Vortex energy and 360° Néel walls in thin‐film micromagnetics

We study the vortex pattern in ultrathin ferromagnetic films of circular crosssection. The model is based on the following energy functional: for in‐plane magnetizations m: B² → S¹ in the unit disc . The avoidance of volume charges ? · m ≠ 0 in B² and surface charges m · ν ≠ 0 on δB² leads to the formation of a vortex in the limit ε → 0. At the level ε > 0 the vortex is regularized by the formation of a 360° Néel wall (a one‐dimensional transition layer with core of scale ε) concentrated along a radius of B². We derive the limiting energy of the vortex by matching upper and lower bounds. Our analysis on the lower bound is based on a dynamical system argument and an interpolation inequality with sharp leading‐order constant, while the upper bound uses the leading‐order energy for 360° Néel walls. © 2010 Wiley Periodicals, Inc.     

New epitaxial thin‐film models and numerical approximation

This paper concerns new continuum phenomenological model for epitaxial thin‐film growth with three different forms of the Ehrlich–Schwoebel current. Two of these forms were first proposed by Politi and Villain 1996 and then studied by Evans, Thiel, and Bartelt 2006. The other one is completely new. Energy structure and properties of the new model are studied. Following the techniques used in Li and Liu 2003, we present rigorous analysis of the well‐posedness, regularity, and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. By using a convex–concave time‐splitting scheme, one can naturally build unconditionally stable semi‐implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published before which indicates that the new model correctly captures the essential morphological states in the thin‐film growth process. Copyright © 2017 John Wiley & Sons, Ltd.     

A dual approach to regularity in thin film micromagnetics

We prove a regularity result in the two-dimensional theory of soft ferromagnetic films. The associated Euler–Lagrange equation is given by a nonlocal degenerate variational inequality involving fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a Hölder estimate for the uniquely determined gradient projection of the magnetization field.     

A relaxation result for micromagnetics in SBV

An integral representation for the functional

A zigzag pattern in micromagnetics

We study a simplified model for the micromagnetic energy functional in a specific asymptotic regime. The analysis includes a construction of domain walls with an internal zigzag pattern and a lower bound for the energy of a domain wall based on an “entropy method”. Under certain conditions, the two results yield matching upper and lower estimates for the asymptotic energy. The combination of these then gives a Γ-convergence result.     

A box‐shaped cyclically reduced operator

We propose a new procedure of partial cyclic reduction, where we apply a 2^d‐color ordering (with d=2, 3 the dimension of the problem), and use different operators for different gridpoints according to their color. These operators are chosen so that the gridpoints can be readily decoupled, and we then eliminate all colors but one. This yields a smaller cartesian mesh and box‐shaped 9‐point (in 2D) or 27‐point (in 3D) operators that are easy to analyze and implement. Multi‐line and multi‐plane orderings are considered, and we perform convergence analysis and numerical experiments that demonstrate the merits of our approach. Copyright © 2010 John Wiley & Sons, Ltd.     

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R²→S² that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S¹-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}_ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-vector fields by S¹-vector fields away from the vortex balls.     

A projection method for phase field models in micromagnetics

Magnetic materials have been finding increasingly wider areas of application in industry and therefore, as indicated by the reviews [1], [2] and [3], there is an increased interest in the efficient modeling of such materials that have an inherent coupling between the magnetic and mechanical characteristics. A particular challenge in the modeling of such materials is the algorithmic preservation of the geometric constraint on the magnetization field, that remains constant in magnitude [4]. In earlier works, [5] and [6], we presented a phase field model within a geometrically exact incremental variational framework where the geometric property of the magnetization director is exactly preserved pointwise by nonlinear rotational updates at the nodes. In the current work however, we present an alternative approach that involves an operator split along with a projection step for the magnetization vector. This method provides significant advantages in terms of speed and ease of implementation at the cost of the maximum time step size used. The current work therefore presents comparative study of the the two methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong convergence for large bodies in micromagnetics

The convexified Landau-Lifshitz minimisation problem in micromagnetics leads to a degenerate variational problem. Therefore strong convergence of finite element approximations cannot be expected in general. This paper introduces a stabilised finite element discretisation which allows for the strong convergence of the discrete magnetisation fields with reduced convergence order for a uniaxial model problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Γ‐convergence result for the two‐gradient theory of phase transitions

The generalization to gradient vector fields of the classical double‐well, singularly perturbed functionals, where W(ξ) = 0 if and only if ξ = A or ξ = B, and A ? B is a rank‐1 matrix, is considered. Under suitable constitutive and growth hypotheses on W, it is shown that I_ε Γ‐converge to where K^* is the (constant) interfacial energy per unit area. © 2002 Wiley Periodicals, Inc.     

