a mesh based partition of unity method for discontinuity modeling this study explores a numerical analysis scheme the manifold method within the framework of the partition of unity method the manifold method is rooted in the discrete element method and has been used in solving discrete continuum interaction problems at its core two meshes are employed in an analysis the mathematical mesh provides the nodes for building a finite covering of the solution domain and the partition of unity functions while the physical mesh provides the domain of integration after providing a geometric interpretation and giving a construct overview the working of the method is illustrated with a number of examples

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

a computational method for frictional contact problem using finite element method a numerical procedure is developed for the solution of the two dimensional frictional contact problems with coulomb s law of friction the formulation for this procedure is reduced to a complementarity problem the contact region is separated into stick and slip regions and the contact stress can be solved systematically by applying the solution technique of the complementarity problem

stable gfem sgfem improved conditioning and accuracy of gfem xfem for three dimensional fracture mechanics abstract in this paper we present an extension of the stable generalized fem sgfem for 3 d fracture mechanics problems numerical experiments show that the use of available singular enrichment bases derived from the 2 d asymptotic crack tip solution leads to a severe loss of accuracy in the sgfem an enrichment scheme based on singular bases and linear polynomials is shown to recover the optimal convergence of the sgfem the proposed sgfem for 3 d fractures delivers the same rate of growth in condition number as the standard fem for proper choice of singular enrichment functions the accuracy and conditioning obtained with the sgfem is compared with the generalized fem gfem for different types of singular enrichment bases a fully 3 d fracture mechanics problem is considered to highlight issues that arise only in 3 d problems the convergence of stress intensity factors sifs extracted from gfem and sgfem solutions and the effect of the size of enrichment sub domains on the accuracy of extracted sifs are also studied it is shown that the accuracy of sifs may deteriorate with mesh refinement when the topological enrichment strategy is adopted the proposed sgfem delivers much better accuracy than the gfem for both geometrical and topological enrichment strategies

on the pre lie algebra of specified feynman graphs we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t

the structure of the ladder insertion elimination lie algebra we continue our investigation into the insertion elimination lie algebra open image in new window of feynman graphs in the ladder case emphasizing the structure of this lie algebra relevant for future applications in the study of dyson schwinger equations we work out the relation to the classical infinite dimensional lie algebra open image in new window and we determine the cohomology of open image in new window

extra dof free and linearly independent enrichments in gfem abstract a gfem without extra dof is developed without the extra dof the resulting linear system becomes independent in size of the order of local approximation the resulting pu approximation becomes free of linear dependence and the resulting stiffness matrix becomes good conditioned numerical studies show the new gfem s excellent convergence properties and excellent stability

stickiness in generic low dimensional hamiltonian systems a recurrence time statistics approach we analyze the structure and stickiness in the chaotic components of generic hamiltonian systems with divided phase space following the method proposed recently in lozej and robnik phys rev e 98 022220 2018 2470 004510 1103 physreve 98 022220 the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid we perform extensive numerical studies of three example systems the chirikov standard map the family of robnik billiards and the family of lemon billiards the filling of the cells is compared to the random model of chaotic diffusion introduced in robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 for the description of transport in the phase spaces of ergodic systems the model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential in generic systems the stickiness induces correlations in the cell visits the distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping we model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times we introduce the variable s which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness we use s to globally assess the distributions of recurrence times we find that in the bulk of the chaotic sea s 1 while s 1 in areas of stickiness we present the results in the form of animated grayscale plots of the variable s in the largest chaotic component for the three example systems included as supplemental material to this paper

a rigorous and unified mass lumping scheme for higher order elements abstract in dynamic analysis with explicit time integration schemes a lumped mass matrix lmm is preferable because lmm can avoid solving the large scale simultaneous algebraic equations mathematically rigorous mass lumping schemes such as the mass lumping by nodal quadrature and the row sum technique are applicable to only linear or bilinear elements for higher order elements such as 8 node serendipity elements the diagonal scaling procedure is the only lumping method that can be recommended to generate positive definite diagonal element mass matrices unfortunately there is no mathematical theory in support of this approach this study proposes a general mass lumping scheme applicable to higher order elements where the virtual work of initial force is integrated over the problem domain that is viewed as the manifold covered by the finite element patches by a series of numerical experiments both free and forced vibration problems it is suggested that even in the implicit time integration scheme the consistent mass matrix cmm can be superseded by the proposed lmm furthermore the proposed lmm has much stronger adaptability to distorted meshes

effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method numerical stability by using certain time integration scheme is a critical issue for accurate simulation of discontinuous deformations of solids to investigate the effects of the time integration schemes on the numerical stability of the numerical manifold method the implicit time integration schemes ie the newmark the hht alpha and the wbz alpha methods and the explicit time integration algorithms ie the central difference the zhai s and chung lee methods are implemented their performance is examined by conducting transient response analysis of an elastic strip subjected to constant loading impact analysis of an elastic rod with an initial velocity and excavation analysis of jointed rock masses respectively parametric studies using different time steps are conducted for different time integration algorithms and the convergence efficiency of the open close iterations for the contact problems is also investigated it is proved that the hilber hughes taylor alpha hht alpha wood bossak zienkiewicz alpha wbz alpha zhai s and chung lee methods are more attractive in solving discontinuous deformation problems involving nonlinear contacts it is also found that the examined explicit algorithms showed higher computational efficiency compared to those implicit algorithms within acceptable computational accuracy

numerical simulations of strain localization in inelastic solids using mesh free methods in this paper a comprehensive account on using mesh free methods to simulate strain localization in inelastic solids is presented using an explicit displacement based formulation in mesh free computations high resolution shear band formations are obtained in both two dimensional 2 d and three dimensional 3 d simulations without recourse to any mixed formulation discontinuous incompatible element or special mesh design the numerical solutions obtained here are insensitive to the orientation of the particle distributions if the local particle distribution is quasi uniform which to a large extent relieves the mesh alignment sensitivity that finite element methods suffer moreover a simple h adaptivity procedure is implemented in the explicit calculation and by utilizing a mesh free hierarchical partition of unity a spectral wavelet adaptivity procedure is developed to seek high resolution shear band formations moreover the phenomenon of multiple shear band and mode switching are observed in numerical computations with a relatively coarse particle distribution in contrast to the costly fine scale finite element simulations copyright 2000 john wiley sons ltd

the mixed cell complex partition of unity method we present a galerkin method for solving partial differential equations which is a blend of ideas from particle based methods on the one side and traditional finite element methods on the other side the method is here named mixed cell complex partition of unity method mccpum it can be arbitrarily considered as being based on a set of scattered particles in the domain and on its boundary or on a delaunay cell decomposition of the domain in contrast to the element free galerkin method and other meshless techniques the partition of unity is not constructed on circular or rectangular supports but rather on overlapping polyhedral supports generated from a voronoi delaunay decomposition of space the mixed cell complex this approach inherits most of the advantages of truly meshless schemes while it greatly facilitates the numerical integration of the weak forms required in galerkin approximations the discretization is exclusively governed by the selection of nodes and the approximation orders associated to the nodes locally here legendre polynomials of arbitrary orders are used the mixed cell complex and the corresponding galerkin discretization are explained numerical examples for the poisson problem in two dimensions are presented and the efficiency of the method is discussed

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

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

an integrated procedure for three dimensional structural analysis with the finite cover method in this paper an integrated procedure for three dimensional 3d structural analyses with the finite cover method fcm is introduced in the pre process of this procedure the geometry of a structure is modelled by 3d cad followed by digitization to have the corresponding voxel model and then the structure is covered by a union of mathematical covers namely a mathematical mesh independently generated for approximation purposes since the mesh topology in the fcm does not need to conform to the physical boundaries of the structure the mesh can be regular and structured thus the numerical analysis procedure is free from the difficulties mesh generation typically poses and in this sense enables us to realize the mesh free analysis after formulating the fcm with interface elements for the static equilibrium state of a structure we detail the procedure of the finite cover modelling including the geometry modelling with 3d cad and the identification of the geometry covered by a regular mesh for numerical integration prior to full 3d modelling and analysis we present a simple numerical example to confirm the equivalence of the performance of the fcm and that of the standard finite element method fem finally representative numerical examples are presented to demonstrate the capabilities of the proposed analysis procedure

