further results for enhanced strain methods with isoparametric elements abstract in this contribution the enhanced strain finite element method is investigated further with particular attention given to the analysis of the method for isoparametric elements it is shown that the results established earlier b d reddy and j c simo stability and convergence of a class of enhanced strain methods siam j numer anal 32 1995 in press for affine equivalent meshes carry over to the case of isoparametric elements that is the method is stable and convergent provided that a set of three criteria is satisfied and for plane elements at least convergence is at the same rate as in the standard method a procedure for recovering the stress is shown to lead to an approximate stress which converges at the optimal rate to the actual stress with regard to computations a new basis for enhanced strains is introduced and numerical results are presented for this basis as well as for existing bases the results for enhanced strain methods are distinguished by the fact that the constant appearing in asymptotic error estimates is much smaller for these methods than for standard approaches

some novel developments in finite element procedures for gradient dependent plasticity summary improved algorithms are proposed for a gradient plasticity theory in which the laplacian of an invariant plastic strain measure enters the yield function particular attention is given to the type of finite elements that can be used within the format of gradient dependent plasticity assuming a weak satisfaction of the yield function mixed finite elements are developed in which the invariant plastic strain measure and the displacements are discretized two families of finite elements are developed one in which the invariant plastic strain measure is interpolated using c continuous polynomials and one in which penalty enhanced c continuous interpolants are used the performance of both families of finite elements is assessed numerically in one dimensional and two dimensional boundary value problems the regularizing effect of the used gradient enhancement in computations of elastoplastic solids is demonstrated both for mesh refinement and for the directional bias of the grid lines

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

well posedness and numerical performances of the strain gap method a mixed method of approximation is discussed starting from a suitably modified expression of the hu washizu variational principle in which the independent fields are displacements stresses and strain gaps defined as the difference between compatible strains and strain fields the well posedness of the discrete problem is discussed and necessary and sufficient conditions are provided the analysis of the mixed method reveals that the discrete problem can be split into a reduced problem and in a stress recovery accordingly the discrete stress solution is univocally determined once an interpolating stress subspace is chosen the enhanced assumed strain method by simo and rifai is based on an orthogonality condition between stresses and enhanced strains and coincides with the reduced problem it is shown that the mixed method is stable and converges computational issues in the context of the finite element method are discussed in detail and numerical performances and comparisons are carried out copyright 2001 john wiley sons ltd

a consistent characteristic length for smeared cracking models a numerical scheme for crack modelling by means of continuous displacement fields is presented in two dimensional problems a crack is modelled as a limiting case of two singular lines with continuous displacements but discontinuous displacement gradients across them which tend to coincide with each other an analysis of the energy dissipated inside the band bounded by both lines allows one to obtain an expression for the characteristic length as the ratio between the energy dissipated per unit surface area fracture energy and the energy dissipated per unit volume specific energy at a point the application of these mathematical expressions to the finite element discretized medium allow one to obtain a general spatial and directional expression for the characteristic length which guarantees the objectivity of the results with respect to the size of the finite element mesh the numerical results presented show the reliability of the proposed expressions

analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials large strain theory for standard dissipative solids we propose a fundamentally new approach to the treatment of shearband localizations in strain softening elastic plastic solids at finite strains based on energy minimization principles associated with microstructure developments the point of departure is a general internal variable formulation that determines the finite inelastic response as a standard dissipative medium consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasi hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables the existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments of finite elasticity furthermore localization phenomena are interpreted as microstructure developments on multiple scales associated with non convex incremental stress potentials in analogy to elastic phase decomposition problems these microstructures can be resolved by the relaxation of non convex energy functionals based on a convexification of the stress potential the relaxed problem provides a well posed formulation for a mesh objective analysis of localizations as close as possible to the non convex original problem based on an approximated rank one convexification of the incremental stress potential we develop a computational two scale procedure for a mesh objective treatment of localization problems at finite strains it constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the microshearing of a rank one laminate aligned to the shear band this problem is sufficiently robust with regard to applications to large scale inhomogeneous deformation processes of elastic plastic solids the performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which report on the mesh independence of the results copyright 2003 john wiley sons ltd

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

localisation in a cosserat continuum under static and dynamic loading conditions abstract localisation studies have been carried out for an unconstrained elasto plastic strain softening cosserat continuum because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement a finite constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions because of the existence of rotational degrees of freedom in a cosserat continuum additional wave types arise and wave propagation becomes dispersive this has been investigated analytically and numerically for an elastic cosserat continuum and an excellent agreement has been found between both solutions

stability and convergence of a class of enhanced strain methods a stability and convergence analysis is presented of a recently proposed variational formulation and finite element method for elasticity which incorporates an enhanced strain field the analysis is carried out for problems posed on polygonal domains in r n the finite element meshes of which are generated by affine maps from a master element the formulation incorporates as a special case the classical method of incompatible modes the problem initially has three variables viz displacement stress and enhanced strain but the stress is later eliminated by imposing a condition of orthogonality with respect to the enhanced strains two other conditions on the choice of finite element spaces ensure that the approximations are stable and convergent some features of nearly incompressible and incompressible problems are also investigated for these cases it is possible to argue that locking will not occur and that the only spurious pressures present are the so called checkerboard modes it is shown th

