## 1 Introduction  
Over time, the progression of acoustic metamaterials has garnered increasing attention due to their significant advantages in sound wave control. These materials find applications in various fields, such as medical treatment, marine applications, aviation, and more [1, 2, 3, 4, 5, 6, 7, 8, 9]. Metamaterials are synthetic composite materials with periodic or aperiodic structures that possess properties determined not only by the constituent materials but also by specially designed architectures [10]. In recent decades, numerous design concepts for acoustic metamaterials have emerged, which can be broadly categorized into resonance and non-resonance approaches. Resonant forms of acoustic metamaterials include local resonance [11; 12; 13], Helmholtz resonance [14], thin film resonance [15; 16], plate resonance [17; 18; 19], as well as their hybrid variation [20]. Resonant metamaterials have been employed to achieve a diverse range of acoustic parameters with distinctive characteristics, including negative mass density, negative volume modulus, and double negative parameters [12; 15; 17]. However, due to the limitations imposed by resonance principles, the frequency band range of effective parameters is often relatively narrow. The equivalent wideband parameters of non-resonant acoustic metamaterials have aroused strong interest, the most representative of which is pentamode metamaterials. These materials possess five zero or very small eigenvalues in their elastic matrix and demonstrate resistance against compression while being susceptible to shear deformation, a property commonly referred to as metafluid. In 1995, [PERSON] proposed a three-dimensional pentamode metamaterial featuring a diamond-like structure [21]. Subsequently, [PERSON] et al. investigated the dispersion characteristics of this pentamode metamaterial under material anisotropy by altering the symmetry of its unit structure [22]. It was discovered that selecting specific cell structure parameters can transform the material from a pentamode into an auxetic material. Due to practical challenges associated with implementing three-dimensional pentamode metamaterials, there has been an increasing focus on two-dimensional counterparts. [PERSON] et al. proposed a hexagonal honeycomb structure for a two-dimensional pentamode acoustic cloak with little shear stiffness [23]. Rectangular counterweights are positioned at each of the six vertices to adjust the density. Subsequently, several researchers have modified this structure and designed various broadband structures based on pentamode materials [24; 25; 26; 27; 28; 29; 30; 31]. [PERSON] et al. [27] developed a two-dimensional pentamode material structure with different structural parameters and investigated the relationship between its effective mechanical properties and acoustic properties. The findings indicate that bending vibration modes in the unit cell band curve influence the acoustic properties of two-dimensional pentamode materials. However, it should be noted that translating mechanical properties into acoustic ones is not straightforward, despite higher ratios of bulk modulus to shear modulus resulting in properties closer to water. [PERSON] et al. [3] systematically designed and explored a series of new isotropic or anisotropic single-phase, two-dimensional pentamode material microstructures through bottom-up topological optimization, enabling customized structural equivalent parameters to be obtained. In 2015, [PERSON] et al., utilizing the excellent broadband characteristics and impedance-matching capabilities of pentamode materials, employed eight different microstructures based on generalized Snell's law principles to design a metasurface, achieving a dispersion-free metasurface across a broad frequency range [25]. In 2019, [PERSON] et al. designed and fabricated a pentamode metamaterial that experimentally validated the broadband conversion from underwater cylindrical waves to plane waves [29]. In 2023, [PERSON] et al.