# Multifunctional Hyperelastic Structured Surface for Tunable and Switchable Transparency
## 2 Materials and Methods  
### Analysis and Design  
The behavior of an HSS is governed by its unit cell characteristics. Accordingly, we start by considering an ensemble of potential unit cell geometries commonly found in the literature [13; 24; 25; 26], and shown in Figure 1a-f. Comparing the potential of these possible geometries in terms of mechanical reconfigurability requires us to solve the expected transmission coefficient of the metasurface when subject to different level of uniaxial deformations, and define a useful performance metric. Let us first describe how we solve this complex coupled-physics situation. We numerically solve the Multiphysics (both mechanics and electromagnetics) problem by finding the mechanical equilibrium of the periodic system when subject to a given level of deformation along x, and then solve Maxwell's equation to obtain the scattering response of the deformed surface when illuminated by a normally incident plane wave polarized along y. These two numerical steps both consider a single unit cell subject to periodic boundary conditions, and are implemented using the commercial Finite Element software, COMSOL  
Figure 1: Percentage of the resonance frequency shift (RFS) under uniaxial deformation with 1% mechanical strain in x direction for the unit cells with different resonator geometries (**a**) split ring with a bar (**b**) modified-I (**c**) split rings (**d**) bow-tie (**e**) four-cut split ring (**f**) parallel bars. Multiphysics [27]. We utilize three different physics: Electromagnetic Waves (Frequency Domain Solver), Structural Mechanics (Stationary Solver), and the Moving Mesh feature. In the electromagnetic analysis, Floquet periodic boundary conditions [28] are applied on the deformed opposite transverse surfaces. In the mechanical analysis, displacements of the opposite transverse surfaces are coupled to enforce periodic boundary conditions. Moreover, air layers on the top and the bottom of the unit cell are handled using the moving mesh feature, and discretized using an arbitrary Lagrangian-Eulerian mesh for easy adaptation to the deformed unit cell [29]. Compared to prior arts, our method fully takes into account the key effects of both nonlinear substrate deformation by considering neo-Hookean mechanical constitutive relations [30], as well as the effect of the microscopic deformation on the electromagnetic response. Defining the same gap size and wire width for all the resonator geometries, we now focus on evaluating the effect of geometry on the deformation sensitivity of these HSSs, defining the following performance metric. We compared the resonance frequency shift (RFS), expressed as a percentage of the natural resonance frequency, for each unit cell under the same uniaxial deformation. Thus, we fixed the average unit cell strain to be 1 percent in the x-direction (\(\bar{\epsilon}_{xx}=0.01\)). All the resonators are made of copper with relative permittivity 1, electrical conductivity 59.98 MS/m, [PERSON]'s modulus 110 GPa, and Poisson's ratio 0.35. The substrate is made of PDMS with relative permittivity 2.8, electrical conductivity 0.025 PS/m, Young's modulus 750 kPa, and Poisson's ratio 0.49. The RFS values are shown below the various designs in the figure. It is seen that parallel bar resonators, Figure 1f, show more sensitivity to the applied deformation compared to the other geometries. The sensitivity of the parallel bar resonators to the number of bars, their width and length and the gap size between them necessitates an optimization in order to obtain the optimal resonator geometry. We implemented a genetic algorithm (GA) to find the optimum choice for the geometrical design parameters of the HSS. The flowchart of this optimization is presented in Figure 2. We consider the number of bars, their length and width, and the gap size between them as design variables that create a so-called chromosome, a potential solution for the optimization problem.