Keywords: topology optimization, transient analysis, waveguide, Helmholtz equation, wave packet, density filter.

The method of topology optimization has previously been applied to a variety of problems that involve steady-state dynamics, e.g. eigenfrequencies, forced vibrations and wave propagation [1]. Recently, there has been a great interest in applying topology optimization to problems that deal with the full transient behavior. In this work we extend results obtained in a recent paper [2] (transient wave propagation for one-dimensional problems) by considering two-dimensional elastic structures. In [2] one-dimensional structures were optimized for transient wave propagation. Bandgap structures were designed as well as structures optimized for prescribed shaping of wave pulses. This included structures that split an input wave pulse into two temporally consecutive pulses. The gradient-based optimization procedure was facilitated by analytical sensitivity analysis developed for transient problems in [3]. Other related studies include optimization of three-dimensional transient vibration problems [4] and three-dimensional antenna design [5]. This work considers transient wave propagation in a two-dimensional elastic medium. We use a finite element discretization of the computational domain and solve the transient problem using implicit or explicit time integration. In order to simulate wave propagation in the domain we implement perfectly matching layers (PML) so that outgoing waves are effectively absorbed and not reflected back into the computational domain. The implementation of time-domain PML is based partly on [6]. In the present paper several examples of structures with optimized wave propagation behavior are given. The problems considered are for example the design of multi-directional bandgap structures and two-dimensional pulse shaping structures including cloaking devices.

1

M.P. Bendsøe, O. Sigmund, "Topology Optimization: Theory, Methods and Applications", Springer Verlag, 2003.

2

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

3

E. Haug, J. Arora, "Design sensitivity analysis of elastic mechanical structures", Comput Methods Appl Mech Eng, 15, 35-62, 1978. doi:10.1016/0045-7825(78)90004-X

4

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

5

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

6

U. Basu, A.K. Chopra, "Perfectly matched layers for transient elastodynamics of unbounded domains", Int J Numer Methods Eng, 59, 1039-1074, 2004. doi:10.1002/nme.896

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