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

element wise a posteriori estimates based on hierarchical bases for non linear parabolic problems we show that the issue of a posteriori estimate the errors in the numerical simulation of non linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right hand side depending on known data of the problem and the computed numerical solution a procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right hand side the residual of the p fem solution is introduced the boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs we prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation when the error being estimated is that of a elliptic problem constant free local lower bounds are obtained the local error estimation procedure is applied to non linear parabolic differential equations in several space dimensions some numerical experiments for both the elliptic and the non linear parabolic cases are provided copyright 2005 john wiley sons ltd

hp discontinuous galerkin time stepping for parabolic problems abstract we consider the hp version of the discontinuous galerkin dg time stepping method for linear parabolic problems with non symmetric elliptic spatial operators we derive new analyticity estimates for the exact solutions by means of semigroup techniques these estimates allow us to show that the hp dg time stepping method can resolve start up singularities at exponential rates of convergence

a multiscale edge detection algorithm based on wavelet domain vector hidden markov tree model abstract the wavelet analysis is an efficient tool for the detection of image edges based on the wavelet analysis we present an unsupervised learning algorithm to detect image edges in this paper a wavelet domain vector hidden markov tree wd vhmt is employed in our algorithm to model the statistical properties of multiscale and multidirectional subband wavelet coefficients of an image with this model each wavelet coefficient is viewed as an observation of its hidden state and the hidden state indicates if the wavelet coefficient belongs to an edge the wd vhmt model can be learned by an expectation maximization algorithm after the model is learned we employ an extended viterbi algorithm to uncover the hidden state sequences according to the maximum a posterior estimation the experiment results of the edge detection for several images are provided to evaluate our algorithm

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

residual analysis for feature detection it is shown that in a very simple form residual analysis achieves results that are at least as good as if not better than those obtained by other techniques there are many ways for extensions of the method for example moving average filters of regularization can be used to obtain the residual images also the strength of the correlation measured by d sub rr o can be used to eliminate noise weak edges etc a more ambitious extension is by considering smoothing filters that leave invariant the function representing the reflectance from smooth surfaces

a posteriori finite element bounds for output functionals of discontinuous galerkin discretizations of parabolic problems we present a neumann subproblem a posteriori finite element procedure for the efficient calculation of constant free sharp lower and upper estimators for linear functional outputs of parabolic equations discretized by a discontinuous galerkin method in time in space a global coarse mesh and a decoupled fine mesh are used to compute the estimators which are shown to converge to the value of the output obtained for a global coupled fine mesh we first formulate the bound procedure with particular emphasis on the proof of the bounding properties we then provide an illustrative numerical example a problem of heat conduction in a composite material

maximum principles for the periodic boundary value problem for impulsive integro differential equations abstract this paper studies a first order periodic boundary value problem for impulsive integro differential equations and establishes some new maximum principles some points of a recent paper are also corrected and extended

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

periodic boundary value problems for impulsive integro differential equations of mixed type by developing a comparison result and using the monotone iterative technique we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first order nonlinear impulsive integro differential equations of mixed type

periodic boundary value problem for the second order impulsive functional differential equations this paper considers the existence of extreme solutions of the periodic boundary value problem for second order functional differential equations by using the method of lower and upper solutions coupled with monotone iterative technique

extremal solutions of periodic boundary value problems for first order integro differential equations of mixed type this paper investigates the maximal and minimal solutions of periodic boundary value problems for first order nonlinear integro differential equations of mixed type by establishing a comparison result and using the monotone iterative technique

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 edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

design of efficient m band coders with linear phase and perfect reconstruction properties this paper presents a new design technique for obtaining m band orthogonal coders where m 2 sup i the structures obtained using the proposed technique have the perfect reconstruction property furthermore all filters that constitute the subband coder are linear phase fir type filters in contrast with conventional design techniques that attempt to find a unitary alias component matrix in the frequency domain we carry out the design in the time domain based on time domain orthonormality constraints that the filters must satisfy the m band design problem is reduced to the problem of finding a suitable lowpass filter h sub 0 n once a suitable lowpass filter is found the remaining m 1 filters of the coder are obtained through the use of shuffling operators on the lowpass filter this approach leads to a set of filters that use the same numerical coefficient values in different shift positions allowing very efficient numerical implementation of the subband coder in addition by imposing further constraints on the lowpass branch impulse response h sub 0 n we are able to construct continuous bases of m channel wavelets with good regularity properties design examples are presented for four eight and 16 band coders along with examples of continuous wavelet bases that they generate

