Using the Sherman-Morrison-Woodbury Formula for Coupling External Circuits With FEM for Simulation of Eddy Current Problems

Simulation of three-dimensional transient eddy current problems is important to numerous applications. The Finite Element Method (FEM) has proven be to be powerful numerical technique for solving the Partial Differential Equations (PDE) describing eddy currents. In order to solve the PDE, boundary conditions must be provided, and in many applications the boundary conditions are not known explicitly but can be provided by a Resistor-Inductor-Capacitor (RLC) circuit model. The emphasis of this paper is on an efficient and exact coupling of the RLC network equations with the FEM equations. The coupling is based on an exact linear algebra identity known as the Sherman-Morrison-Woodbury (SMW) formula. One advantage of this approach is that the FEM matrices are not modified. This is important if a fast ldquoblack-boxrdquo solver is available for the FEM matrices, these solvers typically require that the matrices have certain mathematical properties and these properties are not modified by the SMW approach. A second advantage is that the SMW approach is valid for an arbitrary number of independent external circuits.