## 2 The Design of the Cloak Shell with a Meta-Composite Shell
### A Numerical Simulation of the Cloak Effect in Practice Working Conditions  
The cloak shell was subjected to a numerical simulation using ANSYS 14.0 [31] in 3D in accordance with the specifications listed in Table 1. The real object to be cloaked may present with different kinds of geometries and without fully filling the transformed zone. Based on coordinate transformation only in horizontal plane, previous research [17] on 2D is not suitable for the acoustic cloak of a real 3D object. In this study, the numerical experiment was conducted with a circle-type cloak shell with a cylinder object inside the transformed zone of the cloak shell in three dimensions. The meta-composite structure of the cloak shell was modeled precisely without simplification in order to comply with real working conditions. The plane wave was set with a pressure amplitude of 1 Pa incident on a cylinder object (radius = 185 mm, height = 30 mm) concealed in the center, as shown in Figure 10. The diameter of the cloak shell is 840 mm. We selected plastic as the material for the cloak structure to facilitate its fabrication. The 3D geometry of the object and cloak shell was built and enclosed inside a rectangular acoustic domain (5000 mm \(\times\) 2500 mm \(\times\) 100 mm) for the background medium (air). The density and sound speed of air are 1.2 kg/m\({}^{3}\) and 343 m/s, respectively. As shown in Figure 11, the sound wave is incident on Surface A propagates forward and interacts with the object. By setting a radiation boundary at Surfaces A and B of the acoustic domain, there is no reflected sound wave in the acoustic domain. Aside from Surfaces A and B, other surfaces of the acoustic domain were assumed to be the free surface. Figure 10: Geometry setup used in numerical analysis: plane waves from the right side of the cloak shell are guided along a pre-determined path around the cloak shell away from the cylinder object (radius = 185 mm, height = 30 mm) inside the shell. Figure 9: Spatial distribution of block element tube lengths \(l_{e}\) of the cloak shell (Unit: mm): (**a**) tube length distribution around the cloak shell in the \(e_{u}\) direction, (**b**) in the \(e_{v}\) direction, and (**c**) in the \(e_{w}\) direction. We applied 3D acoustic fluid element (Fluid 221) in the acoustic domain for use in complicated geometry [32]. The 3D geometric model was meshed using a fluid structure interaction (FSI) interface [31] to represent a fluid-structure interaction as plotted in Figure 11. For the sake of accuracy, we used harmonic analysis by the full method to solve all of the system functions directly, rather than using the mode superposition method. The simulation model was solved and evaluated in the post-processing stage, and the pressure field of the whole system was evaluated at the respective frequency. Figures 12-14 presents the total wave field of the uncloaked object, cloaked object, and free field (only incident wave) cases at a 1/3 octave frequency of 800-1250 Hz. As can be observed in Figure 12, in the uncloaked object case, incident waves of 800 Hz resulted in obvious wave scattering as compared to the free field. The sound field of uncloaked object case exhibits strong scattering in both the forward and backward directions. However, in the cloaked object case, the wave field at the same frequency also presented a complex disturbance, and the wave cloaking effect was not apparent. Figure 13 shows a comparison of the field patterns between the uncloaked and cloaked object cases at a cloak shell design frequency of 1000 Hz, with the field patterns of the cloaked object clearly showing that the wave passed around the cloak shell and recovered behind the zone of the cloak shell, reconstructing the original wave field. After passing beyond the cloak shell, the wave fronts of the propagating waves were guided back to their original path by the cloak shell, which prevented the production of intense field disturbances outside the cloak shell.