Anisotropic Finite Element Method Research Articles

Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutions

In this work, anisotropic bilinear finite element and second‐order temporal approximation are adopted to establish a fully discrete scheme for the fractional substantial diffusion equation with variable coefficient. We prove the proposed discrete scheme is unconditionally stable in the senses of ‐norm and ‐norm. By introducing a new projection, the optimal convergence error in ‐norm and the superclose property in ‐norm are obtained under the condition of , where is the exact solution of the problem. Through interpolation post processing technique, we arrive at the global superconvergence of the interpolation. Furthermore, an improved algorithm is proposed to solve the problem with nonsmooth solution. Finally, numerical tests are given to illustrate the validity and efficiency of our theoretical analysis.

A Flexible FEM-BEM-DDM for EM Scattering by Multiscale Anisotropic Objects

In this article, a nonconformal domain decomposition method (DDM) based on the hybrid finite element method (FEM) and boundary element method (BEM) is proposed to solve arbitrary 3-D anisotropic (including bianisotropic and single anisotropic) multiscale composite objects. The object is divided into interior and exterior regions. The interior region is solved by the bianisotropic finite element domain decomposition method (FEM-DDM), and the exterior region is solved by the discontinuous Galerkin (DG) method based on the combined field integral equation (CFIE). In order to decouple the electric and magnetic fields of the bianisotropic object, a new finite element–boundary element–domain decomposition method (FEM-BEM-DDM) formulation is derived. To ensure the continuity of electric current and electric field between anisotropic FEM subdomains, a new auxiliary current is introduced into the interface of different subdomains of FEM and the interface of FEM subdomains with BEM subdomains. The former uses second-order transmission conditions (SOTCs) to ensure continuity, whereas the latter uses Robin transmission conditions (RTCs) to ensure continuity. The BEM dense matrix–vector multiplication is accelerated by a multilevel fast multipole algorithm (MLFMA). The proposed FEM-BEM-DDM allows for nonconformal discretization between any touching subdomains and provides an effective preconditioner. In numerical examples, the accuracy and effectiveness of the method are verified.

Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes

In this paper, an unconditionally stable fully discrete numerical scheme for the two-dimensional (2D) time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The L2-1σ scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization. The unconditional stability and convergence of the proposed scheme are proved rigorously. It is shown that the order O(h2+N−min{rα,2}) can be achieved, where h is the spatial step, N is the number of partition in temporal direction, r is the temporal meshes grading parameter and α is the order of fractional derivative. A numerical example is provided to verify the sharpness of our error analysis.

A Priori Analysis of an Anisotropic Finite Element Method for Elliptic Equations in Polyhedral Domains

Abstract Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces H σ {H^{\sigma}} with σ ∈ [ 0 , 1 ) {\sigma\in[0,1)} . We propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have “smoother” interior data. Numerical tests validate the theoretical analysis.

Local fracture toughness measurements in polycrystalline cubic zirconia using micro-cantilever bending tests

Local fracture toughness of a polycristalline ceramic was characterized with the micro-cantilever bending method. Beams were milled in single grains of cubic zirconia using a Focused Ion Beam (FIB) microscope. Grains with specific crystallographic orientations were chosen using Electron Backscatter Diffraction (EBSD) measurements to test specific plane families. Notched micro-cantilever beams were loaded up to fracture using a nano-indenter. For fracture toughness evaluation, three different methods were considered: an analytical solution, an isotropic Finite Element Method (FEM) calculation, and an anisotropic FEM model. These three methods gave similar toughness values. A good agreement was found with literature data measured on cubic zirconia single crystals at a macroscopic scale. In this work, no significant difference was noticed for fracture toughness between {100}, {110} and {111} crystallographic plane families. Such method could be used to characterize the local fracture properties of a sample showing extensive cracking such as irradiated nuclear fuel.

