properly posed sets of nodes for multivariate lagrange interpolation in c s in this paper we apply techniques from the theory of ideals and varieties in algebraic geometry to study the geometric structure of a properly posed set of nodes or ppsn for short for multivariate lagrange interpolation along an algebraic hypersurface we provide a hyperplane superposition process to construct the ppsn for interpolation along an algebraic hypersurface and as a result we offer a clear understanding of the geometric structure of the ppsn for multivariate lagrange interpolation in cs

computation of rational interpolants with prescribed poles a constructive proof for existence and unicity of the rational r belonging to gm v m 0 n 0 having prescribed n poles and interpolating m 1 hermite data is given it is based upon explicit computation of the confluent cauchy vandermonde determinant in terms of the nodes and the poles an algorithm to compute r z is presented it calculates this value from a triangular field of values of rational interpolants of lowest possible orders the algorithm is based upon a neville aitken interpolation formula and has arithmetical complexity 0 l l max m 1 n

on the sauer xu formula for the error in multivariate polynomial interpolation use of a new notion of multivariate divided difference leads to a short proof of a formula by sauer and xu for the error in interpolation by polynomials of total degree n in d variables at a correct point set

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

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

on the multivariate horner scheme we present an efficient version of the horner scheme for the evaluation of multivariate polynomials and study its stability properties in particular we show that the backward error of evaluation is bounded by a quantity that is linear in the total degree of the polynomial which itself is usually significantly smaller than the number of operations involved in the evaluation process moreover we show that the same type of estimates holds also for the forward error provided that the polynomial basis is chosen suitably

on the fly algorithms and sequential machines frougny has presented a method that generalizes various on the fly operations that have been presented mainly in connection with computer arithmetic first we shall trace the origin of this method to its source which is the celebrated paper of rabin and scott that introduced the notion of nondeterminism and the power set construction second we shall show that an understanding of this origin may lead to great quantitative improvements in applications of the method finally we shall show by a pathological example that the method as originally presented by frougny may result in circuits that are larger in terms of gates per step by two exponentiations than those that are constructed as described in the present paper

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

modeling three dimensional crack propagation a comparison of crack path tracking strategies the development of a new finite element technique for the simulation of discontinuous failure phenomena in three dimensions is the key objective of this study in contrast to the widely used extended finite element technique we apply a purely deformation based strategy based on an independent interpolation of the deformation field on both sides of the discontinuity this method has been applied successfully for two dimensional crack propagation problems in the past however when it comes to three dimensional failure phenomena it faces the same difficulties as the extended finite element method unlike in two dimensions the characterization for the three dimensional failure surface is non unique and the tracking of the discrete crack can be performed in several conceptually different ways in this work we review the four most common three dimensional crack tracking strategies we perform a systematic comparison in terms of standard algorithmic quality measures such as mesh independency efficiency robustness stability and computational cost moreover we discuss more specific issues such as crack path continuity and integratability in commercial finite element packages the features of the suggested crack tracking algorithms will be elaborated by means of characteristic benchmark problems in failure analysis copyright 2008 john wiley sons ltd

vector level sets for description of propagating cracks in finite elements a new level set method is developed for describing surfaces that are frozen behind a moving front such as cracks in this formulation the level set is described in two dimensions by a three tuple the sign of the level set function and the components of the closest point projection to the surface the update of the level set is constructed by geometric formulas which are easily implemented results are given for growth of lines in two dimensions that show the method is very accurate the method combines very naturally with the extended finite element method xfem where the discontinuous enrichment for cracks is best described in terms of level set functions examples of crack growth simulations obtained by combining this level set method with the extended finite element method are given copyright 2003 john wiley sons ltd

