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

modal based goal oriented error assessment for timeline dependent quantities in transient dynamics this article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics in particular a new type of quantities of interest referred as timeline dependent quantities is proposed these quantities are scalar time dependent outputs of the transient solution which are better suited to time dependent problems than the standard scalar ones frozen in time the proposed both the standard scalar and the new timeline dependent quantities of interest the key ingredient is the modal based approximation of the associated adjoint problems which allows efficiently computing and storing the adjoint solution the approximated adjoint solution is readily post processed to produce an enhanced solution requiring only one spatial post process for each vibration mode and using the time harmonic hypothesis to recover the time dependence thus the proposed goal oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem the resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem thus the cost of the error estimate at each time step is very low

error bars and quadratically convergent methods for the numerical simulation of the hartree fock equations this paper presents an a posteriori error analysis of the discretization methods used in computational quantum chemistry on the hartree fock equations upper and lower bounds for the energy are obtained from any discrete approximation strategy of the solution and the estimator proposed is shown to possess further approximation virtues

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

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

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

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

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

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

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

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

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

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

the interior radon transform the interior radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured this problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the x ray dose the corresponding operator mathcal r i is investigated as a mapping between weighted l 2 spaces the main result is a singular value decomposition svd for this operator for functions of unbounded support in mathbb r 2 the proof is based on the construction of intertwining differential operators the techniques used are unified in the sense that svds for other radon transforms with rotational symmetry can easily be derived in the same way consistency conditions are also obtained for mathcal r i for functions of bounded and unbounded support many of the results generalize to higher dimensions they are stated whenever they follow directly from the two dimensional case

strategies for computing goal oriented a posteriori error measures in non linear elasticity we investigate the characteristics and performance of goal oriented a posteriori error measures for a class of non linear elasticity models while restriction is made to small strain theory the chosen error measure of the displacement field can be global or local probing the chosen quantity in a specific spatial point the error is computable with the aid of the solution of a dual problem whose data depend on the error measure the main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution the chosen strategies are compared in terms of accuracy ease of implementation reliability and cost efficiency a well known numerical example the cook s membrane is used for the numerical evaluations

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

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

asymptotic a posteriori finite element bounds for the outputs of noncoercive problems the helmholtz and burgers equations we describe an a posteriori finite element procedure for the efficient computation of lower and upper estimators for linear functional outputs of noncoercive linear and semilinear elliptic second order partial differential equations under a relatively weak hypothesis related to the relative magnitude of the l2 and h1 errors of the reconstructed solution these lower and upper estimators converge to the true output from below and above respectively and thus constitute asymptotic bounds in numerical experiments we find that our hypothesis is satisfied once the finite element triangulation even roughly resolves the structure of the exact solution and thus in practice the bounds prove quite reliable numerical results are presented for the one dimensional helmholtz equation and for the burgers equation

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

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

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

goal oriented adaptivity using unconventional error representations for the multidimensional helmholtz equation in goal oriented adaptivity goa the error in the quantity of interest qoi is represented using the error functions of the direct and adjoint problems this error representation is subsequently bounded above by element wise error indicators that are used to drive optimal refinements in this work we propose to replace in the error representation the adjoint problem by an alternative operator the main advantage of the proposed approach is that when judiciously selecting such alternative operator the corresponding upper bound of the error representation becomes sharper leading to a more efficient goa while the method can be applied to a variety of problems we focus here on two and threedimensional 2d and 3d helmholtz problems we show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p adaptive process we also provide guidelines for finding operators delivering sharp error representation upper bounds we further extend the results to a convectiondominated diffusion problem as well as to problems with discontinuous material coefficients finally we consider a sonic logging while drilling lwd problem to illustrate the applicability of the proposed method

self organized criticality in a forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f 0 provided that the time scales of free growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system there is evidence that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

new sampling scheme for region of interest tomography the wavelet localization technique was previously applied to the study of region of interest roi tomography it achieves a significant saving in the required projections if only a small region of a tomographic image is of interest in this paper we first show that with the same sampling scheme a simple interpolation applied to the samples can give a result at least as good as that using the original wavelet localization approach it implies that the use of the wavelet transform is not the key to the reduction of the sampling requirement in fact the quality of the reconstructed roi is largely determined by the structure of the sampling scheme rather than directly reducing the projection number the use of the wavelet theory permits a clear understanding of how to achieve a good sampling pattern based on an error analysis using the wavelet theory we further suggest a new sampling scheme such that the number of required projections in each angle is reduced in a multiresolution form a new multiresolution interpolation algorithm is then used to interpolate the missing samples to obtain the full projections as a result more than 84 of projections are saved as compared with the traditional approach in reconstructing an roi of 32 spl times 32 pixels in an image of 256 spl times 256 pixels a series of simulations was performed to reconstruct different sizes of the roi all results show that the signal to error ratios of the reconstructed roi are comparable with that using full projection data set