efficient modeling of localized material failure by means of a variationally consistent embedded strong discontinuity approach this paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite strains focus is on localized plastic deformation particularly relevant for slip bands in contrast to already existing models the proposed implementation allows to consider several interacting discontinuities in each finite element based on a proper re formulation of the kinematics an efficient parameterization of the deformation gradient is derived it permits to compute the strains explicitly that improves the performance significantly however the most important novel contribution of the present paper is the advocated variational constitutive update within this framework every aspect is naturally driven by energy minimization i e all unknown variables are jointly computed by minimizing the stress power the proposed update relies strongly on an extended principle of maximum dissipation this framework provides enough flexibility for different failure types and for a broad class of non associative evolution equations by discretizing the aforementioned continuous variational principle an efficient numerical implementation is obtained it shows in addition to its physical and mathematical elegance several practical advantages for instance the physical minimization principle itself specifies automatically and naturally the set of active strong discontinuities copyright 2011 john wiley sons ltd

finite element methods for the multi scale modeling of softening hinge lines in plates at failure abstract this paper presents a multi scale framework for the modeling of softening hinge lines in inelastic plates and its finite element implementation the proposed framework considers the reissner mindlin problem of thick plates as the large scale problem with the hinge introduced locally in small neighborhoods modeling the small scales of the material response the hinges are defined as strong discontinuities of the generalized displacements describing the plate deformations the consideration of the large scale limit of vanishing small scales leads to a large scale model of these localized failures of plates involving the solution of a problem in terms of those large scale generalized displacements with the standard regularity conditions enhanced finite elements define the natural framework for the numerical implementation of these considerations the new finite elements are enhanced with the singular strain fields associated with the discontinuities or hinge lines which given their local character are statically condensed at the element level we present a general strategy for the definition of these local strain fields so the kinematics of the hinge lines are captured correctly allowing in particular their fully softened state with no stresses and hence avoiding the so called stress locking we present several particular finite elements formulated in this framework including triangular and quadrilateral finite elements several numerical simulations are presented illustrating their performance

a formulation for the 4 node quadrilateral element a formulation for the plane 4 node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body incompatible modes are added to the standard displacement field then expressions for gradient operators are obtained from an expansion of the basis functions into a second order taylor series in the physical co ordinates the internal degrees of freedom of the incompatible modes are eliminated on the element level a modified change of variables is used to integrate the element matrices for a linear elastic material the element stiffness matrix can be separated into two parts these are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix the formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials further the approach is suitable for non linear analysis there an application is given for the calculation of inelastic problems in physically non linear elasticity the element is efficient to implement and it is frame invariant locking effects and zero energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion a high accuracy is obtained for numerical solutions in displacements and stresses

an analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids abstract this paper presents an analysis of strong discontinuities in inelastic solids at finite strains solutions exhibiting this type of discontinuities characterized by a discontinuous displacement field are shown to make physical and mathematical sense in a classical multiplicative plasticity continuum model if the softening modulus is reinterpreted as a singular distribution physically the strainsoftening is localized along the discontinuity conditions for the appearance of strong discontinuities in the geometrically nonlinear range are characterized as it is the response of the material during localization in addition these analytical results are exploited in the design of a new class of finite element methods the proposed methods fall within the class of enhanced strain methods and lead to solutions independent of the mesh size and insensitive to mesh alignment without requiring any regularization of the solutions by numerical parameters like a characteristic length

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

variational multiscale enrichment method with mixed boundary conditions for modeling diffusion and deformation problems abstract this manuscript presents the formulation and implementation of the variational multiscale enrichment vme method using canopy shaped microscale enrichment functions obtained through the use of a new family of microscale boundary conditions the purpose of the new enrichment functions and the new boundary condition is to relax the overconstraint imposed by the homogeneous microscale boundary condition e g residual free bubbles commonly employed in the variational multiscale literature the formulation and implementation of the method are presented for diffusion and elasticity problems the performance of the proposed method is assessed by comparing with direct numerical simulations on diffusion and deformation problems a boundary parameter identification approach is proposed to obtain near optimal boundary conditions the identification approach is verified in the context of the deformation response of particle reinforced composites

enhanced assumed strain elements and locking in membrane problems abstract the assumptions which guarantee convergence of eas elements first introduced by simo and rifai can be reduced to two postulates specifically the stability condition refers to a relaxed stability of an associated hu washizu principle we investigate the problem of volume locking of membrane problems estimates of a certain type which for instance imply volume locking of the q1 element are fortunately not present with eas elements finally we verify an equivalence of the eas concept with a hellinger reissner method which was observed in numerical computations by andelfinger and ramm

