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

isospectrality of spherical mhd dynamo operators pseudo hermiticity and a no go theorem the isospectrality problem is studied for the operator of the spherical hydromagnetic 2 dynamo it is shown that this operator is formally pseudo hermitian j symmetric and lives in a krein space based on the j symmetry an operator intertwining ansatz with first order differential intertwining operators is tested for its compatibility with the structure of the 2 dynamo operator matrix an intrinsic structural inconsistency is obtained in the set of associated matrix riccati equations this inconsistency is interpreted as a no go theorem which forbids the construction of isospectral 2 dynamo operator classes with the help of first order differential intertwining operators

eigenvalues of rank one perturbations of unstructured matrices abstract let a be a fixed complex matrix and let u v be two vectors the eigenvalues of matrices a uv r form a system of intersecting curves the dependence of the intersections on the vectors u v is studied

reliable and efficient equilibrated a posteriori finite element error control in elastoplasticity and elastoviscoplasticity with hardening this paper establishes the reliability and efficiency of equilibrated a posteriori estimates for l 2 stress error control of conforming displacement finite element approximations of incremental plasticity and viscoplasticity with hardening explicit expressions for upper bounds of the reliability constant that enters the guaranteed upper error bound illustrate its crucial dependence on the hardening material constants numerical experiments show that adaptive finite element solutions with marking strategy based on the max refinement rule and local equilibrated error estimators lead to optimal empirical convergence rates 2005 elsevier b v all rights reserved

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

on invertibility of nonsquare generalized bezoutians abstract conditions for inverting h bezoutians of nonsquare matrix polynomials are studied necessary and sufficient conditions for invertibility of the anderson jury bezoutian are obtained an application to the inversion of block hankel matrices is given

real time updating of structural mechanics models using kalman filtering modified constitutive relation error and proper generalized decomposition the paper aims at proposing a new strategy for real time identification or updating of structural mechanics models defined as dynamical systems the main idea is to introduce the modified constitutive relation error concept which is a practical tool that enables to efficiently solve identification problems with highly corrupted data into the kalman filtering which is a classical framework for data assimilation furthermore a pgd based model reduction method is performed in order to optimize capabilities of the online updating strategy performances of the proposed approach in terms of robustness gain and computational cost reduction are illustrated on several unsteady thermal applications

eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations abstract udy the perturbation theory of structured matrices under structured rank one perturbations and then focus on several classes of complex matrices generic jordan structures of perturbed matrices are identified it is shown that the perturbation behavior of the jordan structures in the case of singular j hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in 18 28 30 31 thus a generic structured perturbation would not be generic if considered as an unstructured perturbation in other settings of structured matrices the generic perturbation behavior of the jordan structures within the confines imposed by the structure follows the pattern of that of unstructured perturbations

on the control of the load increments for a proper description of multiple delamination in a domain decomposition framework in quasi static non linear time dependent analysis the choice of the time discretization is a complex issue the most basic strategy consists in determining a value of the load increment that ensures the convergence of the solution with respect to time on the base of preliminary simulations in more advanced applications the load increments can be controlled for instance by prescribing the number of iterations of the non linear resolution procedure or by using an arc length algorithm these techniques usually introduce a parameter whose correct value is not easy to obtain in this paper an alternative procedure is proposed it is based on the continuous control of the residual of the reference problem over time whose measure is easy to interpret this idea is applied in the framework of a multiscale domain decomposition strategy in order to perform 3d delamination analysis copyright 2010 john wiley sons ltd

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

constitutive relation error estimators for visco plastic finite element analysis with softening a posteriori error estimators based on constitutive relation residuals have been developed for 20 years in particular at cachan this approach has a strong physical meaning and is quite general here we introduce an extended constitutive relation error estimators family able to measure the quality of finite element computations of structures which exhibit plastic viscoplastic behavior with softening these measures take into account over the studied time interval all the classical error sources involved in the computation the space discretization the mesh the time discretization and the iterative technique used to solve the nonlinear discrete problem

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

error assessment and mesh adaptivity for regularized continuous failure models this paper deals with the adaptive finite element analysis of structural failure a gradient enhanced damage model has been chosen to simulate material degradation since this model is regularized in the post peak regime the finite element solution does not suffer from pathological mesh dependence and thus converges to an objective solution upon mesh refinement however the error analyses have shown that the error in the nonlocal equivalent strain field becomes dominant during the post peak loading stages the accuracy of the nonlocal equivalent strain field and the corresponding damage quantity also greatly influences the accuracy of the quantity of interest two error measures have been proposed the goal oriented error estimates have provided similar error distributions although some small differences have been found in the softening regime objective error estimates together with adaptive criteria have been used to perform automated h adaptivity during computation

