## 4 Conclusions  
In this paper, a numerical method based on the BEM was developed for the SHM of poly-crystalline micro-structures at micro-scale. The proposed method wasused to investigate how the micro-structural features affect the macro-structural properties of poly-crystalline materials, as well as to understand the resulting changes in both wave propagation and wave-material interaction. In investigating the influence of micro-structural features on both macro-structural properties and wave-material interactions, the novel contribution of this work lies in the integration of BEM and UGWs as numerical tools for the SHM characterization of poly-crystalline materials. The method was validated through experimental tests, thereby demonstrating an agreement between numerical and experimental results. The validation confirmed the reliability of the physical measurements and the effectiveness of the numerical simulation in capturing the behavior and characteristics of the investigated system. Through exploring pristine and damaged conditions, the study analyzed the impact of porosity and inter-granular cracks on the overall structural response. The phenomenon called porosity was quantitatively and qualitatively investigated. The influence of the volume fraction of porosity on material macro-properties was analyzed. An increase in the volume fraction of porosity lead to a decrease in the material macro-properties. Furthermore, the detection of voids was considered based on the amplitude and the ToA of the signal that was backscattered by the void itself. This study also investigated inter-granular cracks, which can lead to embrittlement and laceration. Parametric analyses were conducted to examine the effects of crack position/orientation, crack length, and crack distance from a specific reference. The presence of cracks was detected by the BEM model, and longer cracks resulted in a larger amplitude of the backscattered signal. Additionally, the ToA increased with the distance of the crack from the reference point. As a result of these analyses, the BEM method provided insights into wave propagation and wave-material interactions at the micro-scale, for both voids and inter-granular cracks. A framework is proposed for generating Probability of Detection (PoD) curves based on two models using numerical simulations. The PoD curves demonstrated an increase in the Probability of Detection with increasing crack length, thus showing promise for cracks longer than 3 [mm]. Ongoing research aims to refine representation and address the parameter uncertainties for improved PoD predictions within the 0 to 3 [mm] defect length range, where a low PoD value was observed. This study concluded that the BEM and UGW numerical simulations offer efficient and reliable approaches for understanding microstructural features and the wave-material interactions within poly-crystals at micro-scale. Even though each grain has its own material properties, orientation, and crystal orientation, the model assumes isotropic behavior. Hence, future research will implement anisotropic or orthotropic material models for more accurate micro-structure representations. M.M. : methodology, software, investigation, data curation, writing--original draft, and writing--review and editing. Z.S.K. : conceptualization, methodology, supervision, and writing--review and editing. M.H.F.A. : conceptualization, methodology, supervision, and writing--review and editing. All authors have read and agreed to the published version of the manuscript. The research leading to these results has gratefully received funding from the European JTI-CleanSky2 program under grant agreement no. 314768 for the SHERLOC project (sherloc-projectcom, accessed on 10 January 2020).