# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 3 Numerical Examples and Discussions
### The Band Diagram Based on \(\omega(\mathbf{k})\) Formulations  
Then, the numerical simulation for acquiring the real band diagram of a two-dimensional phononic plate with a square lattice is performed based on the \(\omega(\mathbf{k})\) formulations. The unit cell with the corresponding IBZ is shown in Figure 2. The material properties of the components are specified by elastic moduli \(E_{1}\) and \(E_{2}\), densities \(\rho_{1}\) and \(\rho_{2}\), and Poisson's ratios \(\
u_{1}\) and \(\
u_{2}\), and the ratio \(r_{e}\) between the elastic modulus is assumed as 0.001 while the other material parameters remain the same. The material with larger elastic modulus is assumed as steel, the exterior fluid is assumed as water, and its sonic wave speed and density are 1500 m/s and 1025 kg/m\({}^{3}\), respectively. The unit cell length and the thickness of the phononic plate are denoted by \(l\) and \(t\), which are designed as 0.02 m and 0.0002 m, \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Proposed Method**} & \multicolumn{2}{c}{**Reference [36]**} \\ \cline{2-5}  & **System with Fluid** & **System without Fluid** & **System with Fluid** & **System without Fluid** \\ \hline First band gap & \(0.95<\Omega<2.81\) & \(3.09<\Omega<9.08\) & \(0.81<\Omega<3.01\) & \(2.85<\Omega<9.05\) \\ Second band gap & \(9.78<\Omega<15.2\) & \(22.4<\Omega<35.07\) & \(9.78<\Omega<15.17\) & \(22.4<\Omega<35.82\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Locations of band gaps of the periodic plate from [36]. respectively. The height of the fluid domain \(H\) is set as 0.02 m. It is noted that the above parameters are kept unchanged unless otherwise stated. It is obvious that the unit cell possesses the highest-level symmetry and, thus, the complete band gap can be easily obtained by scanning the \(k\)-space along the edges of the IBZ depicted by the grey region in Figure 2b. It is well known that the coordinates of points \(\Gamma\), \(X\), and M are (0,0), (\(\tau\)/\(l\),0), and (\(\tau\)/\(l\), \(\tau\)/\(l\)), respectively. Equation (6) is adopted and solved using a state-space transformation method [40]. To verify the incompressible fluid assumption, the discrete formulation with less computation costs from Equation (10) is also applied to obtain the band diagram. The dispersion curves with high frequencies are ignored; henceforth, only the first to third band regions are given. The fluid-structure coupled system is meshed into 12, 12, 12 and 16, 16, 16 finite element divisions, respectively, in the \(x\), \(y\), and \(z\) directions, respectively. The dispersion relations obtained by Equations (6) and (10) with different mesh schemes are illustrated in Figure 3a. It can be found that the results are merely overlapped, which validates the incompressible fluid assumption used in band gap calculations. Figure 3b provides the band diagram for the pure phononic plate without fluid loading. Obviously, there are two \(\Gamma\)X directional and one complete band gaps in both spatial dispersion plots. The first directional band gap is induced by the local resonance and the second one is generated by the phenomenon of Bragg scattering. In addition, the complete band gap induced by the localized resonant mechanisms is found between the first and second dispersion curves. Figure 3: The real band gap diagram for the phononic plate: (**a**) with external fluid loading; (**b**) without the fluid loading. \(N\) denotes the number of elements. Figure 2: (**a**) The material arrangements of the two-dimensional lattice considered in this work; (**b**) the first Brillouin zone of the 2D periodic lattice; the grey section represents the irreducible region of the Brillouin zone. One can also see that the band structure is strongly shifted to low frequencies and the absolute bandwidths of the stop bands are enlarged by the fluid loading. However, in Ref. [36], the fluid loading is able to narrow the band gap width.