WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

44 Chapter 5 Advanced optimization procedures
by Nomura et al. (2007) for a 3D electromagnetics problem, and by the author,
Ole Sigmund and Jonas Dahl (Dahl et al., 2008) who applied transient topology
optimization to 1D wave propagation problems. The extension to a nonlinear wave
propagation problem in paper [19] is believed to be a novel contribution of this
thesis.
Optimaldesignbasedonatransientsimulationwithsystemparametersthatvary
in time was considered by Chambolle and Santosa (2002) for a single parameter in a
one-dimensional structure. The optimization problem was for the first time treated
as a material distribution problem in space and time in the work by Maestre et al.
(2007) and Maestre and Pedregal (2009). However, it should be emphasized that
the problem of finding the optimal time-evolution of parametrized systems has also
been studied in the context of optimal control problems.
5.1 Padé approximants
As described earlier in this thesis, it is often important to optimize the performance
for ranges of excitation frequencies rather than for single frequencies. This can be
accomplished by considering several frequencies simultaneously, but this is usually
computationally costly. Another drawback is that the performance between the
target frequencies may be poor even with many frequencies considered. This can
partly be resolved by tracking the worst-case frequencies during the optimization
iterations as described in Section 4.1 using Padé approximants.
Another and more efficient approach is to use the Padé approximant directly in
the optimization process. With Padé approximants the frequency response can be
approximated as:
u(Ω) ≈ u(Ω0)+ N i
i=1ai(Ω−Ω0) 1+ N i=1bi(Ω−Ω0) i
(5.1)
inwhichtheunknowncoefficientsbi andai aredetermined basedonthecomputation
offirstandhigherorderderivativesd n u/dΩ n atachosenexpansionfrequencyΩ0. An
accurateapproximationforu(Ω) canbeobtainedwithaproper choiceofthenumber
of expansion functions N. Only one system matrix factorization is needed (for the
center frequency Ω0) and the computational overhead in addition to factorizing the
system matrix is usually small. Additionally, the frequency resolution can be chosen
arbitrarily fine without significant extra computational cost.
Fig. 5.1 illustrates an optimization example that illustrates an application of the
optimization scheme. In Fig. 5.1a is shown a 2D structure subjected to forced vibrations.
Two elastic materials are distributed in the structure in order to minimize
the vibrational response in point A.
Fig. 5.1c-e show three examples of optimized distributions of the two materials.
The structure is optimized for a single frequency (Ω = 1) and for two different
frequencyintervals(Ω = 0.8−1.2andΩ = 0.7−1.3). Forbothfrequencyintervalsthe
optimization procedure is performed with seven expansion functions (N = 7) and a