Waveform Relaxation for the Computational Homogenization of Multiscale Magnetoquasistatic Problems

TL;DR

This paper introduces a waveform relaxation approach for multiscale magnetoquasistatic homogenization, enabling more efficient computation by reducing mesoscale problem solves per Gauss point while maintaining accuracy.

Contribution

It presents a novel weakly coupled waveform relaxation method for multiscale magnetoquasistatic problems, improving computational efficiency over traditional monolithic approaches.

Findings

Reduces the number of mesoscale problems solved per Gauss point

Maintains accuracy of homogenized constitutive laws

Enables separate solution of macroscale and mesoscale problems

Abstract

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 Newton--Raphson scheme. The resolution of many mesoscale problems per Gauss 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 Gauss point.