optimal procedure for the identification of constitutive parameters from experimentally measured displacement fields abstract measured displacement fields constitute a rich database for the identification of mechanical behavior a variety of methods are available today to address this problem a formalism is proposed showing that they all revert to the minimization of a quadratic norm or semi norm between measured and computed displacement fields however they differ in the chosen metric transporting image noise through digital image correlation and identification procedures the uncertainty in the measured displacement fields i e their full covariance and the resulting uncertainty in the estimated constitutive parameters is assessed consequently an optimal choice of metric with respect to image noise is proposed so that the resulting identification uncertainty is minimized

a polynomial approach to nonlinear system controllability uses a polynomial approach to present a necessary and sufficient condition for local controllability of single input single output siso nonlinear systems the condition is presented in terms of common factors of a noncommutative polynomial expression this result exposes controllability properties of a nonlinear system in the input output framework and gives a computable procedure for examining nonlinear system controllability using computer algebra

direct error in constitutive equation formulation for plane stress inverse elasticity problem we present a new computational formulation for inverse problems in elasticity with full field data the formulation is a variant of an error in the constitutive equation formulation but allows direct solution for the modulus field and accommodates discontinuous strain fields the development of the formulation is motivated by the relatively poor performance of current direct formulations reported so far in literature in dealing with discontinuities in the strain and material property distribution the formulation relies on minimizing the error in the constitutive equation and a momentum equation constraint numerical results on model problems show that the formulation is capable handling discontinuous and noisy strain fields and also converging with mesh refinement for continuous and discontinuous material property distributions the application to reconstruct the elastic modulus distribution in solid breast tumors is shown

analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

a posteriori error estimation based on numerical realization of the variational multiscale method this paper presents a numerical realization of the variational multiscale method with the objective of providing a reliable and easy to implement local error estimation technique the variational multiscale framework provides a systematic approach for solution scale decomposition into coarse scales captured by the mesh and fine or subgrid scales in the proposed work the coarse scale errors in the finite element solution are neglected in comparison to the fine scale errors the fine scale variational equation is then localized using a general localization function over an element or a patch of elements to develop a local error estimation technique based on the proposed framework a consistent formulation of a new subdomain error estimator for elliptic problems is derived without the necessity of introducing an error locality assumption the new subdomain error estimator is evaluated numerically within a mesh adaptivity algorithm and it is shown to produce very sharp error estimates that outperform the element residual method estimates

a domain decomposition method for bodies with heterogeneous microstructure basedon material regularization abstract a domain decomposition methodis developed to reduce the computational complexity of boundary value problems associated withthe structural analysis of bodies with arbitrary external geometry loading and linearly elasticmicrostructure the purpose of the method is to augment existing numerical discretizationmethods of analysis such as the finite element method the approach is to partition and decouplethe heterogeneous body into more computationally tractable nonoverlapping subdomains whoseunion forms the entire domain under analysis this is achieved by approximating the subdomainboundary conditions the approximate boundary conditions of displacement or traction type aresupplied from the solution to a relatively computationally inexpensive auxiliary boundary valueproblem characterized by a simple regularized microstructure since the decoupled subdomainsmay then be analyzed separately the memory requirements are reduced and computingprocedures are trivially parallelizable a posteriori error bounds are developed for solutionsgenerated by this process it is shown that in the special case of uniform exterior loading theerror bounds collapse into forms which imply results pertaining to effective property orderingcoinciding with those published by huet 1990

a non intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity problems this paper presents a non intrusive method which yields strict and high quality error bounds of calculated quantities of interest in structural problems solved by the finite element method the focus is on linear viscoelasticity problems described through internal variables in order to solve the adjoint problem in a non intrusive way we use handbook techniques involving enrichment functions introduced through the partition of unity method pum then the mesh and operators used to calculate the reference problem can be reused such a procedure also enables one to address error estimation on pointwise quantities of interest although this implies dealing with infinite energy green functions

reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid uniform grid solvers of the periodic lippmann schwinger equation have been introduced by moulinec and suquet for the numerical homogenization of heterogeneous materials based on the fast fourier transform these methods use the strain as main unknown and usually do not produce displacement fields while this is generally not perceived as a restriction for homogenization purposes some tasks might require kinematically admissible displacement fields in this paper we show how the numerical solution to the periodic lippmann schwinger equation can be post processed to reconstruct a displacement field our procedure is general it applies to any variant of the moulinec suquet solver the reconstruction is formulated as an auxiliary elastic equilibrium problem of a homogeneous material which is solved with displacement based finite elements taking advantage of the periodicity the uniformity of the grid and the homogeneity of the material the resulting linear system is formulated and solved efficiently in fourier space the cost of our procedure is lower than that of one iteration of the lippmann schwinger solver an application of this post processing procedure is proposed in which the reconstructed displacement field is used to compute a rigorous upper bound on the effective shear modulus of some model microstructure this is the pre peer reviewed version of the following article reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid which has been published in final form at 10 1002 nme 5263 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving sections 1 introduction and 5 numerical results have been significantly reworked in the published final version of this article the remainder of this article including sections 3 and 4 where the method itself is presented is essentially unchanged

a strict error bound with separated contributions of the discretization and of the iterative solver in non overlapping domain decomposition methods this paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non overlapping domain decomposition method feti or bdd we propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solver

fast estimation of discretization error for fe problems solved by domain decomposition this paper presents a strategy for a posteriori error estimation for substructured problems solved by non overlapping domain decomposition methods we focus on global estimates of the discretization error obtained through the error in constitutive relation for linear mechanical problems our method allows to compute error estimate in a fully parallel way for both primal bdd and dual feti approaches of non overlapping domain decomposition whatever the state converged or not of the associated iterative solver results obtained on an academic problem show that the strategy we propose is efficient in the sense that correct estimation is obtained with fully parallel computations they also indicate that the estimation of the discretization error reaches sufficient precision in very few iterations of the domain decomposition solver which enables to consider highly effective adaptive computational strategies

discretization error for the discrete kirchhoff plate finite element approximation abstract we provide in this work the discretization error estimates that can guide an adaptive mesh refinement for the discrete kirchhoff plate finite elements the proposed developments are built upon the concept of error estimates for classical elasticity and adapted to suit the kirchhoff plate finite elements we give a detailed illustration of the proposed procedures for the discrete kirchhoff triangular plate element along with several different possibilities for constructing the enhancement of test space needed for error estimates the first novelty concerns the consistent displacement field in terms of the third order polynomial for the discrete kirchhoff triangle whereas the second novelty is the use of the argyris triangle with fifth order polynomials for constructing the enhanced test for error estimates we compare the latter against several alternatives that can be used for kirchhoff plates the results of numerical examples are given to illustrate the effectiveness of proposed discretization error estimates

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

convolutions and multiplier transformations of convex bodies rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to minkowski and blaschke addition are investigated the main focus is on blaschke minkowski homomorphisms we show that such maps are represented by a spherical convolution operator an application of this representation is a complete classification of all even blaschke minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator among further applications is the following result if an even blaschke minkowski homomorphism maps a convex body to a polytope then it is a constant multiple of the projection body operator

optimality and uniqueness of the leech lattice among lattices we prove that the leech lattice is the unique densest lattice in r24 the proof combines human reasoning with computer verification of the properties of certain explicit polynomials we furthermore prove that no sphere packing in r24 can exceed the leech lattice s density by a factor of more than 1 1 65 10 30 and we give a new proof that es is the unique densest lattice in r8

brief analysis of nonlinear time delay systems using modules over non commutative rings the theory of non commutative rings is introduced to provide a basis for the study of nonlinear control systems with time delays the left ore ring of non commutative polynomials defined over the field of meromorphic function is suggested as the framework for such a study this approach is then generalized to a broader class of nonlinear systems with delays that are called generalized roesser systems finally the theory is applied to analyze nonlinear time delay systems a weak observability is defined and characterized generalizing the well known linear result properties of closed submodules are then developed to obtain a result on the accessibility of such systems

