A decoupled, convergent and fully linear algorithm for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects (2309.00605v3)

Abstract: 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.

• The paper introduces a decoupled integrator for magnetoelastic dynamics, rigorously proving unconditional convergence toward weak solutions under small-strain conditions.
• It employs first-order finite element spatial discretization combined with an implicit one-step temporal integration to preserve a discrete energy law that mirrors continuous behavior.
• By eliminating nodal projection and weakly acute mesh restrictions, the algorithm enhances error control and broadens applicability for simulating complex magnetoelastic materials.

A Convergent Decoupled Approach for Magnetoelastic Dynamics

This paper presents a significant advancement in the numerical analysis and simulation of magnetoelastic materials by introducing a novel algorithm for solving the coupled Landau-Lifshitz-Gilbert (LLG) equation and linear momentum conservation laws. The proposed algorithm particularly addresses the magnetoelastic effects in ferromagnetic materials under small-strain conditions—a critical consideration for practical applications such as sensors and actuators.

The authors develop a decoupled integrator using a combination of first-order finite element methods for spatial discretization and an implicit one-step method for temporal integration. A notable contribution of this work is the unconditional convergence of the discrete approximations towards a weak solution as the mesh size and the time-step size tend towards zero. This unconditional convergence is crucial as it removes any dependencies on specific discretization parameter relationships, thus broadening the scope of potential applications.

A key feature of the proposed method is its preservation of a discrete energy law that mirrors the continuous problem's energy behavior, which is essential for maintaining the physical accuracy of simulation over time. Furthermore, the approach is notable for not relying on nodal projection to enforce the unit length constraint on the magnetization, which traditionally requires weakly acute meshes for stability. Thus, the algorithm foregoes the restriction of weakly acute meshes while managing error control relative to the time-step size.

This work also incorporates a more general model encompassing volume and surface forces and a broader representation of magnetostrain, thus providing a comprehensive framework suitable for a wide range of magnetoelastic materials. The assumptions necessary for convergence are theoretically justified, and the practical implementation is demonstrated through a series of numerical experiments on submicron-scale materials. The experiments reinforce the theoretical findings and highlight the robust energy decay characteristics of the proposed method.

Several significant achievements are emphasized:

1. The integrator framework extends previous models by incorporating general body forces and traction, contributing to an enlarged class of magnetoelastic materials for analysis.
2. The algorithm demonstrates a discretized energy law aligning with the theoretical predictions, resulting in an energy inequality for weak solutions under a Courant-Friedrichs-Lewy (CFL) condition.
3. The relaxation of mesh requirements facilitates application across standard finite element packages and introduces improved accessibility for simulations in complex geometries.

The implications of this work are far-reaching both practically and theoretically. The provision of an unconditionally stable numerical scheme with accommodating mesh requirements will stimulate its use in various applied magnetoelastic scenarios. Furthermore, the convergence guarantees possibly spark further research into more generalized settings and nonlinear constraints without the need for projection methods, opening avenues in simulation accuracy and computational efficiency.

Looking forward, further exploration into the relaxation of the time-step related condition is anticipated, potentially unveiling enhanced rates of convergence and broader adaptability. Moreover, with the increasing demand for precise simulations in advanced technological applications, this paper paves the way for rapid developments in reliable computational models for coupled multiphysics problems. Innovation in mesh generation techniques and integration of lower-order effects in the magnetoelastic model would further fortify this research's applicability across additional domains.