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

robust synchronization of uncertain linear multi agent systems this paper deals with robust synchronization of uncertain multi agent networks given a network with for each of the agents identical nominal linear dynamics we allow uncertainty in the form of additive perturbations of the transfer matrices of the nominal dynamics the perturbations are assumed to be stable and bounded in h norm by some a priori given desired tolerance we derive state space formulas for observer based dynamic protocols that achieve synchronization for all perturbations bounded by this desired tolerance it is shown that a protocol achieves robust synchronization if and only if each controller from a related finite set of feedback controllers robustly stabilizes a given single linear system our protocols are expressed in terms of real symmetric solutions of certain algebraic riccati equations and inequalities and also involve weighting factors that depend on the eigenvalues of the graph laplacian for undirected network graphs we show that within the class of such dynamic protocols a guaranteed achievable tolerance can be obtained that is proportional to the quotient of the second smallest and the largest eigenvalue of the laplacian we also extend our results to additive nonlinear perturbations with l2 gain bounded by a given tolerance

robust synchronization of coprime factor perturbed networks this paper deals with robust synchronization of directed and undirected multi agent networks with uncertain agent dynamics given a network with identical nominal dynamics we allow uncertainty in the form of coprime factor perturbations of the transfer matrix of the agent dynamics these perturbations are assumed to be stable and have h infinity norm that is bounded by an a priori given desired tolerance we derive state space equations for dynamic observer based protocols that achieve robust synchronization in the presence of such uncertainty we obtain an achievable interval i e an interval such that for each value of the tolerance contained in this interval there exists a robustly synchronizing protocol c 2016 elsevier b v all rights reserved

the schwarz alternating method in solid mechanics abstract we advance the schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics we prove that the schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well posed i e their corresponding energy density functions are quasi convex it is shown that the use of a newton type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems depending on the treatment of the dirichlet boundary conditions the first kind is a symmetric block diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain i e the off diagonal blocks are all zero and the coupling terms appear only on the right hand side the second kind is a nonsymmetric block system with off diagonal coupling terms several variants of the schwarz alternating method are proposed for the first kind of linear system including one in which the schwarz alternating iterations and the newton iterations are combined into a single scheme this version of the method is particularly attractive as it lends itself to a minimally intrusive implementation into existing finite element codes finally we demonstrate the performance of the proposed variants of the schwarz alternating method on several one dimensional and three dimensional examples

stabilized mixed finite element formulations for materially nonlinear partially saturated two phase media standard and enhanced bbar or eas low order finite elements applied to the problem of consolidation of two phase nonlinear continua do not satisfy the lbb condition when the same interpolation functions are used for both displacement and pore pressure fields strong spatial pressure oscillations are the main consequence of the lbb condition violation the class of direct stabilized methods widely used in the field of fluid mechanics is a powerful tool to circumvent violation of the aforementioned condition three stabilized formulations designed for the problem of consolidation of fully or partially saturated media are presented galerkin least squares gls with the least squares term construction based on the residuum of the local fluid mass conservation equation the pressure stabilized formulation fpl in which the rate of the pore pressure laplacian residual free in the incompressibility limit is added to the fluid mass conservation equation and finally a stabilized formulation in which the stabilization term is constructed based on the rate of the residuum of the local momentum equation an h convergence study and test problems for the three stabilized schemes are discussed

relations between enhanced strain methods and the hr method abstract the three field hu washizu functional is the starting point to develop a comparative analysis of approximate linear elastostatic problems in particular we will analyse the limitation phenomena concerning two mixed methods respectively denoted the strain gap method and the mixed enhanced strain method and the two field hellinger reissner hr finite element formulation accordingly two original limitation principles are contributed we provide the theoretical motivation of the numerical observations provided in computational literature on the coincidence of the displacement and stress solutions between the enhanced assumed strain method and the hr method if different choices of shape functions are made it is further shown that the hr method provides the exact stress solution and the exact nodal displacements with one element the related analytical motivation are thus given the theoretical analyses presented in the paper are supported by numerical examples on tipical benchmarks

a variational multiscale method to model crack propagation at finite strains this contribution presents a hierarchical variational multiscale framework to model propagating discontinuities at finite strains thereby the deformation map is decomposed into coarse scale and fine scale displacements which results in a decoupled system of coarse scale and fine scale equations both are solved numerically by means of the finite element method whereby crack propagation is taken into account at the fine scale growing cracks are numerically handled by the introduction of discontinuous elements a locality assumption on the fine scale solution and an adaptive scheme to resize the fine scale domain are introduced and demonstrated to increase the efficiency of the method copyright 2009 john wiley sons ltd

the finite element method for the mechanically based model of non local continuum summary in this paper the finite element method fem for the mechanically based non local elastic continuum model is proposed in such a model each volume element of the domain is considered mutually interacting with the others beside classical interactions involved by the cauchy stress field by means of central body forces that are monotonically decreasing with their inter distance and proportional to the product of the interacting volume elements the constitutive relations of the long range interactions involve the product of the relative displacement of the centroids of volume elements by a proper distance decaying function which accounts for the decrement of the long range interactions as long as distance increases in this study the elastic problem involving long range central interactions for isotropic elastic continuum will be solved with the aid of the fem the accuracy of the solution obtained with the proposed fem code is compared with other solutions obtained with galerkins approximation as well as with finite difference method moreover a parametric study regarding the effect of the material length scale in the mechanically based model and in the kroner eringen non local elasticity has been investigated for a plane elasticity problem copyright 2011 john wiley sons ltd

conditions for equivalence between the hu washizu and related formulations and computational behavior in the incompressible limit the relationship of the hu washizu mixed formulation to other mixed and enhanced formulations is examined in detail in the context of linear elasticity with a view to presenting a unified framework for such formulations the hu washizu formulation is considered in both its classical form and in a modified form that is suited to establishing well posedness in the incompressible limit recently established theoretical results on uniform well posedness are studied computationally by considering a range of numerical examples which illustrate where appropriate the good behavior predicted by the theory as well as the locking behavior that is evident in those problems in which the conditions for stability and convergence are not satisfied 2005 elsevier b v all rights reserved

