## 2 Materials and Methods
### Elastodynamic Boundary Integral Equation (BIE) in the Laplace Transform Formulation
The matrix \(\bar{A}\) was multiplied by the vector \(\bar{x}\) of unknown transformed displacements and tractions, and \(\bar{B}\), by the vector \(\bar{y}\) of the known transformed boundary conditions, is as follows:  
\[\bar{A}\bar{x}=\bar{B}\bar{y} \tag{5}\]  
The system of equations represented by matrix in Equation (5) was solved to determine the unknown transformed displacements and tractions for a specific Laplace parameter. To obtain the unknown displacements and tractions as functions of time, the transformed variables must be computed for a range of Laplace parameters. The numerical inversion of the Laplace transform can be achieved using [PERSON]'s method, which relies on a Fast Fourier Transform (FFT). The values of a transformed function, denoted as \(\bar{f}(s)\), were calculated for a series of Laplace parameters, denoted as \(s_{k}=a+\frac{2K\pi}{T}\). Here, \(a\) represents a constant, \(i=\sqrt{-1}\), and \(T\) is the relevant time interval of interest. The values of the original function, denoted as \(f(\tau)\), were obtained at periodic time instances \(\tau_{m}=m\Delta t\ (m=1\dots M-1)\), as determined by the following equation:  
\[f(\tau_{m})=\frac{2}{T}\exp{(am\Delta\tau)}\Bigg{\{}-\frac{1}{2}Re\bigg{[} \bar{f}(a)+Re\bigg{[}\sum_{k=0}^{M-1}\ (A(k)+iB(k))W^{mk}\bigg{]}\bigg{]}\Bigg{\}}, \tag{6}\]where:  
\[ \begin{split} A(k)&=\sum_{l=0}^{L}Re\bigg{[}\tilde{f}(a+i (k+lM)\frac{2\pi}{T}\bigg{]}\\ B(k)&=\sum_{l=0}^{L}Im\bigg{[}\tilde{f}(a)+i(k+lM) \frac{2\pi}{T}\bigg{]}\\ W&=\exp\bigg{(}i\frac{2\pi}{M}\bigg{)},\end{split}  \tag{7}\]  
and \(Re\) [ ] and \(Im\) [ ] denote the real and imaginary parts, respectively. To model polycrystalline materials at micro-scale, the multi-region BEM formulation was developed. Let us consider the micro-structure as shown in Figure 2. Two possible boundaries can be present for each grain \(G=1,\dots\)\(N_{g}\), where \(N_{g}\) is the number of grains considered, \(\Sigma_{nc}^{G}\) is the green boundary indicating the intersection between the grain and the domain's boundary (nc = non contact), and \(\Sigma_{c}^{G}\) is the black boundary interface between the adjacent grains (c = contact). Hence, the entire grain boundary is given by the following:  
\[\Sigma^{G} =\Sigma_{nc}^{G}\bigcup\Sigma_{c}^{G}.\] (8) For each grain, the following variables were defined:  
* \(\tilde{u}_{nc}^{G}\tilde{t}_{nc}^{G}\) nodal displacements and tractions at the external boundary;
* \(\tilde{u}_{c}^{G}\tilde{t}_{c}^{G}\) nodal displacements and tractions at the grain-boundary interface. For each grain, the displacement boundary integral Equation (2) can be written as follows:  
\[ \begin{split} C_{ij}^{G}(\mathbf{x}^{\prime})\tilde{u}_{j}^{G}( \mathbf{x}^{\prime},s)&+\int_{\Sigma_{nc}^{G}}\tilde{T}_{ij}^{G}( \mathbf{x}^{\prime},s)\ u_{j}^{G}(x)\ d\Sigma_{nc}^{G}+\int_{\Sigma_{c}^{G}} \tilde{T}_{ij}^{G}(\mathbf{x}^{\prime},s)\ u_{j}^{G}(x)\ d\Sigma_{c}^{G}\\ &=\int_{\Sigma_{nc}^{G}}\tilde{U}_{ij}^{G}(\mathbf{x}^{\prime},s )\ i_{j}^{\widetilde{G}}(x)\ d\Sigma_{nc}^{G}+\int_{\Sigma_{c}^{G}}\tilde{U}_{ ij}^{G}(\mathbf{x}^{\prime},s)\ i_{j}^{\widetilde{G}}(x)\ d\Sigma_{c}^{G},\end{split}  \tag{9}\]  
where the symbol \(\left(\tilde{f}_{i}\right)\) indicates a quantity expressed in the material reference system. Equation (9) can be written in matrix form:  
\[ \begin{split}\left[\tilde{H}_{nc}^{G}\qquad\tilde{H}_{c}^{G} \right]\begin{cases}\tilde{u}_{nc}^{G}\\ \tilde{u}_{c}^{G}\end{cases}=\begin{bmatrix}\tilde{G}_{nc}^{G}\qquad\tilde{G }_{c}^{G}\end{cases}\begin{cases}\tilde{t}_{nc}^{G}\\ \tilde{t}_{c}^{G}\end{cases}.\end{split}  \tag{10}\]  
Interface boundary conditions between two adjacent grains \"p\" and \"q\" were applied as follows:  
\[ \begin{split}&\delta u^{I}=u_{c}^{p}+u_{c}^{q}\\ &t^{I}=t_{c}^{p}=t_{c}^{q}.\end{split}  \tag{11}\]  
Re-arranging the system with unknowns on the left-hand side and known variables on the right-hand side in a similar procedure as Equation (5) gives the following:  
\[