periodic boundary value problems for second order impulsive integro differential equations by developing a new comparison result and using the monotone iterative technique we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second order nonlinear impulsive integro differential equations of mixed type

periodic boundary value problem for the second order impulsive functional differential equations this paper considers the existence of extreme solutions of the periodic boundary value problems for the second order functional differential equations the novelty is that we introduce a new concept for lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid meanwhile we extend previous results

multidirectional and multiscale edge detection via m band wavelet transform in this correspondence the problem of directional and multiscale edge detection is considered orthogonal and linear phase m band wavelet transform is used to decompose the image into m spl times m channels these channels are then combined such that each combination which we refer to as decomposition filter results in zero crossings at the locations of edges corresponding to different directions and resolutions and inherently performs regularization against noise by applying a zero crossing detector on the outputs of the decomposition filters edge maps of desired resolution and direction are obtained in addition with the application of the teager s energy operator at the analysis stage it is possible to obtain a reduction in unwanted zero crossings final edge maps of images are obtained through simple combinations of directional edge maps

remarks on periodic boundary value problems for functional differential equations we extend some results on existence and approximation of solution for a class of first order functional differential equations with periodic boundary conditions we show the validity of the monotone iterative technique under weaker hypotheses and present some examples

periodic boundary value problems for delay differential equations with impulses existence and approximation of solutions for an impulsive delay differential equation with periodic boundary value conditions are presented we use comparison principles and monotone iterative techniques

a posteriori error estimation for the semidiscrete finite element method of parabolic differential equations in this paper three a posteriori error estimators of the error in the semidiscrete finite element solution discrete in space and continuous in time of parabolic partial differential equations are analyzed this approach is based on a posteriori error estimators for the elliptic pdes it is proven that guaranteed resp asymptotically exact a posteriori error estimator for the elliptic problem yields the guaranteed resp asymptotically exact estimator for the parabolic problem

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

numerical model reduction with error control in computational homogenization of transient heat flow numerical model reduction nmr is exploited for solving the finite element problem on a representative volume element rve that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the classical method of spectral decomposition a symmetrized version of the space time variational format is adopted for estimating the error from the model reduction in i energy norm and in ii given quantities of interest this technique which was recently developed in the context of the non selfadjoint stationary diffusion convection problem is novel in the present context of nmr by considering the discrete unreduced model as exact we are able to obtain guaranteed bounds on the error while using only the reduced basis and with minor computational effort the performance of the error estimates is demonstrated via numerical results where the subscale is modeled in both one and three spatial dimensions

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

periodic lienard type delay equations with state dependent impulses abstract conditions are obtained for lienard type equations with delay and state dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation and consequently with lebesgue integrable second derivative the results are applied to josephson s equation and the nonconservative forced pendulum equation

bounds of functional outputs for parabolic problems part ii bounds of the exact solution the paper introduces a methodology to compute upper and lower bounds for linear functional outputs of the exact solutions of parabolic problems in this second part the bounds account for the error both in space and time the assumption stating that the error introduced by the time marching scheme is negligible used in the first part is removed here the bounds are computed starting from an approximation of the exact solution associated with a spatial mesh and a time grid nevertheless the bounds are guaranteed with respect to the exact solution with no reference to any mesh or time discretization

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

periodic boundary value problems for second order impulsive integro differential equations we consider a class of second order impulsive differential equations by developing a new maximum principle and using the monotone iterative technique uniqueness of solution between a lower and an upper solution is presented we give conditions for existence of extremal solutions in an interval delimited by a lower and an upper solution

bounds of functional outputs for parabolic problems part i exact bounds of the discontinuous galerkin time discretization classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest consequently the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh exact bounds that is bounds guaranteed with respect to the exact solution are needed to properly certify the accuracy of the results especially if the meshes are coarse the paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems in this first part the bounds account for the error associated with the spatial discretization the error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error the time discretization is performed using the discontinuous galerkin method both for the primal and adjoint problems in the error estimation procedure equilibrated fluxes at interelement edges are calculated using hybridization techniques

periodic boundary value problem for non lipschitzian impulsive functional differential equations we consider a class of first order impulsive functional differential equations where the functional dependence is not necessarily a lipschitzian function the new maximum principle improves and extends previous results and uniqueness of solution between a lower and an upper solution for a particular nonlinear problem is presented we give conditions for existence of extremal solutions in an interval delimited by a lower and an upper solution

a posteriori analysis of the finite element discretization of some parabolic equations nous considerons la discretisation d equations paraboliques soit lineaires soit semi lineaires par un schema d euler implicite en temps et par elements finis en espace l idee de cet article est de construire des indicateurs d erreur lies a l approximation en temps et en espace et de prouver leur equivalence avec l erreur dans le but de travailler avec des pas de temps adaptatifs et des maillages d elements finis adaptes a la solution