recovery of equilibrium on star patches using a partition of unity technique a technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems the data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model local problems are defined for each star of elements via the introduction of fictitious body forces and strains these problems can be solved independently for equilibrium in a process that can be easily parallelized the quality of the solutions is assessed for two dimensional linear elastic problems by using them to compute bounds of the error of the finite element solutions in terms of both global and local quantities of interest copyright 2009 john wiley sons ltd

variational multiscale a posteriori error estimation for multi dimensional transport problems this paper presents an explicit a posteriori error estimator for the multi dimensional transport equation based on an approximation to the variational multiscale theory the technique is adequate for methods with a local error distribution such as stabilized methods yielding efficiencies close to one for both the hyperbolic and diffusive limit regarding the hyperbolic limit previous work is extended to multi dimensions adequate norms are proposed for the computation of the error and proper error intrinsic time scales are calculated for the bilinear quadrilateral and the linear triangle furthermore the model considers the element interface error along the element edges enabling the error prediction in the diffusive limit the success of the method can be explained by the fact that in stabilized methods the element local problem captures most of the error and the proposed error intrinsic time scales are an approximation to the solution of the dual problem the proposed technology which can be implemented straightforwardly in existing codes is extremely economical because it is a simple explicit postprocessing

dual analysis with general boundary conditions abstract the dual analysis concept was introduced by fraeijs de veubeke 1 as a consequence of upper and lower bounds of the energy as such bounds exist only in those cases where one type of condition is homogeneous it is commonly admitted that a dual error measure does not exist with general boundary conditions this paper presents a re examination of the dual error measure by a way which avoids any use of upper and lower bounds of the energy it is found that such an error measure holds whatever the boundary conditions be furthermore it is not necessary to obtain the approximate solutions by a rayleigh ritz process so that the second analysis which seemed necessary in the original dual analysis concept may be replaced by any admissible approximation this implies the possibility of a dual error measure at a simple post processor level

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

the 3 ball is a local pessimum for packing abstract it was conjectured by ulam that the ball has the lowest optimal packing fraction out of all convex three dimensional solids here we prove that any origin symmetric convex solid of sufficiently small asphericity can be packed at a higher efficiency than balls we also show that in dimensions 4 5 6 7 8 and 24 there are origin symmetric convex bodies of arbitrarily small asphericity that cannot be packed using a lattice as efficiently as balls can be

on the techniques for constructing admissible stress fields in model verification performances on engineering examples robust global goal oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation in the context of computational mechanics the construction of admissible stress fields ie stress tensors which verify the equilibrium equations is required to set up strict and guaranteed error bounds using residual based error estimators and plays an important role in the quality of the error estimates this work focuses on the different procedures used in the calculation of admissible stress fields which is a crucial and technically complicated point the three main techniques that currently exist called the element equilibration technique eet the star patch equilibration technique spet and the element equilibration star patch technique eespt are investigated and compared with respect to three different criteria namely the quality of associated error estimators computational cost and easiness of practical implementation into commercial finite element codes the numerical results which are presented focus on industrial problems they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators which have the same convergence rate as the exact global error is consistent two and three dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches the analysis of the results reveals that the spet is more accurate than eet and eespt methods but the corresponding computational cost is higher overall the numerical tests prove the interest of the hybrid method eespt and show that it is a correct compromise between quality of the error estimate practical implementation and computational cost furthermore the influence of the cost function involved in the eet and the eespt is studied in order to optimize the estimators

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

a posteriori error estimation for variational problems with uniformly convex functionals the objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations we consider variational problems of the form inf f v g v where f v r is a convex lower semicontinuous functional g y r is a uniformly convex functional v and y are reflexive banach spaces and v y is a bounded linear operator we show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and thus can be justified via the duality theory

dual adaptive finite element refinement for multiple local quantities in linear elastostatics in this paper we summarize how dual analysis techniques can be used to determine upper bounds of the discretization error both in terms of global and local outputs we present formulas for the bounds of the error in local outputs based on the approach proposed by greenberg in 1948 and we show that the resulting intervals are the same as those previously presented based on the approach proposed by washizu in 1953 we then explain how the elemental contributions to these bounds can be used to define an adaptive strategy that considers multiple quantities and we present its application to a simple plane stress problem copyright 2010 john wiley sons ltd

