Explicit expressions of dual loads for accurate error estimation and bounding in goal oriented adaptivity

Abstract. Recently, Goal Oriented Adaptivity (GOA) has been an active research area because of its advantages in terms of computational cost and accuracy. This technique consists in solving two Finite Element (FE) problems: the primal one, which is the actual problem and the dual one, which is an auxiliary problem depending on the Quantity of Interest (QoI). To improve the quality of the error estimate in the QoI we consider a recovery-based procedure which enforces local equilibrium for an accurate stress representation. The proposed procedure requires the explicit expressions for the dual loads which, traditionally, are not obtained in the FE framework. Our objective in this paper is to obtain those explicit expressions for the dual problem for the extraction of linear QoI in the context of linear elasticity. The ZZ-type error estimator is used to evaluate the error in the QoI at element level, yielding a high quality, as shown in the numerical tests.

[1]  E. A. W. Maunder,et al.  Adaptive methods for hybrid equilibrium finite element models , 1999 .

[2]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[3]  I. Babuska.  Error-bounds for finite element method , 1971 .

[4]  Pedro Díez,et al.  Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error , 2007 .

[5]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[6]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[7]  Pedro Díez,et al.  Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates , 2003 .

[8]  O. González-Estrada,et al.  Accurate recovery-based upper error bounds for the extended finite element framework , 2010 .

[9]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[10]  Juan José Ródenas,et al.  A recovery‐type error estimator for the extended finite element method based on singular+smooth stress field splitting , 2008 .

[11]  J. Peraire,et al.  A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations , 1997 .

[12]  S.,et al.  " Goal-Oriented Error Estimation and Adaptivity for the Finite Element Method , 1999 .

[13]  Casadei Folco,et al.  Natural quantities of interest in linear elastodynamics for goal oriented error estimation and adaptivity , 2012 .

[14]  E. A. W. Maunder,et al.  Recovery of equilibrium on star patches using a partition of unity technique , 2009 .

[15]  Juan José Ródenas,et al.  Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR‐C technique , 2007 .

[16]  Juan José Ródenas,et al.  Equilibrated patch recovery for accurate evaluation of upper error bounds in quantities of interest , 2012 .

[17]  J. P. Moitinho de Almeida,et al.  A posteriori error estimation for equilibrium finite elements in elastostatic problems , 2000 .

[18]  Juan José Ródenas.  Goal oriented adaptivity: Una introducción a través del problema elástico lineal , 2005 .

[19]  Juan José Ródenas,et al.  Error estimation and error bounding in energy norm based on a displacement recovery technique , 2012 .

[20]  Pierre Ladevèze,et al.  Local error estimators for finite element linear analysis , 1999 .

[21]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

[22]  R. Verfürth.  A review of a posteriori error estimation techniques for elasticity problems , 1999 .

[23]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .