quadrilateral mesh revisited several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent the effect of the bi section condition on the degenerate mesh conditions is also checked

on the relation between strong subadditivity and entanglement strong subadditivity is used to improve the triangular inequality for the entropy of tensorproducts by the amount of entanglement

analysis of three dimensional fracture mechanics problems a two scale approach using coarse generalized fem meshes this paper presents a generalized finite element method gfem based on the solution of interdependent global structural and local crack scale problems the local problems focus on the resolution of fine scale features of the solution in the vicinity of three dimensional cracks while the global problem addresses the macro scale structural behavior the local solutions are embedded into the solution space for the global problem using the partition of unity method the local problems are accurately solved using an hp gfem and thus the proposed method does not rely on analytical solutions the proposed methodology enables accurate modeling of three dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts the boundary conditions for the local problems are provided by the coarse global mesh solution and can be of dirichlet neumann or cauchy type the effect of the type of local boundary conditions on the performance of the proposed gfem is analyzed several three dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance both in terms of problem size and cpu time are presented copyright 2009 john wiley sons ltd

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

structure of states which satisfy strong subadditivity of quantum entropy with equality we give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von neumann entropy by combining a result of petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by koashi and imoto we show that such states will have the form of a so called short quantum markov chain which in turn implies that two of the systems are independent conditioned on the third in a physically meaningful sense this characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the holevo bound on accessible information

crack tip enrichment in the xfem using a cutoff function we consider a variant of the extended finite element method xfem in which a cutoff function is used to localize the singular enrichment surface the goal of this variant is to obtain numerically an optimal convergence rate while reducing the computational cost of the classical xfem with a fixed enrichment area we give a mathematical result of quasi optimal error estimate one of the key points of this paper is to prove the optimality of the coupling between the singular and the discontinuous enrichments finally we present some numerical computations validating the theoretical result these computations are compared with those of the classical xfem and a non enriched method copyright 2008 john wiley sons ltd

the index of weighted singular integral operators with shifts and slowly oscillating data abstract let and be orientation preserving diffeomorphism shifts of r 0 onto itself with the only fixed points 0 and we establish a fredholm criterion and calculate the index of the weighted singular integral operator with shifts a i b u p c i d u p acting on the space l p r where p i s 2 are the operators associated to the weighted cauchy singular integral operator s given by s f t 1 i r t f t d with c satisfying 0 1 p 1 and u u are the isometric shift operators given by u f 1 p f u f 1 p f under the assumptions that the coefficients a b c d and the derivatives of the shifts are bounded and continuous on r and admit discontinuities of slowly oscillating type at 0 and

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

averaging technique for fe a posteriori error control in elasticity part i conforming fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper the reliability of any averaging estimator is shown for low order finite element methods in elasticity theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides and independent of the structure of a shape regular mesh

entanglement bell inequality and all that we start from the geometrical observation that a finite set of pure states correspond to some points on a sphere and their convex span cannot be the whole set of states if we call the left over entangled we can pursue this picture from the simplest case of a two dimensional hilbert space to the usual alice and bob game of entangled states and then move to bigger systems and finely to quantum field theory where almost everything is entangled on the way we encounter more or less known old friends up from the shell structure of states to the monogamy of squashed entanglement we study how entanglement can be concentrated on a small slice and how it depends on the particular factorization of the hilbert space

the partition of unity finite element approach with hp refinement for the stationary fokker planck equation abstract in this paper the stationary fokker planck equation fpe is solved for nonlinear dynamical systems using a local numerical technique based on the meshless partition of unity finite element method pufem the method is applied to stationary fpe for 2 3 and 4 d systems and is argued to be an excellent candidate for higher dimensional problems and the transient problem local refinement is applied by introducing higher order polynomials in selected subdomains local p refinement to keep the problem size small while ensuring high approximation accuracy various local approximations are blended using novel pasting functions that provide any desired order of continuity results are compared with existing global and local techniques local p refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of fpe

integration of singular enrichment functions in the generalized extended finite element method for three dimensional problems a mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized extended finite element method the integration scheme is applicable to 2d and 3d problems including arbitrarily shaped triangles and tetrahedra implementation of the proposed scheme in existing codes is straightforward numerical examples for 2d and 3d problems demonstrate the accuracy and convergence properties of the technique copyright 2008 john wiley sons ltd

