Linearization and return mapping algorithms for elastoplasticity models

The return mapping algorithm is one of the most efficient procedures to solve elasto‐plastic problems. However, a criticism that may be lodged against this method is the difficulty of the practical computation of the consistent tangent matrix when the return is non‐radial. Much research has been done to handle this matrix. In this paper, a unified approach is presented in such a way that a simple closed‐form expression gives the consistent tangent matrix for the classical constitutive relations (von Mises, Tresca, Mohr–Coulomb, Drucker–Prager). The basic ideas are in the properties of eikonal equations appearing in several fields as image treatment, short time computation in elastic waves and others. The same kinds of ideas can be extended to non‐classical models. Copyright © 2003 John Wiley & Sons, Ltd.     

A Comparison of Mapping Algorithms for Hierarchical Adaptive FEM in Finite Elasto-Plasticity

The aim of this contribution is a comparison of different mapping techniques usually applied in the field of hierarchical adaptive FE-codes. The calculation of mechanical field variables for the modified mesh is an important but sensitive aspect of adaptation approaches of the spatial discretization. Regarding non-linear boundary value problems procedures of mesh refinement and coarsening imply the determination of strains, stresses and internal variables at the nodes and the Gauss points of new elements based on the transfer of the required data from the former mesh. The kind of mapping of the field variables affects the convergence behaviour as well as the costs of an adaptive FEM-calculation in a non-negligible manner. In order to improve the stability as well as the efficiency of the adaptive process a comparison of different mapping algorithms is presented and evaluated. Within this context, the mapping methods taken into account are

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation and adaptive refinement technique for fracture analysis of 2-D/3-D crack problems is the state-of-the-art. The objective of the present paper is to propose a new a posteriori error estimator based on strain energy release rate (SERR) or stress intensity factor (SIF) at the crack tip region and to use this along with the stress based error estimator available in the literature for the region away from the crack tip. The proposed a posteriori error estimator is called the K-S error estimator. Further, an adaptive mesh refinement (h-) strategy which can be used with K-S error estimator has been proposed for fracture analysis of 2-D crack problems. The performance of the proposed a posteriori error estimator and the h-adaptive refinement strategy have been demonstrated by employing the 4-noded, 8-noded and 9-noded plane stress finite elements. The proposed error estimator together with the h-adaptive refinement strategy will facilitate automation of fracture analysis process to provide reliable solutions.     

A posteriori error estimator and an adaptive technique in meshless finite points method

In this work, an adaptive technique for application of meshless methods in one- and two-dimensional boundary value problems is described. The proposed method is based on the use of implicit functions for the geometry definition, fixed weighted least squares approximation and an error estimation by means of simple formulas and a robust strategy of refinement based on the own nature of the approximation sub-domains utilised. With all these aspects, the proposed method becomes an attractive alternative for the adaptive solutions to partial differential equations in all scopes of engineering. Numerical results obtained from the computational implementation show the efficiency of the present method.     

A convergent adaptive finite element method for the primal problem of elastoplasticity

The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R‐linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropic‐kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity, the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. Copyright © 2006 John Wiley & Sons, Ltd.     

A priori error estimation of finite element models from first principles

The quality of finite element computational results can be assessed only by providing rational criteria for evaluating errors. Most exercises in this direction are based ona posteriori error estimates, based primarily on experience and intuition. If finite element analysis has to be considered a rational science, it is imperative that procedures to computea priori error estimates from first principles are made available. This paper captures some efforts in this direction.     

Stress intensity factor error index for finite element analysis with singular elements

An error index for the stress intensity factor (SIF) obtained from the finite element analysis results using singular elements is proposed. The index was developed by considering the facts that the analytical function shape of the crack tip displacement is known and that the SIF can be evaluated from the displacements only. The advantage of the error index is that it has the dimension of the SIF and converges to zero when the actual error of the SIF by displacement correlation technique converges to zero. Numerical examples for some typical crack problems, including a mixed mode crack, whose analytical solutions are known, indicated the validity of the index. The degree of actual SIF error seems to be approximated by the value of the proposed index.     

A simple a-posteriori error estimation for adaptive BEM in elasticity

In this paper, the properties of various boundary integral operators are investigated for error estimation in adaptive BEM. It is found that the residual of the hyper-singular boundary integral equation (BIE) can be used for a-posteriori error estimation for different kinds of problems. Based on this result, a new a-posteriori error indicator is proposed which is a measure of the difference of two solutions for boundary stresses in elastic BEM. The first solution is obtained by the conventional boundary stress calculation method, and the second one by use of the regularized hyper-singular BIE for displacement derivative. The latter solution has recently been found to be of high accuracy and can be easily obtained under the most commonly used C ⁰ continuous elements. This new error indicator is defined by a L ₁ norm of the difference between the two solutions under Mises stress sense. Two typical numerical examples have been performed for two-dimensional (2D) elasticity problems and the results show that the proposed error indicator successfully tracks the real numerical errors and effectively leads a h-type mesh refinement procedure.     

A new 3D finite element for adaptive h-refinement in 1-irregular meshes

A new finite element, viable for use in the three-dimensional simulation of transient physical processes with sharply varying solutions, is presented. The element is intended to function in adaptive h-refinement schemes as a versatile transition between regions of different refinement levels, ensuring interelement continuity by constructing a piecewise linear solution at the element boundaries, and retaining all degrees of freedom in the solution phase. Construction of the element shape functions is described, and a numerical example is presented which illustrates the advantages of using such an element in an adaptive refinement problem. The new element can be used in moving-front problems, such as those found in reservoir engineering and groundwater flow applications.     

Recovery based error estimation and adaptivity applied to a modified element-free Galerkin method

In this work, we propose an error estimator of the recovery type, which considers the equilibrium and boundary traction conditions, and an h-refinement procedure that is applied to the modified element-free Galerkin (MEFG) method. The approximate solution obtained by the MEFG method satisfies accurately the essential boundary condition. However, the approximate MEFG stress field presents some discontinuities on a neighborhood of the essential boundary condition and may present some spurious oscillations at regions of high stress gradients or discontinuities. Thus, the h-adaptive scheme is responsible for the reduction not only of the global error but also of the local errors associated with the discontinuities and oscillations of the approximate stress field. In order to validate the proposed procedures, we present some numerical solution for some simple problems and consider the analysis of a complex component.We want to acknowledge the support of the CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico – of Brazil. Grant Numbers: 470426/01-2 and 141806/2000-1.     

A rate-independent elastoplastic constitutive model for biological fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation

 This paper presents a rate-independent elastoplastic constitutive model for (nearly) incompressible biological fiber-reinforced composite materials. The constitutive framework, based on multisurface plasticity, is suitable for describing the mechanical behavior of biological fiber-reinforced composites in finite elastic and plastic strain domains. A key point of the constitutive model is the use of slip systems, which determine the strongly anisotropic elastic and plastic behavior of biological fiber-reinforced composites. The multiplicative decomposition of the deformation gradient into elastic and plastic parts allows the introduction of an anisotropic Helmholtz free-energy function for determining the anisotropic response. We use the unconditionally stable backward-Euler method to integrate the flow rule and employ the commonly used elastic predictor/plastic corrector concept to update the plastic variables. This choice is expressed as an Eulerian vector update the Newton's type, which leads to a numerically stable and efficient material model. By means of a representative numerical simulations the performance of the proposed constitutive framework is investigated in detail. Received: 12 December 2001 / Accepted: 14 June 2002 Financial support for this research was provided by the Austrian Science Foundation under START-Award Y74-TEC. This support is gratefully acknowledged.     

A review of error estimation and adaptivity in the boundary element method

When using the boundary element method, the accuracy of the numerical solution depends critically on the discretization of the boundary into elements (panels). The distribution of the panels is one of the most important decisions taken when analyzing a problem, but still the vast majority of users employ empirical guidelines to distribute the panels. This paper reviews the various adaptive schemes that have been proposed for boundary elements. Numerical results are obtained for infinite fluid flow problems and free surface problems and are used to assess the reliability and effectiveness of each method.     