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

s r decomposition based numerical manifold method the numerical manifold method nmm surmounting the mesh dependence has successfully solved very complicated problems involving small deformation and large movement but had few applications to large deformation and large rotation problems because the false volume expansion and other issues exist in this study it is shown that the false volume expansion in nmm can be excellently resolved by using the s r strain rotation decomposition theorem which can precisely reflect complex physical behaviors occurring in the process of large rotation and large deformation the numerical methods based on the s r decomposition theorem have been limited to the static analysis of large deformations to remove this limitation a new formulation taking into account dynamical features is proposed based on the weak form of momentum conservation law under the framework of nmm the generalized a method is employed to discretize the temporal variables the updates of variables are described using the updated co moving coordinate system thus a new method named s r d based nmm is established the new formulation can be implemented in any other partition of unity based methods as well so as to improve the performances of such methods in the analysis of dynamic large deformations c 2016 elsevier b v all rights reserved

generalized nodes and high performance elements the paper concerns the development of robust and high accuracy finite elements with only corner nodes using a partition of unity based finite element approximation construction of the partition of unity based approximation is accomplished by a physically defined local function of displacements a 4 node quadratic tetrahedral element and a 3 node quadratic triangular element are developed eigenvalue analysis shows that linear dependencies in the partition of unity based finite element approximation constructed for the new elements are eliminable numerical calculations demonstrate that the new elements are robust insensitive to mesh distortion and offer quadratic accuracy while also keeping mesh generation extremely simple copyright 2005 john wiley sons ltd

the enriched space time finite element method est for simultaneous solution of fluid structure interaction the paper introduces a weighted residual based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures the presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple flow immersed solid objects the fluid flow is described by the incompressible navier stokes equations the current configuration of the thin structure of linear elastic material with non linear kinematics is mapped to the flow using the zero iso contour of an updated level set function the formulation of fluid structure and coupling conditions uniformly uses velocities as unknowns the integration of the weak form is performed on a space time finite element discretization of the domain interfacial constraints of the multi field problem are ensured by distributed lagrange multipliers the proposed formulation and discretization techniques lead to a monolithic algebraic system well suited for strongly coupled fluid structure systems embedding a thin structure into a flow results in non smooth fields for the fluid based on the concept of the extended finite element method the space time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions this leads to the present enriched space time est method numerical examples of fluid structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems the test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail copyright 2007 john wiley sons ltd

a fe meshfree quad4 element based on partition of unity a new 4 node quadrilateral element is developed by combining the shape functions of finite element and meshfree methods the objective is to synergise the attractive features of finite element and meshfree shape functions using the concepts of partition of unity pu methods the shape functions of the well known quad4 element are used for pu and the least square point interpolation for local approximation this enables the proposed element inherit the compatibility properties of finite element method and the quadratic polynomial completeness properties of meshfree methods numerical results for the bench mark problems show that the proposed element has a good tolerance to distorted meshes the element also seems to be free from the linear dependence problem which is known to plague some of the pu based elements

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

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

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

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 partition of unity based fe meshfree quad4 element with radial polynomial basis functions for static analyses abstract recently a partition of unity pu based finite element called fe meshfree quad4 element has been published that element employed polynomial basis functions for the local approximation la in the present paper a new fe meshfree quad4 element employing hybrid radial polynomial basis functions for the la is proposed an advantage of radial polynomial basis is the freedom from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions another advantage is the improved accuracy of finite element solution the new element has been applied to several linear and nonlinear test problems the results demonstrate that the new element gives much better performance as compared to the previous fe meshfree quad4 element with pure polynomial basis even with a lower order basis viz with just three or four polynomial terms the present element is capable of giving more accurate solution than the previous element which employs a six term polynomial basis the new element also exhibits a much higher degree of tolerance to mesh distortion than the known quadratic elements moreover the linear dependence problem otherwise associated with many of the partition of unity pu based elements is completely eliminated from the present element