A reduced basis approach to large‐scale pseudospectra computation

For studying spectral properties of a nonnormal matrix $A\in {\mathbb{C}}^{n\times n}$, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε‐pseudospectra, i.e. the level sets of the resolvent norm function $\mathsf{g}(z)=\Vert {(zI?A)}^{?1}{\Vert }_2$. The computation of ε‐pseudospectra requires determining the smallest singular values ${\sigma }_{\min }(zI?A)$ for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A. It incorporates the sampled singular vectors of zI ? A for different values of z, and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.     

A second‐order finite difference scheme for solving the dual‐phase‐lagging equation in a double‐layered nanoscale thin film

This article considers the dual‐phase‐lagging (DPL) heat conduction equation in a double‐layered nanoscale thin film with the temperature‐jump boundary condition (i.e., Robin's boundary condition) and proposes a new thermal lagging effect interfacial condition between layers. A second‐order accurate finite difference scheme for solving the heat conduction problem is then presented. In particular, at all inner grid points the scheme has the second‐order temporal and spatial truncation errors, while at the boundary points and at the interfacial point the scheme has the second‐order temporal truncation error and the first‐order spatial truncation error. The obtained scheme is proved to be unconditionally stable and convergent, where the convergence order in ‐norm is two in both space and time. A numerical example which has an exact solution is given to verify the accuracy of the scheme. The obtained scheme is finally applied to the thermal analysis for a gold layer on a chromium padding layer at nanoscale, which is irradiated by an ultrashort‐pulsed laser. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 142–173, 2017     

Nontopological N‐vortex condensates for the self‐dual Chern‐Simons theory

We prove the existence of nontopological N‐vortex solutions for an arbitrary number N of vortex points for the self‐dual Chern‐Simons‐Higgs theory with 't Hooft “periodic” boundary conditions. We use a shadowing‐type lemma to glue together any number of single vortices obtained as a perturbation of a radially symmetric entire solution of the Liouville equation. © 2003 Wiley Periodicals, Inc.     

Convergent geometric integrator for the Landau-Lifshitz-Gilbert equation in micromagnetics

We consider a lowest-order finite element scheme for the Landau-Lifshitz-Gilbert equation (LLG) which describes the dynamics of micromagnetism. In contrast to previous works, we examine LLG with a total magnetic field which is induced by several physical phenomena described in terms of exchange energy, anisotropy energy, magnetostatic energy, and Zeeman energy. In our numerical scheme, the highest-order term which stems from the exchange energy, is treated implicitly, whereas the remaining energy contributions are computed explicitly. Therefore, only one sparse linear system has to be solved per time-step. The proposed scheme is unconditionally convergent to a global weak solution of LLG. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on relative geometric invariant theory for quasi‐projective varieties

Relative geometric invariant theory studies the behavior of semistable points under equivariant morphisms. More precisely, suppose G is a reductive linear algebraic group over an algebraically closed field k, X and Y are quasi‐projective varieties endowed with G‐actions, is a G‐equivariant projective morphism, the G‐action on Y is linearized in the ample line bundle M, and the G‐action on X is linearized in the φ‐ample line bundle L. For any positive integer n, there is an induced linearization of the G‐action on X in the line bundle . If Y is projective and , the set of points in X that are semistable with respect to this linearization is contained in the preimage under φ of the set of points in Y that are semistable with respect to the given linearization in M. The same statement is trivially also true, if Y is affine and . In this note, we show by means of an example that the statement does not hold for arbitrary quasi‐projective varieties Y. This shows that a claim by Hu of the contrary is not true. Relative geometric invariant theory plays a role in the construction and study of degenerations of moduli spaces.     

Convergence and stability of a numerical method for micromagnetics

The convergence and stability of a numerical method, which applies a nonconforming finite element method and an artificial boundary method to a multi-atomic Young measure relaxation model, for micromagnetics are analyzed. By revealing some key properties of the solution sets of both the continuous and discrete problems, we show that our numerical method is stable, and the solution set of the continuous problem is well approximated by those of the discrete problems. The performance of our method is also illustrated by some numerical examples. The research was supported in part by the Major State Basic Research Projects (2005CB321701), NSFC projects (10431050, 10571006, 10528102 and 10871011) and RFDP of China.     

First order theory for literal‐paraconsistent and literal‐paracomplete matrices

In this paper a first order theory for the logics defined through literal paraconsistent‐paracomplete matrices is developed. These logics are intended to model situations in which the ground level information may be contradictory or incomplete, but it is treated within a classical framework. This means that literal formulas, i.e. atomic formulas and their iterated negations, may behave poorly specially regarding their negations, but more complex formulas, i.e. formulas that include a binary connective are well behaved. This situation may and does appear for instance in data bases (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)