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

a new non intrusive technique for the construction of admissible stress fields in model verification in this paper we investigate a new procedure for constructing admissible stress tensors that are needed in some methods for robust global goal oriented error estimation this procedure is based on properties of the approximate finite element solution which are applied through prolongation conditions it involves the calculation of equilibrated tractions along element boundaries leading to the determination of admissible stress tensors at the element level the main idea of the proposed approach is to use the partition of unity method for the calculation of the equilibrated tractions thus defining local problems on patches of elements which can be solved in an automatic and non intrusive manner therefore this hybrid procedure leads to reasonable computational costs and can be easily implemented into existing finite element codes two dimensional experiments illustrate the capabilities of the method as compared to alternative approaches

when retarded nonlinear time delay systems admit an input output representation of neutral type the paper shows that nonlinear retarded time delay systems can admit an input output representation of neutral type this behaviour represents a strictly nonlinear phenomenon for it cannot happen in the linear time delay case where retarded systems always admit an input output representation of retarded type a necessary and sufficient condition for a nonlinear system to exhibit this behaviour is given and a strategy for finding an input output representation of retarded type is outlined some open problems that arise consequently are discussed as well all the systems considered in this work are time invariant and have commensurable delays

new advances on a posteriori error on constitutive relation in f e analysis this paper deals with error estimators based on residuals on the constitutive relation which have been developed for the past 20 years at cachan especially in the case of elasticity these error estimators can become poor in situations where the material and the mesh are strongly anisotropic the crucial point herein is the construction of equilibrated stress fields from the computed finite element solution therefore in this paper the approach and in particular this last point are re examined several important modifications are introduced in the concepts and basic techniques hence a new generation of error estimators which seem to be extremely robust upon initial applications has been generated

strict bounding of quantities of interest in computations based on domain decomposition abstract this paper deals with bounding the error on the estimation of quantities of interest obtained by finite element and domain decomposition methods the proposed bounds are written in order to separate the two errors involved in the resolution of reference and adjoint problems on the one hand the discretization error due to the finite element method and on the other hand the algebraic error due to the use of the iterative solver beside practical considerations on the parallel computation of the bounds it is shown that the interface conformity can be slightly relaxed so that local enrichment or refinement are possible in the subdomains bearing singularities or quantities of interest which simplifies the improvement of the estimation academic assessments are given on 2d static linear mechanic problems

brief paper state elimination and identifiability of the delay parameter for nonlinear time delay systems the identifiability of the delay parameter for nonlinear systems with a single constant time delay is analyzed we show the existence of input output equations and relate the identifiability of the delay parameter to their form explicit criteria based on rank calculations are formulated the identifiability of the delay parameter is shown not to be directly related to the well characterized identifiability observability of the other system parameters states

a simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems several domain decomposition methods with lagrange multipliers have been recently designed for solving iteratively large scale systems of finite element equations while these methods differ typically by implementational details they share in most cases the same substructure based preconditioners that were originally developed for the feti method the success of these preconditioners is due to the fact that for homogeneous structural mechanics problems they ensure a computational performance that scales with the problem size in this paper we address the suboptimal behaviour of these preconditioners in the presence of material and or discretization heterogeneities we propose a simple and virtually no cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations we consider several intricate structural analysis problems and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities copyright 1999 john wiley sons ltd

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 2 a posteriori error estimation abstract this is the second paper out of a series of papers devoted to model and hpq adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description in our previous publication we investigated the issue of hierarchical models and approximations of such structures we applied 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within the proposed finite element formulation furthermore we assumed that the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q could vary locally i e they could be different in each finite element the a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints the generalized method is applied to the assessment of the total and approximation errors while the modeling error is calculated as the difference between the former two errors the corresponding error controlled adaptive procedures are based on a three step strategy with possible iterations of h and p steps

validation of a posteriori error estimators by numerical approach abstract 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

a posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem the reciprocal theorem of betti and rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities including local variables like single displacements and stresses the proposed error estimator is evaluated solving the set of equations for an additional right hand side simply applying the energy norm error estimators two times furthermore a general h adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables the efficiency of the current approach is demonstrated for plate and shell examples

the equilibrium method for local a posteriori pointwise estimation of the finite element error the equilibrium method for the recovery of finite element errors is extended and given the theoretical foundation necessary to employ it to the pointwise recovery of the finite element function the practical implications of the resulting theory is studied extensively through a computational example

constitutive relation errors for f e analysis considering visco plasticity and damage a posteriori error estimators based on constitutive relation residuals are developed to measure the quality of finite element computations of structures which exhibit plastic viscoplastic behaviour with softening all the classical error sources involved in the computation over the studied time interval are taken into account the space discretization the mesh the time discretization and the iterative technique used to solve the non linear discrete problem copyright 2001 john wiley sons ltd

subdomain based flux free a posteriori error estimators a new residual type flux free error estimator is presented it estimates upper and lower bounds of the error in energy norm the proposed approach precludes the main drawbacks of standard residual type estimators circumvents the need of flux equilibration and results in a simple implementation that uses standard resources available in finite element codes this is specially interesting for 3d applications where the implementation of this technique is as simple as in 2d recall that on the contrary the complexity of the flux equilibration techniques increases drastically in the 3d case the bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs representing quantities of engineering interest the presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest

a posteriori error estimation for elasto plastic problems based on duality theory abstract in this paper we introduce a new approach to a posteriori error estimation for elasto plastic problems based on the duality theory of the calculus of variations we show that in spite of the prevailing view duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts the error in the constitutive law and the error in the equilibrium equations this representation is both physically meaningful and computationally important the duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the galerkin orthogonality condition in particular they encompass the familiar smoothening or gradient averaging techniques commonly used in practice we prove that the duality error estimators are equivalent to the error when the approximate solution is the exact solution of the discretised problem moreover in the case of linear elasticity the known residual type error estimators can be obtained from the duality error estimators numerical results for a model elasto plastic problem show the accuracy of the duality error estimators

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

linearisation via input output injection of time delay systems abstractthis paper deals with the problem of linearisation of systems with constant commensurable delays by input output injection using algebraic control tools based on the theory of non commutative rings solutions for the problem of linearisation free of delays and with delays of an observable nonlinear time delay systems are presented based on the analysis of the input output equation these results are achieved by means of constructive algorithms that use the nth derivative of the output expressed in terms of the state space variables instead of the explicit computation of the input output representation of the system necessary and sufficient conditions are established in both cases by means of an invertible change of coordinates

the disturbance decoupling problem for time delay nonlinear systems in this paper the disturbance decoupling problem ddp for a class of siso nonlinear systems with multiple delays in the input and the state is studied a pioneering mathematical approach is introduced for this class of systems and is claimed to be the cornerstone of the problem necessary and sufficient conditions are given for the existence of a bicausal feedback that solves the ddp sufficient conditions for the existence of a solution within other classes of compensators are included as well

asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method in this paper we analyze an a posteriori error estimator based on the equilibrated residual method we prove that this estimator is asymptotically exact in the energy norm for regular solutions and o h 1 alpha alpha 0 meshes numerical examples are included to illustrate the theoretical results

study of the strong prolongation equation for the construction of statically admissible stress fields implementation and optimization this paper focuses on the construction of statically admissible stress fields sa fields for a posteriori error estimation in the context of verification the recovery of such fields enables to provide strict upper bounds of the energy norm of the discretization error between the known finite element solution and the unavailable exact solution the reconstruction is a difficult and decisive step insofar as the effectiveness of the estimator strongly depends on the quality of the sa fields this paper examines the strong prolongation hypothesis which is the starting point of the element equilibration technique eet we manage to characterize the full space of sa fields satisfying the prolongation equation so that optimizing inside this space is possible the computation exploits topological properties of the mesh so that implementation is easy and costs remain controlled in this paper we describe the new technique in details and compare it to the classical eet and to the flux free technique for different 2d mechanical problems the method is explained on first degree triangular elements but we show how extensions to different elements and to 3d are straightforward