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Explicit time integration of eddy current problems using a selective matrix update strategy

Jennifer Susanne Dutiné (Chair of Electromagnetic Theory, University of Wuppertal, Wuppertal, Germany)
Markus Clemens (Chair of Electromagnetic Theory, Bergische Universitat Wuppertal, Wuppertal, Germany)
Sebastian Schöps (Graduate School of Computational Engineering, Technische Universität Darmstadt, Darmstadt, Germany)



Discretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems.


The resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible.


Numerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation.


The explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.



This work was supported under grant numbers CL143/11-1 and SCHO1562/1-1 of the Deutsche Forschungsgemeinschaft (DFG). The third author is supported by the “Excellence Initiative” of the German Federal and State Governments and the Graduate School of Computational Engineering at the TU Darmstadt.


Dutiné, J.S., Clemens, M. and Schöps, S. (2017), "Explicit time integration of eddy current problems using a selective matrix update strategy", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 36 No. 5, pp. 1364-1371.



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