Finite element analysis of mono- and bicortical mini-implant stability

Loosening and loss rates of monocortical mini-implants are relatively high, therefore the following null hypothesis was tested: 'The local bone stress in mono and bicortically-anchored mini-implants is identical'. Anisotropic Finite Element Method (FEM) models of the mandibular bone, including teeth, periodontal ligaments, orthodontic braces, and mini-implants of varying length, were created. The morphology was based on the Computed Tomography data of an anatomical preparation. All mini-implants with varying insertion depths (monocortical short, monocortical long, bicortical) were typically loaded, and the induced effective stress was calculated in the cervical area of the cortical bone. The obtained values were subsequently analysed descriptively and exploratively using the SPSS 19.0 software. The null hypothesis was rejected, since the stress values of each anchorage type differed significantly (Kruskal-Wallis Test, P < 0.001). Therefore, the lowest effective stress values were induced in bicortical anchorage (mean = 0.65MPa, SD = 0.06MPa) and the highest were induced in monocortical (short) anchorage of the mini-implants (mean = 1.79MPa, SD = 0.29MPa). The Spearman rank correlation was 0.821 (P < 0.001). The deeper the mini-implant was anchored, the lower were the effective stress values in the cervical region of the cortical bone. Bicortical implant anchorage is biomechanically more favourable than monocortical anchorage; therefore, bicortical anchorage should be especially considered in challenging clinical situations requiring heavy anchorage.

Orthodontic bracket debonding: risk of enamel fracture

Until now, it is not clear if various procedures of bracket debonding differ with regard to their risk of enamel fracture. Therefore, the objective of the present study was to compare these procedures biomechanically for assessing the risk of complications. An anisotropic finite element method (FEM) model of the mandibular bone including periodontal ligament, enamel, dentin, and an orthodontic bracket was created. The morphology based on the CT data of an anatomical specimen. Typical loading conditions were defined for each method of bracket debonding (compression, shearing off, twisting off). Shortly before the adhesive's break, the induced stress in enamel, periodontal ligament, and in the alveolar bone was measured. The statistical analysis of the obtained values was performed in SPSS 19.0. Relatively high stresses occurred in the enamel using frontal torque (max. 44.18MPa). With shearing off, the stresses were also high (max. 41.96MPa), and additionally high loads occurred on the alveolar bone as well (max. 11.79MPa). Moderate maximum values in enamel and alveolar bone appeared during the compression of the bracket wings (max. 37.12MPa) and during debonding by lateral torque (max. 35.18MPa). The present simulation results indicate that the risk of enamel fracture may depend on the individual debonding procedure. Further clinical trials are necessary to confirm that. For patients with prior periodontal disease or loosened teeth, a debonding procedure by compression of the bracket wings is recommended, since here the load for the periodontal structures of the tooth is lowest.

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier–Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field. Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.

Investigation of the double shear creep specimens to study the creep behaviour of short fibre metal matrix composites

Double shear creep specimens have been investigated by the transverse anisotropic finite element method, in order to be applied to study the multiaxial creep behaviour of short fibre-reinforced metal matrix composites (MMCs). The creep behaviour was dependent on orientations of the specimens and the transverse anisotropic parameters. The detailed evolution of the stress and strain in the double shear creep specimens shows that there is a steady stress state in the shear zone. The shear stress is near to the applied average shear stress, and independent of orientations of the specimens and the transverse anisotropicic parameters. The double shear specimen can provide a simple multiaxial stress state for MMCs. A procedure has been presented to evolve the parameters for the transversely anisotropic MMC materials, so that the double shear specimens can be tested to obtain the transversely anisotropic material properties, some of which are difficult to get by the usual tensile specimens.

Fatigue crack initiation and propagation in an α-iron polycrystal

The cyclic deformation and crack initiation and propagation of α-iron polycrystals have been studied under stress-control with emphasis on the dependence on the stress amplitudes and orientation of grains. The orientations of grains surrounding cracks were determined by the electron backscatter diffraction (EBSD). It was found that cracks did not originate at grain boundaries but along the slip lines intersecting with the free surface at the corner of the specimen. At lower stress amplitude, the crack propagated in a transgranular mode, but at higher stress amplitude the crack propagated in a mixed mode–transgranular mode and intergranular mode. The stress distribution along the loading axis of grains surrounding the crack was calculated using a three-dimensional anisotropic finite element method (FEM). In the present work, it was shown that the crack initiation mainly depends on three factors: higher stress distribution, abundant dislocation sources and less resistance to dislocation motion.

Investigation of the double shear creep specimens to study the creep behaviour of short fibre metal matrix composites

Double shear creep specimens have been investigated by the transverse anisotropic finite element method, in order to be applied to study the multiaxial creep behaviour of short fibre-reinforced metal matrix composites (MMCs). The creep behaviour was dependent on orientations of the specimens and the transverse anisotropic parameters. The detailed evolution of the stress and strain in the double shear creep specimens shows that there is a steady stress state in the shear zone. The shear stress is near to the applied average shear stress, and independent of orientations of the specimens and the transverse anisotropicic parameters. The double shear specimen can provide a simple multiaxial stress state for MMCs. A procedure has been presented to evolve the parameters for the transversely anisotropic MMC materials, so that the double shear specimens can be tested to obtain the transversely anisotropic material properties, some of which are difficult to get by the usual tensile specimens.

