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Error estimation in quantities of interest by locally. equilibrated superconvergent patch recovery,O A Gonza lez Estrada1 E Nadal 2 J J Ro denas2. P Kerfriden1 S P A Bordas 1 F J Fuenmayor 2,November 24 2012. 1 Institute, of Mechanics and Advanced Materials IMAM Cardiff School of Engineering. Cardiff University Queen s Buildings The Parade Cardiff CF24 3AA Wales UK. e mail estradaoag kerfridenp cardiff ac uk stephane bordas alum northwestern edu. 2 Centro de Investigacio n de Tecnolog a de Veh culos CITV. Universitat Polite cnica de Vale ncia E 46022 Valencia Spain. e mail jjrodena mcm upv es ennaso upv es ffuenmay mcm upv es. Goal oriented error estimates GOEE have become popular tools to quan. tify and control the local error in quantities of interest QoI which are often. more pertinent than local errors in energy for design purposes e g the mean. stress or mean displacement in a particular area the stress intensity factor. for fracture problems These GOEE are one of the key unsolved problems. of advanced engineering firms in the aerospace industry Residual based error. estimators can be used to estimate errors in quantities of interest for finite. element approximations This work presents a recovery based error estima. tion technique for QoIs whose main characteristic is the use of an enhanced. version of the Superconvergent Patch Recovery SPR technique previously. used for error estimation in the energy norm This enhanced SPR technique is. used to recover both the primal and dual solutions It provides a nearly stat. ically admissible stress field that results in accurate estimations of the local. contributions to the discretization error in the QoI and therefore in an ac. curate estimation of this magnitude This approach leads to a technique with. reasonable computational cost that could be easily implemented into already. available finite element codes, KEY WORDS goal oriented error estimation recovery quantities of interest error con. trol mesh adaptivity,1 Introduction, Verification and quality assessment of numerical simulations is a critical area of.

research Techniques to control the error in numerical simulations and verify the. approximate solutions have been a subject of concern for many years Further. development is expected to have a profound impact on the reliability and utility of. simulation methods Effective methods for assessing the global discretization error. in energy have been extensively developed since the late 70s However controlling. the error in quantities of interest to engineers e g the average stress or average. displacement in a given region or the stress intensity factors is not equivalent to. controlling the error in energy Yet the widely stated need of practicing engineers. is to evaluate and minimise the discretisation error in these quantities of interest. Less attention however has been given to the emerging more advanced goal. oriented error estimates This paper is a step in this direction which provides a. simple procedure to evaluate engineeringly relevant quantities of interest. Energy based error estimates can be classified into different families 1 2 residual. based error estimators recovery based error estimators dual techniques etc Residual. based error estimators originally introduced by Babus ka and Rheinboldt 3 are char. acterised by a strong mathematical basis To estimate the error they consider local. residuals of the numerical solution By investigating the residuals occurring in a. patch of elements or even in a single element it is possible to estimate the errors. which arise locally These methods were improved with the introduction of the resid. ual equilibration by Ainsworth and Oden 1 Recovery based error estimates were. first introduced by Zienkiewicz and Zhu 4 and are often preferred by practitioners. because they are robust simple to use and provide an enhanced solution which. may be used as output Further improvements were made with the introduction. of new recovery procedures such as the superconvergent patch recovery SPR pro. posed by the same authors 5 6 and many papers following 7 8 9 10 11 12. Recovery techniques were extended to enriched approximations in 13 14 15 16. and to Hellinger Reissner smoothing based finite elements in 17 and the role of. enrichment and statical admissibility in such recovery procedures discussed in 18. Duality based techniques rely on the evaluation of two different fields one compat. ible in displacements and another equilibrated in stresses 19. Most techniques used to obtain error estimates prior to the mid 90s were aimed. to evaluate the global error in the energy norm Nevertheless as discussed above. the goal of numerical simulations is generally not the determination of the strain. energy alone but the reliable prediction of a particular quantity of interest QoI. which is needed for making decisions during the design process Therefore it is. important to guarantee the quality of such analyses by controlling the error of the. approximation in terms of the QoI rather than the global energy norm Significant. advances in the late 90s introduced a new approach focused on the evaluation of. error estimates of local quantities 1 20 21 In order to obtain an error estimate for. the QoI two different problems are solved the primal problem which is the problem. under consideration and the dual or adjoint problem which is related to the QoI. The adjoint problem is the same as the primal problem except for the loads and. boundary conditions which are determined as a function of the QoI Goal oriented. error estimators have been usually developed from the basis of residual formulations. and the widely used strategy of solving the dual problem 1 22 Although limited. the use of recovery techniques to evaluate the error in quantities of interest can be. found in 23 24 and considering dual analysis in 25. In this paper we propose to extend the recovery techniques presented in 12 26. 15 27 to evaluate accurate error estimates for linear QoIs The recovery technique. requires the analytical expressions of the body loads and boundary tractions which. must be evaluated also for the dual problem in order to consider equilibrium con. ditions In Section 2 we define the problem and the finite element approximation. used Section 3 focuses on the representation of the error in the QoIs and the solu. tion of the dual problem Section 4 deals with the definitions of the dual problem. and describes the expressions of body loads and boundary tractions required for the. stress recovery of the dual problem for different linear QoI In Section 5 we introduce. the nearly equilibrated recovery technique used to obtain enhanced stress fields for. both the primal and dual solutions In Section 6 we present some numerical results. and finally conclusions are drawn in Section 7,2 Problem statement and solution. In this section we briefly present the model for the 2D linear elasticity problem. Denote in vectorial form and as the stresses and strains D as the elasticity. matrix of the constitutive relation D and the unknown displacement field u. which values in R2 u is the solution of the boundary value problem given by. u n t on N 2,u u on D 3, where N and D denote the Neumann and Dirichlet boundaries with. N D and N D b are body loads and t are the tractions imposed. Consider the initial stresses 0 and strains 0 the symmetric bilinear form a. V u V R and the continuous linear form V R are defined by. u v d T u D 1 v d 4,v vT bd vT td T v 0 d T v 0 d 5. With these notations the variational form of the problem reads 28. Find u V u v V a u v v 6, where V is the standard test space for the elasticity problem such that V v v. H 1 2 v D x 0, Let uh be a finite element approximation of u The solution for the discrete.

