Applications of Topology Optimization Techniques in Seismic Design of Structure - Structural Seismic Design Optimization and Earthquake Engineering - page 261

Figure 13. The idealized material model used for
modeling the energy dissipating device

smooth the jagged boundaries of the solutions.
The smoothed versions of the initial and optimal
solution are analyzed and their stress distributions
are compared in Figure 15. It can be seen that the
optimal solution provides an even stress distribu-
tion and the stress concentration areas visible in
the initial design have been dissipated. This even
stress distribution improves the responses of this
design against low cycle fatigue. This has been
verified through experimental tests by Ghabraie
et al. (2010).
Figure 16 compares the force-displacement
curves of the initial and final solutions. The op-
timal solution shows a stiffer response than the
initial design.

is considerably higher (37.4% improvement) than
the initial design.

Example 5. Slit Damper

10. CONCLUSION

We now consider shape optimization of a slit
damper. The initial design is depicted in Figure
14d. In order to preserve the periodicity of the
design, one needs to impose an additional con-
straint to the optimization algorithm. To this end,
we partition the design domain into four cells. To
make the cells identical, the sensitivity numbers
of elements are replaced by the mean value of
sensitivity numbers of corresponding elements
in all cells. Putting this mathematically, we write

This chapter reviewed the application of topol-
ogy optimization techniques in seismic design of
structures. Two established topology optimization
methods, namely SIMP and BESO, have been in-
troduced and their application has been illustrated
using numerical examples.
Eigenfrequency optimization of linear elastic
structures in free vibration has been addressed
using the SIMP method. Sensitivity analysis of
eigenfrequencies has been explained and a simple
solution procedure has been presented based on
optimality criteria. Possible numerical instabili-
ties have been mentioned and possible treatments
have been discussed.
The problems involving multiple eigenfrequen-
cies have been considered and simple approaches
to bypass these problems have been discussed.
The sensitivities of multiple eigenfrequencies
have been calculated and the optimality criteria
have been presented. A simple approach to solve
these problems has been proposed and success-
fully applied to a simple problem. The problem of
maximizing the gap between two eigenfrequen-
cies has also been addressed. This problem is of
practical significance when it is desired to push

N

1

cell

∑

α = α

,

(1.125)

i i j

N

j = 1

cell

where α i is the average (corrected) sensitivity
number of element i in all cells, N cell is the number
of cells, and α i , j is the (original) sensitivity number
of the element i in cell j .
The obtained solution is illustrated in Figure
14e. The evolution history of the objective func-
tion is plotted in Figure 14f. Again a significant
improvement (64.3%) in energy absorption is
observable.
Ghabraie et al. (2010) have used a smooth-
ing postprocessor based on Bézier curves to