An automatic adaptive refinement finite element procedure for 2D elastostatic analysis

An automatic adaptive refinement procedure for finite element analysis is presented. The procedure is applied to two-dimensional elastostatic problems to obtain solutions of prescribed accuracy. Through the combined use of new mesh generator using contour developed by Lo¹ and the concept of strain energy concentration, high-quality graded finite element meshes are generated. The whole process is fully automatic and no user intervention is required during the successive cycles of the mesh refinements. The Zienkiewicz and Zhu² error estimator is found to be effective and has been adopted for the present implementation. In the numerical examples tested, the error estimator gives an accurate error norm estimation and the effectivity index of the estimator converges to a value close to unity.     

Error estimates and adaptive refinement for plate bending problems

The Zienkiewicz–Zhu error estimator is shown to be effective in problems of plate flexure. When used in conjunction with triangular elements and an adaptive mesh generator allowing a prescribed size of elements to be developed, very fast adaptive convergence for results of specified accuracy is achieved.     

Application of new error estimators based on gradient recovery and external domain approaches to 2D elastostatics problems

Two new error estimators for the BEM in 2D potential problems were recently presented by the authors. This work extends these two error estimators for 2D elastostatics problems. The first approach involves a local error estimator based on a gradient recovery procedure in which the error function is based on differences between smoothed and non-smoothed rates of change of boundary variables in the local tangential direction. The second approach is associated with the external problem formulation and gives both local and global measures of the error, depending on a choice of the external evaluation point. These approaches are post-processing procedures. Both estimators show consistency with mesh refinement and give similar qualitative results. The error estimator using the gradient recovery approach presents a more general characteristic as its formulation does not rely on an ‘optimal’ choice of an external parameter, such as in the case of the external domain error estimator. Also, the external domain error estimator can be used only for domains in which an exterior region exists. For example, the external domain error estimator cannot be used for an infinite domain with a crack, because a point in the exterior region (inside the crack) will not be at a finite distance to the crack surface.     

Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions

Adaptive local refinement is one of the main issues for isogeometric analysis (IGA). In this paper, an adaptive extended IGA (XIGA) approach based on polynomial splines over hierarchical T‐meshes (PHT‐splines) for modeling crack propagation is presented. The PHT‐splines overcome certain limitations of nonuniform rational B‐splines–based formulations; in particular, they make local refinements feasible. To drive the adaptive mesh refinement, we present a recovery‐based error estimator for the proposed method. The method is based on the XIGA method, in which discontinuous enrichment functions are added to the IGA approximation and this method does not require remeshing as the cracks grow. In addition, crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors under linear elastic fracture mechanics. The proposed method has been used to analyze numerical examples, and the stress intensity factors results were compared with reference results. The findings demonstrate the accuracy and efficiency of the proposed method.     

New approaches for error estimation and adaptivity for 2D potential boundary element methods

This work presents two new error estimation approaches for the BEM applied to 2D potential problems. The first approach involves a local error estimator based on a gradient recovery procedure in which the error function is generated from differences between smoothed and non‐smoothed rates of change of boundary variables in the local tangential direction. The second approach involves the external problem formulation and gives both local and global measures of error, depending on a choice of the external evaluation point. These approaches are post‐processing procedures. Both estimators show consistency with mesh refinement and give similar qualitative results. The error estimator using the gradient recovery approach is more general, as this formulation does not rely on an ‘optimal’ choice of an external parameter. This work presents also the use of a local error estimator in an adaptive mesh refinement procedure. This r‐refinement approach is based on the minimization of the standard deviation of the local error estimate. A non‐linear programming procedure using a feasible‐point method is employed using Lagrange multipliers and a set of active constraints. The optimization procedure produces finer meshes close to a singularity and results that are consistent with the problem physics. Copyright © 2002 John Wiley & Sons, Ltd.     

