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

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 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

adaptive control for regulation of a quadratic function of the state we propose an adaptive controller for a dynamical system where we need a quadratic function of the state to track a given reference signal this problem appears in the control of generators in weak grid conditions for example while the controller can measure the tracking error the main difficulty arises from the fact that the parameters of the quadratic function itself are not known to the controller our approach consists of simultaneously estimating the quadratic function while tracking the reference signal similar to the approach employed in adaptive control the quadratic structure of the tracking function necessitates however a new adaptive law for estimating the parameters even though estimation and control are in general two contradicting requirements using this new adaptive law and a multilevel controller we prove that the tracking error converges to zero in the absence of measurement noise in the presence of bounded noise we show that the tracking error can be driven to a neighborhood around the origin

robustness of error estimators for finite element solutions of problems with high orthotropy in this paper we address the accuracy of a posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used which are often encountered in engineering computations we present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy the main culprit is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution this is often imposed by the size of the problem domain the capabilities for mesh generation and the employed finite element software and cannot be changed hence there is need for a posteriori error estimation approaches which are robust for coarse meshes and which are constructed using non intrusive approaches with respect to the employed software

some improvements to the flux type a posteriori error estimators abstract this paper presents explicit equilibrated fluxes for the local pure neumann problems used by the a posteriori error estimator in ainsworth and oden m ainsworth j t oden a procedure for a posteriori error estimation for h p finite element methods comput methods appl mech engrg 101 1992 73 96 we modify the target function for minimization to obtain new equilibrated fluxes in a weighted form we investigate the performance of the new equilibrated fluxes in two dimensional spectral hp element approximations of an elliptic equation using conforming triangle elements the new fluxes appear more robust and lead to substantial computational savings compared to the original numerical procedure in ainsworth and oden ainsworth and oden et al 1992

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

adaptive output feedback control of nonlinear systems using neural networks a direct adaptive output feedback control design procedure is developed for highly uncertain nonlinear systems that does not rely on state estimation the approach is also applicable to systems of unknown but bounded dimension in particular we consider single input single output nonlinear systems whose output has known but otherwise arbitrary relative degree this includes systems with both parameter uncertainty and unmodeled dynamics the result is achieved by extending the universal function approximation property of linearly parameterized neural networks to model unknown system dynamics from input output data the network weight adaptation rule is derived from lyapunov stability analysis and guarantees that the adapted weight errors and the tracking error are bounded numerical simulations of an output feedback controlled van der pol oscillator coupled with a linear oscillator is used to illustrate the practical potential of the theoretical results

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 general method for recovering equilibrating element tractions abstract a new geometrical presentation of a general method for recovering equilibrating element tractions is developed for 2d finite element models of problems in structural mechanics the data for the method consists of the prescribed loading and the stress fields resulting from a conventional linear elastic finite element analysis the geometrical presentation utilises the physical concepts of nodal forces and maxwell force diagrams based on nodal equilibrium only small problems have to be solved for each node this presentation complements the original algebraic approach of ladeveze and extends the method to cover the use of hierarchical p elements in regular meshes and general higher order elements in 1 irregular meshes as encountered in h p mesh refinement the same presentation is considered for the recovery of residual tractions as required in the solution of the local residual problem comparison is made with related methods for obtaining residual tractions in particular those proposed by bank and weiser and ainsworth et al detailed examples are included to illustrate the presentation this paper considers only the fundamental aspects of recovering equilibrating tractions the knowledge gained may be used in various applications such as error analysis limit analysis and structural design

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

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

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

fast calculation of energy and mass preserving solutions of schrodinger poisson systems on unbounded domains this paper deals with the numerical solution of the time dependent schrodinger poisson system in the spherically symmetric case since the problem is posed on an unbounded domain one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of a so called discrete transparent boundary condition tbc for a crank nicolson type predictor corrector scheme for solving the schrodinger poisson system this scheme has the property of mass and energy conservation exactly on the discrete level we propose different strategies for the discrete tbc and present an efficient implementation finally a numerical example illustrate the findings and shows the comparison results between the different approaches

computable upper and lower bounds on eigenfrequencies this paper is a revisit of the work ladeveze and pelle int j numer methods engrg 28 1989 where the goal here is to acquire guaranteed accurate and computable bounds of eigenfrequencies through post processing of conventional finite element results to this end a new theoretical quotient is introduced and thereafter a practical way to deal with the new quotient is developed where the constitutive relation error estimation featured with guaranteed bounding property acts as a key role academic numerical examples are performed to check the accuracy of the bounds