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

computational error estimates and adaptive processes for some nonlinear structural problems abstract in earlier papers the authors introduced and analyzed the calculation of reliable a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them while most of this work concerned linear problems this paper is intended to show that the same approaches remain also highly effective in the nonlinear case as a model problem a planar elastic rod is considered involving both material as well as geometrical nonlinearities but as far as possible the discussion is kept general for nonlinear problems of this type interest centers on an analysis of the shape and features of the equilibrium surface after some general remarks about such surfaces and the related computational problems a general continuation process is summarized and some of its extensions for determining limit points and tracing stability paths are discussed then estimators are introduced for the error along the solution paths and of the computed critical values experimental results for the model problem show the effectivity of these estimators and of the adaptive procedures based upon them then some aspects relating to the occurrence and identification of spurious solutions are discussed in order to illustrate desirable generalizations of these results the paper ends with an outline of some results for linear problems of the type that should be achievable also in the nonlinear case

a new multi solution approach suitable for structural identification problems abstract in this paper a multi solution strategy aimed at greatly decreasing the cost of structural identification problems is proposed it is based on the large time increment method and on the possibility offered by this approach to obtain a family of admissible displacement and stress fields in order to solve a nonlinear time dependent problem this technique relies on a time and space decomposition of the solution it has been extended to solve at a low cost the inverse problem of identification for various techniques in particular it has been adapted to the minimization of the gradient of the gap between test and simulation in this first attempt it is applied to the simple case of bending of an elastic viscoplastic beam the issue of the numerical efficiency of this type of strategy is studied

the linearization of boundary eigenvalue problems and reproducing kernel hilbert spaces the boundary eigenvalue problems for the adjoint of a symmetric relation s in a hilbert space with finite not necessarily equal defect numbers which are related to the selfadjoint hilbert space extensions of s are characterized in terms of boundary coefficients and the reproducing kernel hilbert spaces they induce 2001 elsevier science inc all rights reserved

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

displacement structure of generalized inverse matrices abstract it is shown that matrices with a uv displacement structure possess generalized inverses with a vu displacement structure estimations for the displacement rank of the generalized inverses are presented the results apply to matrices of toeplitz vandermonde and cauchy type and provide formulas for generalized inverses which are important for fast algorithms

adaptive fem analysis of selected elastic visco plastic problems abstract elastic visco plastic deformations modeled by the bodner partom constitutive equations s r bodner y partom constitutive equations elastic viscoplastic strain hardening materials j appl mech 42 1975 385 389 have been evaluated by the adaptive finite element method j m bass j t oden adaptive finite element methods for a class of evolution problems in viscoplasticity int j engrg sci 25 623 653 1987 l demkowicz j t oden w rachowicz o hardy toward a universal hp adaptive finite element strategy part 1 constrained approximation and data structure comput methods appl mech engrg 77 1989 79 112 we use the explicit residual error estimate i babuska w c rheinboldt error estimates for adaptive finite element computations int j numer methods engrg 12 1978 1597 1615 supplemented with a special term accounting for the fact that the right hand side of the equations we use is only approximated since it depends on the inelastic strains which are computed only at the gauss integration points we also address the problems of adaptation strategy and the method of solution transfer to a new mesh results of recent tests confirm efficiency and robustness of the approach

changing the spectrum of an operator by perturbation abstract we discuss the following problem how can the spectrum a of a linear operator a be changed upon perturbation of a with a one dimensional operator k in particular we study this question for normal self adjoint and unitary operators

sharp eigenvalue estimates for rank one perturbations of nonnegative operators in krein spaces abstract let a and b be selfadjoint operators in a krein space and assume that the resolvent difference of a and b is of rank one in the case that a is nonnegative and i is an open interval such that a i consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of b in i the general result is illustrated with eigenvalue estimates for singular indefinite sturm liouville problems

perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations abstract for selfadjoint matrices in an indefinite inner product possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation genericity is understood in the sense of algebraic geometry special attention is paid to the perturbation behavior of the sign characteristic typically under such a perturbation for every given eigenvalue the largest jordan block of the eigenvalue is destroyed and in case the eigenvalue is real all other jordan blocks keep their sign characteristic the new eigenvalues i e those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix are typically simple and in some cases information is provided about their sign characteristic if the new eigenvalue is real the main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product a version of the brunovsky canonical form and on general results concerning rank one perturbations obtained

on error estimates and adaptivity in elastoplastic solids applications to the numerical simulation of strain localization in classical and cosserat continua the a posteriori error estimates based on the post processing approach are introduced for elastoplastic solids the standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy this is known to represent a natural metric for the class of elastoplastic problems of evolution in addition the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates a posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem as a model problem the constitutive equations describing a generalization of standard j2 elastoplasticity within the cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post instability region the proposed a posteriori error estimates are appropriately modified to account for the cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems superior behaviour of the cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material resulting in convergent solutions with an h refinement and almost uniform error distribution in all considered error norms

