Regularity of bondgraphs and directed bondgraphs

This article concludes the presentation of a theory for the mathematical foundations that tacitly support the construction and manipulation of a bondgraph model (BGM) of a spatially discrete physical system. A di-bondgraph B¯ consists of an underlying bondgraph B  , a diagram with only junctions and bonds, and directions assigned to the bonds. B¯ generates a pair of integral chain group representations of its dual cycle and cocycle matroids, which encapsulate the structural relationships in the system. The important class of regular (di-)bondgraphs can be identified using the concepts of regular matroids and chain groups. Bond and junction elimination operations are described to create a sub-diagram called a minor, reflecting the matroid operations of contraction and deletion of matroid elements. This provides a device for manipulating a (di-)bondgraph diagram to test for a redundant internal junction, or for a non-regular matroid. A di-bondgraph B¯ induces an orientation on its cycle and cocycle matroid; if B¯ is regular it has the same matroids as its underlying bondgraph B  , and the orientation defined by B¯ is a signing of these. In a BGM the integral cycles and cocycles obtained from the dual chain groups may be used to express spatial constraints between variables, signed quantities being essential to represent polarities required by physical measurements. A discussion is focused on several important issues: a rationale for excluding TF and GY elements in a purely combinatorial analysis, maintaining the symmetry of duality which is so central to the bondgraph concept; the independent influence of bond directions on structure (matroids) and orientation (chain groups); and the relevance of non-regular di-bondgraphs in practical modelling. In conclusion, an argument is made for the essential practical equivalence of bondgraph and graph-theoretic modelling (GTM). Any technique, formulation, computer software, algorithm, or application expressed in terms of one method will have a parallel version in terms of the other. Moreover, conversion between the BGM and GTM views ought to be a conceptually and computationally trivial exercise.