In simulating time-varrying magnetic fields in electromagnetic devices like motors and transformers, it is often necessary to couple the partial differential equation for the magnetic field with a model for the external electrical circuit connections. The electrical circuit is a system of linear equations relating the unknown currents and voltages of the electrical conductors present in the device to known voltage and current sources. Time-varrying sources give rise to magnetically induced currents and voltages in the conductors. The partial differential equation and the circuit are coupled by these magnetically induces quantities.
In low-frequent time-harmonic Maxwell formulations in two dimensions, the partial differential governing the magnetic vector potential is the Helmholtz equation with a complex shift. The finite element discretization of this equation results in sparse, complex symmetric system matrices. Discretized field-circuit coupled problems yield two by two block structured matrices whose diagonal is formed by the discretized partial differential equation and the electrical circuit matrix. The size of the second diagonal block is typically several orders of magnitude smaller than that of the first. The coupling is performed in such a way that no fill-in occurs in the discritized differential equation matrix and that the two by two block matrices are again complex symmetric.
Solving the linear system is the computational bottleneck in simulating technically relevant engineering problems. Motivated by previous experience [1,2], we want to alleviate this bottleneck by the application of algebraic multigrid (AMG) techniques. The straightforward application of AMG is hampered by the presence of the electrical circuit. We develloped a multigrid cycle that takes the circuit into account.
Our multigrid technique is a generalization of a method by Hackbush for solving an elliptic problem augmented by an algebraic equation. We base the AMG setup on the differential equation block of the matrix. The electrical circuit is taking into account in the cycling phase. The resulting algorithm is a black-box solver for general field-circuit coupled problems.
For the implementation of our multigrid technique, we develloped an interface between the GMD-AMG-code by Stueben and PETSc. This interace allows to call the AMG setup on the differential equation block of the system matrix. After the setup, the AMG coarser grid and interpolation operators are available as PETSc matrices. The multigrid cycling is done by PETSc's multigrid components that we extended to be able to treat the electrical circuit.
Our algorithm has been tested on a variety of engineering problems. It has proven to be stable and to deliver a speedup by a factor between five and ten compared to previously existing solvers.