## 4 Stretchable MSP Antenna-Based Strain Sensor
### Optimization  
Once the analyses were conducted for the initial geometry and the necessary geometry points of the antenna patch were determined, a shape optimization was utilized to improve the performance of the antenna. As explained in the introduction, diverse studies on the strain patch antenna have already been conducted by optimization, but most of them have concentrated on the sizing optimization to take updated height, width, or length of the antenna systems. The optimization may fail to boost the productivity of the antenna because the optimized design almost maintains its original shape. Topology optimization has also been employed to develop a new design, but its efficiency is only considerable under expensive additive manufacturing fabrication. In this study, therefore, a shape optimization was used to find the optimal shape of the antenna patch that minimizes the variation of the \(\mathrm{R_{f}}\) and the return loss. It implies that, not only can the possibility of frequency shift induced by structural deformation be decreased, but also the performance of the antenna corresponding to electrical return loss will be maximized. Here, the lower return loss is guaranteed, and the better antenna performance is expected. Unfortunately, however, the explained general equations (Equations (1)-(5)) corresponding to the \(\mathrm{R_{f}}\) variation have an assumption that the substrate should have a constant thickness. This assumption only draws a linear relation between the shift of the \(\mathrm{R_{f}}\) and the strain. With respect to the stretchable strain MSP antenna including the thickness variation of the substrate, the direction of the \(\mathrm{R_{f}}\) shifts could be anticipated by the general equations, but a correct calculation of the shift will not be guaranteed. For those reasons, in this study, a surrogate model to establish a relation between mechanical and electrical behavior of the antenna system was developed, engaging detailed FEA for structural deformation in ANSYS(r) and electrical analysis in HFSS. The \(\mathrm{R_{f}}\) shift and the return loss respecting the structural deformation with additional attention of thickness variation were analyzed with the help of this surrogate model. To establish objective functions, the substrate thickness variation and distorted patch's shape were considered. The function approximation modeled by ANN was combined with the shape optimization process. To escalate the capability of the antenna, an additional objective function was employed to truncate the return loss. \[\mathrm{Minimize}\ \ \ \left[\ \begin{array}{c}R_{ff}(x,f_{i})-R_{fs}(x,f_{i}, \varepsilon)\\ S(x,f_{i},\varepsilon)\end{array} \right]\ \ \ i=1,\ 2. \tag{18}\]  
\[\mathrm{Subject\ to}\ \ \ \left|R_{ff}(x,f_{i})-R_{fs}(x,f_{i},\varepsilon) \right|\ \leq\theta, \tag{19}\]  
\[S(x,f_{i},\varepsilon)+\gamma\leq 0, \tag{20}\]  
\[x_{l}\leq x\leq x_{u}, \tag{21}\]  
\[0\leq\varepsilon\leq 30, \tag{22}\]  
\[f_{1}=2.5\ \mathrm{GHz}\ \textit{and}\ f_{2}=5\ \mathrm{GHz}, \tag{23}\]  
Figure 6: Selected new significant coordinates. where \(R_{\mathit{ff}}\) and \(R_{\mathit{S}}\) stand for R\({}_{\mathit{f}}\) of frequency samples and R\({}_{\mathit{f}}\) under applied strain (\(\varepsilon\)). \(x\), \(\theta\), \(S\), and \(\gamma\) refer to a vector of the DVs, bandwidth regarding a return loss of \(-10\) dB, the return loss, and acceptable return loss, respectively. The bandwidth corresponding to \(-10\) dB was employed to assure enough antenna operation, and the strain of \(30\%\) was established based on the boundary exploited in the FEA. This constraint enables linear static analysis. In order to utilize the benefit of UWB, dual bandwidths, \(2.5\) GHz and \(5\) GHz, which are preferred in a wireless fidelity (Wi-Fi) were considered. Moreover, a spline curve was used to model the geometry of the patch antenna with \(28\) geometry points.