On the stress distribution around a triple junction

Abstract The internal and external stresses around a triple junction of copper have been analysed by a disclination model and by a three-dimensional anisotropic finiteelement method respectively. The results indicate that the internal stress depends on whether the triple line is an I-line or a U-line. At the triple line the elastic strain energy concentrations caused by external stresses depend on both the orientations of the component grains and the loading directions. For a loading direction parallel to the triple line, the elastic energy concentration is usually large. For a loading direction perpendicular to the triple line, the elastic energy concentration is complex; in some cases, elastic energy dispersion can occur but, in other cases, the elastic energy concentration is much larger than that with the loading direction parallel to the triple line.

Finite Element Analysis of Elastic Stress and the Resolved Shear Stress in the Primary Slip System of a Copper Tricrystal

The elastic stress distribution around a triple junction and the resolved shear stress on the primary slip system in each component crystal of a copper tricrystal were analyzed by three-dimensional anisotropic FEM (finite element method). The calculated results were compared with experimental observations.

Distribution of stresses and elastic strain energy in an ideal multicrystal model

A planar array of hexagonal grains was used as a two-dimensional model of the ideal multicrystal, and the anisotropic finite element method was employed to analyse the features about the stress and elastic strain energy distributions. The purpose was to investigate the effects of grain boundary and triple junction on the local mechanical properties of the multicrystal. The results show that the distributions of stresses and elastic strain energy are distinctly inhomogeneous in a macroscopic isotropic multicrystal under uniaxial load. The highest and lowest of the stress and elastic strain energies exist at some grain boundaries and triple junctions. The highest resolved shear stress of each primary slip system in each grain occurs at some grain boundaries, but not at triple junctions. Slip may first initiate from the grain boundary of the stiffer grain.

Characteristics of strain and resolved shear stress in a bicrystal with the grain boundary perpendicular to the tensile axis

The distributions of strain and resolved shear stress in a copper bicrystal with the grain boundary perpendicular to the tensile axis were analyzed by a three-dimensional anisotropic finite element method. The differences of strain and resolved shear stress between two component crystals, between the region close to the grain boundary and the region far from the grain boundary, and between the free surface and the interior region, were studied. This paper presents some general trends about the changes of strain and resolved shear stress near the grain boundary in the bicrystal with a grain boundary perpendicular to the tensile axis.

Modelling of elastic interaction stresses in two-phase materials by FEM

Recently, by considering a bicrystal of α and β phases of Ti alloys with Burgers orientation relationships, it has been shown that elastic interactions play a significant role in the initiation of plastic deformation in two-phase materials. However, it is not known how far these interactions exist away from the interfaces. Further, it is not known how these interactions vary with morphology of phases (i.e. thickness, orientation and distribution). To study these effects systematically, a three-dimensional anisotropic finite element method (FEM) has been employed in this investigation. It is found that the interaction stresses decay from the interface in an exponential form in a distance of one-sixth to one-half of the plate thickness, that large hydrostatic stresses are found within alpha plates in beta matrix and that the anisotropic elastic material properties are not responsible for the anisotropic sliding experimentally observed by earlier investigators. In addition, it has been found that the magnitude of the interaction stresses varies with the morphology of phases.

Multigroup application of the anisotropic FEM code FELTRAN to one, two, three-dimensions and R- Z problems

The multigroup even parity form of the neutron transport equation, with anisotropic scattering, has been solved by the Finite Element Method (FEM). The maximum principle for the even parity equation is employed. Spherical harmonics are used to represent the angular dependence of the flux and scattering while finite elements are used in the spatial domain. A finite element code, FELTRAN, has been developed which solves multigroup anisotropic scattering problems in one, two and three dimensions. Angular integrals which appear in the functional are evaluated in a general and modular form within the code; hence any desired order of anisotropic scattering and moments for the even parity flux can be chosen. In the spatial domain various types of elements are used for both regular and irregular geometries. Lagrangian linear, quadratic and cubic rectangular elements are used in X- Y and R- Z regular geometry, while curved geometries are handled by triangular elements. Solutions may also be obtained for the adjoint form of the even parity equation. Some representative examples in one, two and three dimensions are analysed and comparison is made with discrete-ordinates finite-difference codes ANISN and DOT, together with a 22-group problem intended to demonstrate the capabilities of FELTRAN in solving more practical problems.