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A Mathematical Functional Decomposition Approach Through Granularity Partition Process in Quotient Space

Over the past decades, several methodologies have coalesced around the functional decomposition and partial solution manipulation techniques. These methodologies take designers through steps that help decompose a design problem and build conceptual solutions based on the intended, product functionality. However, this kind of subjective decomposition restricts solutions of conceptual design within designers’ intended the local, rather the whole, solution space. In such cases, the ability for AI-based functional reasoning systems to obtain creative conceptual design solutions is weakened. In this paper, a functional decomposition model based on the domain decomposition theory in quotient space is proposed for carrying out functional decomposition without needing functional reasoning knowledge to support. In this model, the functional decomposition is treated as a granularity partition process in quotient space composed of three variables: the domain granularities, the attribute properties, and the topological structures. The closeness degrees and the attribute properties in fuzzy mathematics are utilized to describe the fuzzy equivalence relations between the granularities in the up-layer and in the lower-layer of the functional hierarchies. According to the order characteristics in the partially sequential quotient space, based on the homomorphism principle, the attribute properties and the topological structures corresponding to the lower-layer of the functional hierarchies are constructed then. Here, the attribute properties are expressed with membership functions pointed to the lower-layer from the up-layer of the functional hierarchies, and the topological structures are expressed with matrixes and the directed function network represent the topological connections among the subfunctions in the lower-layer of the functional hierarchies. Through refining the functional decomposition process step by step, and traversing all tree branches and leaf nodes in the functional decomposition tree, the functional hierarchies are obtained. Since the functional decomposition process not need the user to indicate or manage desired functionality, the model presented in this paper can reduce designers’ prejudices or preconceptions on the functional hierarchies, as well as extend the solution space of conceptual design.