a stable affine approximate finite element method the notion of the affine figure closest to a given quadrilateral can be given a precise mathematical definition the resulting figure is referred to as the equivalent parallelogram associated with a quadrilateral equipped with such a concept it is then feasible to consider finite element approximations in which the original quadrilateral elements are replaced by their equivalent parallelograms the idea is appealing not least because of the resulting economy arising from computations performed on an element generated by an affine map furthermore numerical experiments reported recently indicate that highly efficient and accurate schemes result when such a concept is combined with the enhanced strain method or the method of incompatible modes the purpose of this work is to analyze finite element schemes resulting from approximation of quadrilaterals by their equivalent parallelograms the focus is on low order bilinear elements and the analysis is carried out in the context of linear elasticity for standard approximations as well as for those which use enhanced strains the affine approximation applies only to the element map and the primary unknown the displacement vector in the context of elasticity is approximated by conventional piecewise bilinear functions the analysis confirms convergence at the optimal rate provided that the deviations of the quadrilaterals from their equivalent parallelograms are at most o h

a space time formulation for multiscale phenomena abstract we develop subgrid scale models for a class of nonsymmetric linear evolution operators by applying the variational multiscale method in space time the results generalize those of hughes 14 which were confined to the steady case the subgrid scale models are shown to be a paradigm for bubble function finite element methods and provide a theoretical and practical framework for the development of so called stabilized methods

on multiple burst correcting mds codes it is well known that binary random error correcting mds codes are trivial however the same is not true for burst correcting codes we present properties of mds codes for multiple burst correcting codes over the binary field generalizing those of random error correcting codes

the equivalent parallelogram and parallelepiped and their application to stabilized finite elements in two and three dimensions a new class of quadrilateral and hexahedral elements four and eight noded in two and three dimensions respectively is presented these elements are obtained by combining the concept of the equivalent parallelogram for plane problems and the equivalent parallelepiped for three dimensional problems with the notion of incompatible modes a key feature of the new elements is that integration of the element stiffness matrices is carried out using one point integration the use of affine equivalent elements parallelograms and parallelepipeds permits a closed form eigenvalue analysis which includes the incompatible modes and a stabilization procedure based on the eigenvalue analysis ensures the full rank of the stiffness matrix numerical results for problems in elasticity and plasticity indicate equivalent or superior performance of the new elements when compared with various elements based on enhanced strains or incompatible modes

a variational multiscale approach to strain localization formulation for multidimensional problems abstract the multiscale approach to strain localization problems developed earlier is generalized to higher dimensions the key idea is the identification of the fine scale field with a component of the displacement that has a large gradient the difference between the total displacement and the fine scale is the coarse scale field the formulation is variationally based with the weak form as the point of departure with suitable assumptions on the space of fine scale fields the weak form is separable into coarse and fine scale parts the weak form of the fine scale problem is used to eliminate the fine scales from the formulation this is achieved by way of a projection onto the fine scale space and allows a reformulation of the full problem in terms of coarse scale components alone in this framework fine scale interpolations are identified that ensure sharp resolution of localized displacement irrespective of the underlying finite element mesh it is demonstrated that by correctly accounting for the microstructural fine scale behavior the formulation results in solutions devoid of pathological mesh sensitivity for problems wherein the detailed structure of the localized displacement is of interest recovery of the fine scale components can be performed to reconstruct the entire displacement field several numerical examples are presented that demonstrate the robustness of the formulation

variational multiscale methods to embed the macromechanical continuum formulation with fine scale strain gradient theories a variational basis is presented to link fine scale theories of material behaviour with the classical macromechanical continuum theory the approach is based on the weak form of the linear momentum balance equations and a separation of the weighting function and displacement fields into coarse and fine scale components coarse and fine scale weak forms are defined the latter is used to introduce a strain gradient theory that operates at finer scales of deformation attention is focused upon applications requiring the enhanced physical accuracy of the fine scale strain gradient theory without the computational cost of discretization that spans the range from coarse to fine scales a variationally consistent method is developed to embed the fine scale strain gradient theory in the macromechanical formulation the embedding is achieved by eliminating the fine scale displacement field from the problem two examples demonstrate the numerical efficiency of the method while retaining physical and mathematical properties of the fine scale strain gradient theory copyright 2003 john wiley sons ltd

spectral analysis of the dissipation of the residual based variational multiscale method abstract we analyze the application of the residual based variational multiscale vms method to the solution of a generic partial differential equation we prove that the model terms introduced by this method are dissipative under certain restrictions that are likely to be satisfied by spectral approximations of conservation laws in particular we demonstrate that the fourier spectral approximations of burgers equation and the incompressible navier stokes equations satisfy these conditions hence we show that the residual based vms model for these equations is dissipative even though the form of the model term does not resemble an eddy viscosity we analyze the scale dependence of this dissipation using fully resolved numerical solutions and conclude that the vms model accurately captures the scale dependence of the cross stress terms u u and under predicts the contribution from the reynolds stress terms u u these observations have implications on the use of the residual based vms method in spectral approximations of nonlinear conservation laws

