Project B8 • Theory and numerics of the coupled Maxwell–Landau–Lifshitz–Gilbert equations
Principal investigators
Prof. Dr. Willy Dörfler
(7/2019 - )
Prof. Dr. Michael Feischl
(7/2019 - )
Dr. Ivan Fernandez-Corbaton
(7/2023 - )
Project summary
Magnetic materials play an important role in technologies such as magnetic sensors, actuators, magnetic data storage, and electric motors. For example, in hard disk drives (HDD), a sequence of bits is stored in an array of magnetic elements (see Figure 1). Each element stores one bit with a magnetization that points either upwards (0) or downward (1). Information is effectively written by a "writing head" that slides along the array and generates a magnetic field that flips the magnetizations at selected locations. The density of stored information ($\textrm{bit/mm}^2$) is inversely proportional to the size of the magnetic elements. However, when the device shrinks in size, heat fluctuations become increasingly problematic, up to the point that the thermal energy can reverse the magnetizations, thus corrupting the data.
Figure 1. The arrangement of magnetic elements (grey) stores magnetizations pointing either upwards or downwards to represent bits. The writing head (white) produced a magnetic field that can flip the magnetizations.
This and other applications motivate the study of dynamic micromagnetism. A popular differential model in this field is the Landau–Lifshitz–Gilbert equation, which we now briefly describe. We consider a magnetic body $\Omega\subset \mathbb{R}^3$ and a finite time-horizon $T>0$. The magnetization of $\Omega$ is modeled by a vector field $\boldsymbol{M}: \Omega\times [0,T]\rightarrow \mathbb{R}^3$ with constant unit modulus, i.e., $\vert \boldsymbol{M}(t,x)\vert=1$ everywhere in $\Omega$ and at all times.
More concretely, given an initial magnetization $\boldsymbol{M}_0$, the time evolution of the system is described by the following partial differential equation: \begin{align*} \begin{cases} \partial_t \boldsymbol{M} - \alpha \boldsymbol{M}\times \partial_t \boldsymbol{M} = -\boldsymbol{M} \times \boldsymbol{M}_{\rm eff}(\boldsymbol{M})\quad &\textrm{in } (0,T)\times \Omega,\\ \partial_n \boldsymbol{M} = 0\qquad &\textrm{on } (0,T) \times \partial\Omega,\\ \boldsymbol{M}(0,\cdot) = \boldsymbol{M}_0\qquad &\textrm{in } \Omega, \end{cases} \end{align*} where $\alpha>0$ is the relaxation constant and $n$ denotes the unit normal to $\partial \Omega$. The magnetization is guided by the effective field $\boldsymbol{H}_{\rm eff}(\boldsymbol{M}) = C_e \Delta \boldsymbol{M} + \boldsymbol{H}(\boldsymbol{M}) + \boldsymbol{H}_{\rm ext}$, whose terms model respectively the effect of the magnetization itself, of the induced magnetic field and of a possible external magnetic field.
The electromagnetic field induced by the magnetization, $\boldsymbol{H}(\boldsymbol{M})$ and $\boldsymbol{E}$, is described by the well-known time-dependent Maxwell equations in $\mathbb{R}^3$, which in turn depend on the magnetization $\boldsymbol{M}$. The Maxwell–Landau–Lifshitz–Gilbert (MLLG) system consists of the coupling of these two problems.
Solution of the Maxwell–Landau–Lifshitz–Gilbert system with FEM-BEM coupling
In [Boh21], [BFK23] we study the well-posedness of this problem and propose a provably convergent discretization scheme. While most of the literature assumes the magnetic field to be quasi-static, we do not make this approximation.
We give a rigorous mathematical formulation of the MLLG system as a partial differential equation for $\boldsymbol{M}$, $\boldsymbol{H}$ and $\boldsymbol{E}$ on the bounded domain $\Omega$ and an integral equation for $\boldsymbol{E}, \boldsymbol{H}$ on the exterior $\mathbb{R}^3\setminus \overline{\Omega}$.
