Applications of Topology Optimization Techniques in Seismic Design of Structure - Structural Seismic Design Optimization and Earthquake Engineering - page 253

Figure 11. Minimizing the dynamic compliance of the frame of example 3 under a combination of periodic
loading of Figure 10a and static loading: external static loads (a), and final topologies for different input
frequencies of the periodic load; Ω = 0 (b), Ω = 60 rad/s (c), Ω = 175 rad/s (d), and Ω = 330 rad/s (e)

Including either geometrical or material non-
linearity in problems causes the stiffness matrix
to be load-dependent. The equilibrium equations
in such problems are commonly solved by the
Newton-Raphson method. Finding eigenfrequen-
cies will require a subsequent modal analysis.
Sensitivity analysis of a general displacement-
based functional in a geometrically and materially
non-linear system has been formulated by Jung
and Gea (2004). This formulation can be used to
calculate sensitivities of compliance-like objective
functions. Similar procedure of deriving the sen-
sitivities of an energy functional for a non-linear
system is presented in Section 9.1.
In geometrically non-linear problems, the ex-
tremely soft “void” elements of the SIMP material
model can be troublesome showing zero or even
negative tangent stiffness and causing serious
convergence problems (see e.g. Buhl et al. 2000).
A new approach called Element Connectivity
Parameterization has been proposed by Yoon and
Kim (2005) to address this problem. This approach

proves to be useful in topology optimization of
non-linear structures under dynamic loads (Yoon
2010a, 2011). Another approach to solve this issue
is eliminating the void elements (see Section 9.1).
Considering damping effects will also change
the sensitivities of the objective functions consid-
ered. In the following we update the previously
derived sensitivities in presence of damping.

8.1. Forced Vibration with Damping

The equation of motion under a periodic load
takes the form

 

Mu Cu Ku + + = = p p cos Ω t (1.78)

C

Introducing the complex displacement
z z z

( )

i e Ω , and assuming u to be the real
part of z , we rewrite this equation in the complex
space as

= +

i t

C S