Explicit time integration of transient eddy current problems

Abstract

Abstract. 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.

Multiple right-hand side techniques in semi-explicit time integration methods for transient eddy current problems

“…(5b) A regularization of K n by a grad-div or tree/cotree gauging can be used alternatively [6], [8]. Here, the pseudo-inverse is evaluated using the preconditioned conjugate gradient method (PCG) [9]. The finitely stiff ODE (5a) can be integrated explicitly in time, e.g.…”

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

“…[4], [5], have been proposed but are rarely used in practice. Recently, explicit methods gained interest as computational hardware architectures seem to favor those algorithms [6], [7], [8]. Another approach is time domain parallelization [9].…”

Study of the non-linear eddy-current response in a ferromagnetic plate: Theoretical analysis for the 2D case

“…The standard way of treating this problem is via successive linearisation using either the fixed-point method (also known as polarization or Picard-Banach method) or the Newton-Raphson scheme and the application of a numerical technique for the solution of the resulting linear problem at each iteration. Considerable progress has been made the past years in the development of such solvers based on the finite elements method (FEM) [1,2], the finite integration technique (FIT) [3,4,5] or the integral equation approach [6,7,8]. The relevant literature is vast, and the above list should be understood only as indicative.…”

“…The main inconvenience in using these techniques is that they rely on the application of a volume mesh, with a large number of degrees of freedom (dofs), which results in the repeated inversion of a large (sparse or full, depending on the formulation) system of linear equations. To overcome this drawback, sophisticated techniques using semi-explicit schemes for the minimization of linear system inversions have been proposed [4,5]. Another approach to cope with the raised computational effort is to resort to hardware acceleration e.g.…”