Space-time topology optimization for one-dimensional wave propagation

A space-time extension of the topology optimization method is presented. The formulation, with design variables in both the spatial and temporal domains, is used to create structures with an optimized distribution of material properties that can vary in time. The method is outlined for one-dimensional transient wave propagation in an elastic rod with time dependent Young's modulus. By two simulation examples it is demonstrated how dynamic structures can display rich dynamic behavior such as wavenumber/frequency shifts and lack of energy conservation. The optimization method's potential for creating structures with novel dynamic behavior is illustrated by a simple example; it is shown that an elastic rod in which the optimized stiffness distribution is allowed to vary in time can be much more efficient in prohibiting wave propagation compared to a static bandgap structure. Optimized designs in form of spatio-temporal laminates and checkerboards are generated and discussed. The example lays the foundation for creating designs with more advanced functionalities in future work.