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

analysis of transfer procedures in elastoplasticity based on the error in the constitutive equations theory and numerical illustration the aim of this work is to illustrate a methodology for the assessment of adaptive strategies for the solution of associative rate independent plasticity problems solved by employing the incremental displacement conforming finite element method this is the first step towards a more rational definition of transfer operators in terms of the ensuing error the motivating idea is the observation that change of data and or finite element mesh from one time interval to the other can be both related to a discontinuity jump of the approximate solution across the time instant t n thus reliable a posteriori estimates will have to depend not only on the time step and finite element mesh size but also on the value of the jump a new error estimate based on the error in the constitutive equations is developed which allows characterization of the discontinuity jump

generalized bezoutian and matrix equations abstract a natural generalization of the classical bezout matrix of two polynomials is introduced for a family of several matrix polynomials the main aim of the paper is to show that this generalized bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials m y z l r and the class of linear matrix equations ax xb c each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized bezoutian corresponding to a solution y z of the equation in matrix polynomials is a solution of the matrix equation and conversely any solution x of the matrix equation is a generalized bezoutian corresponding to a certain solution of the equation in matrix polynomials in particular either both equations are solvable or both have no solutions explicit formulas connecting the solutions of the two equations are given also various representation formulas for the generalized bezoutian are derived and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed

bezout and hankel matrices associated with row reduced matrix polynomials barnett type formulas abstract based on the approach introduced by b d o anderson and e i jury in 1976 the definition of finite hankel and bezout matrices corresponding to matrix polynomials is extended to the case where the denominator of the corresponding rational matrix function is not necessarily monic but is row reduced the matrices introduced keep most of the well known properties that hold in the monic case in particular we derive extensions of formulas giving a connection with polynomials in the companion matrix usually called barnett formulas of the inversion theorem and of formulas concerning alternating products of hankel and bezout matrices

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

on inversion of toeplitz and close to toeplitz matrices abstract it is proved that three columns of an inverse of a scalar toeplitz matrix when properly chosen are always enough to reconstruct it a parametrization of invertible toeplitz matrices in terms of the first and a combination of two consecutive columns of the inverse matrix is also given

on the reproducing kernel of a pontryagin space of vector valued polynomials abstract we give necessary and sufficient conditions under which the reproducing kernel of a pontryagin space of d 1 vector polynomials is determined by a generalized nevanlinna pair of d d matrix polynomials

bezoutian and schur cohn problem for operator polynomials abstract a generalized bezout operator bezoutian for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials the location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle infinite dimensional version of the schur cohn problem is expressed via the inertia of a suitable bezoutian an application to the geometric dichotomy problem for difference equations with operator coefficients is given as well

matrix representations of bezoutians abstract an approach is presented to get matrix representations for classical and more general r bezoutians with entries from a noncommutative ring formulas due to ammar and gader barnett gohberg and semencul gohberg and heinig kravitsky and russakovskii lerer and tismentsky ptak and wimmer as well as some generalizations and modifications are obtained in a unified way

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

arbitrary lagrangian eulerian finite element analysis of strain localization in transient problems non local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint the non locality prevents strain localization with zero global dissipation of energy and consequently finite element calculations converge upon mesh refinements to non zero width localization zones one of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the internal length hence the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity this paper deals with the application of the arbitrary lagrangian eulerian ale formulation well known in hydrodynamics and fluid structure interaction problems to transient strain localization in a non local damageable material it is shown that the ale formulation which is employed in large boundary motion problems can also be well suited for non linear transient analysis of softening materials where localization bands appear the remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable two well known one dimensional examples illustrate the capabilities of this technique the first one deals with localization due to a propagating wave in a bar and the second one studies the wave propagation in a hollow sphere

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

on bezoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks abstract the notion of bezoutian of nonsquare matrix polynomials is defined it is used to establish formulae for the inverse of square matrices partitioned into nonsquare blocks in particular a generalization of the gohberg semencul formula for the inverse of a generalized toeplitz matrix is proved

a lookahead algorithm for the solution of block toeplitz systems abstract we give a weakly stable algorithm to solve a block toeplitz system of linear equations if the lookahead steps taken to compute the parameters of the inversion formula for the block toeplitz matrix are small compared to the order n of the matrix the algorithm requires o n2 floating point operations the parameters of the inversion formula are interpreted and computed in a recursive way as solutions of certain interpolation problems given the formal laurent series based on the data of the block toeplitz matrix

a secular equation for the eigenvalues of a diagonal matrix perturbation let d denote a diagonal n n complex matrix and suppose x1 xr and w1 wr are complex n vectors it is shown that there is a rational function f such that if is not an eigenvalue for d then is an eigenvalue for p d x 1w1 x rwr if and only if f 0 this generalizes a well known result for the eigenvalues of a rank one self adjoint perturbation an immediate corollary in the rank one self adjoint case is that the eigenvalues of p and d must interlace if the eigenvalues of d are distinct and the perturbation matrix is irreducible it is shown that in the general case the function f also carries information about the eigenvalues of p for example is an eigenvalue of multiplicity m 0 for p if and only if f f f m 1 0 and f m 0 in the self adjoint case a necessary and sufficient condition for the eigenvalues of p and d to interlace is given and the problem of determining the multiplicities of the eigenvalues of d as eigenvalues of p is studied the formula yields a simple algorithm for determining the characteristic polynomial of a tridiagonal matrix

the theory of v bezoutians and its applications abstract some propositions of the theory of v bezoutians are presented as an application the following problem is considered to construct an appropriate pontryagin space in which a given matrix boundary value problem containing the eigenvalue parameter in the boundary conditions is self adjoint the solution of this problem is given in three special cases with the help of v bezoutians of matrix polynomials entering the boundary conditions

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