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

robust image watermarking based on multiband wavelets and empirical mode decomposition in this paper we propose a blind image watermarking algorithm based on the multiband wavelet transformation and the empirical mode decomposition unlike the watermark algorithms based on the traditional two band wavelet transform where the watermark bits are embedded directly on the wavelet coefficients in the proposed scheme we embed the watermark bits in the mean trend of some middle frequency subimages in the wavelet domain we further select appropriate dilation factor and filters in the multiband wavelet transform to achieve better performance in terms of perceptually invisibility and the robustness of the watermark the experimental results show that the proposed blind watermarking scheme is robust against jpeg compression gaussian noise salt and pepper noise median filtering and con vfilter attacks the comparison analysis demonstrate that our scheme has better performance than the watermarking schemes reported recently

computing bounds for linear functionals of exact weak solutions to poisson s equation we present a method for poisson s equation that computes guaranteed upper and lower bounds for the values of piecewise polynomial linear functional outputs of the exact weak solution of the infinite dimensional continuum problem with piecewise polynomial forcing the method results from exploiting the lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem localization is achieved by lagrangian relaxation and the bounds are computed by appeal to a local dual problem the proposed method computes approximate lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers the computed bounds hold uniformly for any level of refinement and in the asymptotic convergence regime of the finite element method the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution given a finite element solution and its output adjoint solution the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization the elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators as we demonstrate with a numerical example

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

on error controlled numerical model reduction in fe2 analysis of transient heat flow numerical model reduction is exploited for solving the two scale problem that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the method of spectral decomposition a key feature is the construction of a symmetrized version of the space time variational format by which it is possible to estimate the error from the model reduction in i energy norm and in ii a given quantity of interest in previous work by the authors this strategy has been applied to the solution of an individual representative volume element problem whereas the error transport to the macroscale problem in the finite element squared fe2 approach was ignored in this paper such transport of error is included in the error estimate it is remarkable that it is still possible to obtain guaranteed bounds on the error as compared to the fully resolved discrete finite element problem while using only the reduced basis and with minor extra computational effort the performance of the error estimates is demonstrated via numerical results whereby the subscale is modeled in three dimensions whereas the macroscale problem is either one or two dimensional

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

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

a shearlet approach to edge analysis and detection it is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges in this paper we propose a novel approach based on the shearlet transform a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges indeed unlike traditional wavelets shearlets are theoretically optimal in representing images with edges and in particular have the ability to fully capture directional and other geometrical features numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges and outperforms methods based on wavelets as well as other standard methods furthermore the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions

guaranteed a posteriori error estimation for fully discrete solutions of parabolic problems this paper addresses the construction of guaranteed computable estimates for fully discrete solutions of parabolic problems copyright 2003 john wiley sons ltd

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

a level set method based on the bayesian risk for medical image segmentation this paper proposes an alternative criterion derived from the bayesian risk classification error for image segmentation the proposed model introduces a region based force determined through the difference of the posterior image densities for the different classes a term based on the prior probability derived from kullback leibler information number and a regularity term adopted to avoid the generation of excessively irregular and small segmented regions compared with other level set methods the proposed approach relies on the optimum decision of pixel classification and the estimates of prior probabilities thus the approach has more reliability in theory and practice experiments show that the proposed approach is able to extract the complicated shapes of targets and robust for various types of medical images moreover the algorithm can be easily extendable for multiphase segmentation

error estimation including pollution assessment for nonlinear finite element analysis a residual type error estimator for nonlinear finite element analysis is introduced this error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure pollution errors are taken into account by a feedback strategy that is an error estimate based on local computations is used as the input of the pollution analysis this estimator is used in the frame of an adaptive procedure numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions moreover one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms

computing upper and lower bounds for the j integral in two dimensional linear elasticity we present an a posteriori method for computing rigorous upper and lower bounds for the j integral in two dimensional linear elasticity the j integral which is typically expressed as a contour integral is recast as a quadratic continuous functional of the displacement involving only area integration by expanding the quadratic output about an approximate finite element solution the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error the quadratic component is bounded by the energy norm of the error scaled by a continuity constant which is determined explicitly the linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields the method is illustrated with two fracture problems in plane strain elasticity an important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations