A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second-order problems. Part II of the paper deals with the strategy for adaptive refinement and concentrates on the p-convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.     

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.     

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part II: Adaptive refinement

In Part II of this study, an automatic adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D linear boundary value problems is suggested. Based in the theoretical development and the numerical experiments done in Part I of this study, the Zienkiewicz and Zhu (Z–Z) error estimation scheme is combined with a new stress recovery procedure for the a posteriori error estimation of the adaptive refinement procedure. By considering the a priori convergence rate of the RKPM and the estimated error norm, an adaptive refinement strategy for the determination of optimal point distribution is proposed. In the suggested adaptive refinement scheme, the local refinement indicators used are computed by considering the partition of unity property of the RKPM shape functions. In addition, a simple but effective variable support size definition scheme is suggested to ensure the robustness of the adaptive RKPM procedure. The performance of the suggested adaptive procedure is tested by using it to solve several benchmark problems. Numerical results indicated that the suggested refinement scheme can lead to the generation of nearly optimal meshes for both smooth and singular problems. The optimal convergence rate of the RKPM is restored and thus the effectivity indices of the Z–Z error estimator are converging to the ideal value of unity as the meshes are refined.     

hp‐adaptive extended finite element method

This paper discusses higher‐order extended finite element methods (XFEMs) obtained from the combination of the standard XFEM with higher‐order FEMs. Here, the focus is on the embedding of the latter into the partition of unity method, which is the basis of the XFEM. A priori error estimates are discussed, and numerical verification is given for three benchmark problems. Moreover, methodological aspects, which are necessary for hp‐adaptivity in XFEM and allow for exponential convergence rates, are summarized. In particular, the handling of hanging nodes via constrained approximation and an hp‐adaptive strategy are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

An adaptive time integration strategy based on displacement history curvature

This work introduces a time‐adaptive strategy that uses a refinement estimator on the basis of the first Frenet curvature. In dynamics, a time‐adaptive strategy is a mechanism that interactively proposes changes to the time step used in iterative methods of solution. These changes aim to improve the relation between quality of response and computational cost. The method here proposed is suitable for a variety of numerical time integration problems, for example, in the study of bodies subjected to dynamical loads. The motion equation in its space‐discrete form is used as reference to derive the formulation presented in this paper. Our method is contrasted with other ones based on local error estimator and apparent frequencies. We check the performance of our proposal when employed with the central difference, the explicit generalized‐ α and the Chung‐Lee integration methods. The proposed refinement estimator demands low computational resources, being easily applied to several direct integration methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Towards automatic adaptive geometrically non-linear shell analysis. Part I: Implementation of an h-adaptive mesh refinement procedure

Effective methods leading to automated adaptive numerical solutions to geometrically non-linear shell-type problems are studied in this work. In particular, procedures for improving the accuracy, the reliability and the computational efficiency of the finite element solutions are of primary interest here. This is addressed using h-adaptive mesh refinement based on a posteriori error estimation, self-adaptive methods in global incremental/iterative processes, as well as smart algorithms and heuristic approaches based on methods of knowledge engineering. Seemless integration of h-adaptive finite element methods with adaptive step-length control makes it possible to maintain a prescribed accuracy while maintaining the solution efficiency without user intervention throughout the process of a non-linear analysis. Several examples illustrate the merit and potential of the approach studied herein and confirm the feasibility of developing an automatic adaptive environment for geometrically non-linear analysis of shell structures.     

On the development of adaptive random differential quadrature method with an error recovery technique and its application in the locally high gradient problems

An adaptive numerical method called, the adaptive random differential quadrature (ARDQ) method is presented in this paper. In the ARDQ method, the random differential quadrature (RDQ) method is coupled with a posteriori error estimator based on relative error norm in the displacement field. An error recovery technique, based on the least square averaging over the local interpolation domain, is proposed which improves the solution accuracy as the spacing, h → 0. In the adaptive refinement, a novel convex hull approach with the vectors cross product is proposed to ensure that the newly created nodes are always within the computational domain. The ARDQ method numerical accuracy is successfully evaluated by solving several 1D, 2D and irregular domain problems having locally high gradients. It is concluded from the convergence values that the ARDQ method coupled with error recovery technique can be effectively used to solve the locally high gradient initial and boundary value problems.     