counterpart of the variational problem in 6 lies in a subspace V h u V u. associated with a mesh of finite elements of characteristic size h and it is such that. vh V h V a uh vh vh 7,3 Error control,3 1 Error representation. In this section we define a general framework for goal oriented error analysis First. we assume that the finite element discretization error is given by e u uh To. quantify the quality of uh in terms of e different p methods have been proposed. generally based on the energy norm kek induced by a e e and written as. where and h represent the exact and finite element stress fields respectively. Following Zienkiewicz Zhu 4 an estimate of the error kees k can be formulated in. the context of FE elasticity problems by introducing the approximation. kek kees k 9, where represents the recovered stress field which is a better approximation to. the exact solution than h Similarly local element contributions can be evaluated. from 9 considering the domain e of element e, The error estimate measured in the energy norm as given in 9 can overestimate. or underestimate the exact error although it is asymptotically exact1 provided that. has a higher convergence rate than the FE solution In order to improve the. quality of our recovery based estimate it is useful to include the error bounding. property Recently D ez et al 26 and Ro denas et al 27 have presented a. methodology to obtain practical upper bounds of the error in the energy norm. kek using an SPR based approach where equilibrium was locally imposed on each. patch The recovered stresses obtained with this technique were proven to provide. very accurate estimations of the error in the energy norm. Note that we have followed here the definition of error estimates provided in. References 4 7 29 and that we reserve the term error bounding for techniques which. provide upper or lower bounds of the error The semantics used in Mathematics. refer to 9 as an error indicator and any technique providing bounds as an error. 3 2 Error in quantities of interest duality technique. In this section we show how error estimators measured in the energy norm may be. utilised to estimate the error in a particular quantity of interest 1 The strategy. consists in solving a primal problem which is the problem at hand and a dual. problem useful to extract information on the quantity of interest identical to the. primal problem except for the applied boundary conditions and internal loads. From the FE solutions of both problems it is possible to estimate the contribution. of each of the elements to the error in the QoI This error measure allows to adapt. the approximate error tends towards the exact error. the mesh using procedures similar to traditional techniques based on the estimated. error in the energy norm, Consider the linear elasticity problem given in 6 and its approximate FE solu. tion uh For the sake of simplicity let us assume homogeneous Dirichlet boundary. conditions in this problem This problem is related to the original problem to be. solved that henceforth will be called the primal problem. Let us define Q V R as a bounded linear functional representing some. quantity of interest acting on the space V of admissible functions for the problem. at hand The goal is to estimate the error in functional Q when calculated using. the value of the approximate solution uh as opposed to the exact solution u. Q u Q uh Q u uh Q e 10, As will be shown later Q v may be interpreted as the work associated with a.

displacement field v and a distribution of loads specific to each type of quantity of. interest If we particularize Q v for v u this force distribution will allow to. extract information concerning the quantity of interest associated with the solution. of the problem in 6, A standard procedure to evaluate Q e consists in solving the auxiliary dual. problem also called adjoint or extraction problem defined as. Find wQ V w Q v V a v wQ Q v 11, An exact representation for the error Q e in terms of the solution of the dual. problem can be simply obtained by substituting v e in 11 and remarking that. in case w Q 0 for all whQ V h due to the Galerkin orthogonality a e whQ 0. Q e a e wQ a e wQ a e whQ a e wQ whQ a e eQ 12, Therefore the error in evaluating Q u using uh is given by. p hp D 1 d hd d,Q u Q u Q e a e eQ 13, where p is the stress field associated with the primal solution and d is the one. associated with the dual solution, Following 24 we can introduce a first coarse upper bound for the global error.

in the QoI employing the Cauchy Schwarz inequality that results in. Q e a e eQ kekkeQ k 14, Then the evaluation of the error in the QoI is now expressed in terms of the error. in the energy norm for the dual and primal solutions for which several techniques. are already available However this upper bound is rather conservative due to the. use of the Cauchy Schwarz inequality 24 Moreover it does not provide a local. indicator that can be used to guide the adaptivity process. For a discretization with ne elements let us consider the local test space Ve. V and ae Ve Ve R as the associated bilinear form a to an element. e ne e such that,u v V a u v ae u v 15, To obtain sharper error measures and local indicators we can decouple the ele. ment contributions as shown in 24 such that,Q e a e eQ ae e eQ from 12 16a. ae e eQ 16b,kek e keQ k e 16c,kekkeQ k 16d, Note that in 13 and the indicators derived afterwards the error in the QoI is. related to the errors in the FE approximations uh and whQ On that account any of. the available procedures to estimate the error in the energy norm may be considered. e mail festradaoag kerfridenpg cardi ac uk stephane bordas alum northwestern edu 2 Centro de Investigaci on de Tecnolog a de Veh culos CITV Universitat Polit ecnica de Val encia E 46022 Valencia Spain

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