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.     

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.     

A methodology for the formulation of error estimators for time integration in linear solid and structural dynamics

In this article, we present a novel methodology for the formulation of a posteriori error estimators applicable to time‐stepping algorithms of the type commonly employed in solid and structural mechanics. The estimators constructed with the presented methodology are accurate and can be implemented very efficiently. More importantly, they provide reliable error estimations even in non‐smooth problems where many standard estimators fail to capture the order of magnitude of the error. The proposed methodology is applied, as an illustrative example, to construct an error estimator for the Newmark method. Numerical examples of its performance and comparison with existing error estimators are presented. These examples verify the good accuracy and robustness predicted by the analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

A priori error estimator of the generalized‐α method for structural dynamics

An a priori error estimator for the generalized‐α time‐integration method is developed to solve structural dynamic problems efficiently. Since the proposed error estimator is computed with only information in the previous and current time‐steps, the time‐step size can be adaptively selected without a feedback process, which is required in most conventional a posteriori error estimators. This paper shows that the automatic time‐stepping algorithm using the a priori estimator performs more efficient time integration, when compared to algorithms using an a posteriori estimator. In particular, the proposed error estimator can be usefully applied to large‐scale structural dynamic problems, because it is helpful to save computation time. To verify efficiency of the algorithm, several examples are numerically investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

Local polynomial regression smoothers with AR-error structure

Consider the fixed regression model with random observation error that follows an AR(1) correlation structure. In this paper, we study the nonparametric estimation of the regression function and its derivatives using a modified version of estimators obtained by weighted local polynomial fitting. The asymptotic properties of the proposed estimators are studied: expressions for the bias and the variance/covariance matrix of the estimators are obtained and the joint asymptotic normality is established. In a simulation study, a better behavior of the Mean Integrated Squared Error of the proposed regression estimator with respect to that of the classical local polynomial estimator is observed when the correlation of the observations is large. This work has been partially supported by grants PB98-0182-C02-01, PGIDT01PXI10505PR and MCyT Grant BFM2002-00265 (European FEDER support included).     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity

We obtain fully computable a posteriori error estimators for the energy norm of the error in second‐order conforming and nonconforming finite element approximations in planar elasticity. These estimators are completely free of unknown constants and give a guaranteed numerical upper bound on the norm of the error. The estimators are shown to also provide local lower bounds, up to a constant and higher‐order data oscillation terms. Numerical examples are presented illustrating the theory and confirming the effectiveness of the estimator. Copyright © 2009 John Wiley & Sons, Ltd.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

Validation of a posteriori error estimators by numerical approach

A numerical methodology which determines the quality (or robustness) of a posteriori error estimators is described. The methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems, namely, the local geometry of the grid and the structure of the solution. The methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations.     

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.     

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.     

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

This paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)‐scale problems. The local problems focus on the resolution of fine‐scale features of the solution in the vicinity of three‐dimensional cracks, while the global problem addresses the macro‐scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp‐GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three‐dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three‐dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented. Copyright © 2009 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.     

Partition of unity for localization in implicit a posteriori finite element error control for linear elasticity

The partition of unity for localization in adaptive finite element method (FEM) for elliptic partial differential equations has been proposed in Carstensen and Funken (SIAM J. Sci. Comput. 2000; 21 : 1465–1484) and is applied therein to the Laplace problem. A direct adaptation to linear elasticity in this paper yields a first estimator η_L based on patch‐oriented local‐weighted interface problems. The global Korn inequality with a constant C_Korn yields reliability for any finite element approximation u_h to the exact displacement u. In order to localize this inequality further and so to involve the global constant C_Korn directly in the local computations, we deduce a new error estimator µ_L. The latter estimator is based on local‐weighted interface problems with rigid body motions (RBM) as a kernel and so leads to effective estimates only if RBM are included in the local FE test functions. Therefore, the excluded first‐order FEM has to be enlarged by RBM, which leads to a partition of unit method (PUM) with RBM, called P₁+RBM or to second‐order FEMs, called P₂ FEM. For P₁+RBM and P₂ FEM (or even higher‐order schemes) one obtains the sharper reliability estimate . Efficiency holds in the strict sense of . The local‐weighted interface problems behind the implicit error estimators η_L and µ_L are usually not exactly solvable and are rather approximated by some FEM on a refined mesh and/or with a higher‐order FEM. The computable approximations are shown to be reliable in the sense of . The oscillations are known functions of the given data and higher‐order terms if the data are smooth for first‐order FEM. The mathematical proofs are based on weighted Korn inequalities and inverse estimates combined with standard arguments. The numerical experiments for uniform and adapted FEM on benchmarks such as an L‐shape problem, Cook's membrane, or a slit problem validate the theoretical estimates and also concern numerical bounds for C_Korn and the locking phenomena. Copyright © 2007 John Wiley & Sons, Ltd.     