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

implicit upper bound error estimates for combined expansive model and discretization adaptivity abstract a computational methodology for goal oriented combined discretization and expansive refined model adaptivity by overall implicit error control of quantities of interest is presented requiring estimators of primal and dual discretization and model errors in the case of dimensional within model adaptivity prolongations of coarse model solutions into the solution space of a fine model for defining a consistent model error are necessary which can be achieved at the element level by two strategies the first one is an orthogonalized kinematic prolongation of nodal displacements whereas the second one uses prolongations of the external loads which are then used to solve additional local variational problems thus yielding prolongated solutions which a priori fulfill the required orthogonality relations at the element level finally a numerical example of an elastic continuous t beam is presented with comparative results where goal oriented error estimation is applied to linear elasticity with a 2 1 2 d discrete reissner mindlin plate model as the coarse model and the 3d theory as the fine model

the piecewise polynomial partition of unity functions for the generalized finite element methods a partition of unity pu function is an essential component of the generalized finite element method gfem the popular shepard pu functions which are rational functions are easy to construct but have difficulties in dealing with essential boundary conditions and require lengthy computing time for reasonable accuracy in numerical integration in this paper we introduce two simple pu functions the first is a highly regular piecewise polynomial consisting of two distinct polynomials that is effective for uniformly partitioned patches the second is a highly regular piecewise polynomial consisting of three distinct polynomials which is for arbitrary partitioned patches

on the approximability and the selection of particle shape functions particle methods also known as meshless or meshfree methods have become popular in approximating solutions of partial differential equations especially in the engineering community these methods do not employ a mesh or use it minimally in the construction of shape functions there is a wide variety of classes of shape functions that can be used in particle methods in this paper we primarily address the issue of selecting a class of shape functions among this wide variety that would yield efficient approximation of the unknown solution we have also made several comments and observations on the order of convergence of the interpolation error when these shape functions are used specifically we have shown that the interpolation error estimate for certain classes of shape functions may not indicate the actual order of convergence of the approximation error

a stable and optimally convergent generalized fem sgfem for linear elastic fracture mechanics abstract in this paper we investigate the accuracy and conditioning of the stable generalized fem sgfem and compare it with standard generalized fem gfem for a 2 d fracture mechanics problem the sgfem involves localized modifications of enrichments used in the gfem and the conditioning of the stiffness matrix in this method is of the same order as in the fem numerical experiments show that using the sgfem with only the modified heaviside functions which are used as enrichments in the gfem to approximate the solution of fracture problems in 2 d gives inaccurate results however the sgfem using an additional set of enrichment function yields accurate results while not deteriorating the conditioning of the stiffness matrix rules for the selection of the optimal set of enrichment nodes based on the definition of enrichment functions used in the sgfem are also presented this set leads to optimal convergence rates while keeping the number of degrees of freedom equal to or close to the gfem we show that it is necessary to enrich additional nodes when the crack line is located along element edges in 2 d the selection of these nodes depends on the definition of the enrichment functions at the crack discontinuity a simple and yet generic implementation strategy for the sgfem in an existing gfem xfem software is described the implementation can be used with 2 d and 3 d elements it leads to an efficient evaluation of sgfem enrichment functions

