Innocent Niyonzima

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

In this paper, 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. This paper closes with the first demonstration of a simulation of a realistic four-pole induction machine model using Parareal.

Waveform relaxation for the computational homogenization of multiscale magnetoquasistatic problems

This paper proposes the application of the waveform relaxation method to the homogenization of multiscale magnetoquasistatic problems. In the monolithic heterogeneous multiscale method, the nonlinear macroscale problem is solved using the NewtonRaphson scheme. The resolution of many mesoscale problems per Gau point allows to compute the homogenized constitutive law and its derivative by finite differences. In the proposed approach, the macroscale problem and the mesoscale problems are weakly coupled and solved separately using the finite element method on time intervals for several waveform relaxation iterations. The exchange of information between both problems is still carried out using the heterogeneous multiscale method. However, the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per Gau point.

Multiscale Finite Element Modeling of Nonlinear Magnetoquasistatic Problems using Magnetic Induction Conforming Formulations

In this paper we develop magnetic induction conforming multiscale formulations for magnetoquasistatic problems involving periodic materials. The formulations are derived using the periodic homogenization theory and applied within a heterogeneous multiscale approach. Therefore the fine-scale problem is replaced by a macroscale problem defined on a coarse mesh that covers the entire domain and many mesoscale problems defined on finely-meshed small areas around some points of interest of the macroscale mesh (e.g. numerical quadrature points). The exchange of information between these macro and meso problems is thoroughly explained in this paper. For the sake of validation, we consider a two-dimensional geometry of an idealized periodic soft magnetic composite.

ParaExp Using Leapfrog as Integrator for High‐Frequency Electromagnetic Simulations

Recently, ParaExp was proposed for the time integration of linear hyperbolic problems. It splits the time interval of interest into sub-intervals and computes the solution on each sub-interval in parallel. The overall solution is decomposed into a particular solution defined on each sub-interval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method depends on fast approximations of this matrix exponential based on recent results from numerical linear algebra. This paper deals with the application of ParaExp in combination with Leapfrog to electromagnetic wave problems in time-domain. Numerical tests are carried out for a simple toy problem and a realistic spiral inductor model discretized by the Finite Integration Technique.

An application of ParaExp to electromagnetic wave problems

Recently, ParaExp was proposed for the time integration of hyperbolic problems. It splits the time interval of interest into sub-intervals and computes the solution on each sub-interval in parallel. The overall solution is decomposed into a particular solution defined on each sub-interval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method results from fast approximations of this matrix exponential using tools from linear algebra. This paper deals with the application of ParaExp to electromagnetic wave problems in time-domain. Numerical tests are carried out for an electric circuit and an electromagnetic wave problem discretized by the Finite Integration Technique.

Investigation of the time integration methods on the parareal method for field computation of eddy currents problems

In this paper, we investigate the influence of time integration methods on the performance of the Parareal method for the computation of eddy current problems. Parareal is a method that allows parallelization in time domain. The time interval is split into many smaller time intervals (as many as the number of available CPUs) with a fine time grid and solved in parallel allowing to capture the finescale details of the solution. An approximation of initial conditions for these fine problems is obtained by solving a cheap, time-dependent problem defined on a coarse grid for the entire time interval. The method has been successfully implemented for the problem of eddy currents using the implicit backward Euler method. In this paper we will investigate the influence of the time stepping methods on the convergence and the complexity of the Parareal algorithm.