Explicit Time Integration of Transient Eddy Current Problems
This work addresses the computational cost of implicit time integration in transient eddy current problems for computational electromagnetics researchers, but the improvement is incremental as explicit methods are already known and the CFL constraint remains.
The authors applied a generalized Schur-complement to convert the differential-algebraic equation system of transient eddy current problems into an ODE system, enabling explicit Euler time integration. The method was validated on the TEAM 10 benchmark, showing that the cascaded Subspace Extrapolation method reduces PCG iterations, though explicit integration still requires small time steps due to the CFL condition.
For time integration of transient eddy current problems commonly implicit time integration methods are used, where in every time step one or several nonlinear systems of equations have to be linearized with the Newton-Raphson method due to ferromagnetic materials involved. In this paper, a generalized Schur-complement is applied to the magnetic vector potential formulation, which converts a differential-algebraic equation system of index 1 into a system of ordinary differential equations (ODE) with reduced stiffness. For the time integration of this ODE system of equations, the explicit Euler method is applied. The Courant-Friedrich-Levy (CFL) stability criterion of explicit time integration methods may result in small time steps. Applying a pseudo-inverse of the discrete curl-curl operator in nonconducting regions of the problem is required in every time step. For the computation of the pseudo-inverse, the preconditioned conjugate gradient (PCG) method is used. The cascaded Subspace Extrapolation method (CSPE) is presented to produce suitable start vectors for these PCG iterations. The resulting scheme is validated using the TEAM 10 benchmark problem.