High-order hierarchical A- and L-stable integration methods

Numerical integration of stiff first-order systems of differential equations is considered. It is shown how a C⁰-continuous polynomial time discretization, in conjunction with a weighted residual method, can be used to derive methods corresponding to the diagonal and first subdiagonal Padé approximants. In this manner A -stable schemes of order 2k and L-stable schemes of order 2k - l are obtained, at least for k ? 4. The methods are hierarchical in the sense that a scheme with a given accuracy embraces the equations of all lower-order methods that have the same stability type. We show how this feature may be utilized to perform partial refinements in the integration process, and how an error estimation by an embedding approach naturally follows. An adaptive algorithm based on the error estimator is suggested. Some numerical experiments that illustrate the ideas are included.     

Automatic local refinement for irregular rectangular meshes

The use of local mesh refinements for the generation of meshes for the finite element or finite difference methods is studied. A class of rectangular meshes which admit restricted local refinements, referred to as irregular rectangular meshes, is introduced and its representation discussed. Properties of algorithms for mesh refinements are discussed from the viewpoints of termination with a mesh in the specified class, memory utilization, symmetry and fragmentation of the mesh.     

ERROR ESTIMATION AND ADAPTIVITY IN ELASTOPLASTICITY

In this paper, a method is developed to control the parameters of a finite element computation for time-dependent material models. This method allows the user to obtain a prescribed accuracy with a computational cost as low as possible. To evaluate discretization errors, we use a global error measure in constitutive relation based on Drucker's inequality. This error includes, over the studied time interval, the error of the finite element model and the error of the algorithm being used. In order to master the size of the elements of the mesh and the length of the time increments, an error estimator, which permits estimating the errors due to the time discretization, is proposed. These tools are used to elaborate two procedures of adaptivity. Various examples for monotonous or non-monotonous loadings, for 2-D or axisymmetric problems, show the reliability of these procedures.     

Error estimation and mesh adaptivity for the virtual element method based on recovery by compatibility in patches

In this article, a recovery by compatibility in patches (RCP)-based a posteriori error estimator is proposed for the virtual element method (VEM), and it is utilized to drive adaptive mesh refinement processes in two-dimensional elasticity problems. In RCP, recovered stresses are obtained by minimizing the complementary energy of patches of elements over a set of assumed equilibrated stress modes. To this aim, the explicit knowledge of displacements is only needed along the patch boundaries and no knowledge of superconvergent points is required, so making the RCP naturally suitable for the VEM. The a posteriori error estimation is conducted by comparing the stress field of a standard displacement-based VEM solution and the stress field obtained through RCP. The procedure is simple, and it does not require ad hoc modifications for small patches. The capability of this RCP-based error estimator to drive adaptive mesh refinements is successfully demonstrated through various numerical examples.     

A posteriori error estimation in non‐linear FE analyses of shell structures

During the last decade, significant scientific efforts were made in the area of quality assurance of numerical results obtained by means of the finite element method (FEM). These efforts were based on adaptive remeshing controlled by an estimated error. This paper reports on the extension of error estimation to non‐linear shell analysis involving strain‐hardening and softening plasticity. In the context of incremental‐iterative analyses, an incremental error estimator is introduced. It is based on the rate of work. The stress recovery technique proposed by Zienkiewicz and Zhu (Int. J. Numer. Meth. Engng 1992; 33 :1331) is modified to allow for discontinuities of certain stress components in case of localization arising from, e.g. cracking of concrete. The developed error estimator is part of a calculation scheme for adaptive non‐linear FE analysis. If the estimated error exceeds a prespecified threshold value in the course of an adaptive analysis, a new mesh is generated. After mesh refinement the state variables are transferred from the old to the new mesh and the calculation is restarted at the load level which was attained with the old mesh. The performance of the proposed error estimator is demonstrated by means of adaptive calculations of a reinforced concrete (RC) cooling tower. The influence of the user‐prescribed error threshold on the numerical results is investigated. Copyright © 2002 John Wiley & Sons, Ltd.     

An automatic adaptive refinement procedure using triangular and quadrilateral meshes

An automatic adaptive refinement procedure for finite element analysis for two-dimensional stress analysis problems is presented. Through the combined use of the new mesh generator developed by the authors (to appear) for adaptive mesh generation and the Zienkiewicz-Zhu [Int. J. numer. Meth. Engng31, 1331–1382 (1992)] error estimator based on the superconvergent patch recovery technique, an adaptive refinement procedure can be formulated which can achieve the aimed accuracy very economically in one or two refinement steps. A simple method is also proposed to locate the existence and the position of singularities in the problem domain. Hence, little or no a priori knowledge about the location and strength of the singularities is required. The entire adaptive refinement procedure has been made fully automatic and no user intervention during successive cycles of mesh refinements is needed. The robustness and reliability of the refinement procedure have been tested by solving difficult practical problems involving complex domain geometry with many singularities. We found that in all the examples studied, regardless of the types of meshes employed, triangular and quadrilateral meshes, nearly optimal overall convergence rate is always achieved.     