on an equivalence between a discontinuous galerkin method and reduced integration with hourglass stabilization for finite elasticity abstract discontinuous galerkin methods and methods of reduced integration with hourglass stabilization have different points of origin whereas the discontinuous galerkin idea was developed to investigate a wide range of applications among them problems of electrodynamics fluid and solid mechanics the reduced integration technology started out from the observation of the so called locking problem in linear elasticity today the two methodologies are still considered separately from each other although it has already been recognized that discontinuous galerkin methods can be also used to circumvent locking phenomena in elasticity the idea of the present contribution is to find a direct equivalence between a discontinuous galerkin method and a finite element technology based on reduced integration with hourglass stabilization the transfer between the two methods is carried out in the context of finite elasticity thus the derivation is valid for arbitrarily large deformations the main advantage of this analysis is that the choice of the free parameter in nitsche s method often also called stabilization parameter is no longer unclear but prescribed by the connection to the hourglass stabilization further important in this context is the replacement of the scalar parameter by a symmetric 2 2 matrix if two dimensional problems are considered which is here the case the result of the derivation is a discontinuous galerkin formulation which shows very good convergence behaviour not only in the limit of incompressibility but also for bending of thin structures

on the orthogonal subgrid scale pressure stabilization of finite deformation j2 plasticity the use of stabilization methods is becoming an increasingly well accepted technique due to their success in dealing with numerous numerical pathologies that arise in a variety of applications in computational mechanics in this paper a multiscale finite element method technique to deal with pressure stabilization of nearly incompressibility problems in nonlinear solid mechanics at finite deformations is presented a j2 flow theory plasticity model at finite deformations is considered a mixed formulation involving pressure and displacement fields is used as starting point within the finite element discretization setting continuous linear interpolation for both fields is considered to overcome the babuska brezzi stability condition a multiscale stabilization method based on the orthogonal subgrid scale osgs technique is introduced a suitable nonlinear expression of the stabilization parameter is proposed the main advantage of the method is the possibility of using linear triangular or tetrahedral finite elements which are easy to generate and therefore very convenient for practical industrial applications numerical results obtained using the osgs stabilization technique are compared with results provided by the p1 standard galerkin displacements linear triangular tetrahedral element p1 p1 standard mixed linear displacements linear pressure triangular tetrahedral element and q1 p0 mixed bilinear trilinear displacements constant pressure quadrilateral hexahedral element for 2d 3d nearly incompressible problems in the context of a nonlinear finite deformation j2 plasticity model

multiscale stabilized solutions to one dimensional systems of conservation laws abstract we present a variational multiscale formulation for the numerical solution of one dimensional systems of conservation laws the key idea of the proposed formulation originally presented by hughes comput methods appl mech engrg 127 1995 387 401 is a multiple scale decomposition into resolved grid scales and unresolved subgrid scales incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties capable of accurately representing the sharp features of the solution in the formulation developed herein the multiscale split is invoked prior to any linearization of the equations special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity capturing diffusion the methodology is applied to the one dimensional simulation of three phase flow in porous media and the shallow water equations these numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear nearly hyperbolic systems on very coarse grids application of the numerical formulation to multidimensional problems is presented in a forthcoming paper

a variational multiscale method to incorporate strain gradients in a phenomenological plasticity model we treat mathematical and computational issues related to the incorporation of strain gradient terms in a phenomenological plasticity model the strain gradients are associated with incompatibilities due to geometrically necessary dislocations and are quantified by the nye dislocation density tensor when incorporated within a flow rule for the plastic strain rate this tensor fundamentally alters the mathematical structure of the theory several computational complexities also arise as a result these problems are posed in the setting of a variationally based multiscale method it allows the circumvention of some of the mathematical and computational difficulties associated with this model the phenomenological plasticity model its enhancement by strain gradients formulation within a multiscale context and two numerical examples are presented

variational multiscale analysis of elastoplastic deformation using meshfree approximation abstract a variational multiscale method has been presented for efficient analysis of elastoplastic deformation problems severe deformation occurs in plastic region and leads to high gradient displacement therefore solution needs to be refined to properly capture local deformation in plastic region in this work scale decomposition based on variational formulation is presented a coarse scale and a fine scale are introduced to represent global and local behavior respectively the displacement is decomposed into a coarse and a fine scale subsequently the problem is also decomposed into a coarse and a fine scale from the variational formulation each scale variable is approximated using meshfree method adaptivity can easily and nicely be implemented in meshfree method as a method of increasing resolution extrinsic enrichment of partition of unity is used each scale problem is solved iteratively and conversed results are obtained consequently iteration procedure is indispensable for the elastoplastic deformation analysis therefore iterative solution procedure of each scale problem is naturally adequate the proposed method is applied to the prandtl s punch test and shear band problem the results are compared with those of other methods and the validity of the proposed method is demonstrated

analysis of some quadrilateral nonconforming elements for incompressible elasticity in this work some nonconforming elements on arbitrary quadrilateral meshes in solving incompressible elastic equations are analyzed a uniform optimal convergence rate is established at the incompressible limit nu 0 5 for both displacement and stresses or pressure in the case of incompressible flow

multiscale enrichment based on partition of unity a new multiscale enrichment method based on the partition of unity mepu method is presented it is a synthesis of mathematical homogenization theory and the partition of unity method its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible mepu is perfectly suited for enriching the coarse scale continuum descriptions pdes with fine scale features and the quasi continuum formulations with relevant atomistic data numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi continuum formulations copyright 2005 john wiley sons ltd

