Keywords: dynamic materials, topology optimization, wave propagation, transient analysis.

The concept of dynamic materials was first introduced by Lurie (for an overview see for example the recent monograph [1]). The concept implies structures with material properties that can vary in both space and time. Such structures have rich dynamic behavior and may exhibit phenomena not attainable with standard static structures. This paper introduces the concept of time-space topology optimization in order to synthesize and optimize wave propagation in structures with dynamic materials. Topology optimization of transient wave propagation for a one-dimensional structure was recently studied [2]. It was demonstrated that periodic bandgap structures could be synthesized and that structures could be optimized so that they can be used for wave pulse shaping for example by turning a single input pulse into two temporally consecutive output pulses. Transient formulations of the topology optimization problem have been studied previously, for example for three-dimensional elastodynamic vibrations [3]and for three-dimensional electromagnetic wave propagation [4].

The proposed time-space topology optimization formulation introduces a two-dimensional design grid (for a one-dimensional spatial structure). The spatial grid-direction corresponds to the finite element discretization and the temporal grid-direction is divided into equally sized steps. In this way the material property of each structural point can attain different values at different times depending on the corresponding design variable. A gradient-based optimization formulation is used with the sensitivities computed with the adjoint method. The paper shows examples of the analysis of dynamic structures using different numerical techniques and discusses the importance of the choice of a proper time integration algorithm. Examples of topology optimized structures will be given, for example dynamic bandgap structures and pulse shaping structures. The performance of these structures is compared to their static counterparts.

1

K.A. Lurie, "An Introduction to Mathematical Theory of Dynamic Materials", Springer Verlag, 2007.

2

J. Dahl, J.S. Jensen, O. Sigmund, "Topology optimization for transient wave propagation problems in one dimension", Struct Multidisc Optim, 2008. doi:10.1007/s00158-007-0192-5

3

S. Turtletaub, "Optimal non-homogeneous composites for dynamic loading", Struct Multidisc Optim, 30, 101-112, 2005. doi:10.1007/s00158-004-0502-0

4

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, S. Nishiwaki, "Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique", Int J Numer Methods Eng, 71, 1261-1296, 2007. doi:10.1002/nme.1974

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