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

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

on the characterization of compactness in the space of functions of bounded variation in the sense of jordan abstract the main goal of this paper is to prove a compactness criterion for subsets of the banach space of functions of bounded variation in the sense of jordan in comparison to the compactness criterion contained in dunford schwartz s monograph the conditions which appear in our criterion seems to be easier to verify moreover using this criterion we define a quasimeasure of noncompactness in that space some applications of quasimeasures of noncompactness to fixed point theorems are also presented

block scaling with optimal euclidean condition let m denote the set of all complex n n matrices whose columns span certain given linear subspaces the minimal euclidean condition number of matrices in m is given in terms of the canonical angles between the linear subspaces and optimal matrices in m are described the result is also stated in terms of norms of certain projections

computing stable eigendecompositions of matrices abstract if a matrix t is known only to within a tolerance because of measurement or roundoff errors then it may be difficult to compute an eigendecomposition of t since its invariant subspaces are discontinuous functions of its entries in this paper we show how to compute a stable decomposition of an uncertain matrix t which varies continuously and boundedly as t varies in a ball of radius

a unified approach to strict upper and lower bounds of quantities in linear elasticity based on constitutive relation error estimation abstract this paper presents a unified approach to acquiring strict upper and lower bounds of various quantities in linear elasticity the key ingredient lies in a unified representation of linear quantities including displacement integrals stress integrals and even pointwise quantities with the unified representation dual error analysis can be performed easily which results in a unified approximation and thereby a unified error representation via the primal dual equivalence theorem then the constitutive relation error cre estimation featured with the ability to provide strict upper bound of global energy norm error is utilized and strict upper and lower bounds of the quantities are obtainable thereafter moreover two extant bounding approaches to goal oriented error estimation are analyzed and optimized and as a result optimal approximation and bounds are obtained numerical examples are studied to validate the proposed unified approach

a new a posteriori error estimation for nonlinear time dependent finite element analysis abstract in this paper an error on the constitutive law labeled the dissipation error is used to measure the quality of finite element computations of plastic and viscoplastic structures whose behavior is described by internal variables this measure takes into account all the classical sources of error involved in the computation the space discretization the mesh the time discretization and the iterative technique used to solve the nonlinear discrete problem more specifically to quantify the quality of the space and time discretizations two quantities called indicators are introduced the efficiency of both the error and the indicators is shown for several examples

new bounding techniques for goal oriented error estimation applied to linear problems the paper deals with the accuracy of guaranteed error bounds on outputs of interest computed from approximate methods such as the finite element method a considerable improvement is introduced for linear problems thanks to new bounding techniques based on saint venant s principle the main breakthrough of these optimized bounding techniques is the use of properties of homothetic domains which enables to cleverly derive guaranteed and accurate boundings of contributions to the global error estimate over a local region of the domain performances of these techniques are illustrated through several numerical experiments

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

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 finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

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

strict upper bounds of the error in calculated outputs of interest for plasticity problems this paper introduces a new computational technique for deriving guaranteed upper bounds of the error in calculated outputs of interest for plasticity problems under quasi static conditions this approach involves the resolution of some nonstandard additional problems for which resolution techniques are given all sources of error spatial and temporal discretization iteration stepping are taken into account

model verification in dynamics through strict upper error bounds abstract this paper describes new strict error bounds of calculated outputs of interest in linear dynamics all sources of errors including modeling errors are taken into account which makes such error bounds suitable tools for analyzing various approximations in dynamics small displacement problems are included through the classical thermodynamics framework involving internal state variables preliminary numerical illustrations are given

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

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

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

local error estimators for finite element linear analysis local error estimators on stresses displacements etc are derived from constitutive relation error estimators examples show that they can be considered as an upper bound

a posteriori estimation and adaptive control of the pollution error in the h version of the finite element method abstract we studied the pollution error in the h version of the finite element method and its effect on the quality of the local error indicators resp the quality of the derivatives recovered by local postprocessing in the interior of the mesh here we show that it is possible to construct a posteriori estimates of the pollution error in a patch of elements by employing the local error indicators over the entire mesh we also give an adaptive algorithm for the local control of the pollution error in a patch of elements of interest

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

the condition number of equivalence transformations that block diagonalize matrix pencils how ill conditioned must a matrix s be if its columns are constrained to span certain subspaces we answer this question in order to find nearly best conditioned matrices sr and sl that block diagonalize a given matrix pencil t a b i e s l 1 tsr is bloc diagonal we show that the best conditioned sr has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of thus the more nearly the right subspaces overlap the more ill conditioned sr must be the same is true of sl and the left subspaces for the standard eigenproblem t a i sl sr and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number namely the norm of the projection matrix associated with one of the subspaces we apply this result to bound the error in an algorithm to compute analytic functions of matrices for instance exp t

