Anisotropic Case Research Articles

Some explicit solutions of the Landau‐Lifshitz equation for anisotropic ferromagnets

The aim of this paper is to construct explicit solutions of the Landau‐Lifshitz equation for anisotropic ferromagnets. Firstly, we present some explicit solutions for n‐dimensional Landau‐Lifshitz equation with a generalized uniaxial anisotropy field (depending only on time) under a suitable transformation, where the generalized uniaxial anisotropy field contains the classical easy‐axis case and the important higher‐order approximations. Some relevant examples are indicated that contains the cubic system proposed by Landau and Lifshitz in 1960. Secondly, an explicit plane‐wave solution of Landau‐Lifshitz equation for the completely anisotropic case is obtained. Finally, some properties of magnetization for ferromagnets are provided by analysis of the solutions.

Computing interior transmission eigenvalues for homogeneous and anisotropic media

The method of fundamental solutions is investigated in a stabilized version for the computation of interior transmission eigenvalues in two dimensions for homogeneous and anisotropic media without voids. This approach has already proven to be very competitive in practice for the isotropic framework among regular scattering shapes and keeps predominating through its simplicity as being mesh- and integration free. We give a new approximation analysis, present various numerical results and show that the eigenvalue spectrum for isotropic scatterers is generally different from the corresponding anisotropic borderline cases.

Theoretical investigations of infrared optical wave modulation transfer function models in anisotropic marine turbulence.

The previously derived marine turbulence modulation transfer function (MTF) adopts the classical assumption of isotropic and Kolmogorov turbulence. However, more theoretical research shows that marine turbulence demonstrates anisotropic non-Kolmogorov properties. In this study, new long-exposure plane and spherical wave MTF models under anisotropic non-Kolmogorov marine turbulence will be investigated. Comparative analyses are performed for the new models and the existing ones that have been developed under the isotropic marine turbulence and the anisotropic terrestrial turbulence cases. It turns out that the anisotropic marine turbulence influences the electro-optical imaging system less than the isotropic marine turbulence does. Also, under the same turbulence strength condition, due to the humidity fluctuations of marine atmosphere, the anisotropic marine turbulence will reduce more severely the quality of electro-optical imaging systems than the anisotropic terrestrial turbulence. Investigations conducted in this paper are beneficial to understand better the electro-optical imaging mechanism in marine atmospheric turbulence media.

Anisotropic H-k stacking and (revisited) crustal structure in the southeastern margin of Tibet

P receiver functions (PRFs) and the H-k stacking technique are tools often used to constrain the thickness of the crust and the ratio between the velocities of P and S waves. Both the crustal depth and the Poisson's ratio are very important parameters to study the tectonic evolution of the lithosphere, but they can be significantly affected by the presence of crustal anisotropy. In order to address this problem, we have extended the isotropic stacking approach to include six converted P-to-S phases that are generated in the anisotropic case, instead of only three phases considered in the isotropic case, and we have examined the feasibility of the stacking technique using synthetic and real data. Based on PRFs acquired by 108 permanent broadband seismic stations deployed in the southeastern margin of Tibet including the Sichuan and Yunnan areas, we have applied the anisotropic H-k stacking scheme to investigate the crustal thickness and Poisson’s ratio. The results reveal that the crustal thickness varies from ∼60 km in the Songpan-Ganzi fold system and the northern part of the Sichuan-Yunnan diamond-shaped block, near the Eastern Himalayan Syntaxis, to ∼33 km in southern Yunnan, and that the Poisson’s ratio varies mostly from 0.24 to 0.32. The highest values of 0.28–0.30 are found along the axis formed by the Longmenshan fault, the Lijiang-Jinhe fault and the Xiaojiang fault, and are attributed to the accumulation of lower crustal flow in front of the Sichuan Basin. The highest value of ∼0.32 is observed at the Tengchong volcano area to the north of the Indochina block, and is attributed to the upwelling of hot mantle associated to the eastward subduction of the Indian plate, rather than to the expansion of east Tibet. Comparing with the results provided by the isotropic stacking scheme, it is appreciated that the crustal depth determined by the anisotropic stacking method well adjusts the observed pattern of gravity anomaly. We conclude that the results obtained by anisotropic stacking yield a much better constraint on the estimation of the crustal depth and Poisson's ratio when compared to those achieved by isotropic stacking.

Chiral superconductivity in thin films of doped Bi2Se3

Recent experimental evidences point to rotation symmetry breaking superconductivity in doped Bi$_2$Se$_3$, where the relevant order parameter belongs to a two-component odd-parity representation $E_u$ of the crystal point group. The $E_u$ channel admits two possible phases, the chiral phase, that breaks time-reversal symmetry, and nematic phase, that well explains the reported rotation symmetry breaking. In weakly anisotropic three-dimensional systems the nematic phase is the stable one. We study the stability of the nematic phase versus the chiral phase as a function of the anisotropy of the system and the thickness of the sample and show that in the strongly anisotropic case or in thin slabs the chiral phases is favoured. For the extreme 2D limit composed by a single layer of Bi$_2$Se$_3$ the system hosts two chiral Majorana modes flowing at the boundary of the system.

