## 1 Introduction  
With the improvement in requirements for environmental quality, noise as a kind of physical pollution is becoming noticeable. Traditional acoustic materials tend to obey the law of mass and are ineffective at low frequencies. Some of these materials, such as glass wool, are difficult to recycle. Recycling costs can be reduced through natural degradation, but this requires a large periodicity. Therefore, in practical applications, traditional acoustic materials are often limited. The advent of acoustic metamaterials has compensated, to some extent, for the shortcomings of conventional materials by enabling low-frequency noise attenuation through subwavelength or deep-subwavelength dimensions. In many noise scenarios, maintaining ventilation is a primary requirement to facilitate heat exchange by convection. Thus, in this paper, an acoustic metamaterial with ventilation capability is proposed for future application in noise-reduction scenarios such as industrial production plants. At the turn of the century, the concept of metamaterials was gradually clarified [1; 2; 3]. Over time, metamaterials have been fully developed in optics [4; 5] and electromagnetism [6; 7; 8]. In recent years, acoustic metamaterial technology has received widespread attention [9; 10; 11; 12]. These metamaterials have been designed, analyzed, and optimized for different applications [13], such as ducted acoustic metamaterial mufflers [14; 15], ventilated acoustic metamaterial windows [16], and acoustic metamaterial filters [17]. Among them, the study of acoustic metamaterials for ventilation is one of the current research hotspots [18; 19; 20]. [PERSON] [21] proposed an acoustic metamaterial to enhance the natural ventilation and acoustic performance of windows. The metacage window structure reduced noise transmission by an average of 30 dB in the frequency range of 350-5000 Hz, with an opening rate of 33% compared with the whole system surface. This study demonstrates the natural ventilation potential of the metacage window. [PERSON] [22] proposed a deep subwavelength acoustic metasurface unit as a high-performance selective muffler based on the Fano-like interference phenomenon and verified the acoustic performance experimentally. This open metamaterial design for high-performance muffling with a large degree of open area is of great practical value in the field of smart sound barriers and fan or engine noise reduction. Whereas designs based on Fano-like interference can only operate in a narrow frequency range, designing broadband sound barriers remains challenging. In this regard, [PERSON] [23] designed a subwavelength-thickness acoustic ventilation barrier consisting of a central hollow aperture and two spiral channels with different pitches. The structure extends the effective bandwidth of the Fano-like principle. The proposed design was experimentally verified, and the results were consistent with analytical predictions and simulations. [PERSON] [24] proposed a vented acoustic metamaterial with resonant cavities arranged around a central air channel. The underlying mechanism of switchable narrowband sound transmission was revealed. Further, acoustic metamaterials extending to three dimensions were proposed. The results of this work may inspire more exploration of acoustic barriers and multifunctional applications, such as innovative noise reduction for building facades and logic elements for acoustic circuits. [PERSON] [25] proposed a subwavelength labyrinth acoustic meta-structure that exhibits superior sound absorption properties. The meta-structure achieves broadband absorption by dissipation of incident-propagating acoustic waves within the labyrinthine channel. In addition, the unique meta-structure design allows air circulation to promote natural ventilation and sound absorption. This design creates additional possibilities for architectural acoustics and noise shielding for both natural ventilation and noise reduction.