Shape optimization and experiments of a paraboloidal membrane reflector using an improved element-free Galerkin method

This study aims to enable surface accuracy analysis and shape optimization of parabolic membranes for accurate geometric representation. Based on the nonlinear strain-displacement relationship, a mathematical model of the membrane reflector in curvilinear coordinates was developed using the meshless method. Within the element-free Galerkin framework, the moving least squares method was employed to generate shape functions, and the Cartesian transformation method was adopted to enhance the integration strategy, effectively eliminating redundant stiffness matrix calculations. Geometric nonlinearities were addressed through an incremental Newton-Raphson iteration scheme. The proposed optimization model successfully determined the optimal initial focal length of the reflector and the corresponding skirt geometry. Finally, electrostatic testing experiments were conducted on both clamped-edge membrane reflectors and skirted membrane reflectors using thermoformed curved membranes and the designed ECDMA prototype, demonstrating the effectiveness of the proposed analysis and optimization methods.