## 4 Conclusions  
The theoretical and numerical modelling of the metamaterial comprising periodically repeated Swiss cross-shaped holes demonstrated exotic wave effects at high frequencies. As well as beam steering and focusing applications, the structures show potential for elastic energy reservoirs. For example, the long-wavelength envelope wave mode in Figure 8 has an extended lifetime, as can be observed by viewing the videos in the supplementary material, for both the smooth and rough crosses considered here. Therefore, slabs of the microstructured material may be used to store elastic energy and possibly release it on demand to act externally on the homogeneous medium or other slabs of microstructured material.  
Figure 8: Effect of roughness on the scattering of a Gaussian beam with \(\alpha=35^{\circ}\) and \(f=2.78\) MHz. The roughness is defined by \(\delta=5\) μm and \(\lambda_{0}=5\) μm. Panels (**a,b**) show the IFC diagram (and its magnification) for smooth crosses. Panels (**c,d**) show simulation snapshots for the smooth and rough cases at the same time, \(t=93\) μm, with the total time being \(T=120\) μm.  
Bloch-Floquet theory and the resulting isofrequency contour diagrams were used to predict trapped waves, negative refraction, and cloaking phenomena. High-fidelity and rapid finite element simulations in the transient regime illustrated these effects. The use of GPU-driven implementation provides a step change in modelling capability, with run-times hundreds of times faster than with standard CPU-driven software, and the capability to investigate models with several hundred million degrees of freedom. This increase in modelling fidelity and efficiency is crucial for future experimental validation and manufacturing, as well as extensions to three-dimensional space.  
The study here was broadened with a view to future applications and the manufacturing of metamaterial samples. Additive manufacturing methods, such as selective laser melting, produce parts with remnant surface roughness. The cross-shaped holes were also analysed for cases when the smooth edges were replaced with rough edges (characterised by statistical parameters following Gaussian distributions). The wave effects were shown to be robust, with little difference between smooth and rough cases except for small changes in phase due to the finer domain mesh required for the perturbed case.  
These preliminary results indicate that the manufacture of the metamaterial, following refined model-informed design for specific wave phenomena, may be expected to be largely unaffected by surface roughness finish. The access to modelling run-times of minutes and hours, compared with days and weeks, enables a much broader range of parameter value analysis. It is envisaged that extensive Monte Carlo studies, covering wide ranges of frequency and roughness parameters, will identify potential manufacturing bounds.  
The following are available online at [[https://www.mdpi.com/article/10_3390/app11167576/s1](https://www.mdpi.com/article/10_3390/app11167576/s1)]([https://www.mdpi.com/article/10_3390/app11167576/s1](https://www.mdpi.com/article/10_3390/app11167576/s1)).  
Conceptualization, D.T. and S.G.H.; methodology, D.T. and S.G.H.; software, D.T. (COMSOL Multiphysics and MATLAB) and S.G.H. (Pogo); formal analysis, D.T.; investigation, D.T and S.G.H; original draft preparation, D.T. and S.G.H. Both authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Videos of all transient simulations referred to in this article are available in the Supplementary Material.  
DT and SGH acknowledge the support of the University of Liverpool Research Centre in Mathematics and Modelling which provided funds for a research visit of [PERSON] to Liverpool in February 2020, during which the concept of this work was conceived.  
The authors declare no conflict of interest.