Field-Circuit Coupling by Means of the Woodbury Formula

摘要

For the coupling of the magnetic field and the electric circuit equations, there are different approaches. In any case, the flux linkage has to be taken into account by augmenting the finite element system by additional equations. Recently, it has been proposed to eliminate the circuit part by taking the Schur complement, which results in symmetric and positive definite matrices. The rank of the circuit's contribution to the Schur complement equals the number of linear independent coupling variables. Field-circuit coupling therefore introduces a low rank correction into the equations of the field problem. The consequences of this key observation are discussed in the paper. If a direct solution of the finite element system is considered, the circuit coupling can be treated elegantly by using the Woodbury formula. The Woodbury formula gives an explicit expression for the inverse of a matrix with low rank correction in terms of the inverse of the original matrix. In the framework of a preconditioned conjugate gradient solver it turns out that it is sufficient to include the circuit equations into the matrix-by-vector product, while the finite element preconditioner can be retained. These considerations will be illustrated by numerical results that have been obtained from a simple model problem.