NOVELTY - The structure (64) has a set of weight reduction pockets (20) aligned in a primary direction, and a coupling element-holding pocket (18) aligned in a secondary direction. An elastic coupling element (26) i.e. rubber grommet, is secured to the coupling element-holding pocket by a coupling element retaining structure (14) to receive and removably retain a spherical support structure (66) i.e. ball-stud. An over-push structure (16) e.g. protrusion, is provided such that deformation of the coupling element upon receiving the support structure is limited by contact with the over-push structure.    USE - Single-piece molded dual-core attachment structure for attaching a molded shell i.e. plastic engine cover, to a supporting part i.e. engine block, of a vehicle. Can also be used for attaching a molded shell carrying acoustic foams and bumpers, air duct and electrical unit to vehicle frame or battery tray.    ADVANTAGE - The structure attaches the molded shell to the support structure in a cost-effective manner, without sacrificing any functionality, stability or other similarly important design qualities. The structure can be assembled or removed in an easy manner. The structure provides freedom of structure orientation, thus allowing weight-optimized fully function-oriented design of the structure. The structure is easy to manufacture, and feasible for prototyping and low and high volume production.    DETAILED DESCRIPTION - An INDEPENDENT CLAIM is also included for a method for manufacturing an injection molded attachment structure extending from a molded shell base.    DESCRIPTION OF DRAWING(S) - The drawing shows an exploded perspective view of a dual-core attachment structure in partial section.    Coupling element retaining structure (14)    Over-push structure (16)    Coupling element-holding pocket (18)    Weight reduction pockets (20)    Elastic coupling element (26)    Single-piece molded dual-core attachment structure (64)    Spherical support structure (66)