Error estimation for boundary element method

In this paper, six error indicators obtained from dual boundary integral equations are used for local estimation, which is an essential ingredient for all adaptive mesh schemes in BEM. Computational experiments are carried out for the two-dimensional Laplace equation. The curves of all these six error estimators are in good agreement with the shape of the error curve. The results show that the adaptive mesh based on any one of these six error indicators converges faster than does equal mesh discretization.     

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.     

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 simple error estimator and adaptive procedure for practical engineerng analysis

A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes. The estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials. When combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable. Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.     

An error estimator for adaptive mesh refinement analysis based on strain energy equalisation

The two most widely used error estimators for adaptive mesh refinement are discussed and developed in the context of non-linear elliptic problems. The first is based on the work of Babuska and Rheinboldt (1978) where the error norm is a function of the residual and the inter-element discontinuity of the stress field. The discontinuous stress field arises in the Finite Element formulation where C ₀ continuity of the velocity field is assumed. The second error estimator is based on the work of Zienkiewicz and Zhu (1987). This method also uses the discontinuous stress field to measure the error, but results in a more simplified expression for the error norm. In fact, the equivalence between the two error norms has been shown by Zienkiewicz. Finally, an error estimator which is based on the approximation velocity space only is proposed. This estimator has the advantage in that it does not require the a posteriori calculation of the pressure (or stress) field. The method is applied to non-Newtonian Stokes flow which has a similar formulation to non-linear elasticity problems.     

Using mesh adaption for the identification of a spatial field of material properties

It is well known that the solution of an inverse problem is ill‐posed and not unique. To avoid difficulties caused by this, when solving such a problem, Tikhonov's regularization terms are usually added to the norm quantifying the discrepancy between the model's predictions and experimental data. This regularization term however is often inadequate to perform the identification of a field of material properties that varies spatially. This is all the more difficult when dealing with the numerical solution of this inverse problem, for the sought field is spatially discretized and this discretization can influence the result of the identification. We will here examine an overall strategy using classical adaptive meshing methods used to circumvent these drawbacks. The first step consists of using two distinct meshes: one associated with the discretization of the sought spatial field and the other associated with the solution of the mechanical problems (forward and adjoint states). In the second step, we will introduce local error estimators that allow an oriented refinement of the mesh associated with the sought parameters. This general strategy is applied to a practical case study: the detection of underground cavities using experimental data obtained by an interferometric device on a satellite. We will then address the question of how the regularization terms and the error estimator driving the mesh refinement were selected. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive finite element methods for a class of evolution problems in viscoplasticity

Adaptive finite element procedures are presented for the analysis of broad classes of two-dimensional problems in viscoplasticity which involve internal state variables. Several a posteriori error estimates are developed and used as a basis for refinements of meshes of triangular and quadrilateral elements. Algorithms are presented which reduce the computational complexity of the adaptive process and provide for the use of arbitrary local estimates of the approximation error. The results of numerical experiments are given which illustrate the effectiveness of the adaptive algorithms and various error indicators.     

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.     

Statistics of Nadaraya-Watson estimator errors in surrogate-based optimization

For the optimization under uncertainty problem, there has been recent interest in coupling trust-region methods with surrogate surfaces or function approximations. There are many theoretical and statistical issues that must be carefully considered in following such an approach. Herein, the Nadaraya-Watson estimator is used for the smooth function approximation, and the effects of observation noise and random sampling on estimator error are examined. For the fundamental optimization problem where the exact function is quadratic, analytical results are derived for the mean-square error of the difference and gradient of the function. It is also shown how these statistics are related to the trust-region method, how the analytical results can be used to determine the bandwidth of the kernel of the estimator, and how third-order terms can affect the error statistics.