## 4 Simulating Array Acquisitions and Array Imaging
### Finite Element Modelling of Ultrasound  
Time-domain finite element simulations of ultrasound were developed and executed in Pogo, a fast GPU-based explicit elastodynamic solver developed at Imperial College London. The use of FE is motivated by its versatility in representing any geometry, material composition, etc., and the ability to represent the rich physics of wave propagation. Modelling ultrasound with FE is a standard technique in NDE, well-documented, verified, and validated in the literature, both against experiments, analytical theories, and (for Pogo) with equivalent commercial solvers, e.g., [26; 27; 28; 29], so this step is not reported in this paper. Instead, we follow the agreed guidelines for developing FE models by ensuring that a sufficient number of elements per wavelength is included. The geometry of the weld and the transducer setup remained constant across all simulations; the only differing part was the local orientations. Based on the most common defect types of interest, we chose three defect scenarios for further consideration--(i) three single-drilled-holes (SDHs) along the left chamfer, (ii) three SDHs along the right chamfer, and two notches perpendicular to the backwall, close to the weld root. Figure 6 shows the diagram of the numerical configuration with the dimensions and studied defects and the \"nominal\" grain orientation set. Note that while the defects are drawn on the same image in Figure 6, they were considered as three separate defect scenarios in the simulations, as indicated above.  
Weld formation simulations executed at Bay Zoltan Nonprofit Ltd. for Applied Research (BZN) produced local grain orientation over two grids, 0.25 and 2 mm. The latter is a widely accepted pragmatic choice sufficient for supporting inspections. Hence, the delay laws came from ray models over a 2 mm grid. The same 2 mm grid determined the orientation layout in time-domain FE simulations (numerical \"experiments\").  
The domain shown in Figure 6 was meshed using regular quadrilateral elements (CPE4), with an element size of 0.04 mm, which gives more than seventy elements per longitudinal wavelength in the parent material. The time step was 6.21 ns, and simulation time ensured that all signals reflected from the backwall were comfortably captured. We modelled the ultrasonic array as a series of \"element\" excitations; that is, we applied a set of four equal forces acting outward from the element (see Figure 6). Assuming a 64-element array with a pitch of 1 mm and an element width of 0.7 mm, the exciting element was sought in the vicinity of the theoretical centre of each transducer. On reception, the out-of-plane response at a node closest to the theoretical centre of each transducer was recorded. The physical domain was surrounded by absorbing regions along all sides, except for the backwall, to minimise reflections and surface effects not considered in this study. Further, to better expose the effect of modified delay laws on defect signatures, for each considered case, we subtracted the results obtained for an \"intact\" structure from the structure with target defects.