Wavelength-dependent Fresnel beam propagator and migration in VTI media

Ray-based wave propagators are widely applied in seismic migration due to implementation flexibility and computational efficiency. The classic ray theory that under the high-frequency assumption requires sufficiently smooth velocity models, which limits the applications of ray-based methods because seismic waves are band-limited. Besides, it is desirable to extend seismic propagation and migration to general anisotropic case since the reality of subsurface. We adapt a wavelength-dependent smoothing (WDS) operator for transversely isotropic media with a vertical symmetry axis (VTI), which considers both characteristics of band-limited wave propagation and local anisotropic heterogeneity. Frequency-dependent traveltimes are computed with the WDS models by using an anisotropic dynamic programming approach. Then, a wavelength-dependent Fresnel beam propagator is constructed based on the frequency-dependent traveltimes. Analysis of traveltime fields demonstrates that wavelength-dependent Fresnel beam propagator can provide accurate wave propagating directions and traveltimes. We develop a wavelength-dependent Fresnel beam migration (WDFBM) method based on the wavelength-dependent Fresnel beam propagator for VTI media, which generates angle domain common imaging gathers (ADCIGs) efficiently. Numerical examples are included to demonstrate the effectiveness of our approach, which shows its potential application for subsequent velocity estimation via ADCIGs.

Planar least gradient problem: existence, regularity and anisotropic case

We show existence of solutions to the least gradient problem on the plane for boundary data in BV(partial varOmega ). We also provide an example of a function f in L^1(partial varOmega ) backslash (C(partial varOmega ) cup BV(partial varOmega )), for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.

A novel protocol for rapid identification of anisotropic diffusion properties of polymer matrix composite materials with complex texture

The paper focuses on a proposal for a novel protocol for the rapid identification of anisotropic diffusion properties of composite materials with complex texture. The method consists in performing small time gravimetric tests on composites samples and measuring the initial slope of their mass uptake curve with respect to t. The identification is performed by employing the analytical expression of the slope of the curve in the anisotropic case, an extension of the Shen and Springer slope method for isotropic and orthotropic diffusion. Analysis and discussion are dealt within a planar anisotropic diffusion test case issued from the literature.

Effect of stiffness anisotropy on topology optimisation of additively manufactured structures

Additive manufacturing has been receiving attention as it is capable of producing complex geometries previously unmanufacturable, particularly those resulting from topology optimisation. However, additive manufacturing processes like Selective Laser Melting result in materials that have varying ratios of anisotropy. Currently, the majority of topology optimisation algorithms utilise an isotropic assumption. Through a case study, the present study confirms the hypothesis that anisotropy in the stiffness of the material has a significant detrimental effect on the optimisation outcome where (1) increasing ratio of anisotropy; and/or (2) increasing deviation of the building angle away from being parallel or perpendicular to the principal loading direction results in the decrease of volume reduction achievable through topology optimisation. Material with an out-of-plane shear modulus which is very different from the “near isotropic” value also performed poorly. Compared with the 50% weight reduction of an isotropic case, the worst anisotropic case only managed approximately 27% weight saving. However, in the cases where both the building angle is small (<30°) and the degree of anisotropy is low (≤±10%), the impact on resultant volume reduction was small (<10%). This study indicates that, in general, material anisotropy should be considered in topology optimisation for additive manufacturing.

Imaging of isotropic and anisotropic conductivities from power densities in three dimensions

We present numerical reconstructions of anisotropic conductivity tensors in three dimensions, from knowledge of a finite family of power density functionals. Such a problem arises in the coupled-physics imaging modality ultrasound modulated electrical impedance tomography for instance. We improve on the algorithms previously derived in Bal et al (2013 Inverse Problems Imaging 7 353–75); Monard and Bal (2013 Commun. PDE 38 1183–207) for both isotropic and anisotropic cases, and we address the well-known issue of vanishing determinants in particular. The algorithm is implemented and we provide numerical results that illustrate the improvements.

Isotropic LQC and LQC-inspired models with a massless scalar field as generalised Brans–Dicke theories

We explore whether generalised Brans -- Dicke theories, which have a scalar field $\Phi$ and a function $\omega(\Phi)$, can be the effective actions leading to the effective equations of motion of the LQC and the LQC--inspired models, which have a massless scalar field $\sigma$ and a function $f(m) \;$. We find that this is possible for isotropic cosmology. We relate the pairs $(\sigma, f)$ and $(\Phi, \omega)$ and, using examples, illustrate these relations. We find that near the bounce of the LQC evolutions for which $f(m) = sin \; m$, the corresponding field $\Phi \to 0$ and the function $\omega(\Phi) \propto \Phi^2 \;$. We also find that the class of generalised Brans -- Dicke theories, which we had found earlier to lead to non singular isotropic evolutions, may be written as an LQC--inspired model. The relations found here in the isotropic cases do not apply to the anisotropic cases, which perhaps require more general effective actions.

