An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes

Summary. A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only. The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be a useful tool. With its help anisotropic interpolation estimates and subsequently the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes. A numerical example supports the anisotropic error analysis. Received April 6, 1999 / Revised version received July 2, 1999 / Published online June 8, 2000     

各向异性双晶和三晶体晶界附近应力场分析

采用率相关晶体滑移有限元程序对不同取向晶粒构成的双晶体和三晶体在晶界和三晶交点附近的应力集中特性进行了计算分析．双晶体的数值结果表明,不同取向晶粒的晶界附近应力场具有较大的应力梯度,存在应力集中现象;三晶体由于晶界之间的相互作用使得三晶交点可能造成应力集中地,也可能不造成应力集中,晶界附近的应力结构与双晶体晶界附近的应力结构亦不相同,这主要取决于三个晶粒的晶体取向．对双晶体和三晶体的分析说明,不同取向的晶粒具有不同的变形规律．因此研究金属材料的损伤、断裂问题至少需要采用晶体滑移理论从细观的角度分析不同晶粒之间的相互作用．     

An immersed interface method for anisotropic elliptic problems on irregular domains in 2D

This work is a generalization of the immersed interface method for discretization of a nondiagonal anisotropic Laplacian in 2D. This first‐order discretization scheme enforces weakly diagonal dominance of the numerical scheme whenever possible. A necessary and sufficient condition depending on the mesh size h for the existence of this scheme at an interior grid point is found in terms of the anisotropy matrix. A linear programming approach is introduced for finding the weights of the schemes. The method is tested with a parametrized family of anisotropic Poisson equations. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Numerical modeling of fluid flow in anisotropic fractured porous media

A model of double porosity in the case of an anisotropic fractured porous medium is considered (Dmitriev, Maksimov; 2007). A function of fluid exchange between the fractures and porous blocks depending on flow direction is given. The flow function is based on the difference between the pressure gradients. This feature enables one to take into account anisotropic properties of filtration in a more general form. The results of numerical solving a model two-dimensional problem are presented. The computational algorithm is based on a finite-element space approximation and explicit-implicit time approximations.     

Effective Flow Potentials for Anisotropic Polycrystals

In this paper a micromechanically based flow potential for anisotropic fcc polycrystals is derived that takes into account the crystallite orientation distribution function (codf) in terms of tensorial texture coefficients. The effective flow potential is based on a representation theorem for anisotropic scalar functions depending on a 2nd-order tensor. A priori unknown functions in the representation are determined by defining and solving explicitly a minimization problem over SO(3). Important analytical properties of the coefficient matrix of the minimization problem are discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bounds and an isotropically self-consistent singular approximation of the linear elastic properties of cubic crystal aggregates for application in materials design

For crystal aggregates, the orientation distribution of single crystals affects the anisotropic linear elastic properties. In the singular approximation for cubic materials, this influence is reflected by a fourth-order texture coefficient. From this approximation, the statistical bounds of Voigt, Reuss and Hashin-Shtrikman, and an isotropically self-consistent singular approximation can be obtained. Here, an approximation is called isotropically self-consistent, if, for a vanishing texture, it results in the isotropic self-consistent approximation. The isotropically self-consistent singular approximation has the following advantages: i) it lies between the bounds of Voigt, Reuss and Hashin-Shtrikman, ii) it offers a useful approximation of the effective material behavior of textured anisotropic polycrystals, and iii) it can be used for material design purposes tailoring anisotropic properties mainly depending on the crystallographic texture. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori error estimation for the Stokes–Darcy coupled problem on anisotropic discretization

This paper presents an a posteriori error analysis for the stationary Stokes–Darcy coupled problem approximated by finite element methods on anisotropic meshes in or 3. Korn's inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to non‐conforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for non‐conforming case. With the obtained finite element solutions, the error estimators are constructed and based on the residual of model equations plus the stabilization terms. The lower error bound is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so‐called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media. Copyright © 2016 John Wiley & Sons, Ltd.     

An explicit form of the solution of the Cauchy problem for the Barenblatt–Zheltov–Kochina equation

A solution of the Cauchy problem in an isotropic medium and in an anisotropic medium with clearly expressed vertical or horizontal permeability is constructed for the Barenblatt–Zheltov–Kochina model representation of the seepage of a fluid in fissured porous rock by reducing the seepage problems under consideration to solving an abstract Cauchy problem in a Banach space.     

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.     

Development of an elastic-plastic constitutive model for highly anisotropic materials

Homogenization methods are used to obtain the effective properties of highly anisotropic materials such like textile reinforced composites. The state of the art is the utilization of this method to compute elastic properties. But the consideration of inelastic and anisotropic properties requires the application of more advanced techniques such as the FE²-method. Due to the high numerical effort induced by this approach, this paper presents a new method to evaluate inelastic properties of an heterogeneous elastic-plastic material. The parameters describing the inelastic properties require a modification of the return mapping algorithm which is used for the numerical implementation. Finally, the verification shows the accuracy of the results obtained with this new homogenization method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical layer-element solutions of Biot’s consolidation with anisotropic permeability and incompressible fluid and solid constituents

Based on the governing equations of 2D plane-strain Biot’s consolidation, the relationship between generalized displacements and stresses of a single soil layer with anisotropic permeability and incompressible fluid and solid constituents is described by an analytical layer-element, which is deduced in the Laplace–Fourier transform domain by using the eigenvalue approach. Taking the boundary conditions and the continuity of the soil layers into consideration, a global stiffness matrix is subsequently assembled and solved. As to the 3D case, the same derivation is employed after the application of a decoupling transformation. The actual solutions in the physical domain can further be acquired by inverting the Laplace–Fourier transform. Finally, numerical examples are carried out to verify the presented theory and discuss the influence of the anisotropic permeability on the consolidation behavior.     

