# Predicting Electromagnetic Performance Under Wrinkling in Thin-Film Phased Arrays
## 2 Wrinkling of Step-Stiffness Structure and Dual-Domain Displacement Mod
### Surface Deformation of STMA  
The typical shape of a Space Tensioned-Membrane Antenna (STMA) includes two layers: phased array patches are set on the up layer, and a ground patch is set on the down layer. The stiffness step structure easily causes out-of-plane deformation, such as wrinkling deformation and bending deformation. Both the deformations lead to changes in the spacing of the two layers. The spacing of two layers is a design parameter required for electromagnetic performance, so determining the extent to which deformation influences electromagnetic performance is essential for tensioned-membrane antenna. Figure 3 shows the possible deformation types of the up layer. As the membrane antenna is stretched at corners, the stress concentration easily leads to wrinkling deformation. Futhermore, fabrication errors also cause changes. To determine reliable results of out-of-plane deformation of STMA, a simulation and experiment were carried out. Because the effectiveness of the finite-element approach was confirmed in the previous section, we employed finite-element analysis to compute the surface deformation of the more complex structure. The model was built in ABAQUS 2020. To ensure numerical accuracy, the 7 \(\times\) 3 suspension-cable configuration (m \(\times\) n cables represent the number of complete catenaries in the horizontal and vertical directions) uses the following mesh counts: the array-layer film has 105,974 elements, and the radiator layer has 100,252 elements; the supporting frame has 66,474 elements; each support beam has 6534 elements \(\times\) 2 beams; and each connection flange has 4629 elements \(\times\) 4 flanges. The 5 \(\times\) 3 suspension-cable configuration uses 104,211 elements for the array-layer film and 99,886 elements for the radiator layer; its supporting frame again has 66,474 elements, each support beam is 6534 \(\times\) 2, and each flange is 4629 \(\times\) 4. The tension cables are modeled with connector elements, and the copper foil antenna on the film is implemented via a composite lamination. All elements' length settings were verified by increasing the number of elements until the simulation results converged. All membrane regions used the S4R element, and the support beam, frame, and connection flange used the S3 element. The material properties are shown in Table 1.  
The loading method of the structure uses the cooling method on the tension cable, that is, giving a thermal expansion coefficient and calculating the temperature change range corresponding to a given tension force. This method can be used to apply tension to the rope finite element structure. We applied a post-buckling finite-element analysis based on the initial buckling-mode defect method. First, a stress analysis was performed; next, the buckling modes were computed; finally, the primary buckling mode was introduced into the model as an initial imperfection via keyword modification in ABAQUS. This procedure yields the wrinkling pattern analysis of the thin-film structure. Based on existing references, it can be determined that different thicknesses of the membrane were set to analyze material parameters' influence on out-of-plane deformation. Both the thickness and patch size are essential factors for out-of-plane deformation types [27]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Part**} & \multicolumn{3}{c}{**Material Parameters**} \\ \cline{2-4}  & **Material** & **Young’s Modulus** & **Poisson Ratio** \\ \hline Membrane & Polymide & 3 GPa & 0.33 \\ Beam/Frame/Connection flange & Aluminum & 206 GPa & 0.3 \\ Truss & Carbon Fiber & 110 GPa & 0.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Antenna’s material parameters. Figure 3: Possible deformation types of STMA. Figure 4 shows the out-of-plane displacement contours for the 7 \(\times\) 3 suspension-cable configuration. On the left is the array-layer film contour: as the prestress increases, the zones of maximum deformation expand from the four corner tension points to all tension points.