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

analysis of the zienkiewicz zhu a posteriori error estimator in the finite element method an a posteriori error estimator for finite element analysis proposed by zienkiewicz and zhu is analysed and shown to be effective and convergent in addition we analyse wider classes of estimators of which the zienkiewicz zhu estimator is a special case it is shown that some of these estimators will be asymptotically exact numerical evidence is presented supporting the analysis

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

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

anisotropic discretization and model error estimation in solid mechanics by local neumann problems first a survey of existing residuum based error estimators and error indicators is given generally residual error estimators which have at least upper bound in contrast to indicators can be locally computed from residua of equilibrium and stress jumps at element interfaces using dirichlet or neumann conditions for element patches or individual elements rem another equivalent method for error estimation can be derived from a posteriori computed improved boundary tractions which provide exact equilibrium of elements they are computed from local neumann problems and yield improved local solution by testing with higher test functions this method is called posterior equilibrium method pem and yields better efficiency indices than rem for the investigated problem even in case of locking also the important feature of model adaptivity via hierarchical error estimation can only be realized by this strategy global neumann type estimators and examples are presented for elastic and elastoplastic deformations controlling equilibrium in tangential space

variational multiscale a posteriori error estimation for systems application to linear elasticity abstract this paper presents an explicit residual based a posteriori error estimator for linear elasticity the method is derived from the variational multiscale theory where the subgrid scales and the constants of the method are calculated explicitly from fine scale green s functions several numerical examples are presented showing satisfactory local and global efficiencies when the h 1 seminorm is employed the formulation which is applied to plane stress problems can be easily extended to other elasticity problems

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

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

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 posteriori estimation of the error in the error estimate in this paper we address the problem of a posteriori estimation of the error in the error estimate we consider the case of estimates for the error in the derivatives the strains or the stresses which are constructed in terms of locally computed element error indicators of the element residual or the least squares recovery type the estimates of the error in the error estimate have the same structure as the original error estimates and are determined by locally averaging recycling the original error indicators the most accurate indicators of the error in the error indicators are obtained by employing a harmonic basis in the recycling of the indicators namely a basis which locally satisfies the partial differential equation and the boundary conditions

equilibration techniques for solving contact problems with coulomb friction abstract in this paper we consider residual and equilibrated error indicators for contact problems with coulomb friction the contact problem is handled within the abstract framework of saddle point problems more precisely the non penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual lagrange multiplier space thus from the displacement we can easily compute in a local post process the lagrange multiplier which acts as a neumann condition on the possible contact zone having computed the discrete lagrange multiplier we can apply standard error estimators by replacing the unknown neumann data by its approximation as it is shown in 1 this results in an error estimator for a one sided contact problem without friction here we consider more general situations and discuss two additional contact terms which measure the non conformity of the discrete lagrange multiplier numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

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

verification of stochastic models in uncertain environments using the constitutive relation error method abstract this paper deals with local stress errors which occur in linear elastic fe analysis in an uncertain environment material loads the estimation technique described here uses the concept of error in constitutive relation but it can be adapted to other estimators first we review the principle of the method then extend it to the stochastic case this approach yields bounds on the stresses associated with a given confidence level we illustrate this presentation through a test example

a posteriori error estimation for the steady navier stokes equations in random domains we consider finite element error approximations of the steady incompressible navier stokes equations defined on a randomly perturbed domain the perturbation being small introducing a random mapping these equations are transformed into pdes on a fixed reference domain with random coefficients under suitable assumptions on the random mapping and the input data in particular the so called small data assumption we prove the well posedness of the problem we assume then that the mapping depends affinely on l independent random variables and adopt a perturbation approach expanding the solution with respect to a small parameter e that controls the amount of randomness in the problem we perform an a posteriori error analysis for the first order approximation error namely the error between the exact random solution and the finite element approximation of the first term in the expansion with respect to e numerical results are given to illustrate the theoretical results and the effectiveness of the error estimators