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

a note on the singular linear system of the generalized finite element methods we characterize the kernel of the global stiffness matrix in the singular linear system of the generalized finite element methods gfem which uses the classical finite element fe shape functions and local approximation space of harmonic polynomials

symmetric quadrature rules on a triangle abstract we present a class of quadrature rules on triangles in r 2 which somewhat similar to gaussian rules on intervals in r 1 have rapid convergence positive weights and symmetry by a scheme combining simple group theory and numerical optimization we obtain quadrature rules of this kind up to the order 30 on triangles this scheme essentially a formalization and generalization of the approach used by lyness and jespersen over 25 years ago can be easily extended to other regions in r 2 and surfaces in higher dimensions such as squares spheres we present example formulae and relevant numerical results

representation of singular fields without asymptotic enrichment in the extended finite element method summary in this paper we replace the asymptotic enrichments around the crack tip in the extended finite element method xfem with the semi analytical solution obtained by the scaled boundary finite element method sbfem the proposed method does not require special numerical integration technique to compute the stiffness matrix and it improves the capability of the xfem to model cracks in homogeneous and or heterogeneous materials without a priori knowledge of the asymptotic solutions a heaviside enrichment is used to represent the jump across the discontinuity surface we call the method as the extended sbfem numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number a simple code is annexed to compute the terms in the stiffness matrix which can easily be integrated in any existing fem xfem code copyright 2013 john wiley sons ltd

dynamical entropy of c algebras and von neumann algebras the definition of the dynamical entropy is extended for automorphism groups ofc algebras as an example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density

generalized p fem in homogenization summary a new finite element method for elliptic problems with locally periodic microstructure of length varepsilon 0 is developed and analyzed it is shown that the method converges as varepsilon rightarrow 0 to the solution of the homogenized problem with optimal order in varepsilon and exponentially in the number of degrees of freedom independent of varepsilon 0 the computational work of the method is bounded independently of varepsilon numerical experiments demonstrate the feasibility and confirm the theoretical results

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

a posteriori error estimation in large strain elasticity using equilibrated local neumann problems abstract a method to obtain a posteriori an accurate measure of the error in finite element computations is to solve a neumann problem with equilibrated tractions for each finite element we extend this approach to large strain elasticity and mixed finite elements which are well suited for nearly incompressible rubber like materials for an l2 projection of the error in the first piola kirchhoff stress tensor we prove that the estimator provides a guaranteed upper bound without multiplicative constants numerical examples confirm the effectiveness for both linear and ogden material

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

mesh based construction of flat top partition of unity functions a novel idea to construct flat top partition of unity functions in a closed form on a general structured or unstructured finite element mesh is presented an efficient and practical construction method of a flat top partition of unity function is important in the generalized finite element method gfem details on how to construct flat top partition of unity functions on a provided mesh are given the generalized finite element approximation with the use of the new flat top partition of unity function is presented with various numerical examples that demonstrate the effectiveness of proposed flat top partition of unity functions

a posteriori error estimation for extended finite elements by an extended global recovery this contribution presents an extended global derivative recovery for enriched finite element methods fems such as the extended fem along with an associated error indicator owing to its simplicity the proposed scheme is ideally suited to industrial applications the procedure is based on global minimization of the l2 norm of the difference between the raw strain field c 1 and the recovered c0 strain field the methodology engineered in this paper extends the ideas of oden and brauchli int j numer meth engng 1971 3 and hinton and campbell int j numer meth engng 1974 8 by enriching the approximation used for the construction of the recovered derivatives strains with the gradients of the functions employed to enrich the approximation employed for the primal unknown displacements we show linear elastic fracture mechanics examples both in simple two dimensional settings and for a three dimensional structure numerically we show that the effectivity index of the proposed indicator converges to unity upon mesh refinement consequently the approximate error converges to the exact error indicating that the error indicator is valid additionally the numerical examples suggest a novel adaptive strategy for enriched approximations in which the dimensions of the enrichment zone are first increased before standard h and p adaptivities are applied we suggest to coin this methodology e adaptivity copyright 2008 john wiley sons ltd

the intrinsic xfem a method for arbitrary discontinuities without additional unknowns a new method for treating arbitrary discontinuities in a finite element fe context is presented unlike the standard extended fe method xfem no additional unknowns are introduced at the nodes whose supports are crossed by discontinuities the method constructs an approximation space consisting of mesh based enriched moving least squares mls functions near discontinuities and standard fe shape functions elsewhere there is only one shape function per node and these functions are able to represent known characteristics of the solution such as discontinuities singularities etc the mls method constructs shape functions based on an intrinsic basis by minimizing a weighted error functional thereby weight functions are involved and special mesh based weight functions are proposed in this work the enrichment is achieved through the intrinsic basis the method is illustrated for linear elastic examples involving strong and weak discontinuities and matches optimal rates of convergence even for crack tip applications copyright 2006 john wiley sons ltd

effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for gfem xfem stress approximations around brittle cracks abstract the convergence rates of the conventional generalized extended finite element method gfem xfem in crack modeling are similar to the convergence rates of the finite element method fem laborde et al 2005 and bechet et al 2005 unless the crack tip enrichment functions are applied in a subdomain with fixed dimension independently of the mesh parameter h and also demanding some special care for blending along the transition zone chahine et al 2006 and taracon et al 2009 thus to improve convergence rates more degrees of freedom dof are generated due to the larger quantity of enriched nodes this work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh based smooth partitions of unity pou such pou with higher regularity can be built through the so called c k gfem framework following duarte et al 2006 based on a moving least square of degree zero and considering mesh based smooth weighting functions associated with arbitrary polygonal clouds the purpose herein is to investigate some possible advantages of mesh based smooth pou for modeling discontinuities and singularities in two dimensional problems of linear elastic fracture mechanics in such a fashion that the discretization error associated to stress discontinuities inherent in standard c 0 continuous gfem xfem approximations is eliminated the procedure shares features similar to the standard fem regarding domain partition and numerical integration but as neither the pou nor the enrichment functions are defined in natural domains the integrations are performed using only global coordinates the approximation capabilities of c k gfem discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the h and p versions considering global measures in terms of strain energy and l 2 norm of displacements the effects on stability are also verified by analyzing the evolution of the condition number the effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved the smoothness of the pou the different types of enrichment functions and the pattern of enrichment the performance of the smooth approximations is compared to the c 0 counterparts built using conventional c 0 fem based pou it is shown that smoothness in the presence of extrinsically applied singular enrichments is important as the information provided by such enrichment is better captured in addition there are no stress jumps around singularities reducing error propagation beyond the neighborhood of the singularity

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

anisotropic discretization and model error estimation in solid mechanics by local neumann problems first a survey of existing residuum based error estimators and error indicators is given generally residual error estimators which have at least upper bound in contrast to indicators can be locally computed from residua of equilibrium and stress jumps at element interfaces using dirichlet or neumann conditions for element patches or individual elements rem another equivalent method for error estimation can be derived from a posteriori computed improved boundary tractions which provide exact equilibrium of elements they are computed from local neumann problems and yield improved local solution by testing with higher test functions this method is called posterior equilibrium method pem and yields better efficiency indices than rem for the investigated problem even in case of locking also the important feature of model adaptivity via hierarchical error estimation can only be realized by this strategy global neumann type estimators and examples are presented for elastic and elastoplastic deformations controlling equilibrium in tangential space

extended finite element method on polygonal and quadtree meshes in this paper we present mesh independent modeling of discontinuous flelds on polygonal and quadtree flnite element meshes this approach falls within the class of extended and generalized flnite element methods where the partition of unity framework is used to introduce additional enrichment functions within the classical displacement based flnite element approximation for crack modeling a discontinuous function and the two dimensional asymptotic crack tip flelds are used as enrichment functions linearly complete partition of unity approximations are adopted on polygonal convex and nonconvex elements and quadtree meshes excellent agreement with reference solution results is obtained for mixed mode stress intensity factors on benchmark crack problems and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics

fast integration and weight function blending in the extended finite element method two issues in the extended finite element method xfem are addressed efficient numerical integration of the weak form when the enrichment function is self equilibrating and blending of the enrichment the integration is based on transforming the domain integrals in the weak form into equivalent contour integrals it is shown that the contour form is computationally more efficient than the domain form especially when the enrichment function is singular and or discontinuous a method for alleviating the errors in the blending elements is also studied in this method the enrichment function is pre multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains a method for blending step function enrichment with singular enrichments is described it is also shown that if the enrichment is not shifted properly the weighted enrichment is equivalent to the standard enrichment an edge dislocation and a crack problem are used to benchmark the technique the influence of the variables that parameterize the weight function is analyzed the resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing xfem codes

