On the solution of electromagnetic field problems with the aid of graph theory

The method of Summary Representation is a technique developed by G. N. Polozhii in the early 60's. This technique allows the solution of partial differential equations written in finite-difference form to be expressed over the domain of the defining grid in terms of matrix-operations on the values on the boundaries of the grid. When multi-region problems are considered, the problem becomes one of solving for the unknown variables along interior boundary lines. These problems can be expressed in signal-flow graph terms, using matrix operators for the transmittance quantities and vectors for the unknown variables at the nodes, which represent the boundaries. By using the standard graph reduction technique available from graph theory, complex problems can be reduced directly to a simple equivalent problem. This technique allows internal boundaries to be replaced by equivalent systems of boundaries and results in a clearer understanding of the interactions among various parts of the system. An example is presented which shows that boundary conditions for EM problems play a "feedback" role when viewed in a signal flow graph representation.