New bounding techniques for goal‐oriented error estimation applied to linear problems

The paper deals with the accuracy of guaranteed error bounds on outputs of interest computed from approximate methods such as the finite element method. A considerable improvement is introduced for linear problems, thanks to new bounding techniques based on Saint‐Venant's principle. The main breakthrough of these optimized bounding techniques is the use of properties of homothetic domains that enables to cleverly derive guaranteed and accurate bounding of contributions to the global error estimate over a local region of the domain. Performances of these techniques are illustrated through several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

A combined approach for goal‐oriented error estimation and adaptivity using operator‐customized finite element wavelets

We describe how wavelets constructed out of finite element interpolation functions provide a simple and convenient mechanism for both goal‐oriented error estimation and adaptivity in finite element analysis. This is done by posing an adaptive refinement problem as one of compactly representing a signal (the solution to the governing partial differential equation) in a multiresolution basis. To compress the solution in an efficient manner, we first approximately compute the details to be added to the solution on a coarse mesh in order to obtain the solution on a finer mesh (the estimation step) and then compute exactly the coefficients corresponding to only those basis functions contributing significantly to a functional of interest (the adaptation step). In this sense, therefore, the proposed approach is unified, since unlike many contemporary error estimation and adaptive refinement methods, the basis functions used for error estimation are the same as those used for adaptive refinement. We illustrate the application of the proposed technique for goal‐oriented error estimation and adaptivity for second and fourth‐order linear, elliptic PDEs and demonstrate its advantages over existing methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Influence functions and goal‐oriented error estimation for finite element analysis of shell structures

In this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal‐oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over‐estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal‐oriented error estimation procedure shows good effectivity indices. Copyright © 2005 John Wiley & Sons, Ltd.     

Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

This paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non‐linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non‐linear quantities of interest by means of projection properties in order to take into account higher‐order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two‐dimensional applications in the context of elastic fatigue problems. Copyright © 2010 John Wiley & Sons, Ltd.     

A constitutive relation error estimator based on traction‐free recovery of the equilibrated stress

A new methodology for recovering equilibrated stress fields is presented, which is based on traction‐free subdomains' computations. It allows a rather simple implementation in a standard finite element code compared with the standard technique for recovering equilibrated tractions. These equilibrated stresses are used to compute a constitutive relation error estimator for a finite element model in 2D linear elasticity. A lower bound and an upper bound for the discretization error are derived from the error in the constitutive relation. These bounds in the discretization error are used to build lower and upper bounds for local quantities of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

Enhanced goal‐oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

This work focuses on providing accurate low‐cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte‐Carlo method for multi‐dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal‐oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments. Copyright © 2016 John Wiley & Sons, Ltd.     

Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are strict in the sense that they refer to the difference between the displacement computed on the FE mesh and the exact displacement, solution of the continuous equations, rather than to the difference between the displacements computed on two FE meshes, one coarse and one refined. This method is based on the resolution of a series of local problems on patches of elements and does not require the resolution of a previous problem of flux equilibration, as happens with other methods. The paper concentrates more specifically on linear solid mechanics issues, and on the assessment of the energy norm of the error, seen as a necessary tool for the estimation of the error in arbitrary quantities of interest (linear functional outputs). Applications in both 2D and 3D are presented.     

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.     

Modal‐based goal‐oriented error assessment for timeline‐dependent quantities in transient dynamics

This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities of interest (referred as timeline‐dependent quantities) is proposed. These quantities are scalar time‐dependent outputs of the transient solution, which are better suited to time‐dependent problems than the standard scalar ones, frozen in time. The proposed methodology furnishes error estimates for both the standard scalar and the new timeline‐dependent quantities of interest. The key ingredient is the modal‐based approximation of the associated adjoint problems, which allows efficiently computing and storing the adjoint solution. The approximated adjoint solution is readily post‐processed to produce an enhanced solution, requiring only one spatial post‐process for each vibration mode and using the time‐harmonic hypothesis to recover the time dependence. Thus, the proposed goal‐oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem. The resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem. Thus, the cost of the error estimate at each time step is very low. Copyright © 2013 John Wiley & Sons, Ltd.     

Strategies for computing goal‐oriented a posteriori error measures in non‐linear elasticity

We investigate the characteristics and performance of goal‐oriented a posteriori error measures for a class of non‐linear elasticity models, while restriction is made to small strain theory. The chosen error measure of the displacement field can be global or local (probing the chosen quantity in a specific spatial point). The error is computable with the aid of the solution of a dual problem whose data depend on the error measure. The main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution. The chosen strategies are compared in terms of accuracy, ease of implementation, reliability and cost‐efficiency. A well‐known numerical example, the Cook's membrane, is used for the numerical evaluations. Copyright © 2002 John Wiley & Sons, Ltd.     

Guaranteed energy error bounds for the Poisson equation using a flux‐free approach: Solving the local problems in subdomains

A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain‐based a posteriori error estimate which yields self‐equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux‐free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds. Copyright © 2009 John Wiley & Sons, Ltd.     

