A knotless aperture fixation system, and method for use thereof, for securing a flexible material (i.e., suture, cable, fiber tape, or any other suitable flexible material) to a nearby tissue using knotless fixation achieved by a compression fit of the flexible material between two smooth surfaces. The system comprises a cylindrical or spherical flanged sleeve with threaded upper portion and smooth tapered lower portion is used in conjunction with a headless set screw with a bullet-shaped tip to achieve a compression fit of the flexible material between the smooth tapered lower portion of the sleeve and the smooth bullet-shaped tip of the headless set screw. The system may be used in conjunction with an expanded washer with a recessed lip, a flanged washer, or a bone plate with an opening containing a recessed lip.