Multiple Right-Hand Side Techniques in Semi-Explicit Time Integration Methods for Transient Eddy Current Problems

The spatially discretized magnetic vector potential formulation of magnetoquasistatic field problems is transformed from an infinitely stiff differential algebraic equation system into a finitely stiff ordinary differential equation (ODE) system by application of a generalized Schur complement for nonconducting parts. The ODE can be integrated in time using explicit time integration schemes, e.g. the explicit Euler method. This requires the repeated evaluation of a pseudo-inverse of the discrete curl-curl matrix in nonconducting material by the preconditioned conjugate gradient (PCG) method which forms a multiple right-hand side problem. The subspace projection extrapolation method and proper orthogonal decomposition are compared for the computation of suitable start vectors in each time step for the PCG method which reduce the number of iterations and the overall computational costs.Comment: 4 pages, 5 figure

Parallel-In-Time Simulation of Eddy Current Problems Using Parareal

In this contribution the usage of the Parareal method is proposed for the time-parallel solution of the eddy current problem. The method is adapted to the particular challenges of the problem that are related to the differential algebraic character due to non-conducting regions. It is shown how the necessary modification can be automatically incorporated by using a suitable time stepping method. The paper closes with a first demonstration of a simulation of a realistic four-pole induction machine model using Parareal

Explicit time integration of transient eddy current problems

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.Comment: 9 pages, 6 figure

Explicit Time Integration of Transient Eddy Current Problems

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.Comment: 9 pages, 6 figure