## 2 Governing Equations and Methods
### Dispersive Properties of a Swiss-Cross Periodic Structure  
In this manuscript, we are interested in dynamically exciting a metastructure whose building blocks are enclosed within the dashed black square of Figure 1a. The blue region represents plane-strain aluminium (material parameter values are provided in Section 2.3 when describing the finite element models) whereas the white region, reminiscent of a Swiss cross, is void, with zero traction boundary conditions prescribed on its boundaries, i.e.,  
\[\sigma_{ij}n_{j}=0. \tag{3}\]  
The unit vector \(n_{j}\equiv n_{j}(x_{1},x_{2})\) is normal to the line delimiting the void and \(\sigma_{ij}\) are the in-plane components of the plane-strain elastic stress tensor. Bloch-Floquet quasi-periodic boundary conditions [32] for the displacement field are  
\[\mathbf{u}(\mathbf{x}+\zeta_{1}\mathbf{L}_{1}+\zeta_{2}\mathbf{L}_{2})=\exp(i \zeta_{1}^{*}\mathbf{k}\cdot\mathbf{L}_{1}+i\zeta_{2}\mathbf{k}\cdot\mathbf{L }_{2})\mathbf{u}(\mathbf{x}),\ \ \ \text{with}\ \ \mathbf{x}\in\Omega, \tag{4}\]  
where \((\zeta_{1},\zeta_{2})\in\mathbb{Z}^{2}\), \(\mathbf{k}=(k_{x},k_{y})^{\mathrm{T}}\), \(\mathbf{L}_{1}=L(1,0)^{\mathrm{T}}\), \(\mathbf{L}_{2}=L(0,1)^{\mathrm{T}}\), and \(\Omega\) is the region enclosed by the dashed square in Figure 1a. The dispersion surfaces for the unit cell illustrated in Figure 1a are obtained by performing the following computational steps:  
* FE discretisation of the time-harmonic counterpart of the Lame system of Equation (1);  
Figure 1: Panel (**a**) is a schematic representation of a \(2\times 2\) cluster of unit cells comprising Swiss-cross holes. The black dashed square of side-length \(L\) encloses an individual unit cell. Panel (**b**) shows the first Brillouin zone of the unit cell in panel (**a**), where the dashed blue lines indicate its irreducible fraction. * prescription of Bloch-Floquet quasi-periodic conditions as in Equation (4), along with \(\mathbf{F}=\mathbf{0}\) and zero-traction conditions (see Equation (3)) at the boundaries of the Swiss-cross holes;
* solution of the resulting eigenvalue problem for the Bloch frequency \(\omega\) as a function of the Bloch vector \(\mathbf{k}\) in the first Brillouin zone represented in Figure 1b. The high symmetry points in Figure 1b are \[\Gamma= \begin{pmatrix}0\\ 0\end{pmatrix} ,\ \ \ \mathbf{X}=\frac{\pi}{L} \begin{pmatrix}1\\ 0\end{pmatrix} \ \ \text{and}\ \ \ \mathbf{M}=\frac{\pi}{L} \begin{pmatrix}1\\ 1\end{pmatrix} .\] (5)  
In order to obtain a fine resolution of the dispersive properties, the first Brillouin zone was sampled using \(100\times 100\) equally spaced points. The problem was then solved by using the commercial FE software COMSOL Multiphysics. Table 1 summarises the geometric parameter values of the unit cell represented in Figure 1a. In Figure 2a, we present the high-frequency dispersion diagrams projected over the boundaries of the irreducible Brillouin zone. Figure 2b shows some selected eigenmodes associated with the corresponding branches of the dispersion curves. Once the Bloch dispersion surfaces \(\omega(\mathbf{k})\) are obtained, it is possible to evaluate the group velocity of waves travelling parallel to the wave vector \(\mathbf{k}\), as  
\[v_{g}(\mathbf{k})=\
abla_{\mathbf{k}}\omega(\mathbf{k}), \tag{6}\]  
by using standard numerical gradient routines. The group velocity landscape of the structure is presented in Section 3. As originally documented in the seminal book by [PERSON] et al. (2018), focusing on photonic crystals, the comparison of isofrequency contours and group velocity of the bulk homogenous external medium (2) with those of the periodic microstructured medium give invaluable information about the capability of metamaterials to effectively \"mould\" the flow of waves. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(a\) [mm] & \(b\) [mm] & \(c\) [mm] & \(d\) [mm] & \(L\) [mm] \\ \hline \hline \(0.1\) & \(0.3\) & \(0.7\) & \(0.9\) & \(1\) \\ \hline \hline \end{tabular}
\end{table}