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

the generalized product partition of unity for the meshless methods the partition of unity is an essential ingredient for meshless methods named by gfem pufem partition of unity fem xfem extended fem rkpm reproducing kernel particle method rppm reproducing polynomial particle method the method of hp clouds in the literature there are two popular choices for partition of unity a piecewise linear fem mesh and the shepard type partition of unity however the partition of unity pu by a fem mesh leads to the singular or nearly singular matrices and non smooth approximation functions the shepard type partition of unity requires lengthy computing time and its implementation is difficult in order to alleviate these difficulties oh et al introduced the smooth piecewise polynomial pu functions with flat top that lead to small matrix condition numbers and almost everywhere partition of unity that can handle essential boundary conditions nevertheless we could not have the smooth closed form pu functions with flat top for general polygonal patches 2d and general polyhedral patches 3d in this paper we introduce one of the most simple and efficient partition of unity called the generalized product partition of unity the product pu functions constructed by this method are the closed form smooth piecewise polynomials with flat top and could handle background meshes general polygonal patches as well as general polyhedral patches arising in practical applications of meshless methods

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

a recovery explicit error estimator in energy norm for linear elasticity significant research effort has been devoted to produce one sided error estimates for finite element analyses in particular to provide upper bounds of the actual error typically this has been achieved using residual type estimates one of the most popular and simpler in terms of implementation techniques used in commercial codes is the recovery based error estimator this technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm some attempts to remedy this situation provide bounds depending on unknown constants here a new step towards obtaining error bounds from the recovery based estimates is proposed the idea is 1 to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error 2 to add an explicit type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and 3 to efficiently and accurately evaluate the constants appearing in the bounding expressions thus providing asymptotic bounds the numerical tests with h adaptive refinement process show that the bounding property holds even for coarse meshes providing upper bounds in practical applications

linear dependence problems of partition of unity based generalized fems a known problem of partition of unity based generalized finite element methods referred to as gfem is the linear dependence problem which leads to singular global stiffness matrices thus far attempts to eliminate the linear dependence problem have been unsuccessful numerical experiments are carried out among several gfems to investigate the problem based on the numerical experiments simple but effective approaches to eliminating the linear dependence problem are suggested

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

superconvergence in the generalized finite element method in this paper we address the problem of the existence of superconvergence points of approximate solutions obtained from the generalized finite element method gfem of a neumann elliptic boundary value problem gfem is a galerkin method that uses non polynomial shape functions and was developed in babuska et al in siam j numer anal 31 945 981 1994 babuska et al in int j numer meth eng 40 727 758 1997 melenk and babuska in comput methods appl mech eng 139 289 314 1996 in particular we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non linear equations this system does not depend on the solution of the boundary value problem for approximate solutions with second derivatives we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non linear equations we note that smooth generalized finite element approximation is easy to construct

improving the accuracy of xfem crack tip fields using higher order quadrature and statically admissible stress recovery this study is concerned with improving the accuracy of crack tip fields obtained using the extended generalized finite element method xfem first the numerical integration of the element stiffness matrices which guarantees convergence with quadrature of not only the regular nodal displacements but also additional degrees of freedom corresponding to the enrichment functions is studied as the accuracy of the stresses obtained by direct differentiation of the converged with quadrature regular nodal displacements and of the coefficients corresponding to enrichment functions is still not satisfactory a statically admissible stress recovery sar scheme is introduced sar uses basis functions which meet the equilibrium equations within the domain and the local traction conditions on the boundary and moving least squares mls to fit the stresses at sampling points e g quadrature points obtained by the xfem important parameters controlling the accuracy of crack tip fields using the xfem and sar namely the order of quadrature the number of retained terms in the crack tip asymptotic field the number of enriched layers and use of arbitrary branch functions a proper choice of the sampling points in the enriched element and the size of the domain of influence doi of mls are investigated copyright 2005 john wiley sons ltd

