Toward anisotropic mesh construction and error estimation in the finite element method

Directional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so‐called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well‐aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625–648, 2002; DOI 10.1002/num.10023     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

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     

Anisotropic isoparametric hypersurfaces in Euclidean spaces

In this paper, by Nomizu’s method and some technical treatment of the asymmetry of the F-Weingarten operator, we obtain a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is a generalization of the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local anisotropic isoparametric surface constructed by B. Palmer, we find that in general anisotropic isoparametric hypersurfaces have both local and global aspects as in the theory of proper Dupin hypersurfaces, which differs from classical isoparametric hypersurfaces.     

Simulation of Deep Drawing of Carbon Fibre Prepregs

The simulation of prepregs must regard highly anisotropic, viscoelastic and thermal-chemical properties. To this end a constitutive model is split into an anisotropic elastic part, which represents the fibre fraction and an isotropic, viscoelastic part, representing the matrix. The second part also contains curing, causing a dependency on time and temperature. During real deep-drawing processes large deformations up to 50 % occur, which is considered in a formulation at large strains. This model contains an anisotropic elastic part based on a Neo-Hooke law enhanced by an anisotropic part. A viscoelastic part is added using Hencky-strains and the work-conjugate Hill-stress to transfer a model for small strains into large strains. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic Meshless Frames on ℝ n

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of ℝ ⁿ . The wavelets we construct are C ^∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C ^∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay.     

Anisotropic shearlet transforms for L2

In this paper, we present a new anisotropic generalization of the continuous shearlet transformation. This is achieved by means of an explicit construction of a family of reproducing Lie subgroups of the symplectic group. We study the properties of this new family of anisotropic shearlet transformations. In particular, we provide an analog of the Calderón admissibility condition for anisotropic shearlet reproducing functions.     

Sur la magnétohydrodynamique anisotrope relativiste

Summary This paper is concerned with the relativistic phenomenological theory of anisotropic magnetohydrodynamics. An anisotropic fluid scheme is defined and studied. The main system of anisotropic magnetohydroldynamics is deduced. This system may describe a collisionless anisotropic plasma embedded in a strong magnetic field. The main system is shown to yield to three types of waves as in isotropic (perfect) magnetohydrodynamics: the entropic waves, the magnetosonic waves and the Alfven waves. For the rays associated respectively to the magnetosonic and Alfven waves the fundamental property concerning the propagation of infinitesimal discontinuities of variables is established. The conditions under which the velocities of propagation of magnetosonic and Alfven waves are real are derived: these conditions imply as in the classical theory the absence of fire hose and mirror instabilities in the fluid. The study of wave cones allows, on the one hand to point out some particularities of the propagation of waves in anisotropic magnetohydrodynamics, and on the other hand to clear up the hyperbolicity character of differential operators associated to various waves.

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Entrata in Redazione il 23 aprile 1975.     

Anisotropic elasto-plastic damage formulation for finite strains using a damage structural tensor and the effective stress concept

Using the strain equivalence principle and the effective stress concept an anisotropic finite strain damage model is proposed as a direct extension of the classical isotropic LEMAITRE damage model to the anisotropic finite strain case. The damage tensor is included as a structural tensor in the complementary energy potential. This approach allows to consider a wide range of anisotropic damage phenomena on the level of continuum mechanics. (© 2005 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.     

Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem on anisotropic meshes

In this paper, we derive robust a posteriori error estimates for conforming approximations to a singularly perturbed reaction-diffusion problem on anisotropic meshes, since the solution in general exhibits anisotropic features, e.g., strong boundary or interior layers. Based on the anisotropy of the mesh elements, we improve the a posteriori error estimates developed by Cheddadi et al., which are reliable and efficient on isotropic meshes but fail on anisotropic ones. Without the assumption that the mesh is shape-regular, the resulting mesh-dependent error estimator is shown to be reliable, efficient and robust with respect to the reaction coefficient, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming one, like the piecewise linear finite element one. Our estimates are based on the usual H(div)-conforming, locally conservative flux reconstruction in the lowest-order Raviart-Thomas space on a dual mesh associated with the original anisotropic simplex one. Numerical experiments in 2D confirm that our estimates are reliable, efficient and robust on anisotropic meshes.     

A note on the area and coarea formulas for general volume densities and some applications

We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. We apply these formulas to derive an anisotropic tube formula for hypersurfaces in \({{\mathbb{R}}^n}\) and to give a short euclidean proof of the anisotropic Sobolev inequality. A discussion about the first variation of the anisotropic area is also included.     

The anisotropic integrability regularity criterion to 3D magnetohydrodynamics equations

In this article, we establish sufficient conditions for the regularity of solutions of 3D MHD equations in the framework of the anisotropic Lebesgue spaces. In particular, we obtain the anisotropic regularity criterion via partial derivatives, and it is a generalization of the some previous results. Besides, the anisotropic integrability regularity criteria in terms of the magnetic field and the third component of the velocity field are also investigated. Copyright © 2017 John Wiley & Sons, Ltd.     

Two-Level-Split Decomposition of Anisotropic Besov Spaces

Anisotropic Besov spaces (B-spaces) are developed based on anisotropic multilevel ellipsoid covers (dilations) of ℝ ⁿ . This extends earlier results on anisotropic Besov spaces. Furthermore, sequences of anisotropic bases are constructed and utilized for two-level-split decompositions of the B-spaces and nonlinear m-term approximation.     

Separable anisotropic elliptic operators and applications

This paper focuses on boundary value problems for anisotropic differential-operator equations of high order with variable coefficients in the half plane. Several conditions are obtained which guarantee the maximal regularity of anisotropic elliptic and parabolic problems in Banach-valued L _p -spaces. Especially, it is shown that this differential operator is R-positive and is a generator of an analytic semigroup. These results are also applied to infinite systems of anisotropic type partial differential equations in the half plane.     

DECAY OF ENERGY FOR A DISSIPATIVE ANISOTROPIC ELASTIC SYSTEM

In this article, we study the large-time behavior of energy for a N-dimensional dissipative anisotropic elastic system. By means of multiplicative techniques, energy method, and Zuazua’s estimate technique, we prove the decay property of energy for anisotropic elastic system.     

New anisotropic models from isotropic solutions

We establish an algorithm that produces a new solution to the Einstein field equations, with an anisotropic matter distribution, from a given seed isotropic solution. The new solution is expressed in terms of integrals of known functions, and the integration can be completed in principle. The applicability of this technique is demonstrated by generating anisotropic isothermal spheres and anisotropic constant density Schwarzschild spheres. Both of these solutions are expressed in closed form in terms of elementary functions, and this facilitates physical analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonstationary love waves of the SH-type in an anisotropic elastic medium. A kinematic approach

In this paper, the propagation of Love waves in anisotropic elastic media is studied. These waves are a similar to the transverse surface SH waves in the isotropic case. Necessary conditions for the existence of Love waves of this polarization type near the surface Σ of an anisotropic elastic body are deduced. The algorithm developed here makes it possible to find the direction (s) of transverse surface wave propagation (at every point on the surface Σ). The algorithm employed is illustrated by some special anisotropic cases. The space-time method is used to construct the asymptotics of Love waves for those types of anisotropic media the eikonal equation of which is valid on the surface of an elastic body. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 262–276 Translated by Z. A. Yanson     

The scalar Oseen operator − Δ + ∂/2202<Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript> in ℝ<Superscript>2</Superscript>

This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in L ^p theory.