Page 389 - Finite Element Modeling and Simulations with ANSYS Workbench

374 Finite Element Modeling and Simulation with ANSYS Workbench

FIGURE 11.1
Topology optimization of a bridge structure: (a) The original design space and (b) the optimized layout with an
80% weight reduction target.

11.3 Parametric Optimization
In the final design stage, a design’s performance is greatly influenced by its shape and size.
Parametric optimization can be used to help designers determine the optimal shape and
dimensions of a structure. In parametric optimization, the independent variables whose
values can be changed to improve a design are called design variables. Design variables are
usually geometric parameters such as length, thickness, or control point coordinates that
control a design’s shape. Responses of the design to applied loads are known as state vari-
ables, which are functions of the design variables. Examples of state variables are stresses,
deformations, temperatures, frequencies, and so on. The restrictions placed on the design
are design constraints. In general, parametric optimization involves minimizing an objec-
tive function of the design variables subject to a given set of design constraints [17].
For example, consider the parametric optimization of a stiffened aluminum panel with
clamped edges. Stiffened panels are suited for weight-sensitive designs and are widely
used in ship decks, air vehicles, and offshore structures. The main drawback is that they
are light-weight structures with low natural frequencies, leading to a greater risk of reso-
nance. For the design shown in Figure 11.2, the variables that are allowed to change, that
is, the design variables, are the stiffener height h, the plate thickness t, the longitudinal
stiffener thickness t long , and the lateral stiffener thickness t . To reduce the panel’s vulner-
lat
ability to vibration-induced movement, its fundamental frequency f base , that is, the state
variable in the study, is to be set above, say, 20 Hz. Suppose your objective is to minimize
the panel’s overall weight. You may set up a parametric optimization study solving for the
optimum values of the four design variables so as to minimize the panel’s weight while
satisfying the design constraint of f base > 20 Hz.

11.4 Design Space Exploration for Parametric Optimization
A “black box” model shown in Figure 11.3 can be used to illustrate the parametric optimi-
zation schematic, as a direct relationship between the design variables and responses is
generally unknown. The schematic includes the use of a parametric finite element model
for finding the effects of design variables (inputs) to the responses (outputs). After multiple
response datasets are collected from finite element simulations, a mapping relationship can