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The advent of photonic and phononic crystals, and electromagnetic and acoustic metamaterials, has attracted extensive attention and profoundly affected researchers, since such do not appear in nature and exhibit novel characteristics unavailable with natural materials. Phononic crystals and acoustic metamaterials were developed following on photonic crystals and electromagnetic metamaterials by conceived with different forms of waves, and they respectively exhibit analogous behavior. Acoustic metamaterials are regarded as a new type of artificial composite acoustic material that is closely related to the concept of phononic crystals at a subwavelength scale. The fact that the unit cell dimensions in acoustic metamaterials is roughly one tenth that of the acoustic wavelength, or even smaller, makes these materials especially promising in a broad range of applications, since their size is far smaller than that in phononic crystals which are impractically huge and unapplicable in real life, and triggering intensive research on a variety of topics since Liu et al. (2000). They first engineered a cubic array of bi-layer coated spheres in water that show a complete band gap as a result of effective negative elastic constants. The other two novel properties of acoustic metamaterials include a negative mass density and a negative refractive index. Abstract Much effort has been made to experimentally fabricate acoustic metamaterials that display novel properties, such as single or double negativity, negative refractive index. The novel properties displayed by acoustic metamaterials and their deep subwavelength nature, whereas the size of phononic crystals is in a comparable scale to wavelength, facilitate the development of real-world applications, from sound attenuation to superlenses, and acoustic cloaking. Compared with a conventional trial and error method, a systematic structural design methodology such as topology optimization is useful to achieve optimal designs of acoustic metamaterials. This paper proposes a topology optimization method for the design of acoustic metamaterials with the property of negative mass density. We consider mass-in-mass system that consists of a solid inclusion coated with a soft materials embedded in matrix material to produce a local dipole resonance that demonstrate negative mass density. The optimization is conducted to find optimal configurations of a locally resonant unit cell structure which achieves negative mass density at a certain desired frequency. The S-parameter retrieval method is efficaciously applied to describe the acoustic metamaterial according to its effective mass density. A level set-based method is employed to obtain the optimal configurations that have clear boundaries without gray scales, which would otherwise limit fabricability of the obtained acoustic metamaterial designs. The sensitivity analysis is performed by the adjoint variable method, and a reaction-diffusion equation is used to update the level set function. Several numerical examples are provided to demonstrate how the algorithm works, and the obtained optimal configurations illustrate the achievement of acoustic metamaterials that have negative mass density at certain prescribed frequencies.

Topology optimization of acoustic metamaterials with negative mass density using a level set-based method