An approach for dealing with high local stresses in finite element analyses

Highly localised through-thickness stress concentrations, higher than the strength of the material, may occur when a linear elastic finite element analysis of a composite structure is performed. Such stresses may be caused by real geometrical or material discontinuities or by artefacts in the model. The objective of this paper is to present a validated approach to determine when these high stresses will not lead to failure by delamination or matrix cracking and can therefore be ignored. Named as the High Stress Concentration (HSC) method, the approach presented in this paper is found to provide good results when applied to several finite element analyses, and is also in agreement with experimental data.     

An enhanced layerwise finite element using a robust solid-shell formulation approach

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios, usually employs plate or facet shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work, a different approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This 8-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to resolve several locking pathologies coming from the high aspect ratios of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, which is based on the finite element variables transformation matrix. The new finite element is tested and the implemented numerical remedies are verified. The results for a soft core sandwich plate are hereby presented to demonstrate the proposed finite element applicability and robustness.     

An a posteriori error estimate for the generalized finite element method for transient heat diffusion problems

We propose the study of a posteriori error estimates for time‐dependent generalized finite element simulations of heat transfer problems. A residual estimate is shown to provide reliable and practically useful upper bounds for the numerical errors, independent of the heuristically chosen enrichment functions. Two sets of numerical experiments are presented. First, the error estimate is shown to capture the decrease in the error as the number of enrichment functions is increased or the time discretization refined. Second, the estimate is used to predict the behaviour of the error where no exact solution is available. It also reflects the errors incurred in the poorly conditioned systems typically encountered in generalized finite element methods. Finally, we study local error indicators in individual time steps and elements of the mesh. This creates a basis towards the adaptive selection and refinement of the enrichment functions. Copyright © 2016 John Wiley & Sons, Ltd.     

A posteriori error indicators for the p-version of the finite element method

The existence of local a posteriori error indicators for the p-version of the finite element method is demonstrated through numerical examples. The optimal sequence of p-distributions can be closely followed on the basis of the indicators.     

Singular stress analyses of V-notched anisotropic plates based on a novel finite element method

A super singular wedge tip element whose stiffness matrix is based on numerical eigensolutions is incorporated into standard hybrid-stress finite elements to study singular stress fields around the vertex of anisotropic multi-material wedge. The numerical eigensolutions are obtained by an ad hoc finite element eigenanalysis method. To demonstrate the validity of the method, singular stresses for some typical anisotropic single-material/bimaterial wedges are investigated. All numerical results show present finite element method converges rapidly to available solutions with few elements. The present method is applicable to dealing with the problems with more complex geometries.     

Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

This paper presents a new stress recovery technique for the generalized/extended finite element method (G/XFEM) and for the stable generalized FEM (SGFEM). The recovery procedure is based on a locally weighted L² projection of raw stresses over element patches; the set of elements sharing a node. Such projection leads to a block-diagonal system of equations for the recovered stresses. The recovery procedure can be used with GFEM and SGFEM approximations based on any choice of elements and enrichment functions. Here, the focus is on low-order 2D approximations for linear elastic fracture problems. A procedure for computing recovered stresses at re-entrant corners of any internal angle is also presented. The proposed stress recovery technique is used to define a Zienkiewicz-Zhu (ZZ) a posteriori error estimator for the G/XFEM and the SGFEM. The accuracy, computational cost, and convergence rate of recovered stresses together with the quality of the ZZ estimator, including its effectivity index, are demonstrated in problems with smooth and singular solutions.     

An embedded statistical method for coupling molecular dynamics and finite element analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with finite element (FE) nodes at their common interface, necessarily requiring that the FE mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modelling to two‐dimensional material domains due to difficulties in simulating full three‐dimensional material processes. In the present work, a new approach to MD–FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and FE nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. Thus, the method lends itself for use with any FEM or MD code. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three‐dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied. Published in 2009 by John Wiley & Sons, Ltd.     

