An efficient approach based on a hybrid combination of numerical and uniform geometrical theory of diffraction (UTD) methods is developed to predict the performance of large conformal array antennas on a locally convex portion of an otherwise complex and electrically large metallic platform. The metallic platform is assumed to be modeled by a perfect electric conductor (PEC). Such conformal array antennas may either be a simple slot array in a convex PEC surface, or be an array of complex antenna elements slightly recessed in a cavity structure just below the skin line of a convex PEC boundary, and covered by a flush-mounted radome. Electrically large complex platforms comprising of a locally convex portion could be an aircraft, a spacecraft, an unmanned aerial vehicle (UAV), etc.