# Predicting Electromagnetic Performance Under Wrinkling in Thin-Film Phased Arrays
## 1 Introduction  
Spacecraft increasingly demand high-gain, beam-steerable antennas that fit within tight volume and mass budgets. Deployable reflectors typically fall into three main categories: rigid-surface designs such as Sunflower and SSDA (Solid Surface Deployable Antenna) [1, 2, 3], MESH-surface antennas [4], and thin-film approaches. Among these, tensioned thin-film antennas have emerged as promising candidates for next-generation satellite platforms, offering lightweight, compact configurations and large stowage-to-deployment ratios.  
Phased array antennas, in particular, are seeing rapid development across satellite communications, microwave remote sensing, and Earth observation [5]. For spaceborne synthetic aperture radar (SAR), the increasing demands for wider bandwidths and finer resolution have led to the adoption of large-scale active planar phased arrays [6, 7]. Compared with conventional rigid-shell structures, flexible thin-film phased arrays enable higher aperture efficiency and lighter weight, offering advantages for stowage and on-orbit deployment.  
Nevertheless, wrinkling in thin membranes is a ubiquitous structural phenomenon that severely affects both mechanical accuracy and electromagnetic performance [8]. To address this, researchers have proposed a range of analytical, numerical, and experimental models to investigate wrinkle formation mechanisms [9, 10, 11]. Mainstream approachesinclude stress/strain-based criteria and energy competition models [12]. However, most prior studies focus on homogeneous membranes [6; 13; 14; 15], and few explicitly consider the impact of spatially varying stiffness due to surface reinforcement. [PERSON] et al. [16] examined wrinkling in membranes with embedded rigid squares, but a general modeling approach for reinforced thin-film structures remains underdeveloped.  
In parallel, antenna electrical performance under structural disturbance has been widely studied through electromechanical-thermal coupling models [17; 18; 19]. These efforts include applications of topology optimization, multiphysics finite element modeling, and electromechanical co-simulation frameworks to improve deployable antenna systems [20; 21; 22]. Several compensation schemes have been proposed to mitigate radiation distortion due to fabrication or mechanical errors, including those based on Fast Fourier Transforms (FFTs) and advanced computational electromagnetic tools like all-dielectric superstructures and EBG-enhanced designs (Electromagnetic Band Gap-enhanced designs) [23; 24; 25; 26]. However, these methods are often tailored to rigid or mesh-based antennas, while integrated electromechanical modeling of thin-film patch antennas remains scarce. In most cases, experimental verification dominates, and predictive modeling for deformed flexible surfaces is lacking [27].  
To bridge this gap, we propose a wrinkle-aware electromechanical coupling model that captures the effect of non-uniform stiffness in stepped-thickness membrane structures. Specifically, we formulate a wrinkling analysis framework for tensioned membranes with embedded reinforcements, derive the resulting displacement field, and extract representative unit-cell deformation patterns. These are then fitted into a dual-domain displacement model, in which the antenna patch is decomposed into a stiffened region (e.g., copper-coated layer) and the surrounding compliant membrane. A key parameter introduced is the stiffness ratio, defined as the ratio of bending stiffness between the reinforced and base regions. This model is integrated into full-wave electromagnetic simulations to evaluate antenna performance under various wrinkling conditions. By systematically varying the stiffness ratio, we analyze its influence on out-of-plane deformation, wrinkle suppression, and gain preservation, thus providing a unified framework for structure-electromagnetic performance prediction in flexible thin-film phased array systems.