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

polygonal finite elements for topology optimization a unifying paradigm in topology optimization literature the parameterization of design is commonly carried out on uniform grids consisting of lagrangian type finite elements e g linear quads these formulations however suffer from numerical anomalies such as checkerboard patterns and one node connections which has prompted extensive research on these topics a problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members leading to mesh dependent sub optimal designs thus to address the geometric features of the spatial discretization we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions more specifically we consider polygonal meshes constructed from voronoi tessellations which in addition to possessing higher degree of geometric isotropy allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities copyright 2009 john wiley sons ltd

conforming polygonal finite elements summary in this paper conforming finite elements on polygon meshes are developed polygonal finite elements provide greater flexibility in mesh generation and are better suited for applications in solid mechanics which involve a significant change in the topology of the material domain in this study recent advances in meshfree approximations computational geometry and computer graphics are used to construct different trial and test approximations on polygonal elements a particular and notable contribution is the use of meshfree natural neighbour nn basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons this numerical formulation enables the construction of conforming approximation on n gons n 3 and hence extends the potential applications of finite elements to convex polygons of arbitrary order numerical experiments on second order elliptic boundary value problems are presented to demonstrate the accuracy and convergence of the proposed method copyright 2004 john wiley sons ltd

on the approximation in the smoothed finite element method sfem this letter aims at resolving the issues raised in the recent short communication int j numer meth engng 2008 76 8 1285 1295 doi 10 1002 nme 2460 and answered by int j numer meth engng 2009 doi 10 1002 nme 2587 by proposing a systematic approximation scheme based on non mapped shape functions which both allows to fully exploit the unique advantages of the smoothed finite element method sfem comput mech 2007 39 6 859 877 doi 10 1007 s00466 006 0075 4 commun numer meth engng 2009 25 1 19 34 doi 10 1002 cnm 1098 int j numer meth engng 2007 71 8 902 930 comput meth appl mech engng 2008 198 2 165 177 doi 10 1016 j cma 2008 05 029 comput meth appl mech engng 2007 submitted int j numer meth engng 2008 74 2 175 208 doi 10 1002 nme 2146 comput meth appl mech engng 2008 197 13 16 1184 1203 doi 10 1016 j cma 2007 10 008 and resolve the existence linearity and positivity deficiencies pointed out in int j numer meth engng 2008 76 8 1285 1295 we show that wachspress interpolants a rational basis for function approximation academic press inc new york 1975 computed in the physical coordinate system are very well suited to the sfem especially when elements are heavily distorted obtuse interior angles the proposed approximation leads to results that are almost identical to those of the sfem initially proposed in comput mech 2007 39 6 859 877 doi 10 1007 s00466 006 0075 4 these results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of sfems copyright 2009 john wiley sons ltd

improved implementation and robustness study of the x fem for stress analysis around cracks numerical crack propagation schemes were augmented in an elegant manner by the x fem method the use of special tip enrichment functions as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh which is of industrial interest especially when a numerical model for crack propagation is desired this paper improves the implementation of the x fem method for stress analysis around cracks in three ways first the enrichment strategy is revisited the conventional approach uses a topological enrichment only the elements touching the front are enriched we suggest a geometrical enrichment in which a given domain size is enriched the improvements obtained with this enrichment are discussed second the conditioning of the x fem both for topological and geometrical enrichments is studied a preconditioner is introduced so that off the shelf iterative solver packages can be used and perform as well on x fem matrices as on standard fem matrices the preconditioner uses a local nodal cholesky based decomposition third the numerical integration scheme to build the x fem stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme a 2d benchmark problem is designed to show the improvements and the robustness

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

optimal convergence analysis for the extended finite element method we establish some optimal a priori error estimate on some variants of the extended finite element method xfem namely the xfem with a cut off function and the standard xfem with a fixed enrichment area the results are established for the lame system homogeneous isotropic elasticity and the laplace problem the convergence of the numerical stress intensity factors is also investigated we show some numerical experiments which corroborate the theoretical results

piecewise polynomial spaces and geometric continuity of curves we say that a curve has geometric continuity if its curvatures and frenet frame are continuous in this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity we show that the basic theoretical properties of ordinary spline functions also hold for these spaces these results extend and unify recent work on beta splines and nu splines which are used as a design tool in computer aided geometric design of free form curves and surfaces