symmetric numerical integral formulas for regular polygons the theory of interpolatory integration formulas in n dimensions is combined with group theory to classify and generate symmetric integration formulas in a systematic way the points are computed as the common zeros of a set of polynomials these polynomials constitute a canonical basis for the real ideal belonging to the formula for a symmetric formula they can be chosen to span a representation of the symmetry group for regions with high symmetry and formulas of low degree this leads to only a few distinct possibilities and the existence of the corresponding formulas is easily checked in general however the canonical basis is not completely determined by symmetry and degree and a suitable choice has to be made the application to regular polygons in two dimensions is discussed and formulas are presented for polygons with three five six seven and eight vertices with varying degrees of exactness up to order 27 for the hexagon

a numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions we present a numerical algorithm for the construction of efficient high order quadratures in two and higher dimensions quadrature rules constructed via this algorithm possess positive weights and interior nodes resembling the gaussian quadratures in one dimension in addition rules can be generated with varying degrees of symmetry adaptable to individual domains we illustrate the performance of our method with numerical examples and report quadrature rules for polynomials on triangles squares and cubes up to degree 50 these formulae are near optimal in the number of nodes used and many of them appear to be new

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

the structure of renormalization hopf algebras for gauge theories i representing feynman graphs on bv algebras we study the structure of renormalization hopf algebras of gauge theories we identify certain hopf subalgebras in them whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms this can be understood physically as wave function renormalization and renormalization of the coupling constants respectively after taking into account the slavnov taylor identities for the couplings as generators of a hopf ideal we find hopf subalgebras in the corresponding quotient as well in the second part of the paper we explain the origin of these hopf ideals by considering a coaction of the renormalization hopf algebras on the batalin vilkovisky bv algebras generated by the fields and couplings constants the so called classical master equation satisfied by the action in the bv algebra implies the existence of the above hopf ideals in the renormalization hopf algebra finally we exemplify our construction by applying it to yang mills gauge theory

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

on generation of lumped mass matrices in partition of unity based methods summary the partition of unity based methods such as the extended finite element method and the numerical manifold method are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials in the dynamic analysis of cracked bodies using an explicit time integration algorithm as a result huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses where we inversely utilize the from local to global skill the proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix but has more excellent dynamic property than the latter meanwhile the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes a very excellent dynamic property copyright 2017 john wiley sons ltd

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

an enriched finite element method and level sets for axisymmetric two phase flow with surface tension a finite element method for axisymmetric two phase flow problems is presented the method uses an enriched finite element formulation in which the interface can move arbitrarily through the mesh without remeshing the enrichment is implemented by the extended finite element method x fem which models the discontinuity in the velocity gradient at the interface by a local partition of unity it provides an accurate representation of the velocity field at interfaces on an eulerian grid that is not conformal to the weak discontinuity the interface is represented by a level set which is also used in the construction of the element enrichment surface tension effects are considered and the interface curvature is computed from the level set field the method is demonstrated by several examples copyright 2003 john wiley sons ltd

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

accurate fracture modelling using meshless methods the visibility criterion and level sets formulation and 2d modelling fracture modelling using numerical methods is well advanced in 2d using techniques such as the extended finite element method xfem the use of meshless methods for these problems lags somewhat behind but the potential benefits of no meshing particularly in 3d prompt continued research into their development in methods where the crack face is not explicitly modelled as the edge of an element for instance two procedures are instead used to associate the displacement jump with the crack surface the visibility criterion and the diffraction method the visibility criterion is simple to implement and efficient to compute especially with the help of level set coordinates however spurious discontinuities have been reported around crack tips using the visibility criterion whereas implementing the diffraction method in 3d is much more complicated than the visibility criterion in this paper a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved the formulation is based on the use of level set coordinates and the element free galerkin method and is generally applicable for single or multiple crack problems in 2d or 3d the paper explains the formulation and provides verification of the method against a number of 2d crack problems copyright 2011 john wiley sons ltd