Hydrogen transport and large strain elastoplasticity near a notch in alloy X-750

The finite element method is used to solve the coupled large strain elastoplasticity boundary value problem and transient hydrogen diffusion initial boundary value problem. As an example, solutions are obtained in the neighborhood of a rounded notch in a 4-point bend specimen of alloy X-750 at two temperatures under plane strain deformation conditions. The model accounts for the dilatational strain caused by the presence of hydrogen in the lattice and the hydrostatic stress induced drift of hydrogen. The hydrogen population profiles in both normal interstitial lattice sites (NILS) and trapping sites are calculated and conditions for the predominance of the total amount of hydrogen by either of the populations are studied. The competition between hydrostatic stress and plastic strain in the enhancement of local hydrogen concentrations is investigated. The effect of different types of traps on the relative level of trapped hydrogen as a portion of the total hydrogen is examined. The numerical analysis in conjunction with current experimental evidence suggests a specifically designed line of experiments that will isolate the parameters crucial to hydrogen induced material degradation in X-750.     

A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method

In References 1 and 2 we showed that the error in the finite-element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error-indicators and the quality of the derivatives recovered by local post-processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error-indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest.     

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.     

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

This paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

A posteriori error estimation for finite element solutions of Helmholtz' equation—Part II: estimation of the pollution error

In part I of this investigation, we proved that the standard a posteriori estimates, based only on local computations, may severely underestimate the exact error for the classes of wave-numbers and the types of meshes employed in engineering analyses. We showed that this is due to the fact that the local estimators do not measure the pollution effect inherent to the FE-solutions of Helmholtz' equation with large wavenumber. Here, we construct a posteriori estimates of the pollution error. We demonstrate that these estimates are reliable and can be used to correct the standard a posteriori error estimates in any patch of elements of interest. © 1997 John Wiley & Sons, Ltd.     

A posteriori error estimation for extended finite elements by an extended global recovery

This contribution presents an extended global derivative recovery for enriched finite element methods (FEMs), such as the extended FEM along with an associated error indicator. Owing to its simplicity, the proposed scheme is ideally suited to industrial applications. The procedure is based on global minimization of the L₂ norm of the difference between the raw strain field (C_?1) and the recovered (C₀) strain field. The methodology engineered in this paper extends the ideas of Oden and Brauchli (Int. J. Numer. Meth. Engng 1971; 3 ) and Hinton and Campbell (Int. J. Numer. Meth. Engng 1974; 8 ) by enriching the approximation used for the construction of the recovered derivatives (strains) with the gradients of the functions employed to enrich the approximation employed for the primal unknown (displacements). We show linear elastic fracture mechanics examples, both in simple two‐dimensional settings, and for a three‐dimensional structure. Numerically, we show that the effectivity index of the proposed indicator converges to unity upon mesh refinement. Consequently, the approximate error converges to the exact error, indicating that the error indicator is valid. Additionally, the numerical examples suggest a novel adaptive strategy for enriched approximations in which the dimensions of the enrichment zone are first increased, before standard h‐ and p‐adaptivities are applied; we suggest to coin this methodology e‐adaptivity. Copyright © 2008 John Wiley & Sons, Ltd.     

A postprocessed error estimate and an adaptive procedure for the semidiscrete finite element method in dynamic analysis

In the paper we present a postprocessed type of a posteriori error estimate and a h-version adaptive procedure for the semidiscrete finite element method in dynamic analysis. In space the super-convergent patch recovery technique is used for determining higher-order accurate stresses and, thus, a spatial error estimate. In time a postprocessing technique is developed for obtaining a local error estimate for one step time integration schemes (the HHT-α method). Coupling the error estimate with a mesh generator, a h-version adaptive finite element procedure is presented for two-dimensional dynamic analysis. It updates the spatial mesh and time step automatically so that the discretization errors are controlled within specified tolerances. Numerical studies on different problems are presented for demonstrating the performances of the proposed adaptive procedure.