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

on the conformity of strong regularized embedded and smeared discontinuity approaches for the modeling of localized failure in solids abstract once strain localization occurs in softening solids inelastic loading behavior is restricted within a narrow band while the bulk unloads elastically accordingly localized failure in solids can be approached by embedding or smearing a traction based inelastic discontinuity band within an equivalent elastic matrix along a specific orientation in this context the conformity of the strong regularized and embedded smeared discontinuity approaches are investigated regarding the strategies dealing with the kinematics and statics on one hand the traction continuity condition imposed in weak form results in the strong and regularized discontinuity approaches with respect to the approximation of displacement and strain discontinuities in addition to the elastic bulk consistent plastic damage cohesive models for the discontinuities are established the conformity between the strong discontinuity approach and its regularized counterpart is shown through the fracture energy analysis on the other hand the traction continuity condition can also be enforced point wisely in strong form so that the standard principle of virtual work applies in this case the static constraint resulting from traction continuity can be used to eliminate the kinematic variable associated with the discontinuity band at the material level this strategy leads to embedded and smeared discontinuity models for the overall weakened solid which can also be cast into the elastoplastic degradation framework with a different kinematic decomposition being equivalent to the kinematic constraint guaranteeing stress continuity upon strain localization mohr s maximization postulate is adopted for the determination of the discontinuity orientation closed form results are presented in plane stress conditions with the classical rankine mohr coulomb von mises and drucker prager criteria as illustrative examples the orientation of the discontinuity band and the stress based failure criteria consistent with the given traction based counterparts are derived finally a generic failure criterion of either elliptic parabolic or hyperbolic type appropriate for the modeling of mixed mode failure is analyzed in a unified manner furthermore a novel method is proposed to calibrate the involved mesoscopic parameters from available macroscopic test data which is then validated against willam s numerical test

on the rate of convergence for the approximation of nonlinear problems in this paper we present an abstract theory to obtain error estimates in various norms for the approximation of solution branches of nonlinear equations by the aid of known estimates for corresponding linear problems

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

a finite element method for the computational modelling of cohesive cracks the present contribution is concerned with the computational modelling of cohesive cracks in quasi brittle materials whereby the discontinuity is not limited to interelement boundaries but is allowed to propagate freely through the elements in the elements which are intersected by the discontinuity additional displacement degrees of freedom are introduced at the existing nodes therefore two independent copies of the standard basis functions are used one set is put to zero on one side of the discontinuity while it takes its usual values on the opposite side and vice versa for the other set to model inelastic material behaviour a discrete damage type constitutive model is applied formulated in terms of displacements and tractions at the surface some details on the numerical implementation are given concerning the failure criterion the determination of the direction of the discontinuity and the integration scheme finally numerical examples show the performance of the method

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

on lagrange and hermite interpolation in rk a method for the construction of a set of data of interpolation in several variables is given the resulting data which are either function values or directional derivatives values give rise to a space of polynomials in such a way that unisolvence is guaranteed the interpolating polynomial is calculated using a procedure which generalizes the newton divided differences formula for a single variable

an improved stable xfem is xfem with a novel enrichment function for the computational modeling of cohesive cracks abstract owing to the solution space enriched with a priori known knowledge through the partition of unity the extended or generalized finite element method xfem gfem has been widely applied to the modeling of localized failure in solids as far as cohesive cracks are concerned in this work on the one hand the xfem enriched with heaviside function or the shifted one with respect to nodal values is of optimal accuracy but is prone to the issue of ill conditioned or even singular system matrix for arbitrary crack propagation on the other hand the stable xfem s xfem circumvents the ill conditioning issues by introducing the residual heaviside function with respect to its linear interpolant at the cost of losing accuracy e g spurious stress locking for discontinuities with non uniform displacement jumps aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix this work addresses a simple yet effective improved stable xfem is xfem with a novel enrichment function both the xfem and s xfem are recovered as its particular cases with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale in the proposed is xfem as the enrichment basis functions are not contained in the standard polynomial space conditioning of the resulting system matrix is nearly insensitive to the mesh discontinuity configuration even if the crack path gets close to element nodes furthermore the is xfem with a small stabilization parameter is almost of the same accuracy as the xfem particularly pathological numerical results polluted by spurious stress locking exhibited in the s xfem are not observed across a fully softened discontinuity with non uniform displacement jumps as only the enrichment function is modified the proposed is xfem together with appropriate crack propagation criterion and tracking algorithm can be implemented easily in the existing xfem codes several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed is xfem regarding both accuracy of the solution and conditioning of the system matrix

