imposing dirichlet boundary conditions in the extended finite element method this paper is devoted to the imposition of dirichlet type conditions within the extended finite element method x fem this method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces imposing neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh observed numerically in earlier papers on the contrary much less work has been devoted to dirichlet boundary conditions for the x fem or the limiting case of stiff boundary conditions in this paper we introduce a strategy to impose dirichlet boundary conditions while preserving the optimal rate of convergence the key aspect is the construction of the correct lagrange multiplier space on the boundary as an application we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding rtm process while avoiding remeshing the case of inner conditions is also discussed as well as two important practical cases material interfaces and phase transformation front capturing copyright 2006 john wiley sons ltd

a non intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity problems this paper presents a non intrusive method which yields strict and high quality error bounds of calculated quantities of interest in structural problems solved by the finite element method the focus is on linear viscoelasticity problems described through internal variables in order to solve the adjoint problem in a non intrusive way we use handbook techniques involving enrichment functions introduced through the partition of unity method pum then the mesh and operators used to calculate the reference problem can be reused such a procedure also enables one to address error estimation on pointwise quantities of interest although this implies dealing with infinite energy green functions

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

hitting probabilities and large deviations let fyngn2z be a sequence of random variables in r d and let a r d then pfyn 2 a for some ng is the hitting probability of the set a by the sequence fyng we consider the asymptotic behavior as m 1 o fpfyn 2 ma some ng pfhitting mag whenever 1 the probability law of yn n satises the large deviation principle and 2 the central tendency of yn n is directed away from the given set a for a particular function i w e showpfyn 2 ma some n ge m i a

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

interpolation of hardy spaces of dirichlet series abstract we initiate the study of interpolation of the hardy spaces of dirichlet series we prove that there are no variants of the classical interpolation theorems in the setting of hardy spaces on the countably infinite polytorus t as in the case of the hardy spaces on the finite dimensional torus as a byproduct of our results we get a similar conclusion for the hardy spaces of dirichlet series by using the famous bohr lift our proofs rely on a general method based on interpolation estimates for finite dimensional banach spaces this method is applied to interpolation of injective tensor products and also of spaces of m homogeneous polynomials

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

erdős renyi laws for gibbs measures can one detect a phase transition from a single large sample of a gibbs measure what information does one get on the other gibbs distributions with the same potential we approach these questions via erd s renyi laws in particular we prove almost sure limit theorems for sets of empirical distributions of sub samples of the given one for suitable sub samples size this set converges to the set of stationary gibbs measures moreover we formulate erd s renyi laws for general families of random variables with suitable large deviation principles

on the techniques for constructing admissible stress fields in model verification performances on engineering examples robust global goal oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation in the context of computational mechanics the construction of admissible stress fields ie stress tensors which verify the equilibrium equations is required to set up strict and guaranteed error bounds using residual based error estimators and plays an important role in the quality of the error estimates this work focuses on the different procedures used in the calculation of admissible stress fields which is a crucial and technically complicated point the three main techniques that currently exist called the element equilibration technique eet the star patch equilibration technique spet and the element equilibration star patch technique eespt are investigated and compared with respect to three different criteria namely the quality of associated error estimators computational cost and easiness of practical implementation into commercial finite element codes the numerical results which are presented focus on industrial problems they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators which have the same convergence rate as the exact global error is consistent two and three dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches the analysis of the results reveals that the spet is more accurate than eet and eespt methods but the corresponding computational cost is higher overall the numerical tests prove the interest of the hybrid method eespt and show that it is a correct compromise between quality of the error estimate practical implementation and computational cost furthermore the influence of the cost function involved in the eet and the eespt is studied in order to optimize the estimators

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

a 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

a numerical method with a posteriori error estimation for determining the path taken by a propagating crack the problem of predicting the path taken by a propagating crack under general loading in a two dimensional domain is considered an efficient numerical procedure for computing this path is given and compared to a standard method the model used to predict the direction of crack growth is the principle of local symmetry that is the crack propagates along such a path that the mode ii stress intensity factor is zero along the path unlike many methods for computing crack paths the procedure presented here uses smooth curves to model the crack the theory behind the method is summarized and an a posteriori error estimate for the difference between the computed crack and the true crack is developed this allows one to perform an adaptive procedure for finding crack paths the numerics use a finite element implementation to solve the linear elasticity equations and the contour integral method is used to compute the stress intensity factors

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

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

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

local error estimators for finite element linear analysis local error estimators on stresses displacements etc are derived from constitutive relation error estimators examples show that they can be considered as an upper bound