error bounds for asymptotic solutions of second order linear difference equations abstract error bounds for asymptotic solutions of second order linear difference equation y n 2 a n y n 1 b n y n 0 where a n b n have asymptotic expansions a n s 0 a s n s b n s 0 b s n s n b0 0 are discussed in detail in this paper

bounds for quantities of interest and adaptivity in the element free galerkin method a novel approach to implicit residual type error estimation in mesh free methods and an adaptive refinement strategy are presented this allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest the proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux equilibration and resulting in a simple implementation that avoids integrals on edges sides of a domain decomposition mesh this is especially interesting for mesh free methods the adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision copyright 2008 john wiley sons ltd

explicit bounds for second order difference equations and a solution to a question of stevic this note gives explicit applicable bounds for solutions of a wide class of second order difference equations with nonconstant coefficients among the applications is an affirmative answer to a recent question of stevic

strict and effective bounds in goal oriented error estimation applied to fracture mechanics problems solved with xfem in this work we analyze a method that leads to strict and high quality local error bounds in the context of fracture mechanics we investigate in particular the capability of this method to evaluate the discretization error for quantities of interest computed using the extended finite element method xfem the goal oriented error estimation method we are focusing on uses the concept of constitutive relation error along with classical extraction techniques the main innovation in this paper resides in the methodology employed to construct admissible fields in the xfem framework which involves enrichments with singular and level set basis functions we show that this construction can be performed through a generalization of the classical procedure used for the standard finite element method thus the resulting goal oriented error estimation method leads to relevant and very accurate information on quantities of interest that are specific to fracture mechanics such as mixed mode stress intensity factors the technical aspects and the effectiveness of the method are illustrated through two dimensional numerical examples

a reduced basis method with exact solution certificates for steady symmetric coercive equations abstract we introduce a reduced basis method that computes rigorous upper and lower bounds of the energy associated with the infinite dimensional weak solution of parametrized steady symmetric coercive partial differential equations with piecewise polynomial forcing and operators that admit decompositions that are affine in functions of parameters the construction of the upper bound appeals to the standard primal variational argument the construction of the lower bound appeals to the complementary variational principle we identify algebraic conditions for the reduced basis approximation of the dual variable that results in an exact satisfaction of the dual feasibility conditions and hence a rigorous lower bound the formulation permits an offline online computational decomposition such that in the online stage the approximation and exact certificates can be evaluated in complexity independent of the underlying finite element discretization we demonstrate the technique in two numerical examples a one dimensional reaction diffusion problem with a parametrized diffusivity constant a planar linear elasticity problem with a geometry deformation we confirm in both cases that the method produces guaranteed upper and lower bounds of the energy at any parameter value for any finite element discretization and for any reduced basis approximation

full asymptotic expansions of the landau constants via a difference equation approach abstract we derive full asymptotic expansions for the landau constants g n as n some of the expansions are not new yet all the coefficients of the expansions are given iteratively in an explicit manner and are more efficiently evaluated as compared with the known results we obtain the asymptotic formulas old and new by applying the theory of wong and li for second order linear difference equations in deriving the expansions we have also confirmed a conjecture made by nemes and nemes

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

the quartic oscillator the quantum quartic oscillator is treated on the basis of a singularity analytic approach to the central two point connection problem of the triconfluent case of heun s differential equation we split off the asymptotic factors by means of a specific linear transformation of the independent variable and represent the solution in terms of a jaffe expansion the result is a fourth order linear difference equation of poincare perron type the asymptotic behavior of which is significant for the connection problem this is investigated by means of the birkhoff set of the difference equation and leads to the exact eigenvalue condition of the problem

