Topology optimization of anisotropic broadband double-negative elastic metamaterials

As the counterpart of electromagnetic and acoustic metamaterials, elastic metamaterials are artificial periodic elastic composite materials offering the possibility to manipulate elastic wave propagation in the subwavelength scale through different mechanisms. For the promising superlensing in the medical ultrasonic detection, double-negative metamaterials possessing the negative effective mass density and elastic modulus simultaneously can be utilized as the ideal superlens for breaking the diffraction limit. In this paper, we present a topology optimization scheme to design the two-dimensional (2D) single-phase anisotropic elastic metamaterials with broadband double-negative effective material properties and demonstrate the superlensing effect at the deep-subwavelength scale. We also discuss the impact of several design parameters adopted in the objective function and constraints on the optimized results. Unlike all previously reported mechanisms, the present optimized structures exhibit the novel quadrupolar or multipolar resonances for the negative effective mass density and negative effective elastic modulus. In addition, negative refraction of the transverse waves in a single-phase material is observed. Most optimized structures in this paper can serve as the anisotropic zero-index metamaterials for the longitudinal or transverse waves at a certain frequency. The cloaking effect is demonstrated for both the longitudinal and transverse waves. Moreover, with the particular constraints in the optimization procedure, a super-anisotropic metamaterial exhibiting the double-negative and hyperbolic dispersions in two principal directions within two different frequency ranges is obtained. The developed optimization scheme provides a robust computational tool for negative-index engineering of elastic metamaterials and may guide the design and optimization of other types of metamaterials, including the electromagnetic and acoustic metamaterials. The unusual properties of our optimized structures can inspire new ideas and novel applications including the low-frequency vibration attenuation, flat lens and ultrasonography for elastic waves.