The mimetic finite difference method for the Landau–Lifshitz equation

Kim, Eugenia Hail; Lipnikov, Konstantin Nikolayevich

doi:10.1016/j.jcp.2016.10.016

Title: The mimetic finite difference method for the Landau–Lifshitz equation

Abstract

The Landau–Lifshitz equation describes the dynamics of the magnetization inside ferromagnetic materials. This equation is highly nonlinear and has a non-convex constraint (the magnitude of the magnetization is constant) which poses interesting challenges in developing numerical methods. We develop and analyze explicit and implicit mimetic finite difference schemes for this equation. These schemes work on general polytopal meshes which provide enormous flexibility to model magnetic devices with various shapes. A projection on the unit sphere is used to preserve the magnitude of the magnetization. We also provide a proof that shows the exchange energy is decreasing in certain conditions. The developed schemes are tested on general meshes that include distorted and randomized meshes. As a result, the numerical experiments include a test proposed by the National Institute of Standard and Technology and a test showing formation of domain wall structures in a thin film.

Authors:

Kim, Eugenia Hail ^[1]; Lipnikov, Konstantin Nikolayevich ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Univ. of California, Berkeley, CA (United States)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Sun Jan 01 00:00:00 EST 2017

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1331275

Alternate Identifier(s):: OSTI ID: 1397763

Report Number(s):: LA-UR-16-21341
Journal ID: ISSN 0021-9991

Grant/Contract Number:: AC52-06NA25396; AC05-06OR23100

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 328; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; micromagnetics, Landau-Lifshitz equation, mimetic finite differences, polygonal meshes

Citation Formats


                    Kim, Eugenia Hail, and Lipnikov, Konstantin Nikolayevich. The mimetic finite difference method for the Landau–Lifshitz equation.  United States: N. p., 2017. 
Web.  doi:10.1016/j.jcp.2016.10.016.

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                    Kim, Eugenia Hail, & Lipnikov, Konstantin Nikolayevich. The mimetic finite difference method for the Landau–Lifshitz equation.  United States.  https://doi.org/10.1016/j.jcp.2016.10.016

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                    Kim, Eugenia Hail, and Lipnikov, Konstantin Nikolayevich. Sun .  
"The mimetic finite difference method for the Landau–Lifshitz equation".  United States.  https://doi.org/10.1016/j.jcp.2016.10.016.  https://www.osti.gov/servlets/purl/1331275.

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@article{osti_1331275,

  title        = {The mimetic finite difference method for the Landau–Lifshitz equation},

  author       = {Kim, Eugenia Hail and Lipnikov, Konstantin Nikolayevich},

  abstractNote = {The Landau–Lifshitz equation describes the dynamics of the magnetization inside ferromagnetic materials. This equation is highly nonlinear and has a non-convex constraint (the magnitude of the magnetization is constant) which poses interesting challenges in developing numerical methods. We develop and analyze explicit and implicit mimetic finite difference schemes for this equation. These schemes work on general polytopal meshes which provide enormous flexibility to model magnetic devices with various shapes. A projection on the unit sphere is used to preserve the magnitude of the magnetization. We also provide a proof that shows the exchange energy is decreasing in certain conditions. The developed schemes are tested on general meshes that include distorted and randomized meshes. As a result, the numerical experiments include a test proposed by the National Institute of Standard and Technology and a test showing formation of domain wall structures in a thin film.},

  doi          = {10.1016/j.jcp.2016.10.016},

  journal      = {Journal of Computational Physics},

  number       = C,

  volume       = 328,

  place        = {United States},

  year         = {Sun Jan 01 00:00:00 EST 2017},

  month        = {Sun Jan 01 00:00:00 EST 2017}

}

Copy to clipboard

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Works referencing / citing this record:

Multiscale approach for magnetization dynamics: unraveling exotic magnetic states of matter
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Méndez, É.; Poluektov, M.; Kreiss, G.
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Title: The mimetic finite difference method for the Landau–Lifshitz equation

Abstract

Citation Formats

Numerical methods for the stray-field calculation: A comparison of recently developed algorithms journal, January 2013

A Fast Finite-Difference Method for Micromagnetics Using the Magnetic Scalar Potential journal, March 2012

Convergence of a Finite Element Discretization for the Landau–Lifshitz Equations in Micromagnetism journal, February 2006

A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation journal, February 2014

A convergent finite element approximation for Landau–Lifschitz–Gilbert equation journal, May 2012

Convergence of an Implicit Finite Element Method for the Landau–Lifshitz–Gilbert Equation journal, January 2006

Nonlinear Magnetization Dynamics under Circularly Polarized Field journal, January 2001

Using multipoles decreases computation time for magnetostatic self-energy journal, November 1991

A Family of Mimetic Finite Difference Methods on Polygonal and Polyhedral Meshes journal, October 2005

Finite element modeling of unbounded problems using transformations: a rigorous, powerful and easy solution journal, March 1992

Error estimates for a semi-implicit numerical scheme solving the Landau–Lifshitz equation with an exchange field journal, July 2005

A Survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism journal, September 2007

Convergence result for the constraint preserving mid-point scheme for micromagnetism journal, June 2009