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

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

on the smoothed finite element method recently liu et al proposed the smoothed finite element method by using the non mapped shape functions and then introducing the strain smoothing operator when evaluating the element stiffness in the framework of the finite element method however the theories and examples by liu et al are not sufficient for general quadrilateral elements this paper shows that the non mapped shape functions used in the smoothed finite element have disadvantages in existence linearity non negativity and patch test copyright 2008 john wiley sons ltd

on the construction of blending elements for local partition of unity enriched finite elements for computational efficiency partition of unity enrichments are preferably localized to the sub domains where they are needed it is shown that an appropriate construction of the elements in the blending area the regionwhere the enriched elements blend to unenriched elements is often crucial for good performance of local partition of unity enrichments an enhanced strain formulation is developed which leads to good performance the optimal rate of convergence is achieved for polynomial enrichments it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence the methods are illustrated by several examples the examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element but other enrichments are also considered results show that both methods provide optimal rates of convergence

high order extended finite element method for cracked domains the aim of the paper is to study the capabilities of the extended finite element method xfem to achieve accurate computations in non smooth situations such as crack problems although the xfem method ensures a weaker error than classical nite element methods the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity the di culty can be overcome by modifying the enrichment of the nite element basis with the asymptotic crack tip displacement solutions as well as with the heaviside function numerical simulations show that the modi ed xfem method achieves an optimal rate of convergence i e like in a standard nite element method for a smooth problem

integrating strong and weak discontinuities without integration subcells and example applications in an xfem gfem framework partition of unity methods such as the extended finite element method allows discontinuities to be simulated independently of the mesh int j numer meth engng 1999 45 601 620 this eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re meshing as the discontinuity evolves however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted in this paper we use a simple integration technique proposed for polygonal domains int j nuttier meth engng 2009 80 1 103 134 doi 10 1002 nme 2589 to suppress the need for element subdivision numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi material problem show that the proposed method yields accurate results owing to its simplicity the proposed integration technique can be easily integrated in any existing code copyright c 2010 john wiley sons ltd

derivative recovery and a posteriori error estimate for extended finite elements this paper is the first attempt at error estimation for extended finite elements the goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method xfem in each element the local estimator is the l2 norm of the difference between the raw xfem strain field and an enhanced strain field computed by extended moving least squares xmls derivative recovery obtained from the raw nodal xfem displacements the xmls construction is tailored to the nature of the solution the technique is applied to linear elastic fracture mechanics in which near tip asymptotic functions are added to the mls basis the xmls shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity the result is a very smooth enhanced strain solution including the singularity at the crack tip results are shown for two and three dimensional linear elastic fracture mechanics problems in mode i and mixed mode the effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near tip problem it is also shown that for the linear elastic fracture mechanics problems treated the proposed estimator outperforms one of the superconvergent patch recovery technique of zienkiewicz and zhu which is only c0 parametric studies of the general performance of the estimator are also carried out

stable generalized finite element method sgfem abstract the generalized finite element method gfem is a partition of unity method pum where the trial space of standard finite element method fem is augmented with non polynomial shape functions with compact support these shape functions which are also known as the enrichments mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully used to solve a variety of problems with complicated features and microstructure however the stiffness matrix of gfem is badly conditioned much worse compared to the standard fem and there could be a severe loss of accuracy in the computed solution of the associated linear system in this paper we address this issue and propose a modification of the gfem referred to as the stable gfem sgfem we show that sgfem retains the excellent convergence properties of gfem does not require a ramp function in the presence of blending elements and the conditioning of the associated stiffness matrix is not worse than that of the standard fem moreover sgfem is very robust with respect to the parameters of the enrichments we show these features of sgfem on several examples

