## 2 Materials and Methods
### Elastodynamic Boundary Integral Equation (BIE) in the Laplace Transform Formulation  
The Boundary Element Method (BEM) is an established method for a range of problems in solid mechanics, including fracture and elastic wave propagation problems [2; 22; 29; 31; 32]. The BEM requires only the discretization of the boundaries without discretizing the domain. For poly-crystalline materials, only the external domains and grain boundaries need to be discretized, thus reducing the size of the problem substantially in comparison to domain-type methods [33; 34; 35; 30; 36]. In the context of a homogeneous and isotropic linear elastic body that is within the spatial domain \(\Phi\) and bounded by the boundary \(\Sigma\), and when subjected to dynamic forces, let us consider the scenario where initial displacements, velocities, and body forcesare all assumed to be zero. Under the assumption of small displacement theory, the Laplace transform of the governing equation for a specific point \(\mathbf{x}\) within the body can be expressed as follows:  
\[(c_{1}^{2}-c_{2}^{2})\bar{u}_{ij}(\mathbf{x},s)+c_{2}^{2}\bar{u}_{j,ii}(\mathbf{ x},s)=s^{2}\bar{u}_{i}(\mathbf{x},s), \tag{1}\]  
where \(c_{1}\) and \(c_{2}\) are the velocities of longitudinal and shear waves, respectively. \(\bar{u}_{i}(\mathbf{x},s)\) is the component of the transformed displacement of a point \(\mathbf{x}\) and \(s\) is the Laplace parameter. The displacement of a point \(\mathbf{x}^{\prime}\) can be determined from the transformed dynamic equivalent of Somigliana's identity [30; 33; 34; 35; 36]:  
\[C_{ij}\big{(}\mathbf{x}^{\prime}\big{)}\,\bar{u}_{j}\big{(}\mathbf{x}^{\prime },s\big{)}=\,\int_{\Sigma}\bar{U}_{ij}\big{(}\mathbf{x}^{\prime},\mathbf{x},\,s \big{)}\,\bar{t}_{j}(\mathbf{x},\,s)\ d\Sigma(\mathbf{x})-\int_{\Sigma}\bar{T}_ {ij}\big{(}\mathbf{x}^{\prime},\mathbf{x},\,s\big{)}\,\bar{u}_{j}(\mathbf{x}, \,s)\ d\Sigma(\mathbf{x})\ for\ i,j=1,2, \tag{2}\]  
where \(\bar{u}_{j}(\mathbf{x},\,s)\) and \(\bar{t}_{j}(\mathbf{x},\,s)\) are the Laplace transform of the displacement and tractions, respectively. At the boundary, \(\bar{U}_{ij}(\mathbf{x}^{\prime},\mathbf{x},\,s)\) and \(\bar{T}_{ij}(\mathbf{x}^{\prime},\mathbf{x},\,s)\) are the Laplace transform of the fundamental solutions of elastodynamics (they are given in Appendix A). \(\mathbf{x}^{\prime}\) is the collocation point, which can be a domain or a boundary point; \(\mathbf{x}\) is the integration point (boundary point); and \(C_{ij}(\mathbf{x}^{\prime})\) is a constant, depending on the position of the collocation point. The spatial variation of the transformed boundary displacements and tractions was approximated by dividing the boundary \(\Sigma\) into \(N_{e}\) boundary linear elements, and the variation of boundary values was approximated using constant elements. Hence, the discretized form of Equation (2) is as follows:  
\[C_{ij}\big{(}\mathbf{x}^{\prime}\big{)}\,\bar{u}_{j}(s)=\,\sum_{n=1}^{N_{e}} \bigg{[}\bar{t}_{j}^{n}(s)\int_{-1}^{1}\bar{U}_{ij}(\mathbf{x}(\eta),\,s)^{n} (\eta)\ d\eta-\ \bar{u}_{j}^{n}(s)\int_{-1}^{1}\bar{T}_{ij}(\mathbf{x}(\eta),\,s) \big{]}^{n}(\eta)\ d\eta\bigg{]} \tag{3}\]  
where \(J^{n}(\eta)\) is the Jacobian of the transformation, and \(\eta\) is the local non-dimensional coordinates \((-1\leq\eta\leq+1)\). The set of boundary equations, applied to all the boundary nodes, can be written in matrix form as follows:  
\[\bar{H}\bar{u}=\bar{G}\bar{t} \tag{4}\]  
where \(\bar{u}\) and \(\bar{t}\) contain the nodal values of the transformed displacement and tractions, respectively. \(\bar{H}\) and \(\bar{G}\) depend upon integrals of the transformed fundamental solutions, as well as the Jacobian and the interpolating functions. The matrices \(\bar{H}\) and \(\bar{G}\) were reordered according to the boundary conditions to give the new matrices \(\bar{A}\) and \(\bar{B}\).