A robust preconditioning technique for the extended finite element method

The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill‐conditioned. In that case iterative solvers need a large number of iterations to obtain an acceptable solution. In this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments. A domain decomposition is employed and the algorithm is very well suited for parallel computations. The method was tested in numerical experiments to show its effectiveness. The experiments have been conducted for structures containing cracks and material interfaces. We show that the corresponding enrichments can result in arbitrarily ill‐conditioned matrices. The method proposed here, however, provides well‐conditioned matrices and can be applied to any sort of enrichment. The complexity of this approach and its relation to the domain decomposition is discussed. Computation times have been measured for a structure containing multiple cracks. For this structure the computation times could be decreased by a factor of 2. Copyright © 2010 John Wiley & Sons, Ltd.     

On using different recovery procedures for the construction of smoothed stress in finite element method

The performance of three different stress recovery procedures, namely, the superconvergent patch recovery technique (SPR), the recovery by equilibrium in patches (REP) and a combined method known as the LP procedure is reviewed. Different order of polynomials and various patch formation strategies have been employed in the numerical studies for the construction of smoothed stress fields. Two 2-D elastostatic problems of different characteristics are used to assess the behaviour of the stress recovery procedures. The numerical results obtained indicate that when the order of polynomial used in the recovery procedure is equal to that of the finite element analysis, the behaviours of all three recovery procedures are very similar and all of them are adequate to provide a reliable recovered stress field for error estimation. In case that the order of polynomial of the recovered stress is increased, the LP procedure seems to give a more stable recovery matrix and a more reliable recovered stress field than the REP procedure. © 1998 John Wiley & Sons, Ltd.     

An a posteriori error estimator for the p‐ and hp‐versions of the finite element method

An a posteriori error estimator is proposed in this paper for the p‐ and hp‐versions of the finite element method in two‐dimensional linear elastostatic problems. The local error estimator consists in an enhancement of an error indicator proposed by Bertóti and Szabó (Int. J. Numer. Meth. Engng. 1998; 42 :561–587), which is based on the minimum complementary energy principle. In order to obtain the local error estimate, this error indicator is corrected by a factor which depends only on the polynomial degree of the element. The proposed error estimator shows a good effectivity index in meshes with uniform and non‐uniform polynomial distributions, especially when the global error is estimated. Furthermore, the local error estimator is reliable enough to guide p‐ and hp‐adaptive refinement strategies. Copyright © 2004 John Wiley & Sons, Ltd.     

A posteriori pointwise upper bound error estimates in the finite element method

An error estimate for the finite element method is presented in this paper. The error is identified as the response to a set of residual forces, and a complementary analysis provides an upper bound estimate of the global energy of the error. The inequality proposed by Babu?ka and Miller¹ is then employed to bound the error in stress and displacement at a point. The formula is derived for two-dimensional elasticity, but the procedure is general; and can be applied to three-dimensional and other problems. Numerical experiments using the procedure are carried out and the results are given for the four-node bilinear compatible element and plane stress.     

The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method

It is demonstrated that the residual in a compatible (displacement) finite element solution can be partitioned into local self-equilibrating systems on each element. An a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved.     

Element by element a posteriori error estimation of the finite element analysis for three-dimensional elastic problems

An a posteriori error estimation method for finite element solutions for three-dimensional elastic problems is presented based on the theory developed by the authors for two-dimensional problems.¹ The error is estimated for the finite element solutions obtained using three-dimensional 8-node elements with a linear interpolation function in an arbitrary hexahedron. The method is successfully applied to three-dimensional elastic problems. In order to decrease computing time and memory use, the error is estimated element by element. The major difficulty in the element-wise error estimation technique is satisfying the self-equilibrium condition of applied forces, especially in three-dimensional problems. These forces are mainly due to traction discontinuity on the element boundaries. The difficulty is circumvented by employing an element-wise optimal procedure. It is also shown that a very accurate stress solution can be obtained by adding estimated error to the original finite element solutions.     

