WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1626 J. S. JENSEN
details and have a similar overall appearance and performance. The presented designs are the best
that could be obtained with the specific initial design, however, it is perfectly possible that these
are not global optima.
5. CONCLUSIONS
This paper proposes an alternative approach to topology optimization of dynamics problems.
The method addresses the need for considering the response in frequency ranges rather than for
single frequencies. With a traditional approach the system matrix needs to be factorized for each
considered frequency which leads to high computational costs and the alternative modal approach
typically leads to non-smooth optimization problems.
The proposed method uses PAs to rapidly compute frequency response functions with high
accuracy over a wide range of frequencies. Formulas for the sensitivity of the response with
respect to design variables are derived using analytical methods. The derivation is based on the
adjoint approach which provides the design sensitivities for a large number of design variables at
low computational cost. This makes the formulation particularly suited for topology optimization
that typically involves thousands of design variables.
The accuracy of the PAs is studied for simple 1-D and 2-D systems. It is shown that the numerical
procedure yields an accurate approximation to the frequency response also for a relatively a small
number of terms in the Padé expansion. Due to ill-conditioning the effect of numerical precision
in the computations is critical and if many terms are included in the expansion an implementation
of the algorithm with higher precision numerics (e.g. quad precision) should be considered.
Two examples of topology optimization of forced vibrations are included to demonstrate the
proposed optimization procedure. In the first example, a 2-D elastic body is optimized by distributing
a limited amount of reinforcement material so that the average response in medium-range
frequency intervals is minimized. By optimizing for frequency intervals rather than single frequencies
a good broadband performance can be obtained rather than a narrow band performance
that is very sensitive to slight frequency de-tuning. In the second example, structural optimization
of a tip-loaded cantilever is considered and the distribution of solid and void optimized so that
the average dynamic response is minimized in the low-frequency range. Different optimization
frequency intervals result in qualitatively different optimized designs that rely on qualitatively
different mechanisms to obtain a low value of the objective function.
APPENDIX A
In this Appendix it is shown that u ′ can be expressed in the following form:
u ′ = N+1
i=0
where () ′ is the derivative w.r.t. a design variable.
To determine Di and Ei the PA is written as
u = A
B
(Diu ′ i + Eiū ′ i ) (A1)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme
(A2)