some geometrical results and estimates for quadrilateral finite elements abstract the concept of the equivalent parallelogram associated with a quadrilateral is introduced the quadrilateral may be regarded as a perturbation of this parallelogram which is most conveniently described by making use of properties of the isoparametric map which defines the quadrilateral the equivalent parallelogram generates a natural means of defining a regular family of quadrilaterals this definition is used together with other properties to obtain in a relatively simple manner estimates in appropriate seminorms or norms of the isoparametric map and its jacobian for use in the determination of finite element interpolation error estimates

numerical integration with taylor truncations for the quadrilateral and hexahedral finite elements for general quadrilateral or hexahedral meshes the finite element methods require evaluation of integrals of rational functions instead of traditional polynomials it remains as a challenge in mathematics to show the traditional gauss quadratures would ensure the correct order of approximation for the numerical integration in general however in the case of nested refinement the refined quadrilaterals and hexahedra converge to parallelograms and parallelepipeds respectively based on this observation the rational functions of inverse jacobians can be approximated by the taylor expansion with truncation then the gauss quadrature of exact order can be adopted for the resulting integrals of polynomials retaining the optimal order approximation of the finite element methods a theoretic justification and some numerical verification are provided in the paper

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

reduced order variational multiscale enrichment method for elasto viscoplastic problems abstract this manuscript presents the formulation and implementation of a novel reduced order variational multiscale enrichment rovme method for elasto viscoplastic problems this method provides a hierarchical model order reduction technique based on the eigenstrain concept to approximate the fine scale response resolved at subdomains of interest by eliminating the requirement of direct fine scale discretization the computational effort associated with the variational multiscale enrichment vme method is significantly reduced the model order reduction is achieved in the scale coupled inelastic problem by automatically satisfying the microscale equilibrium state through the eigenstrain concept and coarse discretization of inelastic strain fields within the microscale domain the inelastic material behavior is idealized with coupled perzyna type viscoplasticity and flow stress evolution based on the johnson cook model numerical verifications are performed to assess the capabilities of the proposed methodology against the direct vme method with detailed fine scale resolutions the verification results demonstrate high accuracy and computational efficiency of the reduced order vme framework for elasto viscoplastic problems with material heterogeneity

a variational multiscale method for inelasticity application to superelasticity in shape memory alloys this paper presents a variational multiscale method for developing stabilized finite element formulations for small strain inelasticity the multiscale method arises from a decomposition of the displacement field into coarse resolved and fine unresolved scales the resulting finite element formulation allows arbitrary combinations of interpolation functions for the displacement and the pressure fields and thus yields a family of stable and convergent elements specifically equal order interpolations that are easy to implement but violate the celebrated babuska brezzi condition become stable and convergent an important feature of the present method is that it does not lock in the incompressible limit a nonlinear constitutive model for the superelastic behavior of shape memory alloys is integrated in the multiscale formulation numerical tests of the performance of the elements are presented and representative simulations of the superelastic behavior of shape memory alloys are shown 2005 elsevier b v all rights reserved

improved versions of assumed enhanced strain tri linear elements for 3d finite deformation problems abstract improved three dimensional tri linear elements for finite deformation problems are developed based on an assumed enhanced strain methodology which in the linear regime incorporates the classical method of incompatible modes as a particular case three crucial modifications of a recently proposed element which reduces to wilson s brick in the linear regime are introduced to prevent locking response in distorted configurations and to maintain proper rank while preserving excellent performance in bending dominated and localization problems i a modified quadrature rule ii an additional enhancement of the divergence term and iii a modification of the isoparametric shape function derivatives for the three dimensional problem moreover these modified shape function derivatives are shown to improve the performance of the standard tri linear brick in distorted configurations in addition a strategy is described to circumvent the memory storage requirements in the static condensation procedure of the enhanced strain parameters the excellent performance of the improved methodology is illustrated in representative numerical simulations

finite element simulation of strain localization with large deformation capturing strong discontinuity using a petrov galerkin multiscale formulation the finite element model for strain localization analysis developed in a previous work is generalized to the finite deformation regime strain enhancements via jumps in the displacement field are captured and condensed on the material level leading to a formulation that does not require static condensation to be performed on the element level a general evolution condition is first formulated laying out conditions for the continued activation of a strong discontinuity then a multiscale finite element model is formulated to describe the post bifurcation behavior highlighting the key roles played by the continuous and conforming deformation maps on the characterization of the finite deformation kinematics of a localized element traced by a strong discontinuity the resulting finite element equation exhibits the features of a petrov galerkin formulation in which the gradient of the weighting function is evaluated over the continuous part of the deformation map whereas the gradient of the trial function is evaluated over the conforming part of the same map in the limit of infinitesimal deformations the formulation reduces to the standard galerkin approximation described in a previous work numerical examples are presented demonstrating absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions 2002 elsevier science b v all rights reserved

