## 3 Results and Discussion
### Probability of Detection (PoD) with Inter-Granular Cracks  
The Probability of Detection (PoD) curve method was extensively employed for evaluating the reliability of NDT techniques. The PoD represents the capability of an NDT method to detect a flaw of specific characteristics, such as type and size. PoD curves are constructed based on the characteristic dimensions of a certain defect, such as depth, length, orientation. These curves are influenced by various factors, including material properties, geometric features, defect types, operator skills, and environmental conditions [55; 56]. Generating PoD curves typically requires a significant amount of experimental data, which can be expensive and time-consuming [57; 58]. Consequently, there has been growing interest in utilizing simulation methods to reduce the reliance on extensive experimental testing. The concept of Model-Assisted and/or Simulation-Supported PoD has emerged in the past two decades, in which the aim is to leverage analytical or numerical models to enhance the efficiency and accuracy of PoD curve determination [59; 60; 61]. Building upon the framework of Model-Assisted Probability of Detection (MAPoD), which has been utilized in numerous studies exploring simulation-based approaches to replace or supplement model-based approaches [60; 61; 62], this research endeavors to investigate the feasibility of employing BEM simulations for generating PoD curves in the ultrasonic inspection of poly-crystalline micro-structures. To incorporate the influence of variability factors into the simulation, as well as to subsequently calculate the MAPoD, it is necessary to consider the potential sources of variability. For simplicity in this study, the sources of variability considered were the number of grains and grain morphologies, the material property variations, as well as crack length, crack orientation, and crack position. Specifically, the number of grains considered were 30, 50, and 100. For each number of grains, five different grains morphologies were considered, where the Young's modulus varied within a range of 5% around the mean value for the corresponding material. The material under investigation was aluminum. The crack length was varied from 0.5 to 7 [mm], in increments of 0.5 [mm]. Both the location and orientation of the cracks were randomly distributed. Thus, 10 cracks located and orientated randomly for each crack length and for each grain morphology were considered, thereby resulting in a total of 50 combinations of variability sources per crack length. Consequently, a total of 700 simulations were conducted. Upon completion of all the simulations, the resulting scatter plot of the flaw responses was utilized to construct the PoD curve. The averaged vertical displacement at the upper edge was selected as the output signal. Snapshots from the BEM simulations that considered different scenarios are presented in Figures 24-28. Figure 23: Effect of the crack length. Figure 24 illustrates the averaged pristine signal obtained at the upper edge of the structure for varying numbers of grains, specifically 30, 50, and 100. As depicted in the figure, the signals exhibited excellent agreement in terms of both amplitude and the ToA. Figure 25 showcases a comparison of the signals generated from different grain morphologies while maintaining a fixed number of grains (i.e., 100 grains in this particular case). Notably, these five signals exhibited slight variations, particularly in terms of the ToA, because of the presence of distinct grain morphologies and their associated material properties. Figure 26 presents the signals obtained under the damaged condition, specifically with a 7 [mm] crack, for varying numbers of grains. As observed in the figure, there was a noticeable increase in the signal amplitude that originated from the crack as the crack length increased. Consequently, the amplitude of the backscattered signal from the upper edge decreased, thereby indicating that long cracks significantly impact the overall signal amplitude.