## 1 Introduction  
In recent years, the structures embedded with acoustic black hole (ABH) have received wide attention in the application of vibration reduction or noise attenuation, and it also contributes to light-weighting of the structures [1; 2; 3]. Above the characteristic frequency of the structure, the ABH can effectively gather the flexural wave in the structure, and while below the characteristic frequency, multiple acoustic black holes are able to improve the low-order modal loss of the structure [4]. Thus, the vibration reduction and noise attenuation can be achieved in a wide frequency domain with the application of ABHs. Lots of literatures have focused on the acoustic or dynamic properties of ABH structures. [PERSON] proposed an acoustic black hole pipe structure. By gradually increasing the effective admittance of the pipe along the axial direction, the velocity of the sound wave propagating along the pipe is gradually reduced to zero [5]. [PERSON] et al. conducted experimental studies on sound radiation of vibrating rectangular plates containing tapered indentations of power-law profile [6] and found that when a very thin damping material is attached to an acoustic black hole region, the radiated sound power of the structure is reduced by 10 to 18 dB compared with the uniform thickness plate in the frequency range of 1 to 3 kHz [7]. [PERSON] et al. also carried out an experimental study into the damping of flexural vibrations in turbofan blades with trailing edges tapered according to a power-law profile and found that trailing edges of power-law profile with appropriate damping layers are efficient in reduction of the airflow-excited vibrations of the fan blades [8]. [PERSON] experimentally analyzed the influence of the diameter of the damping layer on the radiated sound of rectangular plates with \(4\times 5\) array of embedded ABHs in the full frequency range [9]. Then, he used experimental and numerical methods to investigate the transmission loss (TL) of plates with embedded ABHs [10] and also calculated the vibro-acoustic coupling properties of the ABH system based on wavenumber transformation method [11]. [PERSON] et al. established a two-dimensional daubechies wavelet (DW) model for the sound radiation prediction of plates embedded with a circular ABH indentation [12] and they proposed a general method for optimizing the layout of the surface coating damping layer of ABH plates by topology optimization method to minimize the sound radiation of ABH plates to free space in a given frequency or frequency band [13]. [PERSON] et al. presented analytical solutions of sound radiation of a beam with a wedge-shaped edge embedding ABH by using the transfer matrix method (TMM) [14]. [PERSON] et al. established a numerical finite element model of acoustic radiation considering plate-cavity coupling and proposed that the damping enhancement and the coupling strength decreasing between the plate and the cavity are two mechanisms for the intracavity noise reduction [15]. [PERSON] et al. proposed a wavenumber domain method (WNDM) for the analysis of vibro-acoustic coupling and internal noise reduction mechanism of a pentahedral cavity enclosed by a flexible plate with a two-dimensional ABH indentation [16]. [PERSON] et al. numerically investigated the vibration transmission properties of finite-size plate with embedded different number of ABHs [17] and [PERSON] et al. proposed an active acoustic black hole with piezoelectric patch actuators and unconstrained feedforward control to control the low frequency vibration of a beam [18]. Transmission loss (TL) is an important physical parameter that measures the sound insulation performance of a partition, wall, or other structures. The sound insulation of a plate is a complex vibro-acoustic coupling process, in which the sound wave propagates to the surface of the plate to generate a flexural wave inside the structure and the propagation of the flexural wave in the plate also causes the plate to vibrate and radiate sound outward.