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

An hp‐adaptive finite element method for scattering problems in computational electromagnetics

An hp‐adaptive finite element (FE) approach is presented for a reliable, efficient and accurate solution of 3D electromagnetic scattering problems. The radiation condition in the far field is satisfied automatically by approximation with infinite elements (IE). Near optimal discretizations that can effectively resolve local rapid variations in the scattered field are sought adaptively by mesh refinements blended with graded polynomial enrichments. The p‐enrichments need not be spatially isotropic. The discretization error can be controlled by a self‐adaptive process, which is driven by implicit or explicit a posteriori error estimates. The error may be estimated in the energy norm or in a quantity of interest. A radar cross section (RCS) related linear functional is used in the latter case. Adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for a number of generic problems are given and compared to previously published results. Copyright © 2004 John Wiley & Sons, Ltd.     

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part I: Stress recovery and a posteriori error estimation

In this study, an adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D elastostatic problems is suggested. This adaptive refinement procedure is based on the Zienkiewicz and Zhu (ZZ) error estimator for the a posteriori error estimation and an adaptive finite point mesh generator for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the RKPM. The proposed error estimator is different from most recovery type error estimators suggested previously in such a way that, rather than using the least-squares fitting approach, the recovery stress field is constructed by an extraction function approach. Numerical studies using 2D benchmark boundary value problems indicated that the recovered stress field obtained is more accurate and converges at a higher rate than the RKPM stress field. In Part II of the study, concentration will be shifted to the development of an adaptive refinement algorithm for the RKPM.     

Error estimation and h adaptive boundary elements

This paper deals with error estimation of the boundary element method (BEM) and the h adaptive boundary elements. The various error sources in the BEM are discussed, and the upper bound of the BEM solution error is derived by means of the interpolation error of the BEM solution. A new error estimator is presented in the paper by using post-processing data from the standard BEM solutions. Two adaptive algorithms, the standard h adaptive and the h-hierarchical adaptive, are implemented based on direct boundary element method for two-dimensional elasticity problems. A few numerical examples are used to compare the accuracy of the proposed adaptive algorithm, as well as the error estimator. The stability of the BEM system matrix, which may deteriorate due to the introduction of h-hierarchical interpolation functions, has also been studied.     

3D adaptive finite element modeling of non-planar curved crack growth using the weighted superconvergent patch recovery method

In this paper, an adaptive finite element analysis is presented for 3D modeling of non-planar curved crack growth. The fracture mechanical evaluation is performed based on a general technique for non-planar curved cracks. The Schollmann’s crack kinking criterion is used for the process of crack propagation in 3D problems. The Zienkiewicz-Zhu error estimator is employed in conjunction with a weighted SPR technique at each patch to improve the accuracy of error estimation. Applying the proposed technique to 3D non-planar curved crack growth problems shows significant improvements particularly at the boundaries and near crack tip regions. Several numerical examples are presented to illustrate the robustness of the proposed technique.     

An adaptive interface‐enriched generalized FEM for the treatment of problems with curved interfaces

An adaptive refinement scheme is presented to reduce the geometry discretization error and provide higher‐order enrichment functions for the interface‐enriched generalized FEM. The proposed method relies on the h‐adaptive and p‐adaptive refinement techniques to reduce the discrepancy between the exact and discretized geometries of curved material interfaces. A thorough discussion is provided on identifying the appropriate level of the refinement for curved interfaces based on the size of the elements of the background mesh. Varied techniques are then studied for selecting the quasi‐optimal location of interface nodes to obtain a more accurate approximation of the interface geometry. We also discuss different approaches for creating the integration sub‐elements and evaluating the corresponding enrichment functions together with their impact on the performance and computational cost of higher‐order enrichments. Several examples are presented to demonstrate the application of the adaptive interface‐enriched generalized FEM for modeling thermo‐mechanical problems with intricate geometries. The accuracy and convergence rate of the method are also studied in these example problems. Copyright © 2015 John Wiley & Sons, Ltd.     

An adaptive finite element approach for the free surface seepage problem

A posteriori error estimates and adaptive mesh refinements are now on a rigorous mathematical foundation for linear, elliptic boundary-value problems of second order. Yet, for non-linear problems only a few results have been obtained till now. In this paper we consider as a non-linear model problem the two-dimensional fluid flow with free surface and show how results from linear a posteriori theory can be used to control the non-linear iteration and to refine the mesh adaptively. A numerical example shows that, similar to linear problems, considerable improvement of the accuracy is obtained by an adaptive mesh refinement and that the influence of singularities on the order of convergence disappears.     