new strategies for some issues of numerical manifold method in simulation of crack propagation summary aiming to solve in a unified way continuous and discontinuous problems in geotechnical engineering the numerical manifold method introduces two covers namely the mathematical cover and the physical cover in order to reach the goal some issues in the simulation of crack propagation have to be solved among which are the four issues to be treated in this study 1 to reduce the rank deficiency induced by high degree polynomials as local approximation a new variational principle is formulated which suppresses the gradient dependent dofs 2 to evaluate the integrals with singularity of 1 r a new numerical quadrature scheme is developed which is simpler but more efficient than the existing duffy transformation 3 to analyze kinked cracks a sign convention for argument in the polar system at the crack tip is specified which leads to more accurate results in a simpler way than the existing mapping technique and 4 to demonstrate the mesh independency of numerical manifold method in handling strong singularity a mesh deployment scheme is advised which can reproduce all singular locations of the crack with regard to the mesh corresponding to the four issues typical examples are given to demonstrate the effectiveness of the proposed schemes copyright 2013 john wiley sons ltd

generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method new gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method for discontinuous functions we construct gauss like quadrature rules over arbitrarily shaped elements in two dimensions without the need for partitioning the finite element a point elimination algorithm is used in the construction of the quadratures which ensures that the final quadratures have minimal number of gauss points for weakly singular integrands we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques

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

almost everywhere partition of unity to deal with essential boundary conditions in meshless methods abstract due to overlapping nature of supports of partition of unity functions and the lack of the kronecker delta property of meshless shape functions it is difficult to deal with essential boundary conditions in meshless methods in this paper in order to alleviate this difficulty we introduce almost everywhere partition of unity that is a partition of unity except a few points along boundary in two dimensional case actually the gradient of partition of unity functions become infinitely large at these exceptional points however we prove that the presence of these bad points does not change the convergence rates of computed solutions comparing with the computed solutions obtained by the lagrange multiplier method the penalty method and the nitche s method we demonstrate the proposed method is more effective in dealing with essential boundary conditions in meshless methods

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

strong and weak arbitrary discontinuities in spectral finite elements methods for constructing arbitrary discontinuities within spectral finite elements are described and studied we use the concept of the extended finite element method xfem which introduces the discontinuity through a local partition of unity so there is no requirement for the mesh to be aligned with the discontinuities a key aspect of the implementation of this method is the treatment of the blending elements adjacent to the local partition of unity we found that a partition constructed from spectral functions one order lower than the continuous approximation is optimal and no special treatment is needed for higher order elements for the quadrature of the galerkin weak form since the integrand is discontinuous we use a strategy of subdividing the discontinuous elements into 6 and 10 node triangles the order of the element depends on the order of the spectral method for curved discontinuities several numerical examples are solved to examine the accuracy of the methods for straight discontinuities we achieved the optimal convergence rate of the spectral element for the curved discontinuity the convergence rate in the energy norm error is suboptimal we attribute the suboptimality to the approximations in the quadrature scheme we also found that modification of the adjacent elements is only needed for lower order spectral elements copyright 2005 john wiley sons ltd

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

accurate numerical integration of singular boundary element kernels over boundaries with curvature use of the green function for the solution of boundary value problems frequently results in singular integral equations algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method bem they are based upon the use of appropriately weighted gaussian quadrature formulae together with numerical geometrical transformations of the region of integration the use of high order subdomain expansion functions for interpolation over nonplanar elements allows boundary curvature to be accommodated in particular the handling of green functions with logarithmic and r 1 behaviour are detailed volume integrals with r 2 singularity are outlined operations are performed on a simplex thus resulting in generality and ease of automation this scheme has been incorporated into boundary element method software and successfully applied to a variety of problems

a new partition of unity finite element free from the linear dependence problem and possessing the delta property partition of unity based finite element methods pufems have appealing capabilities for p adaptivity and local refinement with minimal or even no remeshing of the problem domain however pufems suffer from a number of problems that practically limit their application namely the linear dependence ld problem which leads to a singular global stiffness matrix and the difficulty with which essential boundary conditions can be imposed due to the lack of the kronecker delta property in this paper we develop a new pu based triangular element using a dual local approximation scheme by treating boundary and interior nodes separately the present method is free from the ld problem and essential boundary conditions can be applied directly as in the fem the formulation uses triangular elements however the essential idea is readily extendable to other types of meshed or meshless formulation based on a pu approximation the computational cost of the present method is comparable to other pufem elements described in the literature the proposed method can be simply understood as a pufem with composite shape functions possessing the delta property and appropriate compatibility

