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.
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