h‐Adaptive FE methods applied to single‐ and multiphase problems

A major problem in using the finite element method for solving numerous engineering problems in the framework of single‐ and multiphase materials is the assessment of discretization errors and the design of suitable meshes. To overcome this problem, adaptive finite element methods have been developed. Based on the error indicator by Zienkiewicz and Zhu, it is the goal of the present paper to present a new error indicator which is especially designed for multiphase problems. Furthermore, efficient h‐adaptive strategies concerning both the generation of new meshes in the framework of independent and hierarchical remeshing strategies and the data transfer between old and new meshes are pointed out. Finally, numerical examples are given to exhibit the efficiency and the quality of the presented h‐adaptive methods and to compare the different strategies to each other. Copyright © 2002 John Wiley & Sons, Ltd.     

A posteriori error estimates and an adaptive scheme of least‐squares meshfree method

A posteriori error estimates and an adaptive refinement scheme of first‐order least‐squares meshfree method (LSMFM) are presented. The error indicators are readily computed from the residual. For an elliptic problem, the error indicators are further improved by applying the Aubin–Nitsche method. It is demonstrated, through numerical examples, that the error indicators coherently reflect the actual error. In the proposed refinement scheme, Voronoi cells are used for inserting new nodes at appropriate positions. Numerical examples show that the adaptive first‐order LSMFM, which combines the proposed error indicators and nodal refinement scheme, is effectively applied to the localized problems such as the shock formation in fluid dynamics. Copyright © 2003 John Wiley & Sons, Ltd.     

An algorithm for adaptive refinement of triangular element meshes

A simple algorithm is developed for adaptive and automatic h refinement of two-dimensional triangular finite element meshes. The algorithm is based on an element refinement ratio that can be calculated from an a posteriori error indicator. The element subdivision algorithm is robust and recursive. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several example problems are presented to illustrate the utility of the approach.     

A combined r-h adaptive strategy based on material forces and error assessment for plane problems and bimaterial interfaces

An r-h adaptive scheme has been proposed and formulated for analysis of bimaterial interface problems using adaptive finite element method. It involves a combination of the configurational force based r-adaption with weighted laplacian smoothing and mesh enrichment by h-refinement. The Configurational driving force is evaluated by considering the weak form of the material force balance for bimaterial inerface problems. These forces assembled at nodes act as an indicator for r-adaption. A weighted laplacian smoothing is performed for smoothing the mesh. The h-adaptive strategy is based on a modifed weighted energy norm of error evaluated using supercovergent estimators. The proposed method applies specific non sliding interface strain compatibility requirements across inter material boundaries consistent with physical principles to obtain modified error estimators. The best sequence of combining r- and h-adaption has been evolved from numerical study. The study confirms that the proposed combined r-h adaption is more efficient than a purely h-adaptive approach and more flexible than a purely r-adaptive approach with better convergence characteristics and helps in obtaining optimal finite element meshes for a specified accuracy.     

A posteriori analysis and adaptive error control for operator decomposition solution of coupled semilinear elliptic systems

In this paper, we develop an a posteriori error analysis for operator decomposition iteration methods applied to systems of coupled semilinear elliptic problems. The goal is to compute accurate error estimates that account for the combined effects arising from numerical approximation (discretization) and operator decomposition iteration. In an earlier paper, we considered ‘triangular’ systems that can be solved without iteration. In contrast, operator decomposition iterative methods for fully coupled systems involve an iterative solution technique. We construct an error estimate for the numerical approximation error that specifically addresses the propagation of error between iterates and provide a computable estimate for the iteration error arising because of the decomposition of the operator. Finally, we develop an adaptive discretization strategy to systematically reduce the discretization error.Copyright © 2013 John Wiley & Sons, Ltd.     

Error controlled adaptive multiscale XFEM simulation of cracks

The accurate and efficient prediction of the interaction of microcracks with macrocracks has been a challenge for many years. In this paper a discretization error controlled adaptive multiscale technique for the accurate simulation of microstructural effects within a macroscopic component is presented. The simulation of cracks is achieved using the corrected XFEM. The error estimation procedure is based on the well known Zienkiewicz and Zhu method extended to the XFEM for cracks such that physically meaningful stress irregularities and non-smoothnesses are accurately reflected. The incorporation of microstructural features such as microcracks is achieved by means of the multiscale projection method. In this context an error controlled adaptive mesh refinement is performed on the fine scale where microstructural effects may lead to highly complex mechanical behavior. The presented method is applied to a few examples showing its validity and applicability to arbitrary problems within fracture mechanics.