## 3 FEA Theory and Methodology
### Design of FEA Model  
The model followed the geometry of a U-pile as illustrated in Table 1 and Figure 2. Two transducer arrays, for excitation and receiving of signal data in pulse-echo configuration, were modelled as point sources as shown in Figure 5. The design of modelled transducer arrays is explained in the Array Topology subsection.  
Excitation points were positioned 1 m from one end of the pile. In order to transmit the shear horizontal mode, the point load was applied in the direction along the width  
Figure 5: (a) 12 \(\times\) 2 configuration shear transducer array for the web of the pile and (b) 7 \(\times\) 2 configuration shear transducer array for the flange of the pile.  
of the pile. At each point was applied a 10-cycle sine wave modulated using the Hann window function in Equation (7) below:  
\[\mathcal{U}(\ell)=0.5\sin(2\pi\ell\ell)\left[1-\cos(\frac{2\pi\ell\ell}{n})\right] \tag{7}\]  
where \(\ell\) is the excitation central frequency, \(\ell\) is time and \(n\) is the number of cycles.  
When setting up a dynamic transient simulation to model wave propagation throughout a structure, it is essential for the mesh to be optimal. In order to avoid potential inaccuracies, it is necessary for meshing and time stepping within the solver to complement each other; as required by the wave equations. Many researchers have recommended using a minimum of 8 mesh elements per wavelength as shown in Equation (8) [17; 19; 32]:  
\[\Delta\pi\ \leq\ \frac{\lambda_{min}}{8} \tag{8}\]  
where \(\Delta\pi\) is the element size and \(\lambda_{min}\) is the minimum wavelength within in the signal bandwidth. Meshing was kept the same throughout the structure to capture mode conversions and coherent noise generated from reflections.  
The duration of the numerical models was decided based on the expected time of arrival (\(\mathcal{T}\mathcal{O}\mathcal{A}\)) of the signal at the excitation location after reflecting from the simulated defects. The expected \(\mathcal{T}\mathcal{O}\mathcal{A}\) was calculated based on the velocity of the \(\mathcal{S}\mathcal{H}0\) mode and the distance travelled by the \(\mathcal{S}\mathcal{H}0\) mode to the end of the pile and back to the excitation location, as shown in Equation (9) below:  
\[\mathcal{T}\mathcal{O}\mathcal{A}=\frac{\beta}{\mathcal{S}\mathcal{H}0} \tag{9}\]  
where \(\beta\) is the total distance travelled and \(\mathcal{e}_{\mathcal{S}\mathcal{H}0}\) is the velocity of \(\mathcal{S}\mathcal{H}0\) mode at the excitation frequency, obtained from the group velocity dispersion curves generated (Figure 4) for the U-pile.  
It is computationally efficient to manually set up the time steps based on the maximum mesh element size. The time steps are chosen to resolve the wave equally over time, whereas meshing resolves the wave equally over the model. Shorter than optimal time steps increase the computation time without significantly improving the results, and longer time steps do not utilise the mesh optimally [33]. The relationship between time step \(\Delta\mathrm{t}\) and mesh size \(\Delta\pi\) is shown in Equation (10):  
\[\Delta\mathrm{t}=\frac{\mathrm{CFL}\left(\Delta\pi\right)}{e} \tag{10}\]  
where \(\mathrm{CFL}\) is a stability condition. The 0.1 value was selected as it was considered to be near-optimal.