## 1 Introduction  
Poly-crystalline materials [1] are commonly used in many engineering structures as they are able to resist severe environmental conditions thanks to their high mechanical and thermal properties. The poly-crystalline structure at the micro-scale is characterized by the random spatial distribution of crystals, also known as grains. Defects such as inter-granular cracks, which occur on the grain boundary; trans-granular cracks, which occur inside the grain; and voids/pores, which lead to porosity often, decrease the material properties of the structure and are the main cause of failure at macro-scale. To improve macroscopic properties of such materials from their design phase up to their potential failure, it is essential to analyze and inspect microscopic features [2]. Structural Health Monitoring (SHM) using Ultrasonic Guided Waves (UGWs) can be an effective technique for detecting micro flaws in poly-crystalline materials. SHM [3; 4; 5] is an advanced form of Non-Destructive Testing (NDT) used to assess the reliability and safety of components. SHM is the process of monitoring the state of an engineering structure through the response of permanently integrated sensors to detect any changes in the state of the structure. The structure is analyzed, localized, and predicted in a way that NDT becomes an integral part of the structure [6; 7]. UGWs are acoustic waves that are generated if the characteristic frequency exceeds a certain threshold (typically above 20 [kHz]), and they propagate through various structures such as plates, sheets, and tubes [8]. The UGW technique is well suited in the SHM framework because the waves undergo absorption or scattering when they encounter the boundaries of a structure or the surfaces of flaws, such as cracks or voids/pores. By determining the velocity of the propagating wave and calculating the time it takes for the scattered or reflected wave to travel, it becomes possible to identify the boundaries of the structure or defects [8; 9]. Therefore, utilizing wave-material interactions is possible to characterize material properties, as well as to detect and localize defects in poly-crystals in the NDT-SHM framework [10; 11; 12]. However, because of the experimental challenges in acquiring detailed information about micro-structure architecture and defect characterization, different numerical and computational methods have been proposed to represent the randomness of the micro-structure and the presence of defects in poly-crystalline materials [1; 2; 13; 14; 15]. In recent years, various numerical methods based on elastodynamics theory have been used to study wave propagation and scattering. These methods include the Finite Element Method (FEM) [16; 17], Finite Difference Method (FDM) [18; 19; 20], Finite Volume Method (FVM) (also known as Finite Integration Technique (FIT) [21]), and Boundary Element Method (BEM) [9; 22]. Studies have shown that ultrasonic scattering or attenuation can be used to characterize poly-crystalline micro-structures and to localize flaws [17; 23; 24]. Establishing the relationship between ultrasonic scattering/attenuation and material parameters enables reliable and high-quality material testing [25; 26]. The FEM has been used to model wave-material interactions in poly-crystalline materials [20; 25; 26; 11; 27]. However, FEM presents limitations. Indeed, it is often difficult to embed inter-granular and trans-granular cracks, voids/pores, and other influential features. Furthermore, the computational expense associated with utilizing FEM to model cracks and voids is significant because of the necessity of employing a fine mesh. Additionally, accurate results necessitate a substantial number of elements per wavelength for high-frequency waves. Another method, the FDM, discretizes elastodynamic equations on a staggered grid but suffers from accuracy issues [20; 28]. In contrast, the BEM has gained significance in SHM assessment.