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.     

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.     

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

The paper introduces a methodology to compute strict upper and lower bounds for linear‐functional outputs of the exact solutions of the advection–diffusion–reaction equation. The bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution. The new methodology extends the a posteriori error estimates yielding bounds for the standard Galerkin formulation to be able to obtain bounds for stabilized formulations. This methodology is combined with both hybrid‐flux and flux‐free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the flux‐free technique. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive methods for thermomechanical coupled contact problems

In this paper an adaptive method for the analysis of thermomechanical coupled multi‐body contact problems is presented. The method is applied to non‐linear elastic solids undergoing finite (thermal) deformations. The contact model considers non‐linear pressure‐dependent heat flux as well as frictional heating in the interface. A time–space‐finite element discretization of the governing equations is formulated including unilateral constraints due to contact. A staggered solution algorithm has been constructed that allows an independent spatial discretization of the coupled subproblems. A posteriori projection‐based error estimators, which enforce implicitly the special boundary conditions due to thermal contact, are used to control the spatial discretization as well as the adaptive time stepping. Numerical examples are presented to corroborate the applicability of the adaptive algorithm to the considered problem type. Copyright © 2004 John Wiley & Sons, Ltd.     

A goal‐oriented hp‐adaptive discontinuous Galerkin approach for biharmonic problems

A goal‐oriented algorithm is developed and applied for hp‐adaptive approximations given by the discontinuous Galerkin finite element method for the biharmonic equation. The methodology is based on the dual problem associated with the target functional. We consider three error estimators and analyse their properties as basic tools for the design of the hp‐adaptive algorithm. To improve adaptation, the combination of two different error estimators is used, each one at its best efficiency, to guide the tasks of where and how to adapt the approximation spaces. The performance of the resulting hp‐adaptive schemes is illustrated by numerical experiments for two benchmark problems. Copyright © 2013 John Wiley & Sons, Ltd.     

An Optimal Weight Predictor for Hierarchical Regression Models

We consider the polynomial regression model. In this model a hierarchical structure, or natural ordering, in the parameter space can be assumed.

Maximum likelihood estimators may be found for the parameters of each order model in the hierarchy. We introduce the class of estimators given by weighted combinations of these maximum likelihood estimators under certain restrictions. This class is obtained by considering a Bayes estimator class and contains the subset regression estimator as a special case.

The optimal weights which minimize the predictive mean squared error are obtained exactly, using an alternative method to that of Kanda (1985). The estimated weights which minimize the estimated predictive mean square error in a similar way to the Mallow’s C_p-statistic are also exactly presented and some numerical examples are shown.

     

Goal-Oriented Error Estimates Based on Different FE-Spaces for the Primal and the Dual Problem with Applications to Fracture Mechanics

The objective of this paper is to derive goal-oriented a posteriori error estimators for the error obtained while approximately evaluating the nonlinear J-integral as a fracture criterion in linear elastic fracture mechanics (LEFM) using the finite element method. Such error estimators are based on the well-established technique of solving an auxiliary dual problem. In a straightforward fashion, the solution to the discretized dual problem is sought in the same FE-space as the solution to the original (primal) problem, i.e. on the same mesh, although it merely acts as a weight of the discretization error only. In this paper, we follow the strategy recently proposed by Korotov et al. [J Numer Math 11:33–59, 2003; Comp Lett (in press)] and derive goal-oriented error estimators of the averaging type, where the discrete dual solution is computed on a different mesh than the primal solution. On doing so, the FE-solutions to the primal and the dual problems need to be transferred from one mesh to the other. The necessary algorithms are briefly explained and finally some illustrative numerical examples are presented.     

Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

In this paper we present two types of local error estimators for the primal finite‐element‐method (FEM) by duality arguments. They are first derived from the (explicit) residual error estimation method (REM) and then—as a new contribution—from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post‐processing with local Neumann problems, with applications in elastic problems. For the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses. Furthermore—for better numerical efficiency—the residua are projected energy‐invariant onto reference elements, where the local Neumann problems have to be solved. Comparative examples between REM‐ and PEM‐type local estimators show superior effectivity indices for the latter one. Copyright © 2001 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.     