blending in the extended finite element method by discontinuous galerkin and assumed strain methods summary in the extended finite element method xfem errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain a discontinuous galerkin dg formulation is developed which circumvents this source of error a patch based version of the dg formulation is developed which decomposes the domain into enriched and unenriched subdomains continuity between patches is enforced with an internal penalty method an element based form is also developed where each element is considered a patch the patch based dg is shown to have similar accuracy to the element based dg for a given discretization but requires significantly fewer degrees of freedom the method is applied to material interfaces cracks and dislocation problems for the dislocations a contour integral form of the internal forces that only requires integration over the patch boundaries is developed a previously developed assumed strain as method is also developed further and compared with the dg method for weak discontinuities and linear elastic cracks the dg method is shown to be significantly more accurate than the standard xfem for a given element size and to converge optimally even where the standard xfem does not the accuracy of the dg method is similar to that of the as method but requires less application specific coding copyright q 2007 john wiley sons ltd

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

uniform bivariate hermite interpolation i coordinate degree on considere des schemas d interpolation d hermite uniforme bivariable a partir d espaces de polynomes de degre coordonne m n on se pose la question de la presque regularite de tels schemas c est a dire celle de l existence d ensembles de n uds pour lesquels avec une constellation de derivees donnees l interpolation est possible on montre que dans le cas affirmatif le schema en question peut etre utilise pour presque tous les ensembles de n uds l etude est basee sur des techniques de reduction

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

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

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

on bivariate hermite interpolation with minimal degree polynomials a newton type approach is used to deal with bivariate polynomial hermite interpolation problems when the data are distributed in the intersections of two families of straight lines as a generalization of regular grids the interpolation operator is degree reducing and the interpolation space is a minimal degree space integral remainder formulas are given for the lagrange case then extended to the hermite case and finally used to obtain error estimates

a multiscale finite element method with embedded strong discontinuity model for the simulation of cohesive cracks in solids abstract a multiscale finite element method with the embedded strong discontinuity model is proposed to simulate the cohesive cracks in solids in the proposed method the kinematic descriptions of the strong discontinuity and space discretization are considered based on the fine scale with the strong discontinuity approach then in order to correctly and conveniently deliver the discontinuous information between the coarse scale and fine scale an enhanced coarse element strategy is proposed to construct the multiscale base functions that can well capture the discontinuous characteristics and preserve an adequate accuracy for the unit cells exhibiting a strong discontinuity the main idea is that the coarse nodes of the enhanced coarse element can be dynamically added according to the identification of the intersection between the crack path and the boundaries of the unit cell during the computational procedure the strategy overcomes the deficiency that the traditional coarse elements in the multiscale finite element method cannot well characterize the displacement jump property on the boundary of the unit cell moreover to accurately obtain the microscopic displacement the displacement decomposition technique is adopted to modify the downscale computations by adding the perturbation solutions numerical examples of normal tension and bending tests are presented to validate the proposed method by comparing the results with the analytical or fine finite element solutions finally the three point bending and four point bending benchmarks are performed to further demonstrate the effectiveness and high efficiency of the method

hyperinterpolation on the square we show that hyperinterpolation at near minimal cubature points for the product chebyshev measure along with xu compact formula for the corresponding reproducing kernel provide a simple and powerful polynomial approximation formula in the uniform norm on the square the lebesgue constant of the hyperinterpolation operator grows like log 2 of the degree as that of quasi optimal interpolation sets recently proposed in the literature moreover we give an accurate implementation of the hyperinterpolation formula with linear cost in the number of cubature points and we compare it with interpolation formulas at the same set of points