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

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

evaluation and verification of an hsdt layerwise generalized finite element formulation for adaptive piezoelectric laminated plates a formulation for the bending analysis of composite laminated plates with piezoelectric layers is implemented using the generalized finite element method this formulation is derived from a mechanical description based on higher order shear deformation theory which allows for the use of c0 continuous approximation functions on the domain on the other hand a layerwise theory is employed for interpolation of electric potential across the thickness of piezoelectric layers in such a way that the kinematical hypotheses result in a mixed model the paper presents an analysis of the approximation capability of the proposed numerical model for static analysis using c0 continuous partition of unity and polynomial enrichments to span the approximation spaces by assessment of convergence analytical solutions obtained from the same kinematical hypotheses are used as references results for relative error in the energy norm considering p and h refinements for regular and distorted meshes in addition to a pointwise evaluation of the stresses and electric field are presented the evaluations show that the numerical methodology is a very effective tool for improving the solution through the enrichment even for pointwise values across the thickness and is robust to mesh distortions moreover the results furnish insight about the physical modeling for both active and sensory modes for thick and thin plates

on error estimator and p adaptivity in the generalized finite element method this paper addresses the issue of a p adaptive version of the generalized finite element method gfem the technique adopted here is the equilibrated element residual method but presented under the gfem approach i e by taking into account the typical nodal enrichment scheme of the method such scheme consists of multiplying the partition of unity functions by a set of enrichment functions these functions in the case of the element residual method are monomials and can be used to build the polynomial space one degree higher than the one of the solution in which the error functions is approximated global and local measures are defined and used as error estimator and indicators respectively the error indicators calculated on the element patches that surrounds each node are used to control a refinement procedure numerical examples in plane elasticity are presented outlining in particular the effectivity index of the error estimator proposed finally the p adaptive procedure is described and its good performance is illustrated by the last numerical example

accurate recovery based upper error bounds for the extended finite element framework this paper introduces a recovery type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method xfem the paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a fem framework the upper bound property requires the recovered solution to be equilibrated and continuous the proposed technique consists of using a recovery technique especially adapted to the xfem framework that yields equilibrium at a local level patch by patch then a postprocess based on the partition of unity concept is used to obtain continuity the result is a very accurate but only nearly statically admissible recovered stress field with small equilibrium defaults introduced by the postprocess sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults as demonstrated by the numerical tests

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

error estimation for crack simulations using the xfem summarythe extended nite element method xfem is by now well established for crack calculations in linear elas tic fracture mechanics an advantage of this method is its discretization independence for crack simulations nevertheless discretization errors occur when using the xfem in this paper a simple recovery based errorestimator for the discretization error in xfem calculations for cracks is presented the method is basedon the zienkiewicz and zhu error estimator enhanced smoothed stresses incorporating the discontinuitiesand singularities because of the cracks are recovered to enable the error estimation for arbitrary distributedcracks this approach also allows the consideration of materials with generally inelastic behaviour theenhanced stresses are computed by means of a least square t problem to assess the quality of the error esti mator global norms and the effectivity index for the global energy norm for examples with known analyticalsolutions are presented copyright 2012 john wiley sons ltd

an algebra of pseudodifferential operators with slowly oscillating symbols let v mathbb r denote the banach algebra of absolutely continuous functions of bounded total variation on mathbb r and let mathcal b p be the banach algebra of bounded linear operators acting on the lebesgue space l p mathbb r for 1 p infty we study the banach algebra mathfrak a subset mathcal b p generated by the pseudodifferential operators of zero order with slowly oscillating v mathbb r valued symbols on mathbb r boundedness and compactness conditions for pseudodifferential operators with symbols in l infty mathbb r v mathbb r are obtained a symbol calculus for the non closed algebra of pseudodifferential operators with slowly oscillating v mathbb r valued symbols is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals as a result the quotient banach algebra mathfrak a pi mathfrak a mathcal k where mathcal k is the ideal of compact operators in mathcal b p is commutative and involutive an isomorphism between the quotient banach algebra mathfrak a pi of pseudodifferential operators and the banach algebra widehat mathfrak a of their fredholm symbols is established a fredholm criterion and an index formula for the pseudodifferential operators a in mathfrak a are obtained in terms of their fredholm symbols