Geometrical integration of Landau–Lifshitz–Gilbert equation based on the mid-point rule journal, November 2005

Fast stray field computation on tensor grids journal, April 2012

Micromagnetic modelling - the current state of the art journal, July 2000

Hybrid method for computing demagnetizing fields journal, March 1990

Finite difference scheme for the Landau–Lifshitz equation journal, December 2011

Improved gauss-seidel projection method for micromagnetics simulations journal, May 2003

Accurate numerical methods for micromagnetics simulations with general geometries journal, January 2003

Adaptive Mesh Refinement for Micromagnetics Simulations journal, June 2006

Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on $${\mathbb R^2}$$ journal, August 2010

Calculation of Demagnetizing Field Distribution Based on Fast Fourier Transform of Convolution journal, December 1996

Approximation of boundary conditions for mimetic finite-difference methods journal, September 1998

Cayley transforms in micromagnetics journal, December 2001

Beyond first-order finite element schemes in micromagnetics journal, January 2014

Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism journal, January 2006

Geometric integration on spheres and some interesting applications journal, February 2003

Manipulation of skyrmions in nanodisks with a current pulse and skyrmion rectifier journal, June 2013

Mimetic finite difference method journal, January 2014

Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems journal, April 2011

Nonuniform grid algorithm for fast calculation of magnetostatic interactions in micromagnetics journal, April 2009

Fast Fourier transform on multipoles for rapid calculation of magnetostatic fields journal, February 2006

Applications of -Matrix Techniques in Micromagnetics journal, December 2004

Computational Micromagnetism book, January 2001

Finite elements in numerical micromagnetics journal, December 1999

Multigrid methods for computation of magnetostatic fields in magnetic recording problems journal, June 1998

Numerical Methods for the Landau--Lifshitz Equation journal, January 2000

A Gauss–Seidel Projection Method for Micromagnetics Simulations journal, July 2001

Fast adaptive algorithms for micromagnetics journal, September 1992

A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation journal, February 2014

Nonlinear Magnetization Dynamics under Circularly Polarized Field text, January 2001

Multiscale approach for magnetization dynamics: unraveling exotic magnetic states of matter journal, January 2020

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Numerical methods for the stray-field calculation: A comparison of recently developed algorithms
journal, January 2013

A Fast Finite-Difference Method for Micromagnetics Using the Magnetic Scalar Potential
journal, March 2012

Convergence of a Finite Element Discretization for the Landau–Lifshitz Equations in Micromagnetism
journal, February 2006

A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation
journal, February 2014

A convergent finite element approximation for Landau–Lifschitz–Gilbert equation
journal, May 2012

Convergence of an Implicit Finite Element Method for the Landau–Lifshitz–Gilbert Equation
journal, January 2006

Nonlinear Magnetization Dynamics under Circularly Polarized Field
journal, January 2001

Using multipoles decreases computation time for magnetostatic self-energy
journal, November 1991

A Family of Mimetic Finite Difference Methods on Polygonal and Polyhedral Meshes
journal, October 2005

Finite element modeling of unbounded problems using transformations: a rigorous, powerful and easy solution
journal, March 1992

Error estimates for a semi-implicit numerical scheme solving the Landau–Lifshitz equation with an exchange field
journal, July 2005

A Survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism
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Convergence result for the constraint preserving mid-point scheme for micromagnetism
journal, June 2009

Geometrical integration of Landau–Lifshitz–Gilbert equation based on the mid-point rule
journal, November 2005

Fast stray field computation on tensor grids
journal, April 2012

Micromagnetic modelling - the current state of the art
journal, July 2000

Hybrid method for computing demagnetizing fields
journal, March 1990

Finite difference scheme for the Landau–Lifshitz equation
journal, December 2011

Improved gauss-seidel projection method for micromagnetics simulations
journal, May 2003

Accurate numerical methods for micromagnetics simulations with general geometries
journal, January 2003

Adaptive Mesh Refinement for Micromagnetics Simulations
journal, June 2006

Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on $${\mathbb R^2}$$
journal, August 2010

Calculation of Demagnetizing Field Distribution Based on Fast Fourier Transform of Convolution
journal, December 1996

Approximation of boundary conditions for mimetic finite-difference methods
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Cayley transforms in micromagnetics
journal, December 2001

Beyond first-order finite element schemes in micromagnetics
journal, January 2014

Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism
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Geometric integration on spheres and some interesting applications
journal, February 2003

Manipulation of skyrmions in nanodisks with a current pulse and skyrmion rectifier
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Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems
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Nonuniform grid algorithm for fast calculation of magnetostatic interactions in micromagnetics
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Fast Fourier transform on multipoles for rapid calculation of magnetostatic fields
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journal, December 2004

Computational Micromagnetism
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Finite elements in numerical micromagnetics
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Multigrid methods for computation of magnetostatic fields in magnetic recording problems
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Numerical Methods for the Landau--Lifshitz Equation
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A Gauss–Seidel Projection Method for Micromagnetics Simulations
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Fast adaptive algorithms for micromagnetics
journal, September 1992

A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation
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Nonlinear Magnetization Dynamics under Circularly Polarized Field
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