interpolation lattices in several variables principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions a principal lattice can be described as the set of intersection points of n 1 pencils of parallel hyperplanes using a projective point of view lee and phillips extended this situation to n 1 linear pencils of hyperplanes in two recent papers two of us have introduced generalized principal lattices in the plane using cubic pencils in this paper we analyze the problem in n dimensions considering polynomial exponential and trigonometric pencils which can be combined in different ways to obtain generalized principal lattices we also consider the case of coincident pencils an error formula for generalized principal lattices is discussed

an embedded strong discontinuity model for cracking of plain concrete abstract a numerical model formulated within the framework of a nonsymmetric strong discontinuity approach for fracture simulations of plain concrete is presented the model is based on the fixed crack concept and makes use of the concept of the elements with embedded discontinuities discontinuity segments of individual elements are considered to form a c0 continuous path enforcement of continuity of the crack path across adjacent elements is established by a partial domain crack tracking algorithm orientation of individual crack segments is derived from a nonlocal strain field the capabilities of the model are shown by means of numerical examples

assumed enhanced strain and the extended finite element methods a unification of concepts abstract we present strong and weak forms of boundary value problems for a solid with a cohesive frictional crack two commonly used techniques for enhancing the finite element fe interpolation are considered the assumed enhanced strain aes and the extended fe methods we compare and contrast the fe approximations involved in these techniques the aes method employs a piecewise constant interpolation of slip that is discontinuous across element boundaries in contrast the extended fe method interpolates slip continuously across element boundaries through numerical examples we discuss the implications of these approximations to the calculated overall deformation field in the regime of infinitesimal deformation with only a heaviside enrichment we find the aes method to predict larger slip i e softer response compared to the extended fe solution we ascribe the discrepancy to the lack of higher order crack tip enhancement in the extended fe solution finally we compare the slip predicted by the above fe enhancement techniques to that calculated by classical nonlinear contact mechanics algorithm for the case where the crack traverses the element sides

a comment on the article a finite element method for simulation of strong and weak discontinuities in solid mechanics by a hansbo and p hansbo comput methods appl mech engrg 193 2004 3523 3540

polynomial interpolation of minimal degree minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which lagrange interpolation with respect to the given points is uniquely solvable and degree reducing this is a generalization of the concept of least interpolation introduced by de boor and ron this paper investigates the behavior of lagrange interpolation with respect to these spaces giving a newton interpolation method and a remainder formula for the error of interpolation moreover a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view

on birkhoff interpolation by polynomials in several variables chui and lai 1987 have discussed a kind of multivariate polynomial interpolation problem defined on the straight line type node configuration c sltncc in this paper we define general birkhoff interpolation problems for the sltncc and show under some restrictions that these interpolation problems are unisolvent also we give some generalizations of the sltncc

evaluation of multivariate polynomials and their derivatives an extension of homer s algorithm to the evaluation of m variate polynomials and their derivatives is obtained the schemes of computation are represented by trees because this type of graph describes exactly in which order the computations must be done some examples of algorithms for one and two variables are given it is well known that horner s algorithm for evaluation of univariate poly nomials and their derivatives is based upon the nested reformulation of the polynomial our aim is to extend this idea to multivariate polynomials let k x be the ring of m variate polynomials on a field k and p be the set of polynomials of exact total degree 1 consider p g k x written in the form n

towards a generalization of a discrete strong discontinuity approach a generalized discrete strong discontinuity approach gsda is presented in which non homogeneous jumps are embedded in parent finite elements the new formulation presents similar kinematics as for interface elements accurately simulating both rigid body motion and stretching induced by the opening of a discontinuity this generalized discrete strong discontinuity approach is built within the framework of strong embedded discontinuity formulations as a consequence it is mesh independent and avoids the progressive remeshing needed with interface elements the gsda can be viewed as a generalization of other embedded formulations for which particular assumptions are adopted regarding both the jump transmission and the variational principle structural examples including mode i mode ii and mixed mode fracture are computed and compared with other embedded formulations and experimental results