For the LLG problem, we use finite elements of order $r$ in space and an order two BDF (backward difference formula) in time. The Maxwell system on $\Omega$ is approximated with a discontinuous Galerkin method of order $r$ in space and a second order leap-frog scheme in time. Finally, in the exterior the Maxwell problem is solved with boundary elements of order $r$ and second order convolution quadrature. Under the (weak) stability condition that the mesh-size should scale at least as the time-step, the convergence orders were verified by numerical experiments. The corresponding codes are provided here.
High-order time-stepping for the Landau-Lifshitz-Gilbert equation
In [AFKL21], we give the first example of high-order times-stepping schemes for LLG. We prove high order convergence under regularity assumptions on the solutions as well as discrete energy dissipation properties that, under mild regularity assumptions, guarantee the scheme to be at least as good as first order schemes.
The physical energy decay of these schemes is obtained mimicking the orthogonality relation $\partial_t \boldsymbol{M} \cdot \boldsymbol{M} = 0$ of the exact solution (recall that $\vert \boldsymbol{M}\vert=1$). This is enforced on the numerical solution in a weak sense, i.e. $\int_{\Omega} \boldsymbol{M}_h\cdot \partial_t \boldsymbol{M}_h \nu_h \textrm{d} x = 0$, for appropriate test functions $\nu_h$. This choice is shown to be superior to enforcing the orthogonality pointwise on the mesh nodes, a widely spread practice which however rules out convergence rates of more than the first order.
Towards optimal adaptive time-stepping
In [Fei22], we show that inf-sup stability of the discretization of PDEs is essential in order to obtain optimal adaptive mesh refinement algorithms. This observation replaces a major technical hurdle in proving optimal convergence of mesh refinement algorithms and enables us to provide optimal adaptive algorithms for the Stokes problem as well as transmission problems with transparent boundary conditions. Moreover, we propose the first, provably optimal, adaptive time-stepping algorithm for a heat equation. This is rendered possible due to a new stability bound for the LU-factorization of matrices. Optimal adaptivity for transparent boundary conditions and for time-stepping algorithms is particularly useful for the proposed research in this project as it will serve as an important stepping stone for the LLG equation.
Treatment of stochastic noise via adaptive collocation
Another problem of interest is the Stochastic Landau–Lifshitz–Gilbert problem, which additionally models the effect of heat fluctuations on the magnetization $\boldsymbol{M}$. We aim at producing a reduced-order model of the stochastic magnetization based on few samples. This task proves to be challenging because each sample is equivalent to solving one (deterministic) LLG problem.
Since the literature on the topic is sparse, in [FS21] we begun by studying a Stochastic Poisson problem and its approximation with adaptive collocation in the stochastic parameter space and adaptive finite elements in physical space. We propose a method that combines adaptive collocation and adaptive finite elements to not only choose effective collocation samples, but also speed up the sampling process itself. In this work, we give the first proof of convergence of a method of its class and verify numerically its increased convergence speed compared to classical methods.
Preliminary work on neural network approximation
The recent work [DF21] proposes a new algorithm which decouples the training of neural networks from the data and thus allows us to efficiently optimize neural network weights almost independently of the size of the dataset. This is possible by using low-discrepancy pointsets, so-called digital nets, to compress the training data and optimize the network on the compressed data only. Under certain assumptions on the setting, we can show that this approximation converges almost linearly in the number of points, and thus delivers an efficient surrogate training objective that can be evaluated independently of the size of the dataset.
In the work [BF21], we show that recurrent neural networks can learn to produce optimal adaptive finite element meshes for a large class of PDE problems and thus can replace traditional adaptive mesh refinement algorithms. While both works do not treat LLG at the moment, they build the foundation for the extension of neural network approaches to the tasks in the current funding period.