Closed form stiffness matrices and error estimators for plane hierarchic triangular elements

Closed form expressions for the stiffness matrix and a simple error estimator and error indicator are derived for plane straight sided triangular finite elements in elasticity problems. The calculation of the error estimator is performed on an element by element basis, and is found to be very accurate and efficient. In general, the solutions for benchmark problems using the error indicators for selective refinement of the regions show accelerated convergence when compared to the convergence rate of solutions using uniform mesh refinement. Evaluation of the stiffness matrices and error estimators using explicit formulations is found to be several times faster than numerical integration.     

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.     

A pollution controlled hybrid interior error estimator for linear elastostatics

A hybrid error estimator using a priori interior region estimates in an a posteriori framework is presented for linear elastostatics problems in FEA. It is shown that local rates of convergence are augmented by this technique and global rates are not adversely affected. The effects of pollution for this estimator are explained and a pollution error estimator is derived using the concept of error loads. It is shown that pollution error estimation can improve the performance of both the conventional a posteriori and the hybrid techniques. A series of numerical results are presented which demonstrate the superior performance of the proposed method over previously published interior error estimation techniques. © 1998 John Wiley & Sons, Ltd.     

Edge stabilization and consistent tying of constituents at Neumann boundaries in multi‐constituent mixture models

A mixture‐theory‐based model for multi‐constituent solids is presented where each constituent is governed by its own balance laws and constitutive equations. Interactive forces between constituents that emanate from maximization of entropy production inequality provide the coupling between constituent‐specific balance laws and constitutive models. The deformation of multi‐constituent mixtures at the Neumann boundaries requires imposing inter‐constituent coupling constraints such that the constituents deform in a self‐consistent fashion. A set of boundary conditions is presented that accounts for the non‐zero applied tractions, and a variationally consistent method is developed to enforce inter‐constituent constraints at Neumann boundaries in the finite deformation context. The new method finds roots in a local multiscale decomposition of the deformation map at the Neumann boundary. Locally satisfying the Lagrange multiplier field and subsequent modeling of the fine scales via edge bubble functions result in closed‐form expressions for a generalized penalty tensor and a weighted numerical flux that are free from tunable parameters. The key novelty is that the consistently derived constituent coupling parameters evolve with material and geometric nonlinearity, thereby resulting in optimal enforcement of inter‐constituent constraints. Various benchmark problems are presented to validate the method and show its range of application. Copyright © 2016 John Wiley & Sons, Ltd.     

A new exact,closed‐form a priori global error estimator for second‐ and higher‐order time‐integration methods for linear elastodynamics

In our recent papers, we suggested a new two‐stage time‐integration procedure for linear elastodynamics problems and showed that for long‐term integration, time‐integration methods with zero numerical dissipation are very effective for all linear elastodynamics problems, including structural dynamics, wave propagation and impact problems. In this paper, we have derived a new exact, closed‐form a priori global error estimator for time integration of linear elastodynamics by the trapezoidal rule and the high‐order time continuous Galerkin (TCG) methods with zero numerical dissipation (these methods correspond to the diagonal of the Padé approximation table). The new a priori global error estimator allows the selection of the size (the number) of time increments for the indicated time‐integration methods at the prescribed accuracy as well as the comparison of the effectiveness of the second‐ and high‐order TCG methods at different observation times. A numerical example shows a good agreement between theoretical and numerical results. Copyright © 2011 John Wiley & Sons, Ltd.     

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.