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

meshfree analysis of softening elastoplastic solids using variational multiscale method a meshfree multiscale method is presented for efficient analysis of elastoplastic solids in the analysis of softening elastoplastic solids standard finite element methods or meshfree methods typically yield mesh dependent results the reason for this well known effect is the loss of ellipticity of the boundary value problem in this work the scale decomposition is carried out based on a variational form of the problem a coarse scale is designed to represent global behavior and a fine scale to represent local behavior a fine scale region is detected from the local failure analysis of an acoustic tensor to indicate a region where deformation changes abruptly each scale variable is approximated using a meshfree method meshfree approximation is well suited for adaptivity as a method of increasing the resolution a partition of unity based extrinsic enrichment is used in particular fine scale approximations are designed to appropriately represent local behavior by using a localization angle moreover the regularization effect through the convexification of non convex potential is embedded to represent fine scale behavior each scale problem is solved iteratively the proposed method is applied to shear band problems in the results of analysis about pure shear and compression problems straight shear bands can be captured and mesh insensitive results are obtained curved shear bands can also be captured without mesh dependency in the analysis of indentation problem

fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition abstract in recent years a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials where the complexity of solving numerically all microscale details is not feasible in this paper we present a multiscale framework based on the decomposition of the displacement field into coarse macro and fine micro scales earlier proposed in the variational multiscale approach or the hp d method the novelty of this work lies in solving the microscale step where a multidimensional parametrized model of a generic rve is solved by means of the multidimensional model reduction technique named as proper generalized decomposition pgd as result of this previous and off line step the displacement field over the rve is obtained by simple algebraic operations for any combination of parameters boundary condition material properties loads etc allowing a significant reduction in computational cost when solving the macro scale problem for problems where the detailed structure of the localized displacement strain and stress fields are of interest recovery of the fine scale components can be performed immediately the basic concepts and several 1 d and 2 d linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving

a study of strain localization in a multiple scale framework the one dimensional problem abstract this work approaches strain localization by recognizing the multiple scales inherent in the problem a component associated with the region of localized strain typically one with high gradient is referred to as the fine scale it represents the microstructure the field obtained by removing the fine scale from the total solution is referred to as the coarse scale the aim of the multiple scale method advanced here is to derive a model for the coarse scale field that accounts for the finite scale this process eliminates the fine scale from the problem yet retains its effect in applying this framework to the nonlinear problems with which localized strains are associated a crucial step is a first order approximation of the relevant relations the fully nonlinear problem is solved by an iterative scheme by accounting directly for the microstructure the multiple scale model recovers the regularizing effects of various alternative formulations for softening strain localization it thus presents itself as a unifying framework for such models numerical solutions are shown to be invariant with respect to the discretization furthermore for cases in which the displacements assume a distinct profile within the localization band the multiple scale model provides a resolution that can be made as accurate as desired even with the coarsest mesh possible the model is applied to strain localization problems that arise in inviscid and viscoplastic solids numerical simulations are presented that demonstrate the efficacy of the approach

an algebraic variational multiscale multigrid method based on plain aggregation for convection diffusion problems a new computational framework referred to as algebraic variational multiscale multigrid method is proposed representing the initial step of our work on merging the variational multiscale method with algebraic multigrid projection methods this new approach allows for a separation of resolved scales into various scale groups in a purely algebraic way that is with no need to utilize further discretizations beyond the basic one by this means it conveniently facilitates the application of modeling terms either to all scale groups or only to selected scale groups for instance only to the fine scales of the problem variational multigrid techniques form the basis of our new approach after presenting the cornerstones of the framework for an abstract problem formulation the close relationship of the variational multiscale method to projection methods in general and to variational multigrid techniques are discussed the algebraic variational multiscale multigrid method is then particularly analyzed for convection diffusion problems the present initial study focuses on exploiting the methodical aspects of the new framework by developing a fine scale discontinuity capturing approach to diminish oscillations at sharp layers our technique including a fine scale discontinuity capturing term is applied to numerical examples of strongly convection dominated convection diffusion problems the results demonstrate that it enables the diminuation of local oscillations at parabolic boundary and interior layers by about 60 80 percent without any notable smearing effect

plane strain finite element analysis of pressure sensitive plasticity with strong discontinuity numerical simulations of localized deformation in solids should capture the structural phenomenon of localization and associated loss of material body strength in a manner independent of spatial discretization many regularization techniques have been proposed to address the ill posedness associated with rate independent softening plasticity that leads to mesh dependent numerical simulations one approach to alleviating mesh dependence is the strong discontinuity approach which represents localized deformation as a slip surface within a plasticity model a strong discontinuity is a discontinuous displacement field this approach is used in this paper to formulate a plane strain pressure sensitive nonassociative plasticity model with strong discontinuity and to implement the model along with an enhanced bilinear quadrilateral element via an assumed enhanced strain method numerical examples demonstrate mesh independence of the method for pressure sensitive materials

on the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation it is shown in this paper that the optimal approximation property of bilinear or trilinear finite elements would be retained when the affine mapping is used to replace the q1 mapping on each element if the grids are refined nestedly the new method truncates the quadratic and cubic terms in reference mappings and produces constant jacobians and jacobian matrices this would avoid a shortcoming of the quadrilateral and hexahedral elements where the integrals of rational functions have to be computed or approximated numerical tests verify the analysis

