WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

5.3 Space-time topology optimization 47
f(t)
Figure5.3 Finiteone-dimensional mass-springstructureconsistingofanumberofmasses
with attached nonlinear oscillators. Viscous dampers are added in the ends to simulate
absorbing boundaries. From paper [19].
mass ratio and nonlinear stiffness in a structure with 25 attached oscillators. The
damping ratio is not shown since it is found it should always attain the minimum
possible value in order to minimize the transmission of the waves. In this way the
attached oscillators maximize their motion and have the largest possible effect on
the system. The remaining parameters attain optimized values which do not immediately
offer a physical interpretation. However, the nonlinear stiffness parameter
increases along the length of the structure, which in Section 2.5 was shown to be
beneficial for reducing the wave transmission. Notice, that the mass ratio is constrained
so that the mean value must not exceed 0.1. If this constraint is omitted,
the mass parameters attain their maximum value in order to maximize the effect of
the attached oscillators.
5.3 Space-time topology optimization
Usual applications of topology optimization produce static material distributions
also for dynamic problems. However, as demonstrated here, the methodology can be
extended to create designs in which the optimized material distribution also change
in time. With time-dependent designs additional functionalities can be obtained
such as further enhancement of vibration and wave quenching or tailoring of the
dynamic response of structures.
The extension of the design procedure is facilitated by adding the discretized
time as an extra dimension to the existing design space. This implies that if a onedimensional
structure is to be designed the procedure results in a two-dimensional
design grid. Since the time dimension is included in the procedure the optimization
formulation is based on a transient simulation of the model equations. The
sensitivity analysis is extended without great difficulties and importantly it can be
performed as efficiently as for a standard ”static” optimization procedure based on
transient simulation.
An application of the method is illustrated by the example shown in Fig. 5.5.
A one-dimensional elastic structure is subjected to a propagating Gauss-modulated
sinusoidal pulse. The aim of the optimization procedure is to distribute two elastic
materials in the design domain so that the transmission of the pulse through the
domain is minimized. In this case the optimal static structure is a composite with
5 inclusion layers of the stiffer (but with equal density) material – thus resulting in