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

anisotropic linear elastic parameter estimation using error in the constitutive equation functional a modified error in the constitutive equation based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored following an alternating minimization procedure associated with the underlying optimization problem the new strategy results in an explicit material parameter update formula for general anisotropic material this immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction we consider a few common materials to derive such conditions then we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space dependent material orientations finally we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest

asymptotic expansions for second order linear difference equations abstract formal series solutions are obtained for the difference equation y n 2 a n y n 1 b n y n 0 where a n and b n have asymptotic expansions of the form a n s 0 a s n s and b n s 0 b s n s for large values of n and b0 0 these solutions are characterized by the roots of the characteristic equation 2 a0 b0 0 our discussion is divided into three cases according to whether the roots are distinct or equal and do not satisfy the auxiliary equation a1 b1 0 or equal and do satisfy the auxiliary equation the last case is further divided into three subcases according to whether the roots of the indicial equation 1 2 a1 a2 b2 0 do not differ by a nonnegative integer or differ by a positive integer or are equal in all cases the formal series solutions will be shown to be asymptotic our approach is based on the method of successive approximations

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

solutions to the discrete airy equation application to parabolic equation calculations in the case of the equidistant discretization of the airy differential equation discrete airy equation the exact solution can be found explicitly this fact is used to derive a discrete transparent boundary condition tbc for a schrodinger type equation with linear varying potential which can be used in parabolic equation simulations in underwater acoustics and for radar propagation in the troposphere we propose different strategies for the discrete tbc and show an efficient implementation finally a stability proof for the resulting scheme is given a numerical example in the application to underwater acoustics shows the superiority of the new discrete tbc

adaptive control of a class of nonaffine systems using neural networks a neural control synthesis method is considered for a class of nonaffine uncertain single input single output siso systems the method eliminates a fixed point assumption and does not assume boundedness on the time derivative of a control effectiveness term one or the other of these assumptions exist in earlier papers on this subject using lyapunov s direct method it is shown that all the signals of the closed loop system are uniformly ultimately bounded and that the tracking error converges to an adjustable neighborhood of the origin simulation with a van der pol equation with nonaffine control terms illustrates the approach

asymptotic expansions for second order linear difference equations with a turning point a turning point theory is developed for the second order difference equation where the coefficients a n and b n have asymptotic expansions of the form 0 being a real number in particular it is shown how the airy functions arise in the uniform asymptotic expansions of the solutions to this three term recurrence relation as an illustration of the main result a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the freud weight exp x 4 x

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

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

effective and practical h p version adaptive analysis procedures for the finite element method two practical and effective h p type finite element adaptive procedures are presented the procedures allow not only the final global energy norm error to be well estimated using hierarchic p refinement but in addition give a nearly optimal mesh the design of this is guided by the local information computed on the previous mesh the desired accuracy can always be obtained within one or at most two h p refinements the rate of convergence of the adaptive h p version analysis procedures has been tested for some examples and found to be very strong the presented procedures can easily be incorporated into existing p or h type code structures

a theory for local a posteriori pointwise residual based estimation of the finite element error abstract residual based a posteriori fem error estimation is based on the formulation and solution of local boundary value problems for the error the error problem is inherently global but it is split into local problems utilizing a recovery of the local boundary conditions for the error on single elements or patches of elements recovery of the fluxes or splitting of the jumps approximation decisions are involved in the splitting of the global problem into local problems and the actual estimator depends heavily on the type of approximations performed to provide a foundation for the decision process and to understand the various versions of residual estimators in use today it is essential to know the underlying mathematical theory behind the error estimation problem and the restrictions that is put on the problems in this work such a theory is presented and in light of the theory various error estimators are developed

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

explicit error bounds for a nonconforming finite element method let u be the solution of the following model left begin array l mbox find u in h 1 0 omega such that 3pt delta u f quad mbox in omega end array right where f is a given function in l2 omega and omega a bounded open set in mathbb r 2 with a polygonal boundary let uh be a finite element approximation of u the goal of this paper is to suggest an explicit bound of the error u uh furthermore this bound is obtained without complex computations the basic idea is to define locally an admissible vector field for the dual model

a new approach in non linear mechanics the large time increment method a new algorithm family taking into account the entire loading process in a single large time increment is proposed to compute structures with physical non linearities and is tested on some examples in elastoplasticity the method considerably reduces the number of transfers between local and global levels hence the numerical cost of calculation is also diminished