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

bernstein and kantorovich polynomials diminish the λ variation abstract we prove the variation diminishing property of the bernstein and kantorovich polynomials next we apply this result to characterize the space c b v c as the closure of the space of polynomials in the norm a new proof of the separability of c b v c is given

the computation of bounds for linear functional outputs of weak solutions to the two dimensional elasticity equations we present a method for the computation of upper and lower bounds for linear functional outputs of the exact solutions to the two dimensional elasticity equations the method can be regarded as a generalization of the well known complementary energy principle the desired output is cast as the supremum of a quadratic linear convex functional over an infinite dimensional domain using duality the computation of an upper bound for the output of interest is reduced to a feasibility problem for the complementary or dual problem in order to make the problem tractable from a computational perspective two additional relaxations that preserve the bounding properties are introduced first the domain is triangulated and a domain decomposition strategy is used to generate a sequence of independent problems to be solved over each triangle the lagrange multipliers enforcing continuity are approximated using piecewise linear functions over the edges of the triangulation second the solution of the adjoint problem is approximated over the triangulation using a standard galerkin finite element approach a lower bound for the output of interest is computed by repeating the process for the negative of the output reversing the sign of the computed upper bound for the negative of the output yields a lower bound for the actual output the method can be easily generalized to three dimensions however a constructive proof for the existence of feasible solutions for the outputs of interest is only given in two dimensions the computed bound gaps are found to converge optimally that is at the same rate as the finite element approximation an attractive feature of the proposed approach is that it allows for a data set to be generated that can be used to certify and document the computed bounds using this data set and a simple algorithm the correctness of the computed bounds can be established without recourse to the original code used to compute them in the present paper only computational domains whose boundary is made up of straight sided segments and polynomially varying loads are considered two examples are given to illustrate the proposed methodology

computable bounds of functional outputs in linear visco elastodynamics this work presents a new technique yielding computable bounds of quantities of interest in the framework of linear visco elastodynamics a novel expression for the error rep resentation is introduced alternative to the previous ones using the cauchy schwarz inequal ity the proposed formulation utilizes symmetrized forms of the error equations to derive error bounds in terms of energy error measures the practical implementation of the method is based on constructing admissible fields for both the original problem and the adjoint problem asso ciated with the quantity of interest here the flux free technique is considered to compute the admissible stress fields the proposed methodology yields estimates with better quality than the ones based on the cauchy schwarz inequality in the studied examples the bound gaps obtained are approximately halved that is the estimated intervals of confidence are reduced

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

on integral operators and nonlinear integral equations in the spaces of functions of bounded variation abstract in this paper we introduce new conditions on a kernel of a linear fredholm integral operator which turn out to be sufficient and necessary for that operator to map the space of functions of bounded variation in the sense of jordan into itself furthermore we apply those conditions to deal with the problem of the existence of solutions to nonlinear hammerstein equations in that space to achieve our goals we will use a topological degree approach as well as a fixed point approach

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

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

a posteriori estimation and adaptive control of the error in the quantity of interest part i a posteriori estimation of the error in the von mises stress and the stress intensity factor in this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution as an example we consider the plane elasticity problem with the von mises stress and the stress intensity factor as the quantities of interest the estimates of the error in the von mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point the local and the pollution errors and by constructing separate estimates for each component the estimates of the error in the stress intensity factors are constructed by employing an extraction method we demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees in part ii we will address the problem of the adaptive control of the error in the quantity of interest the goal of the computation and the construction of goal adaptive meshes for one or multiple goals

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

optimal block diagonal scaling of block 2 cyclic matrices abstract we describe a class of optimal block diagonal scalings preconditionings of a symmetric positive definite block 2 cyclic matrix generalizing a result of forsythe and strauss 1 for point 2 cyclic matrices

a posteriori error estimation and adaptive strategy for pgd model reduction applied to parametrized linear parabolic problems abstract we define an a posteriori verification procedure that enables to control and certify pgd based model reduction techniques applied to parametrized linear elliptic or parabolic problems using the concept of constitutive relation error it provides guaranteed and fully computable global goal oriented error estimates taking both discretization and pgd truncation errors into account splitting the error sources it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of pgd approximations the focus of the paper is on two technical points i construction of equilibrated fields required to compute guaranteed error bounds ii error splitting and adaptive process when performing pgd based model reduction performances of the proposed verification and adaptation tools are shown on several multi parameter mechanical problems

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

bounds on history dependent or independent local quantities in viscoelasticity problems solved by approximate methods this paper introduces new strict error bounds of outputs of interest for linear viscoelasticity structural problems involving internal state variables and calculated by the finite element method using extraction techniques and the concept of dissipation error we show that we can get both strict and effective bounds by using convexity properties and by taking effects of error accumulation into account