## 1 Introduction
BEM provides a more efficient alternative to previous numerical techniques for modeling grain boundaries and the reflection of UGWs. This is because of its ability to discretize solely the boundaries of the micro-structure and the flaws, thereby requiring fewer elements to accurately represent the UGWs [2; 29; 30]. This study presents the development of the multi-region Elastodynamic BEM in the Laplace Domain to investigate the effects of defects, material properties, and grain variations on wave propagation. The focus of this work is to characterize poly-crystalline materials with the application of the BEM as a tool through which to help, guide, and direct Non-Destructive Testing, such as SHM (see Figure 1). The main idea of this work is to establish a numerical tool/model based on the BEM for the SHM of poly-crystalline structures with these specific aims and objectives:  
* To develop a computational framework for modeling dynamic waves propagation in poly-crystalline materials;
* To investigate the influence of grain morphology, size, and distribution;  
Figure 1: Numerical approach/model. * To investigate the possible scattering of UGWs due to the formation of voids and cracks;
* To assess the applicability of this computational framework when utilized to conduct the SHM of poly-crystalline materials. The characterization of poly-crystalline materials with the application of the BEM has the following advantages:  
1. Aids in improving and better understanding wave-material interactions and the material micro- and macro-scale features, including random defects (those that are very difficult or impossible to assess with experiments);
2. Developing numerical models can lead to reducing the number of experiments; hence, reducing cost and time;
3. Numerical models being used to complement experiments to investigate micro-structural features can possibly improve the safety and lifespan of engineering structures in aerospace, oil and gas, and power plant fields. To demonstrate the accuracy and validity of the proposed method, numerical simulations were carried out and the results are presented, thereby taking into consideration the most common micro-structural features present in poly-crystals and how to detect them. The porosity (presence of voids) was investigated both quantitatively and qualitatively. Different crystal morphologies were considered such as cubic, rhombic, and truncated octahedral. The influence of the volume fraction of porosity on the material macro-properties was analyzed. Furthermore, the detection of voids was considered based on the amplitude and the Time of Arrival (ToA) of the signal that was backscattered by the void itself. Intergranular cracks that cause the embrittlement of micro-structures lead to a phenomenon called laceration. Thus, it is of high interest to be able to model their failure mechanisms and, consequently, the UGWs' material interactions. For this reason, parametric analyses were conducted studying three main effects: the position/orientation of the cracks, the length of the crack, and the crack distance from a specific reference. All the effects were examined considering the amplitude and the ToA of the signal that was backscattered by the crack itself. The structure of the paper is as follows: Section 1 outlines a brief introduction to the research field and some fundamental concepts, such as SHM, UGWs, and the types of defects that can be found in poly-crystalline materials. Section 2 explains the methodology/mathematical model implemented (a focus on the BEM and the Elastodynamic BEM, Voronoi tessellation, and implementation of missing grains). Subsequently, Section 3 is dedicated to the numerical results (pristine condition, damaged condition, comparison of BEM-FEM, and Probability of Detection (PoD) analysis). That section then concludes with a comparative analysis between the experimental tests (in pristine condition) and the numerical simulations in order to validate the effectiveness and accuracy of the employed numerical method. Section 4 outlines the conclusions.