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

critical phenomena for sequence matching with scoring consider two independent sequences x 1 x n and y 1 y n suppose that x 1 x n are i i d x and y 1 y n are i i d y where x and y are distributions on finite alphabets x and y respectively a score f x x y r is assigned to each pair x i y j and the maximal nonaligned segment score is m n max 0 i j n 0 l 1 f x i l y j l our result is that m n log n x y a s with determined by a tractable variational formula moreover the pair empirical measure of x i l y j l during the segment where m n is achieved converges to a probability measure which is accessible by the same formula these results generalize to x i y j taking values in any polish space to intrasequence scores under shifts to long quality segments and to more than two sequences

validation of a posteriori error estimators by numerical approach abstract a numerical methodology which determines the quality or robustness of a posteriori error estimators is described the methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems namely the local geometry of the grid and the structure of the solution the methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations

a posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem the reciprocal theorem of betti and rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities including local variables like single displacements and stresses the proposed error estimator is evaluated solving the set of equations for an additional right hand side simply applying the energy norm error estimators two times furthermore a general h adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables the efficiency of the current approach is demonstrated for plate and shell examples

on the interpolation of injective or projective tensor products of banach spaces abstract we prove a general result on the factorization of matrix valued analytic functions we deduce from it that if e 0 e 1 and f 0 f 1 are interpolation pairs with dense intersections then under some conditions on the spaces e i f i we have e 0 bo f 0 e 1 bo f 1 e 0 e 1 bo f 0 f 1 0 we find also conditions on the spaces e i f i so that the following holds e 0 bo f 0 e 1 bo f 1 e 0 e 1 bo f 0 f 1 0 some applications of these results are also considered

error estimation of stress intensity factors for mixed mode cracks this paper presents an a posteriori error estimator for mixed mode stress intensity factors in plane linear elasticity a surface integral over an arbitrary crown is used for the separate calculation of the combined mode s stress intensity factors the error in the quantity of interest is based on goal oriented error measures and estimated through an error in the constitutive relation copyright 2006 john wiley sons ltd

goal oriented a posteriori error estimates in linear elastic fracture mechanics goal oriented a posteriori error estimators are presented in this paper for the error obtained while approximately evaluating the j integral in linear elastic fracture mechanics using the finite element method the j integral provides a criterion for crack propagation of a pre existing crack therefore its reliable and accurate evaluation is of considerable importance in engineering applications to prevent failure of structures this can be achieved by goal oriented adaptive finite element methods based on the well established strategy of solving an auxiliary dual problem in order to control the error of a non linear functional here the j integral in this paper two principally different goal oriented adaptive strategies are presented the first strategy rests upon equilibrated residual error estimators based on local neumann problems whereas the second one makes use of averaging techniques the paper is concluded by a numerical example that illustrates our theoretical results

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

a posteriori error estimation for generalized finite element methods in this paper we address the problem of a posteriori error estimation for generalized finite element methods based on the partition of unity method the computational results focus on the question of the reliability of the error estimators and its assessment

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

error estimation and adaptivity for discontinuous failure discontinuous failure is simulated via the introduction of a geometrical discontinuity the cohesive zone is modelled via the use of the partition of unity property of the finite element interpolation by this approach a crack can pass through the elements without any restriction to the underlying mesh despite such a feature it has been confirmed that a sufficiently fine mesh discretization still needs to be ensured in order to obtain a correct crack path and mechanical response the p adaptive scheme driven by error in an energy norm measure or a goal oriented measure has been examined due to its implementational simplicity the results have shown that if considering only increasing the polynomial degree the p approach can greatly improve the results copyright 2008 john wiley sons ltd

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

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

a posteriori estimation and adaptive control of the error in the quantity of interest part i a posteriori estimation of the error in the von mises stress and the stress intensity factor in this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution as an example we consider the plane elasticity problem with the von mises stress and the stress intensity factor as the quantities of interest the estimates of the error in the von mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point the local and the pollution errors and by constructing separate estimates for each component the estimates of the error in the stress intensity factors are constructed by employing an extraction method we demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees in part ii we will address the problem of the adaptive control of the error in the quantity of interest the goal of the computation and the construction of goal adaptive meshes for one or multiple goals

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

some monotonicity properties of schur powers of matrices and related inequalities abstract a series of inequalities are developed relating the spectral radius a b of the schur product a b of two nonnegative matrices a and b with those of a a and b b yielding a b a a b b 1 2 as a corollary it is proved that the spectral radius of the schur powers r a r a r a a a r factors satisfies 1 r log r is decreasing while 1 r 1 log r is increasing the latter provided a is a stochastic matrix the entropy of a finite stationary markov chain is identified with d r dr r 1 a number of majorization comparisons for the spectral radius of schur powers is given

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

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

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

bounds on history dependent or independent local quantities in viscoelasticity problems solved by approximate methods this paper introduces new strict error bounds of outputs of interest for linear viscoelasticity structural problems involving internal state variables and calculated by the finite element method using extraction techniques and the concept of dissipation error we show that we can get both strict and effective bounds by using convexity properties and by taking effects of error accumulation into account