arXiv subjects

Hywel Normington

Publications and source records attributed to Hywel Normington.

3 recordsLinked to original sources

arXiv2026

A decoupled, stable, and second-order integrator for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects

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.

math.NA↗

arXiv2025

A decoupled, convergent and fully linear algorithm for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects

We consider the coupled system of the Landau--Lifshitz--Gilbert equation and the conservation of linear momentum law to describe magnetic processes in ferromagnetic materials including magnetoelastic effects in the small-strain regime. For this nonlinear system of time-dependent partial differential equations, we present a decoupled integrator based on first-order finite elements in space and an implicit one-step method in time. We prove unconditional convergence of the sequence of discrete approximations towards a weak solution of the system as the mesh size and the time-step size go to zero. Compared to previous numerical works on this problem, for our method, we prove a discrete energy law that mimics that of the continuous problem and, passing to the limit, yields an energy inequality satisfied by weak solutions. Moreover, our method does not employ a nodal projection to impose the unit length constraint on the discrete magnetisation, so that the stability of the method does not require weakly acute meshes. Furthermore, our integrator and its analysis hold for a more general setting, including body forces and traction, as well as a more general representation of the magnetostrain. Numerical experiments underpin the theory and showcase the applicability of the scheme for the simulation of the dynamical processes involving magnetoelastic materials at submicrometer length scales.

math.NA↗

arXiv2023

Convergent finite element methods for antiferromagnetic and ferrimagnetic materials

We consider the numerical approximation of a continuum model of antiferromagnetic and ferrimagnetic materials. The state of the material is described in terms of two unit-length vector fields, which can be interpreted as the magnetizations averaging the spins of two sublattices. For the static setting, which requires the solution of a constrained energy minimization problem, we introduce a discretization based on first-order finite elements and prove its $Γ$-convergence. Then, we propose and analyze two iterative algorithms for the computation of low-energy stationary points. The algorithms are obtained from (semi-)implicit time discretizations of gradient flows of the energy. Finally, we extend the algorithms to the dynamic setting, which consists of a nonlinear system of two Landau-Lifshitz-Gilbert equations solved by the two fields, and we prove unconditional stability and convergence of the finite element approximations toward a weak solution of the problem. Numerical experiments assess the performance of the algorithms and demonstrate their applicability for the simulation of physical processes involving antiferromagnetic and ferrimagnetic materials.

math.NA↗