# Explicit Time Integration of Transient Eddy Current Problems

## TL;DR

This paper introduces an explicit time integration approach for transient eddy current problems by transforming the system into ODEs, utilizing the CFL criterion, and employing PCG with CSPE for efficient computation, validated on a benchmark.

## Contribution

It presents a novel explicit integration scheme using a generalized Schur-complement and CSPE to improve efficiency in transient eddy current simulations.

## Key findings

- The scheme is validated successfully on the TEAM 10 benchmark.
- Explicit Euler method with CFL stability is feasible for these problems.
- The use of CSPE accelerates PCG convergence in the method.

## 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.