Existing computational design technologies are insufficiently developed to integrate the design activities during the product lifecycle. The main reason is three fold: inhomogeneous requirement, function and form definition in the general design context, ill nature of function-to-form mapping and difficulties of design synthesis with qualitative information. Although computational design technology has progressed in the past two decades, there have been little radical changes from a designer's point of view. Therefore, this paper introduces a function-oriented, axiom based generalized mapping scheme, including generalized mapping among requirement domain, function domain and form domain, etc. The mapping issues and the default logic reasoning on recognition and reconstitution of product geometric and topologic information are explored in this paper. The proposed methodology is proved to be effective in the development of common use design tools. A mechanical hand and a water recycling apparatus design examples show a 20 times increase in efficacy with the developed enhancement modules based on this methodology on a general 3D CAD plate form.