Subdomain-based error techniques for generalized finite element approximations of problems with singular stress fields

This paper considers four types of error measures, each tailored to the generalized finite element method. Particular attention is given to two-dimensional elasticity problems with singular stress fields. The first error measure is obtained using the equilibrated element residual method. The other three estimators overcome the necessity of equilibrating the residue by employing a subdomain strategy. In this strategy, the partition of unity (PoU) property is used to decompose the error problem into local contributions over each patch of elements. The residual functional of the error problem is the same for the subdomain estimators, but the bi-linear form is different for each one of them. In the second estimator, the bi-linear form is weighted by the PoU functions associated with the patch over which the error problem is stated. No weighting appears in the bi-linear form of the third estimator. The fourth measure is proposed as an alternative strategy, in which the products of the PoU functions and test functions are introduced as weights in the weighted integral statement of the differential equation describing the error problem. The linear form of the local error problem is then identical to that of the other subdomain techniques, while the bi-linear form is stated differently, with the PoU functions directly multiplying the test functions. The goal of this study is to investigate the performance of the four estimators in two-dimensional elasticity problems with geometries that produce singularities in the stress field and concentration of the error in the numerical solution.     

On the finite element method for calculating stress intensity factors with a modified elliptical model

A finite element method based on the modification of the elliptical displacement function model developed earlier by the authors is presented for the determination of stress intensity factors in cracked bodies. The modified elliptical model not only retains the simplicity of the original method in describing stress conditions at the crack tips but also extends the application of the method to cases where the elliptical shape of the crack surface is not entirely preserved. The present method avoids the need for successive computations of several strain energies in a cracked body as required by the strain energy approach and of high concentration of very fine elements at the crack tip by the conventional stress approach.

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Résumé On présente une méthode par éléments finis basée sur la modification du modèle de la fonction de déplacements elliptiques développée précédemment par les auteurs, pour déterminer les facteurs d'intensité des contraintes dans des solides fissurés. Le modèle modifié conserve la simplicité de la méthode originale pour décrire les conditions de contraintes aux extrémités d'une fissure. En outre, il permet d'étudier son application à des cas où la forme de la surface de la fissure n'est pas rigoureusement elliptique.La méthode proposée évite de devoir procéder à des calculs en série de plusieurs énergies de déformation dans le corps fissuré, ainsi que le requiert l'approche basée sur l'énergie de déformation. Elle permet aussi de ne pas devoir calculer des concentrations importantes d'éléments à très petites mailles à l'extrémité de la fissure, ainsi que l'exige l'approche concentionnelle basée sur les contraintes.

     

An edge-based smoothed finite element method for visco-elastoplastic analyses of 2D solids using triangular mesh

An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper, the ES-FEM is extended to more complicated visco-elastoplastic analyses using the von-Mises yield function and the Prandtl–Reuss flow rule. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic and linear kinematic hardening. The formulation shows that the bandwidth of stiffness matrix of the ES-FEM is larger than that of the FEM, and hence the computational cost of the ES-FEM in numerical examples is larger than that of the FEM for the same mesh. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the ES-FEM is more efficient than the FEM.     

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

Parametric analyses of stitched composite T-joints by the finite element method

Numerical methods were employed to perform a detailed parametric study on composite T-joints with transverse stitching using the finite element method. This analysis was accomplished to discern the effects of key joint parameters including fiber insertion tow modulus, fiber insertion filament count, fiber insertion depth, and resin-rich interface zone thickness on T-joint displacement and damage initiation load. T-joint load conditions included flexure, tension, and shear. Significant results of the parametric finite element analysis indicate that under flexural loading, increasing the fiber insertion tow modulus and tow filament count increases the T-joint damage initiation load; increasing the fiber insertion depth reduces T-joint deflection; and reducing the web-to-flange interface thickness reduces the T-joint deflection. Fiber insertion tow filament count and modulus have a negligible effect on T-joint deflection under tension and initial damage load under shear.