discipline involved. They may be empirical models, such as
a regression analysis of aircraft prices, theoretical models, such as from
computational fluid dynamics, or reduced-order models of either of these.
In choosing the models the designer must trade off fidelity with analysis
time. The multidisciplinary nature of most design problems complicates
model choice and implementation. Often several iterations are necessary
between the disciplines in order to find the values of the objectives and
constraints. As an example, the aerodynamic loads on a wing affect the
structural deformation of the wing. The structural deformation in turn
changes the shape of the wing and the aerodynamic loads. Therefore, in
analyzing a wing, the aerodynamic and structural analyses must be run a
number of times in turn until the loads and deformation converge.
Standard form: Once the design variables, constraints, objectives,
and the relationships between them have been chosen, the problem can
be expressed in the following form: find x that minimizes J(x) subject to
g(x) ≤ 0, h(x) = 0 and �x ≤ x ≤ x where J is an objective, x is a vector
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