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

hermite interpolation by neural networks in this paper we study hermite interpolation by single layer feedforward networks we provide upper bound estimates for the minimum number of hidden units that can solve the hermite interpolation problem exactly we consider both the univariate and the multivariate cases

superposition interpolation process in cn in this paper we deeply research lagrange interpolation by polynomial in several variables and give an application of cayley bacharach theorem for it particularly on the sufficiently intersected algebraic manifold or siam for short we introduce a general method of constructing properly posed set of nodes or ppsn for short for lagrange interpolation namely the superposition interpolation process then we give an equivalent condition about a ppsn along a siam further we introduce a relation between the sufficiently intersected algebraic hypersurfaces and h basis at the end of this paper we use the extended cayley bacharach theorem to resolve some problems of lagrange interpolation along the zero dimensional and one dimensional algebraic manifolds

on the fly conversion of redundant into conventional representations an algorithm to convert redundant number representations into conventional representations is presented the algorithm is performed concurrently with the digit by digit generation of redundant forms by schemes such as srt division it has a step delay roughly equivalent to the delay of a carry save adder and simple implementation the conversion scheme is applicable in arithmetic algorithms such as nonrestoring division square root and on line operations in which redundantly represented results are generated in a digit by digit manner from most significant to least significant

comparison of fast boundary element methods on parametric surfaces we compare fast black box boundary element methods on parametric surfaces in r3 these are the adaptive cross approximation the multipole method based on interpolation and the wavelet galerkin scheme the surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels nonetheless parametric surface representations are easily accessible from computer aided design cad and are recently topic of the studies in isogeometric analysis especially we can apply two dimensional interpolation in the multipole method a main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry this results in a superior compression of the far field compared to other cluster methods

a geometrically regularized gradient damage model with energetic equivalence abstract aiming to bridge the gap between damage and fracture mechanics for modeling localized failure in solids this paper addresses a novel geometrically regularized gradient damage model with energetic equivalence for cracking evolution with the free energy potential defined as usual in terms of the local strain and damage fields the constitutive relations are derived consistently from the standard framework of thermodynamics upon the sharp crack topology geometrically regularized by the damage localization band with a length scale the ensuing energetic equivalence naturally yields the damage evolution law of gradient type and the associated boundary condition of neumann type compared to other gradient damage models no extra assumptions like the nonlocal energy residual and insulation condition are introduced moreover the damage gradient physically accounting for microscopic nonlocal interactions is fully dissipative as expected in line with the unified phase field theory recently proposed by the author wu 2017 during failure processes the material behavior is uniquely characterized by two constitutive functions i e the degradation function defining the free energy potential of the bulk and the geometric function regularizing the sharp crack topology in particular optimal constitutive functions defining an equivalent cohesive zone model of general softening laws are postulated with the involved parameters calibrated from standard material properties the proposed model is numerically implemented into the multi field finite element method and applied to several benchmark tests of concrete under mode i and mixed mode failure it is found that the incorporated length scale can be regarded either a numerical parameter or a material property for the former considered in this work the length scale has negligible if not no effects on the global responses so long as the sharp crack topology and the damage field of high gradients within the localization band are well resolved furthermore the localization bandwidth does not exhibit spurious widening but rather it approaches to a finite value proportional to the length scale comparison between the numerical and experimental results regarding the curve of load versus displacement and crack path confirms validity of the proposed gradient damage model for characterizing localized failure in quasi brittle solids

local approximation by certain spaces of exponential polynomials approximation order of exponential box splines and related interpolation problems local approximation order to smooth complex valued functions by a finite dimensional space 7b h spanned by certain products of exponentials by polynomials is investigated the results obtained together with a suitable quasi interpolation scheme are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline the analysis of a typical 7b h is based here on the identification of its dual with a certain space 7b p of multivariate polynomials this point of view allows us to solve a class of multivariate interpolation problems by the polynomials from 7b p

