Applications of Topology Optimization Techniques in Seismic Design of Structure - Structural Seismic Design Optimization and Earthquake Engineering - page 242

the objective function and suboptimal solutions
as shown in the following example.

modal sensitivities of Equation (1.18), decreases
the objective function instead of increasing it.
This produces oscillation after the point of coalesce
and the algorithm converges at a suboptimal solu-
tion (Figure 5b).

Example 2: Clamped-Clamped Beam

Consider the problem of maximization of the
fundamental frequency of a clamped-clamped
beam with volume fraction of 50% and material
properties similar to example 1. All algorithmic
parameters are similar to example 1. The structure
has been discretized into 30×240 identical 4-node
bi-linear square elements. The initial and the fi-
nal solutions and the evolution of the first three
eigenfrequencies are shown in Figure 5.
It can be seen that after 29 iterations, the first
two eigenfrequencies coalesce. Using the single

5.1. Simple Approaches to Avoid
Multiple Eigenfrequencies

A number of simple approaches can be used to
avoid multiple eigenfrequencies. Kosaka and
Swan (1999) proposed a symmetry reduction ap-
proach in which a symmetry condition is imposed
on the design variables to ensure a symmetric
solution. In their paper, Kosaka and Swan (1999)
noted that in a symmetric structure, the multiple

Figure 5. Example 2 without treatment: initial design (a), final solution (b), and evolution of the eigen-
frequencies (c)