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

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

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

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

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

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 derivation of guaranteed and p robust a posteriori error estimates for the helmholtz equation we propose a novel a posteriori error estimator for conforming finite element discretizations of two and three dimensional helmholtz problems the estimator is based on an equilibrated flux that is computed through the solve of patchwise mixed finite element problems we show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase we also derive a fully computable upper bound on this prefactor for several common settings of domains and boundary conditions leading to a guaranteed estimate in all wavenumber regimes without any assumption on the mesh size or the polynomial degree we finally demonstrate that the estimator is locally efficient as soon as the mesh exhibits sufficiently many degrees of freedom per wavelength in such a regime the estimator is also robust with respect to the polynomial degree we present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp

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

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

applications of statistical mechanics to natural hazards and landforms abstract the concept of self organized criticality was introduced to explain the behavior of the cellular automata sandpile model a variety of multiple slider block and forest fire models have been introduced which are also said to exhibit self organized critical behavior it has been argued that earthquakes landslides forest fires and extinctions are examples of self organized criticality in nature the basic forest fire model is particularly interesting in terms of its relation to the critical point behavior of the site percolation model in the basic forest fire model trees are randomly planted on a grid of points periodically sparks are randomly dropped on the grid and if a spark drops on a tree that tree and the adjacent trees burn in a model fire in the forest fire model there is an inverse cascade of trees from small clusters to large clusters trees are lost primarily from model fires that destroy the largest clusters this quasi steady state cascade gives a power law frequency area distribution for both clusters of trees and smaller fires the site percolation model is equivalent to the forest fire model without fires in this case there is a transient cascade of trees from small to large clusters and a power law distribution is found only at a critical density of trees the earth s topography is an example of both statistically self similar and self affine fractals landforms are also associated with drainage networks which are statistical fractal trees a universal feature of drainage networks and other growth networks is side branching deterministic space filling networks with side branching symmetries are illustrated it is shown that naturally occurring drainage networks have symmetries similar to diffusion limited aggregation clusters

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

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

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

a kalman filtering approach to stochastic global and region of interest tomography we define two forms of stochastic tomography in global tomography the goal is to reconstruct an object from noisy observations of all of its projections in region of interest roi tomography the goal is to reconstruct a small portion of an object an roi from noisy observations of its projections densely sampled in and near the roi and sparsely sampled away from the roi we solve both problems by expanding the object and its projections in a circular harmonic fourier series in the angular variable so that the radon transform becomes abel transforms of integer orders applied to the harmonics the algorithm has three major components first we fit state space models to each order of abel transform and thus represent the radon transform operation as a parallel bank of systems each of which computes the appropriate abel transform of a circular harmonic a variable transformation here allows either the global or roi problem to be solved second the object harmonics are modeled as a brownian branch this is a two point boundary value system which is markovianized into a form suitable for the kalman filter finally a parallel bank of kalman smoothing filters independently estimates each circular harmonic from the noisy projection data numerical examples illustrate the proposed procedure

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

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

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

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

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

a posteriori finite element output bounds with adaptive mesh refinement application to a heat transfer problem in a three dimensional rectangular duct numerical simulations based on an a posteriori finite element bound method with adaptive mesh refinement are presented for the three dimensional convection diffusion equation the bound method provides relevant quantitative inexpensive and rigorous lower and upper bounds to the output on a very fine discretization truth discretization at a cost close to the coarse mesh calculation working discretization to achieve a desired bound gap i e difference between upper and lower bounds at the lowest cost an adaptive mesh refinement technique is used to refine the mesh only where needed an optimal stabilization parameter is also applied to improve the sharpness of the bound gap in this paper the output of a heat transfer problem in a rectangular duct with a given velocity field is investigated the average temperature at one section of the duct is bounded for a given inlet temperature and heat flux for this problem the adaptive mesh refinement strategy provides the same bound gap with only half the number of elements required by an uniform mesh refinement strategy the hybrid flux calculation on the coarse mesh introduced for the domain decomposition approach is compared with the hybrid flux calculation on the fine mesh to analyze the contribution of the hybrid flux to the bound gap

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

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

new results for self similar trees with applications to river networks in a little known series of papers beginning in 1966 tokunaga introduced an infinite class of tree graphs based on the strahler ordering scheme as recognized by tokunaga 1984 these trees are characterized by a self similarity property so we will refer to them as self similar trees or ssts ssts are defined in terms of a generator matrix which acts as a blueprint for constructing different trees many familiar tree constructions are absorbed as special cases however in tokunaga s work an additional assumption is imposed which restricts from ssts to a much smaller class we will refer to this subclass as tokunaga s trees this paper presents several new and unifying results for ssts in particular the conditions under which ssts have well defined horton strahler stream ratios are given as well as a general method for computing these ratios it is also shown that the diameters of ssts grow like m where m is the number of leaves in contrast to many other tree constructions here need not equal 1 2 thus ssts offer an explanation for hack s law finally it is demonstrated that large discrepancies exist between the predictions of shreve s well known model and detailed measurements for large river networks while other ssts fit the data quite well other potential applications of the sst framework include diffusion limited aggregation dla lightning bronchial passages neural networks and botanical trees