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

bivariate hermite birkhoff polynomial interpolation with asymptotic conditions some asymptotic conditions along prescribed directions are added to the usual interpolation data in bivariate problems these asymptotic conditions are written in terms of interpolation and then the new problem is studied in the frame of the interpolation systems introduced by gasca and maeztu some years ago a newton type interpolation formula is obtained for the enlarged problem and then some particular cases are studied

a thermodynamically consistent plastic damage framework for localized failure in quasi brittle solids material model and strain localization analysis aiming for the modeling of localized failure in quasi brittle solids this paper addresses a thermodynamically consistent plastic damage framework and the corresponding strain localization analysis a unified elastoplastic damage model is first presented based on two alternative kinematic decompositions assuming infinitesimal deformations with the evolution laws of involved internal variables characterized by a dissipative flow tensor for the strong or regularized discontinuity to form in such inelastic quasi brittle solids and to evolve eventually into a fully softened one a novel strain localization analysis is then suggested a kinematic constraint more demanding than the classical discontinuous bifurcation condition is derived by accounting for the traction continuity and the loading unloading states consistent with the kinematics of a strong or regularized discontinuity more specifically the strain jumps characterized by maxwell s kinematic condition have to be completely inelastic energy dissipative reproduction of this kinematics implies vanishing of the aforesaid dissipative flow tensorial components in the directions orthogonal to the discontinuity orientation this property allows naturally developing a localized plastic damage model for the discontinuity band with its orientation and the traction based failure criterion consistently determined a posteriori from the given stress based counterpart the general results are then particularized to the 2d conditions of plane stress and plane strain it is found that in the case of plane stress strain localization into a strong or regularized discontinuity can occur at the onset of strain softening contrariwise owing to an extra kinematic constraint in the condition of plane strain some continuous inelastic deformations and substantial re orientation of principal strain directions in general have to take place in the softening regime prior to strain localization the classical rankine mohr coulomb von mises j2 and drucker prager criteria are analyzed as illustrative examples in particular both the closed form solutions for the discontinuity angles validated by numerical simulations and the corresponding traction based failure criteria are obtained

extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids abstract this paper addresses novel extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids on one hand based upon the unified multiscale kinematics of strong discontinuities standard finite elements at the coarse scale are enriched with fine scale kinematics in which non uniform discontinuity modes are incorporated on the other hand the traction continuity condition across the discontinuity is enforced in the statically optimal form as the fine scale statics the admissible discontinuity modes satisfying a priori the traction continuity condition are derived in addition to the relative rigid body motions translations and rotations more general relative stretching modes that induce discontinuous strain stress fields in the bulk are obtained with no limit of a vanishing poisson s ratio the proposed model then particularizes to 2d quadrilateral elements the displacement jumps at the discontinuity nodes are selected as the enrichment parameters and regarded as global variables shared by neighboring elements the resulting model not only is intrinsically stress locking free for a fully softened discontinuity but also automatically guarantees global continuity of the displacement jumps another benefit is that the numerical instability suffered under the unfavorable element discontinuity configuration can be easily circumvented furthermore the vanishing displacement jumps at the discontinuity tip can be enforced trivially as conventional essential boundary conditions and those complex strategies introduced in existing models are avoided together with the failure criterion related to the element average stress and the propagation orientation based on a simplified nonlocal stress a local crack tracking algorithm is adopted to guarantee global continuity of the discontinuity path representative numerical simulations of element and structure benchmark tests are presented to verify the performance of the proposed approach in the modeling of arbitrary discontinuity propagation in solids