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 h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 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.     

B-spline goal-oriented error estimators for geometrically nonlinear rods

We consider goal-oriented a posteriori error estimators for the evaluation of the errors on quantities of interest associated with the solution of geometrically nonlinear curved elastic rods. For the numerical solution of these nonlinear one-dimensional problems, we adopt a B-spline based Galerkin method, a particular case of the more general isogeometric analysis. We propose error estimators using higher order “enhanced” solutions, which are based on the concept of enrichment of the original B-spline basis by means of the “pure” k-refinement procedure typical of isogeometric analysis. We provide several numerical examples for linear and nonlinear output functionals, corresponding to the rotation, displacements and strain energy of the rod, and we compare the effectiveness of the proposed error estimators.     

Adaptive component mode synthesis in linear elasticity

Component mode synthesis (CMS) is a classical method for the reduction of large‐scale finite element models in linear elasticity. In this paper we develop a methodology for adaptive refinement of CMS models. The methodology is based on a posteriori error estimates that determine to what degree each CMS subspace influence the error in the reduced solution. We consider a static model problem and prove a posteriori error estimates for the error in a linear goal quantity as well as in the energy and L² norms. Automatic control of the error in the reduced solution is accomplished through an adaptive algorithm that determines suitable dimensions of each CMS subspace. The results are demonstrated in numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

A novel four‐node quadrilateral smoothing element for stress enhancement and error estimation

A four‐node, quadrilateral smoothing element is developed based upon a penalized‐discrete‐least‐squares variational formulation. The smoothing methodology recovers C¹‐continuous stresses, thus enabling effective a posteriori error estimation and automatic adaptive mesh refinement. The element formulation is originated with a five‐node macro‐element configuration consisting of four triangular anisoparametric smoothing elements in a cross‐diagonal pattern. This element pattern enables a convenient closed‐form solution for the degrees of freedom of the interior node, resulting from enforcing explicitly a set of natural edge‐wise penalty constraints. The degree‐of‐freedom reduction scheme leads to a very efficient formulation of a four‐node quadrilateral smoothing element without any compromise in robustness and accuracy of the smoothing analysis. The application examples include stress recovery and error estimation in adaptive mesh refinement solutions for an elasticity problem and an aerospace structural component. Copyright © 1999 John Wiley & Sons, Ltd.     

Generalization of robustness test procedure for error estimators. Part I: formulation for patches near kinked boundaries

In this part of paper we shall extend the formulation proposed by Babu?ka and co‐workers for robustness patch test, for quality assessment of error estimators, to more general cases of patch locations especially in three‐dimensional problems. This is performed first by finding an asymptotic finite element solution at interior parts of a problem with assumed smooth exact solution and then adding a correction part to obtain the solution near a kinked boundary irrespective of other boundary conditions at far ends of the domain. It has been shown that the solution corresponding to the correction part may be obtained in a spectral form by assuming a suitable proportionality relation between the nodal values of a mesh with repeatable pattern of macro‐patches. Having found the asymptotic finite element solution, the performance of error estimators may be examined. Although in this paper we focus on the asymptotic behaviour of error estimators, the method described in this part may be used to obtain finite element solution for two/three‐dimensional unbounded heat/elasticity problems with homogeneous differential equations. Some numerical results are presented to show the validity and performance of the proposed method. Copyright © 2005 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.     

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.     

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.     

Superconvergence recovery technique and a posteriori error estimators

A new superconvergence recovery technique for finite element solutions is presented and discussed for one dimensional problems. By using the recovery technique a posteriori error estimators in both energy norm and maximum norm are presented for finite elements of any order. The relation between the postprocessing and residual types of energy norm error estimators has also been demonstrated.