Mapping of the process ontology to EngMath is depicted in the right-hand side of Figure 9. Informally, the mapping states that for each energy flow there are two time-dependent physical quantities, one for the effort and one for the flow. The domain of the energy flow determines the dimension of the quantities. For instance, an electrical effort quantity has the dimension energy/electrical-current (voltage) and the flow quantity the electrical-current dimension. For each mechanism there is a mathematical relation that relates the values of the physical quantities of the energy flows connecting the mechanism to each other. The mapping also imposes constraints on the relation. These constraints only depend on the mechanism type and they are independent of the domains of the energy flows. The mathematical relation in Figure 9 belongs to a dissipator mechanism. The constraints on such a relation are that it is a continuous function that lies in the first and third quadrant and that r(0) = 0. For an electrical energy flow, this can be an instantiation of Ohm's law whereas in the mechanical domain it can model some kind of friction with .
Figure 11: Excerpt from the second part of the PHYSSYS ontology where physical processes are projected onto mathematical relations. This is an example of a domain ontology that only contains formalizations of the interdependencies between the viewpoints it includes.
Figure 11 shows an excerpt of the second part of PHYSSYS, the part that performs the process to mathematics projection described above. Only a part of the definition of the relation mech.mathrel, the relation that relates a mathematical relation to a process, is shown. Axiom 7a states that for every dissipator a relation between the effort and flow quantities of the energy flow to the dissipator must exist. In the axiom the relations ef.effortq and ef.flowq are used. These relations link each energy flow to physical quantities for the effort and the flow. Axioms not incorporated in Figure 11 ensure that these quantities have the proper physical dimension. The constraints on the relation for dissipators have been formalized by stating that the effort is zero if and only if the flow is zero and that the product of effort and flow, i.e. the energy flow, must be positive. In other words, the dissipator must dissipate energy. Furthermore, it is probably needless to say that PHYSSYS contains axioms like axiom 7a for each type of mechanism defined in the process view.