on the singularity of multivariate hermite interpolation in this paper we study the singularity of multivariate hermite interpolation of type total degree we present two methods to judge the singularity of the interpolation schemes considered and by methods to be developed we show that all hermite interpolation of type total degree on m d k points in r d is singular if d 2k and then we solve the hermite interpolation problem on email protected d 3 nodes completely precisely all hermite interpolations of type total degree on email protected d 1 points with d 2 are singular only three cases for m d 2 and one case for m d 3 can produce regular hermite interpolation schemes respectively besides we also present a method to compute the interpolation space for hermite interpolation of type total degree

newton basis for multivariate birkhoff interpolation multivariate birkhoff interpolation is the most complex polynomial interpolation problem and people know little about it so far in this paper we introduce a special new type of multivariate birkhoff interpolation and present a newton paradigm for it using the algorithms proposed in this paper we can construct a hermite system for any interpolation problem of this type and then obtain a newton basis for the problem w r t the hermite system

digit set conversions generalizations and applications the problem of digit set conversion for fixed radix is investigated for the case of converting into a non redundant as well as into a redundant digit set conversion may be from very general digit sets and covers as special cases multiplier recodings additions and certain multiplications we generalize known algorithms for conversions into non redundant digit sets as well as apply conversion to generalize the o log n time algorithm for conditional sum addition using parallel prefix computation and a comparison is made with standard carry lookahead techniques examples on multi operand addition are used to illustrate the generality of this approach o 1 time algorithms for converting into redundant digit sets are generalized based on a very simple lemma which provides a framework for all conversions into redundant digit sets applications in multiplier recoding and partial product accumulation are used as exemplifications

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

unified analysis of enriched finite elements for modeling cohesive cracks abstract aiming to provide a justified theoretical ground based on which different enriched finite elements can be systematically compared we developed in this work a unified framework for modeling cohesive cracks using the variational multiscale method the kinematics i e coarse and fine scale displacement and strain fields and statics i e coarse and fine scale equilibrium equations are thoroughly investigated in both continuum and discrete settings with respect to the fine scale kinematics and statics adopted in the embedded finite element method efem and the extended finite element method xfem we mainly discuss four groups of enriched finite elements with non uniform discontinuity modes i e the kinematically optimal symmetric efem and xfem as well as the kinematically and statically optimal non symmetric efem and xfem in all these methods the enrichment parameters can be regarded either as element wise local or continuous global variables the enriched finite elements with material element node enrichments are then exemplified in an increasing order of their respective capabilities in representing the fine scale kinematics and the interrelations between them are also discussed owing to this unified framework we are able to clarify some existing points of view related to efem and xfem it is found that if the same fine scale statics is used and the same local global property is assumed for the involved enrichment parameters xfem can be regarded as a kinematically enhanced efem since it accounts for a more general fine scale kinematics than that i e relative rigid body motions and self stretching considered in efem finally several simple but sufficiently representative numerical examples are presented to show the significance in appropriately reflecting the fine scale kinematics and statics in an enriched finite element for modeling cohesive cracks

a finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation abstract this paper presents a finite element model for strain localization analysis of elastoplastic solids subjected to discontinuous displacement fields based on standard galerkin approximation strain enhancements via jumps in the displacement field are captured and condensed on the material level leading to a formulation that does not require static condensation to be performed on the element level the mathematical formulation revolves around the dual response of a macroscopic point cut by a shear band which requires the satisfaction of the yield condition on the band as the same stress point unloads elastically just outside the band precise conditions for the appearance of slip lines including their initiation and evolution are outlined for a rate independent strain softening drucker prager model and explicit analytical expressions are used to describe the orientation of the slip line in a plane strain setting at post localization the stress point integration algorithm along the band is exact and amenable to consistent linearization numerical examples involving simple shearing of elastoplastic solids with deviatoric plastic flow as well as plane strain compression of dilatant cohesive frictional materials are presented to demonstrate absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions

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

a unified approach to evaluation algorithms for multivariate polynomials we present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the bezier multinomial or taylor lagrange and newton bases this unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of l bases a class of bases that include the bezier multinomial and a rich subclass of lagrange and newton bases all of the known evaluation algorithms can be generated either by considering recursive evaluation algorithms for l bases or by examining change of basis algorithms for l bases for polynomials of degree n in s variables the class of recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity o n s 1 a nested multiplication algorithm with computational complexity o n s lgn and a ladder recurrence algorithm with computational complexity o n s these algorithms also generate a new generalization of the aitken neville algorithm for evaluation of multivariate polynomials expressed in terms of lagrange l bases the second class of algorithms based on certain change of basis algorithms between l bases include a dual nested multiplication algorithm with computational complexity o n s and a divided difference algorithm a forward difference algorithm and a lagrange evaluation algorithm with computational complexities o n s o 1 and o n per point respectively for the evaluation of multivariate polynomials at several points

3d modelling of strong discontinuities in elastoplastic solids fixed and rotating localization formulations this paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials based on the strong discontinuity approach sda computational mechanics 1993 12 277 296 mesh objective 2d and 3d finite element formulations are developed using linear and quadratic 2d elements as well as 8 noded 3d elements in the formulation of the finite element model proposed in the paper the analogy with standard formulations is emphasized the parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique this approach results in linearized constitutive equations formally identical to continuum models therefore the standard return mapping algorithm is used to solve the non linear equations in analogy to concepts used in continuum smeared crack models a rotating formulation of the sda is proposed in addition to the standard concept of fixed discontinuities it is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions the applicability of the proposed sda finite element model as well as its numerical performance is investigated by means of a three dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force copyright 2003 john wiley sons ltd

on the computation of determinants arising in some bivariate rational interpolation problems abstract bivariate cauchy vandermonde determinants arise in some bivariate rational interpolation problems these determinants are easily computed by means of the technique of interpolation systems numer math 39 1 14 1982 when it is applied to prove the unisolvence of those interpolation problems the solution of the problem is given in a newton form

a recursive algorithm for hermite interpolation over a triangular grid a recursive algorithm for hermite interpolation of bivariate data over triangular grids is presented this interpolation algorithm has a dynamic programming flavor and it computes a single polynomial that interpolates the full set of data the data we interpolate are partial derivatives and mixed partials up to some fixed order at the nodes of the grid the interpolant is a polynomial with minimal degree bound when the order is identical for all nodes the proposed interpolation algorithm is affinely invariant has at least linear precision is symmetric with respect to the grid directions and can reuse existing computations if points are added to the grid

stable monomial basis for multivariate birkhoff interpolation problems for a given node set z and the corresponding interpolation conditions one of the key problems in multivariate birkhoff interpolation is determining a monomial basis that spans the interpolation space however a slight perturbation of the node set will result in a different monomial basis we propose a more general multivariate birkhoff interpolation scheme and provide a numerical algorithm that can determine the stable monomial basis to ensure that the interpolation polynomial always exists in the space spanned by the monomial basis for any perturbation of the node set within a given error e

a comparative study on finite elements for capturing strong discontinuities e fem vs x fem a comparative study on finite elements for capturing strong discontinuities by means of elemental e fem or nodal enrichments x fem is presented based on the same constitutive model continuum damage and linear elements triangles and tetrahedra optimized implementations of both types of enrichments in the same non linear code are tested for a representative set of 2d and 3d crack propagation examples it is shown that both methods provide the same qualitative and quantitative results for enough refined meshes for the performed tests e fem exhibited in general a higher accuracy mostly for coarse meshes whereas convergence rate with mesh refinement which is super linear showed slightly higher for x fem as for the computational costs for single crack modelling x fem showed depending on the case from 1 1 to about 2 5 times more expensive than e fem for multiple cracks the computational cost of e fem keeps constant whereas the cost associated to x fem increases linearly with the number of modelled cracks