Dual variational formulation of the electrostatic problem in an inhomogeneous anisotropic dielectric

The use of developed prospective dielectric materials in various modern electrotechnical and electrophysical devices requires reliable forecast of attainability of the required level of final characteristics depending on properties of those materials. Such forecast is based (among others) on solution of electrostatic problems in an inhomogeneous anisotropic medium allowing one to estimate the ability to satisfy the qualifying standards for effective characteristics of created materials. A dual variational formulation of an electrostatic problem in an inhomogeneous anisotropic dielectric is used to solve these problems.     

One class of nonscattering acoustic shells for a model of anisotropic acoustics

We consider a model of linear acoustics describing diffraction of an acoustic wave on a local anisotropic inhomogeneity. Under study is the existence of inhomogeneities which do not scatter the incident acoustic fields that are generated by any exterior compactly supported sources. We prove the existence of the above-mentioned inhomogeneities. Some constructive way is given of constructing a class of anisotropic media depending on an arbitrary function of one variable. We obtain some explicit formulas that define the main parameters of the shell.     

The Universality of an Absolutely Convergent Series on Short Intervals

Siberian Mathematical Journal - Under consideration is a Dirichlet series depending on a parameter and absolutely convergent in the right half of the critical strip. We prove that...     

New a priori estimates of solutions to anisotropic elliptic equations

Under consideration is the Dirichlet problem for singular anisotropic elliptic equations with a nonlinear source. Some new a priori estimates are obtained, implying that the solvability of the Dirichlet problem in the class of bounded solutions essentially depends on the dimension of the domain of the problem.     

Materials design of elastic properties of multiphase polycrystalline composites using model functions

For a chosen combination of materials for a multiphase polycrystalline composite, a crystallite orientation distribution function for each phase is formulated as a superposition of analytic central model von Mises-Fisher functions. The chosen central model functions allow the analytic integration of orientation averages of arbitrarily anisotropic fourth-order tensors. Based on this result, the first-order bounds of Voigt and Reuss and the geometric average of the stiffness can be expressed explicitly in closed forms for arbitrarily anisotropic polycrystalline materials and number of phases depending on the volume fractions and the influence of the crystallographic texture of each constituent. These expressions can then be used for the materials design aiming the determination of volume fractions and orientation distribution of selected materials for prescribed effective properties. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

退化抛物-双曲方程动力学解的唯一性

主要研究系数显含有时间和空间变量的退化抛物-双曲型方程柯西问题动力学解的唯一性.首先推广了这种类型方程的动力学公式,在给定系数适当的光滑性条件下,得到了动力学解的唯一性.     

Films courbés minces ferromagnétiques

We consider a thin curved ferromagnetic film not submitted to an external magnetic field. The behavior of the film is described by an energy depending on the magnetization of the film verifying the saturation constraint. The energy is composed of an induced magnetostatic energy and an energy term with density including the exchange energy and the anisotropic energy. We study the behavior of this energy when the thickness of the curved film goes to zero. We show with Γ-convergence arguments that the minimizers of the free energy converge to the minimizers of a local energy depending on a two-dimensional magnetization. To cite this article: H. Zorgati, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Nonlinear estimation in anisotropic multi-index denoising

In the framework of denoising a function depending of a multidimensional variable (for instance an image), we provide a nonparametric procedure which constructs a pointwise kernel estimation with a local selection of the multidimensional bandwidth parameter. Our method is a generalization of the Lepski's method of adaptation, and roughly consists in choosing the “coarsest” bandwidth such that the estimated bias is negligible. However, this notion becomes more delicate in a multidimensional setting. We will particularly focus on functions with inhomogeneous smoothness properties and especially providing a possible disparity of the inhomogeneous aspect in the different directions. We show, in particular that our method is able to exactly attain the minimax rate or to adapt to unknown degree of anisotropic smoothness up to a logarithmic factor, for a large scale of anisotropic Besov spaces. Received: 11 November 1999 / Revised version: 14 November 2000 / Published online: 24 July 2001     

Stability of anisotropic electroactive polymers with application to layered media

The stability of anisotropic electroactive polymers is investigated. A general criterion for the onset of instabilities under plane-strain conditions is introduced in terms of a sextic polynomial whose coefficients depend on the instantaneous electroelastic moduli. In a way of an example, the stable domains of layered neo-Hookean dielectrics are determined. It is found that depending on the direction of the electrostatic excitation field relative to the lamination direction, the critical stretch ratios at which instabilities may occur can be either larger or smaller than the ones for the purely mechanical case.