Practical methods for a posteriori error estimation in engineering applications

This work presents an extension of the goal‐oriented error estimation technique to the engineering analysis of three‐dimensional linear elastic bodies. In the series of examples shown, the errors are estimated with respect to local displacement and stress components. The paper also introduces novel means to compute lower bounds on the error in the energy norm based on a cost‐effective postprocessing of the upper bound error estimates. The numerical results indicate that the method can be used effectively for complex engineering applications. Copyright © 2003 John Wiley & Sons, Ltd.     

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.     

Dual‐based adaptive FEM for inelastic problems with standard FE implementations

Goal‐oriented error estimation allows to refine meshes in space and time with respect to arbitrary quantities. The required dual problems that need to be solved usually require weak formulations and the Galerkin method in space and time to be established. Unfortunately, this does not obviously leads to structures of standard finite element implementations for solid mechanics. These are characterized by a combination of variables at nodes (e.g. displacements) and at integration points (e.g. internal variables) and are solved with a two‐level Newton method because of local uncoupled and global coupled equations. Therefore, we propose an approach to approximate the dual problem while maintaining these structures. The primal and the dual problems are derived from a multifield formulation. Discretization in time and space with appropriate shape functions and rearrangement yields the desired result. Details on practical implementation as well as applications to elasto‐plasticity are given. Numerical examples demonstrate the effectiveness of the procedure. Copyright © 2015 John Wiley & Sons, Ltd.     

Monitoring a PGD solver for parametric power flow problems with goal‐oriented error assessment

The parametric analysis of electric grids requires carrying out a large number of power flow computations. The different parameters describe loading conditions and grid properties. In this framework, the proper generalized decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to (1) iterating algebraic solver, (2) number of terms in the separable greedy expansion, and (3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal‐oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end‐user. The paper discusses how to compute the goal‐oriented error estimates. This requires linearizing the error equation and the quantity of interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Error bounds for the reliability index in finite element reliability analysis

This work presents an extension of the goal‐oriented error estimation techniques to the reliability analysis of a linear elastic structure. We use a first‐order reliability method in conjunction with a finite element analysis (FEA) to compute the failure probability of the structure. In such a situation the output of interest that is computed from the FEA is the reliability index β. The accuracy of this output, and thus of the reliability analysis, depends, in particular, on the accuracy of the FEA. In this paper, upper and lower bounds of the reliability index are proposed, as well as simple bounds of the failure probability. An application to linear fracture mechanics is presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Error estimation of stress intensity factors for mixed‐mode cracks

This paper presents an a posteriori error estimator for mixed‐mode stress intensity factors in plane linear elasticity. A surface integral over an arbitrary crown is used for the separate calculation of the combined mode's stress intensity factors. The error in the quantity of interest is based on goal‐oriented error measures and estimated through an error in the constitutive relation. Copyright © 2006 John Wiley & Sons, Ltd.     

A goal‐oriented hp‐adaptive finite element method with electromagnetic applications. Part I: electrostatics

We describe the development and application of a finite element (FE) self‐adaptive hp goal‐oriented algorithm for elliptic problems. The algorithm delivers (without any user interaction) a sequence of optimal hp‐grids. This sequence of grids minimizes the error of a prescribed quantity of interest with respect to the problem size. The refinement strategy is an extension of a fully automatic, energy‐norm based, hp‐adaptive algorithm. We illustrate the efficiency of the method with 2D numerical results. Among other problems, we apply the goal‐oriented hp‐adaptive strategy to simulate direct current (DC) resistivity logging instruments (including through casing resistivity tools) in a borehole environment and for the assessment of rock formation properties. Copyright © 2005 John Wiley & Sons, Ltd.     

Error estimation and model adaptation for a stochastic‐deterministic coupling method based on the Arlequin framework

The paper deals with the issue of accuracy for multiscale methods applied to solve stochastic problems. It more precisely focuses on the control of a coupling, performed using the Arlequin framework, between a deterministic continuum model and a stochastic continuum one. By using residual‐type estimates and adjoint‐based techniques, a strategy for goal‐oriented error estimation is presented for this coupling and contributions of various error sources (modeling, space discretization, and Monte Carlo approximation) are assessed. Furthermore, an adaptive strategy is proposed to enhance the quality of outputs of interest obtained by the coupled stochastic‐deterministic model. Performance of the proposed approach is illustrated on 1D and 2D numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Strict lower bounds with separation of sources of error in non‐overlapping domain decomposition methods

This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element and domain decomposition methods. In addition to a fully parallel computation, the proposed lower bounds separate the algebraic error (due to the use of a domain decomposition iterative solver) from the discretization error (due to the finite element), which enables the steering of the iterative solver by the discretization error. These lower bounds are also used to improve the goal‐oriented error estimation in a substructured context. Assessments on 2D static linear mechanic problems illustrate the relevance of the separation of sources of error and the lower bounds' independence from the substructuring. We also steer the iterative solver by an objective of precision on a quantity of interest. This strategy consists in a sequence of solvings and takes advantage of adaptive remeshing and recycling of search directions. Copyright © 2016 John Wiley & Sons, Ltd.