## 1 Introduction
Essentially, these materials are engineered in a way such that external loads cause changes in their topology, resulting in alterations to the width and locations of the bandgaps. To achieve this functionality, programmable periodic metamaterials utilize factors such as specific or varying Poisson's ratios and thermally-- responsive elastic moduli [11; 13; 14; 15; 16]. Another recent improvement involves creating novel designs for periodic metamaterials. Researchers started with an existing base design and systematically applied geometric modification techniques to generate a new class of periodic metamaterial with unique bandgap behavior. The goal is to create these new--class materials that exhibit desired bandgaps within specific frequency ranges. Geometric modifications involve incorporating curvatures, like sinusoidal curves, spline curves, Koch fractals, and hierarchical orders onto the straight ligaments of the base design, which were found to develop new designs of metamaterials with enhanced bandgaps' characteristics [17; 18; 19; 20; 21; 22]. Additionally, altering connections between rods and cores in geometric modification methods was proposed for creating optimized periodic metamaterials with enhanced bandgaps [23; 24; 25; 26]. Following the success of periodic metamaterials in applications targeting vibration attenuation, researchers have turned their attention to exploring the potential use of these materials for creating acoustic bandgaps that can attenuate sound waves. Two designs configurations have been proposed: one involves using shell cores and slits as repeating elements in the periodic metamaterial, which act as resonators for acoustic waves and result in different resonating frequencies and bandgaps, while the other approach involves adjusting Co--Continuous complex structures to achieve bandgaps at various frequency ranges. Previous studies focused on this first design configuration to create periodic metamaterials capable of attenuating acoustic waves within the audible frequency range [27; 28; 29; 30; 31; 32; 33]. The variations among these studies primarily stemmed from differences in the shape configuration of shell cores and distributions of slits across the repeating elements of the periodic metamaterial. A parametric analysis was conducted to verify the tunability of acoustic bandgaps with varying parameters controlling the size of the shell--coreand slits in these earlier mentioned works. The second proposed design configuration involves the development of a periodic metamaterial that is based on Triply Periodic Minimal Structures (TPMS). These TPMS--based structures have complex architectures with intricate arrangements of pores, resulting in unique interactions with acoustic waves. The performances of different TPMS--based structures, such as simply cubic, body--centered cubic, face--centered cubic, primitive, I--Wrapped Package, and Newious structures, were analyzed and compared in terms of their bandgap behavior [34; 35]. By varying the rate of porosity in each TPMS--based structure, the relationship between topology and bandgaps was investigated. In order to advance the field of acoustic metamaterials, researchers have proposed further enhancements in bandgap programmability and novel design creation. The primary aim of the aforementioned design configurations was to enhance acoustic bandgaps by developing a new class of acoustic periodic metamaterial. This falls under the category of novel design creation. Another approach is the development of programmable acoustic metamaterials that allow for dynamic tunability of bandgaps through an external tuning mechanism dependent on external loads. By incorporating repeating elements with foldability and compressibility characteristics, it has been possible to achieve programmable periodic acoustic metamaterials, where changes in topology result in variations in both the width and location of the acoustic bandgaps, similar to how topological changes affect mechanical wave attenuation in programmable periodic metamaterials [36; 37].