Functionality of artifacts as well as structure and behavior plays crucial roles in problem-solving such as design. This article discusses automatic identifications of functional structures of artifacts from given behavioral models of components and their connection information (called functional understanding). We propose a functional concept ontology which provides a rich vocabulary representing functions together with clear definitions grounded on behavior. The ontology enables the automatic identification system to make the search in the functional space tractable and to screen out meaningless interpretations. Furthermore, the ontology includes a new category of functional concepts named meta-functions representing conceptual categories of interdependency between functions. It plays a crucial role in consolidation of functions to give criteria of grouping functions, that is, identity of consolidated functions. It enables the identification system to generate such functional hierarc...