Multi-vortex crystal lattices in Bose-Einstein condensates with a rotating trap.

We consider vortex dynamics in the context of Bose-Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross-Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.

On Weighted Dirac Operators and Their Fundamental Solutions for Anisotropic Media

The heat transfer problem in isotropic media has been studied extensively in Clifford analysis, but very little in the anisotropic case for this setting. As a first step in this way, we introduce in this work Dirac operators with weights belonging to the Clifford algebra $${\mathcal {A}}_n$$ , which factor the second order elliptic differential operator $$ {\tilde{\Delta }}_n= div (B \,\nabla ), $$ where $$B \in \mathbb {R}^{n \times n}$$ is a symmetric and positive definite matrix. For these weighted Dirac operators we construct fundamental solutions and get a Borel–Pompeiu and Cauchy integral formula.

Remarks on uniaxial solutions in the Landau–de Gennes theory

We study uniaxial solutions of the Euler–Lagrange equations for a Landau–de Gennes free energy for nematic liquid crystals, with a fourth order bulk potential, with and without elastic anisotropy. These uniaxial solutions are characterised by a director and a scalar order parameter. In the elastic isotropic case, we show that (i) all uniaxial solutions, with a director field of a certain specified symmetry, necessarily have the radial-hedgehog structure modulo an orthogonal transformation, (ii) the “escape into third dimension” director cannot correspond to a purely uniaxial solution of the Landau–de Gennes Euler–Lagrange equations and we do not use artificial assumptions on the scalar order parameter and (iii) we use the structure of the Euler–Lagrange equations to exclude non-trivial uniaxial solutions with ez as a fixed eigenvector i.e. such uniaxial solutions necessarily have a constant eigenframe. In the elastic anisotropic case, we prove that all uniaxial solutions of the corresponding “anisotropic” Euler–Lagrange equations, with a certain specified symmetry, are strictly of the radial-hedgehog type, i.e. the elastic anisotropic case enforces the radial-hedgehog structure (or the degree +1-vortex structure) more strongly than the elastic isotropic case and the associated partial differential equations are technically far more difficult than in the elastic isotropic case.

General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. A general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion {0<α,β≤2} based on the fractional derivative, has not been obtained, where α and β are time and space derivative orders. It is essential to derive a general PFG signal attenuation expression including the FGPW effect for PFG anisotropic anomalous diffusion research. In this paper, two recently developed modified-Bloch equations, the fractal differential modified-Bloch equation and the fractional integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. The theory and the CTRW simulation agree with each other. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI.

The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems.

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay-Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Control Properties for Second-Order Hyperbolic Systems in Anisotropic Cases with Applications in Inhomogeneous and Anisotropic Elastodynamic Systems

We consider control properties for second-order hyperbolic systems in anisotropic cases with variable coefficient matrices in a bounded domain in ${\mathbb{R}}^n$ with $C^{2}$-boundary. Assuming th...

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semigeodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.

Regularity of anisotropic refinable functions

This paper presents a detailed regularity analysis of anisotropic wavelet frames and subdivision. In the univariate setting, the smoothness of wavelet frames and subdivision is well understood by means of the matrix approach. In the multivariate setting, this approach has been extended only to the special case of isotropic refinement with the dilation matrix all of whose eigenvalues are equal in the absolute value. The general anisotropic case has resisted to be fully understood: the matrix approach can determine whether a refinable function belongs to $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$, but its H\"older regularity remained mysteriously unattainable. It this paper we show how to compute the H\"older regularity in $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$. In the anisotropic case, our expression for the exact H\"older exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We provide an efficient algorithm for determining the finite set of the restricted transition matrices whose spectral properties characterize the H\"older exponent of the corresponding refinable function. We also analyze the higher regularity, the local regularity, the corresponding moduli of continuity, and the rate of convergence of the corresponding subdivision schemes. We illustrate our results with several examples.

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

We consider an anisotropic hyperbolic equation with memory term:∂t2u(x,t)=∑i,j=1n∂i(aij(x)∂ju)+∫0t∑|α|⩽2bα(x,t,η)∂xαu(x,η)dη+R(x,t)f(x)for and , which is a simplified model equation for viscoelasticity. The main result is a both-sided Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor of the force term . The proof is based on a Carleman estimate and due to the anisotropy, the existing transformation technique does not work and we introduce a new transformation of u in order to treat the integral terms.