# Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity
## 3 Numerical Examples
### Numerical Verification. Infinite Plate Subjected to a Transverse Harmonic Force  
The literature has largely studied the problem of a thin plate subjected to harmonic loads, and closed-form solutions can be found in many classical books on vibrations, e.g., [34]. This numerical example is introduced here to verify the method described in Section 2 and show its applicability to continuous structures.  
An aluminium plate of thickness \(h=1\) mm is considered. Flexural waves in the plate are excited by a central transverse harmonic force of magnitude \(F/\left(Eh^{2}\right)=1.4\cdot 10^{-3}\), and the complex out of plane displacement is evaluated at a distance \(\overline{r}/h=300\) from the plate centre, viz. origin of the polar coordinates.  
In this case, the structure is continuous and uniform, and it can be studied as a periodic structure with arbitrary radial and circumferential periodicity: the choice of the period, viz. length \(\overline{R}\) of the unit cell, is arbitrary under standard FE assumptions to avoid element distortion and dispersion errors [38]. Therefore, the number of segments that are used to approximate the slice from the centre to \(\overline{r}\) is arbitrarily chosen. The discretisation must be refined until it does not produce a negligible change in the solution. In this example, only three segments have been found sufficient for convergence.  
In order to simplify the WFE model, a small circle around the point force is removed and the left nodes of the first cell are shifted at a distance \(R_{0}/h=r_{1}=10\) from the point force. In this example, the WFE approximation is obtained for a slice of angle \(\theta=6^{\circ}\). Waves are induced only in the positive radial direction and therefore \(\mathbf{a}_{1}^{-}=0\) is assumed. Periodicity in each segment is further exploited. Here, the number of \"sub-period\" in each segment is chosen by a simple algorithm that optimises the mesh according to the slice's dimension and the number of segments. No significant differences are noticed decreasing the distance \(R_{0}\) further.  
Figure 3 shows the absolute value and the phase of the transverse displacement of the plate. A comparison between the numerical and analytical solutions is shown. The latter is evaluated using the analytical equation given in [34]: \(w=\frac{-i\overline{F}}{8 Dk^{2}}\big{(}H_{0}^{2}(kr)-i\frac{2}{\pi}K_{0}(kr) \big{)}\), where \(k=\sqrt[4]{\omega^{2}\rho h/D}\) is the flexural wavenumber, \(H_{0}^{2}(kr)\) is the zero-order Hankel function of the second kind and \(K_{0}(kr)\) is the zero-order modified Bessel function of the second kind.  
Results obtained using the FE model of the plate with PMLs are, as seen in Figure 3, in excellent agreement with the WFE results (small discrepancies are due to the differences in the FE models). However, this numerical study also demonstrates the computational superiority of the WFE method over the standard FE + PML analysis: the CPU time used by Matlab(r) to solve the WFE model of the plate was approximately four hundred times less than the CPU time used by COMSOL Mutiphysics(r) to solve the correspondent FE + PML model. One of the main reasons for such a difference in the computational cost is the very small size of the WFE matrices compared to the number of DOFs involved in standard FE analysis. The WFE model for this numerical example was realised using one shell element, resulting in 6 DOFs after the WFE reduction, Equation (5), while the size of the FE + PML model was 676,422 DOFs.