properties of discontinuous bifurcation solutions in elasto plasticity explicit expressions for the spectral properties of the bifurcation problem involving discontinuities for general elastic plastic materials are presented it then follows that the classical value of the critical hardening modulus derived by rice 1976 proc 14th iutam congr delft the netherlands pp 207 220 north holland amsterdam is in fact the only possible one furthermore from the spectral analysis it follows in a very straightforward fashion that bifurcation displaying elastic unloading on one side of the singular surface can never precede bifurcation with plastic loading on both sides of this surface explicit analytical results for the critical bifurcation directions and the corresponding hardening modulus are derived for non associated volumetric flow rules while the devialoric portion is associated the considered yield and potential functions may depend on all three stress invariants and may involve mixed isotropic and kinematic hardening the result obtained by rudnicki and rice 1975 j mech phys solids 23 371 394 for a drucker pruger material appear as a special case other criteria that are investigated arc those of coulomb and rankine less

three dimensional isogeometric solutions to general boundary value problems of toupin s gradient elasticity theory at finite strains abstract we present to the best of our knowledge the first complete three dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity we have chosen for our work toupin s theory toupin 1962 one of the more general formulations of strain gradient elasticity our framework has three crucial ingredients the first is isogeometric analysis hughes et al 2005 which we have adopted for its straightforward and robust representation of c 1 continuity the second is a weak treatment of the higher order dirichlet boundary conditions in the formulation which control the development of strain gradients in the solution the third ingredient is algorithmic automatic differentiation which eliminates the need for linearization by hand of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains we present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions we discuss length scale effects the role of higher order boundary conditions and perhaps most importantly the relevance of the framework to problems with elastic free energy density functions that are non convex in strain space

a two scale finite element relaxation analysis of shear bands in non convex inelastic solids small strain theory for standard dissipative materials abstract we propose a fundamentally new approach to the treatment of shear band localizations in strain softening elastic plastic solids at small strains based on energy minimization principles associated with micro structure developments the point of departure is a general internal variable formulation that determines the inelastic response as a standard dissipative medium consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasihyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables the existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments in finite elasticity furthermore localization phenomena are interpreted as micro structure developments on multiple scales associated with non convex incremental stress potentials in analogy to elastic phase decomposition problems these micro structures can be resolved by the relaxation of non convex energy functionals based on a convexification of the stress potential the relaxed problem provides a well posed formulation for a mesh objective analysis of localizations as close as possible to the non convex original problem we develop based on an approximated rank one convexification of the incremental stress potential a computational two scale procedure for a mesh objective treatment of localization problems it constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the micro shearing of a rank one laminate aligned to the shear band this problem is sufficiently robust with regard to applications to large scale inhomogeneous deformation processes of elastic plastic solids the performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which illustrate the mesh independence of the results

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

fe formulation of a nonlocal plasticity theory a nonlocal continuum plasticity theory is presented the nonlocal field introduced here is defined as a certain weighted average of the corresponding local field taken over all the material points in the body hereby a quantity with the dimension of length occurs as a material parameter when this so called internal length is equal to zero the local classical plasticity theory is regained in the present model the yield function will depend on a nonlocal field the consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers the integral equations are classified as fredholm equations of the second kind and the existence of a solution will be commented upon after discretization a matrix equation is obtained and an algorithm for finding the solution is proposed for a generalized von mises material a plane boundary value problem is solved with a fe method since the nonlocal quantities are integrals c0 continuous elements are sufficient the solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations in the numerical simulations shear band formation is analysed and the results display mesh insensitivity

a burst error correcting algorithm for reed solomon codes it is known that for a burst error environment the error correcting capability of reed solomon codes can be extended beyond the singleton bound with a high degree of confidence this is significant in that an n k code with an arbitrarily small probability of a miscorrection can correct more than n k 2 errors a decoding algorithm for correcting a burst of length greater than n k 2 is presented

burst decoding of cyclic codes based on circulant parity check matrices an error burst correcting algorithm is developed based on a circulant parity check matrix of a cyclic code the proposed algorithm is more efficient than error trapping if the code rate is less than about 2 3 it is shown that for any n k cyclic code there is an n n circulant parity check matrix such that the algorithm applied to this matrix corrects error bursts of lengths up to the error burst correction limit of the cyclic code this same matrix can be used to efficiently correct erasure bursts of lengths up to n k the error burst correction capabilities of a class of cyclic low density parity check ldpc codes constructed from finite geometries are also considered

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

novel burst error correction algorithms for reed solomon codes in this paper we present three novel burst error correcting algorithms for an n k reed solomon code the algorithmic complexities are of the same order as error and erasure decoding o rn where r n k in particular their hardware implementation shares elements of blahut error and erasure decoding in contrast all existing single burst error correcting algorithms which are equivalent to the proposed first algorithm have complexity o r2n the first algorithm corrects the shortest single burst with length f up to r 1 the algorithm follows the key characterization that the ending locations of all candidate bursts can be purely determined by the roots of a polynomial which is a linear function of syndromes and moreover the shortest burst is associated with the longest sequence of consecutive roots the algorithmic miscorrection probability is bounded by q r 1 f where q denotes the field size the second algorithm extends the first one to correct the shortest burst with length f r 3 and additionally a random symbol error the algorithmic miscorrection probability is bounded by q r 3 f the third algorithm probabilistically corrects the shortest burst with length f r 1 2 and additionally a small constant random symbol errors the algorithmic miscorrection and failure probabilities are both bounded by q r 1 2 f our simulation results for 60 40 and 30 16 shortened reed solomon codes verify that the miscorrection probability for three algorithms and the failure probability for the third algorithm all decay exponentially at the rate of q 1 with respect to the length of burst