Foundations of decomposition for manufacturing geometrical products

Geometrical tolerancing was developed to improve the weakness of previous tolerance systems to handle imperfect form and ambiguous references and was primarily developed for assembly of components. Although there have been recent modifications to increase tolerance zone's utility, it is still basically a go/no-go system with components being in or out of tolerance. The use of tolerance zones is causing real industrial problems in the specification of high valued precision products. For example: in healthcare, the specification of the geometrical shape of the cup in total hip-joint replacements by a simple tolerance zone is allowing some cups to fail, by dislocating out of position, prematurely. A new design of non-spherical head is beginning to appear and the market requires improved specification. Further mathematical decomposition for the specification of the tolerancing zone is required to distinguish between good and failing functional geometries [1, 2]. The connection with filtration is explored. In particular the definitions for 'primary mapping' contained in ISO 16610-1 [3] is developed to include the foundations of decomposition in terms of structures called 'complete lattices'. One very useful property of complete lattices is the existence of a smallest subset of lattice elements (called lattice generators) that can reconstruct the whole of the lattice using just joint (or alternatively meet) operations. These lattice generators constitute a basis (or frame) of the lattice and form a measurement scale for the nesting index of the associated decomposition/filter. This definition of decomposition, given here, is universal and goes beyond just decomposition of geometrical products. Examples will be given to illustrate the utility of the definition to other aspects of smart manufacturing, including: formal concept analysis [4] for smart information systems.