  Aiming at the promising superlensing for the medical ultrasonic and
detection, the double-negative metamaterials which possess the negative mass
density and elastic modulus simultaneously can be acted as the ideal superlens
for breaking the diffraction limit. In this paper, we use topology optimization
to design the two-dimensional single-phase anisotropic elastic metamaterials
with broadband double-negative indices and numerically demonstrate the
superlensing at the deep-subwavelength scale. We also discuss the impact of
several parameters adopted in the objective function and constraints on the
optimized results. Unlike all previous reported mechanisms, our optimized
structures exhibit the new quadrupolar or multipolar resonances for the
negative mass density, negative longitudinal and shear moduli. In addition, we
observe the negative refraction of transverse waves in a single-phase material.
Most structures can serve as the anisotropic zero-index metamaterials for the
longitudinal or transverse wave at a certain frequency. The cloaking effect is
demonstrated for both the longitudinal and transverse waves. Moreover, with the
particular constraints in optimization, we design a super-anisotropic
metamaterial exhibiting the double-negative and hyperbolic dispersions along
two principal directions, respectively. Our optimization work provides a robust
computational approach to negative index engineering in elastic metamaterials
and guides design of other kinds of metamaterials, including the
electromagnetic and acoustic metamaterials. The unusual properties of our
optimized structures are likely to inspire new ideas and novel applications
including the low-frequency vibration attenuation, flat lens and
ultrasonography for elastic waves in the future.