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

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

goal oriented adaptivity using unconventional error representations for the 1d helmholtz equation in this work the error of a given output functional is represented using bilinear forms that are different from those given by the adjoint problem these representations can be employed to design novel h p and h p energy norm and goal oriented adaptive algorithms numerical results in 1d show that for wave propagation problems the advantages of this new representation are notorious when selecting the laplace equation as the dual problem specifically the computed upper bounds of the new error representation are sharper than the classical ones used in both energy norm and goal oriented adaptive methods especially when the dispersion pollution error is significant

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

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

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

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

wavelet localization of the radon transform the authors develop an algorithm which significantly reduces radiation exposure in x ray tomography when a local region of the body is to be imaged the algorithm uses the properties of wavelets to essentially localize the radon transform this algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image not the image module the nullspace of the local tomography operator or the lambda transform of the image this is possible because the authors do not truly invert the interior radon transform but rather sample the radon transform sparsely away from the local region of interest much attention in the field has been directed towards localized tomography the authors believe that this technique represents a significant contribution towards this effort

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

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

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

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

a quasi minimal residual method for simultaneous primal dual solutions and superconvergent functional estimates the adjoint solution has found many uses in computational simulations where the quantities of interest are the functionals of the solution including design optimization error estimation and control in those applications where both the solution and the adjoint are desired the conventional approach is to apply iterative methods to solve the primal and dual problems separately however we show that there is an advantage associated with iterating the primal and dual problem simultaneously since this enables the construction of iterative methods where both the primal and the dual iterates may be chosen so that they provide functional estimates that are superconvergent in that the error converges at twice the order of the optimal global solution error norm in particular we show that the structure of the lanczos process allows for this superconvergence property and propose a modified qmr method which uses the same lanczos process to simultaneously solve the primal and dual problems thus both the primal and the dual systems are solved at essentially the same computational cost as the conventional qmr method applied to the primal problem alone numerical experiments show that our proposed method does indeed exhibit superconvergence behavior

combinatorial tools for the analysis of ramified patterns tree like patterns appear in many domains of physics and the quantitative description of their morphology raises an interesting problem to analyze their topological structure we introduce combinatorial concepts the bifurcation and length ratios and the ramification matrix which generalize ideas originating in hydrogeology two dimensional diffusion limited aggregation dla patterns are studied along these lines and their statistical combinatorial properties are compared to those of random and growing binary trees and to experimental data for injection of water in clay

a hankel transform approach to tomographic image reconstruction a relatively unexplored algorithm is developed for reconstructing a two dimensional image from a finite set of its sampled projections the algorithm referred to as the hankel transform reconstruction htr algorithm is polar coordinate based the algorithm expands the polar form fourier transform f r theta of an image into a fourier series in theta calculates the appropriately ordered hankel transform of the coefficients of this series giving the coefficients for the fourier series of the polar form image f p phi resolves this series giving a polar form reconstruction and interpolates this reconstruction to a rectilinear grid the htr algorithm is outlined and it is shown that its performance compares favorably to the popular convolution backprojection algorithm

a posteriori finite element output bounds in three space dimensions using the feti method abstract a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solution to the convection diffusion equation is presented the novelty is to utilize a domain decomposition method with lagrange multipliers and an iterative interface solution scheme the finite element tearing and interconnecting feti procedure to extend the bound method to three space dimensions numerical results show that the bounds are sharp their calculation consumes only a fraction of the memory required by a sparse solver and cpu usage is optimal i e proportional to the total number of d o f

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

region of interest tomography using exponential radial sampling the authors combine several ideas including nonuniform sampling and circular harmonic expansions into a new procedure for reconstructing a small region of interest roi of an image from a set of its projections that are densely sampled in the roi and coarsely sampled outside the roi specifically the radial sampling density of both the projections and the reconstructed image decreases exponentially with increasing distance from the roi the problem and data are reminiscent of the recently formulated local tomography problem however the authors algorithm reconstructs the roi of the image itself not the filtered version of it obtained using local tomography the new algorithm has the added advantages of speed it can be implemented entirely using the fft and parallelizability each image harmonic is computed independently numerical examples compare the new algorithm to filtered backprojection

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

an inverse cascade model for self organized critical behavior we introduce an inverse cascade model to explain self organized critical behavior this model is motivated by the forest re model in the forest re model trees are randomly planted on a grid sparks are also dropped on the grid resulting in res in which trees are lost in the inverse cascade model single trees are introduced and these combine to form larger and larger clusters this is the inverse cascade and gives a power law fractal frequency size distribution of clusters model res eliminate trees from all cluster sizes but signicant numbers of trees are lost only from the largest clusters and this loss terminates the power law scaling finally our model illustrates important dierences between critical and self organized critical behavior c 1999 elsevier science b v all rights reserved

the background of error estimation and adaptivity in finite element computations the paper summarizes the present state of the art concerning error estimation and adaptive remeshing recent work shows that methods based on recovery result on most accurate error estimators

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985