The ability of a material containing a periodic arrangement of second-phase inclusions to prevent transmission of waves in certain frequency ranges is well known. This is true for all types of waves including acoustic, electromagnetic, and elastic. These forbidden regions are called band gaps. They arise as incident waves are effectively attenuated by interference among the scattered wave fields. Indeed much of current semiconductor technology revolves around band-gap engineering with regard to electron flow in the periodic potentials resulting from atoms in their lattice positions. The phenomena are also being heavily explored in the context of light via the development of photonic crystals. Things become more interesting if instead of thinking of periodic arrangements, one selectively removes some of the inclusions in the periodic geometry creating defects. If done right, this can result in a material microstructure that can guide waves through the material. Advances in nano and micromanufacturing technologies in the last couple of years have opened up the possibility to fabricate heterogeneous material systems with precise positional control of the constituent materials. For example, it is now possible to place thin-film materials precisely at a resolution of fractions of a micron. Depending on how it is done, one can envision designing a material so that a wave will be guided to a particular location and/or away from another and as a result damping or amplifying the wave locally. In this work we develop a topology optimization approach to design such nanostructured materials. We demonstrate the approach through the design of three multifunctional phononic composite materials composed of silicon and aluminum: i) a grating designed to stop wave propagation at a specified frequency, ii) a waveguide that bends the propagation path of an elastic wave, and iii) an elastic switch that switches an input signal between two output ports based on the state of the input signal.

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