A decoupled, stable, and second-order integrator for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects
This work addresses the simulation of magnetoelastic effects in materials, which is incremental as it builds on existing methods for the Landau-Lifshitz-Gilbert equation.
The authors tackled the numerical approximation of a nonlinear PDE system modeling magnetostriction by proposing a fully discrete scheme using finite elements and a decoupled time discretization, resulting in a second-order, stable method validated through numerical experiments.
We consider the numerical approximation of a nonlinear system of partial differential equations modeling magnetostriction in the small-strain regime consisting of the Landau--Lifshitz--Gilbert equation for the magnetization and the conservation of linear momentum law for the displacement. We propose a fully discrete numerical scheme based on first-order finite elements for the spatial discretization. The time discretization employs a combination of the classical Newmark-$β$ scheme for the displacement and the midpoint scheme for the magnetization, applied in a decoupled fashion. The resulting method is fully linear and formally of second order in time. We derive the discrete energy law satisfied by the approximations and prove the stability of the scheme. Finally, we assess the performance of the proposed method in a collection of numerical experiments.