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

the orthonormalized generalized finite element method ogfem efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions abstract an extension of the generalized finite element method gfem is proposed with which we efficiently reduce approximation errors the new method constructs a stiffness matrix with a conditioning that is significantly better than the stable generalized finite element method sgfem and the finite element method fem accordingly the risk of a severe loss of accuracy in the computed solution which burdens the gfem is prevented furthermore the computational cost of the inversion of the associated stiffness matrix is significantly reduced the gfem employs a set of enriched basis functions which is chosen to improve the rate at which the approximation converges to the exact solution the stiffness matrix constructed from these basis functions is often ill conditioned and the accuracy of the solution cannot be guaranteed we prevent this by orthonormalizing the basis functions and refer to the method as the orthonormalized generalized finite element method ogfem because the ogfem has the flexibility to orthonormalize either a part or all of the basis functions the method can be considered as a generalization of the gfem the method is applicable with single or multiple global and or local enrichment functions problems in blending elements are avoided by a modification of the enrichment functions the method is demonstrated for the one dimensional modified helmholtz and poisson equations and compared with the fem gfem and sgfem

complementarity problem arising from static growth of multiple cracks and mls based numerical manifold method abstract the full response of a brittle structure containing multiple cracks to loading under the servo control is of vital importance in the evaluation of properties of the structure during crack growth the fracture toughness condition at all the crack tips as well as the equilibrium condition should be obeyed leading to a nonlinear complementarity problem ncp the vector valued function in the ncp depends implicitly on the cracking increments which in turn determine the stress field the stress field can be obtained through solving a mixed variational problem since the degrees of freedom in the discrete variational problem vary with cracking not only in magnitude but also in number the jacobian matrix of the ncp is hard to compute and it cannot be expected to be solved by the newton methods that involve the calculation of jacobian matrices therefore a well scaled projection contraction algorithm is designed the proposed procedure is able to simulate growth of multiple cracks in a natural way allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments through the analysis of some examples that have been widely tested many interesting and profound phenomena are found which have never been revealed

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

prediction of rank deficiency in partition of unity based methods with plane triangular or quadrilateral meshes in the partition of unity pu based methods the global approximation is built by multiplying a partition of unity by local approximations within this framework high order approximations are achieved by directly adopting high order polynomials as local approximations and therefore nodes along sides or inside elements which are usually adopted in the conventional finite element methods are no more required however the pu based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom in this paper the origin of the rank deficiency in the pu based approximation space is first dissected at an element level and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency finally examples are investigated to validate the present approach the current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix it can be resolved in its own manner

a new way to treat material discontinuities in the numerical manifold method abstract a new technique to describe the material discontinuities within the numerical manifold method framework is presented a new concept of weak discontinuous physical covers is introduced special local approximations which are continuous in the approximation but discontinuous in its first derivative are tailored for weak discontinuous physical covers this new method takes full advantage of the partition of unity several typical examples are presented results have shown the accuracy and convenience of the proposed method in modeling material interfaces especially composite materials with multiple holes inclusions

structure size and statistical properties of chaotic components in a mixed type hamiltonian system we perform a detailed study of the chaotic component in mixed type hamiltonian systems on the example of a family of billiards introduced by robnik in j phys a math gen 16 3971 1983 jphac50305 447010 1088 0305 4470 16 17 014 the phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times the structure of the chaotic component is discerned from the cells visited by the chaotic orbit the fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box counting method the cell filling dynamics is compared to a model of uncorrelated motion the so called random model robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 and deviations attributed to sticky objects in the phase space are found the statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run the stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times the recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed the standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter lastly the decay of correlations and the diffusion of momenta is analyzed

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

allman s triangle rotational dof and partition of unity a simplification of the 1984 allman triangle one of historically most important elements with rotational dofs is presented it is found that this old element takes a typical form of the partition of unity approximation the allman s rotation presented in the partition of unity form offers merits and convenience in formulation and practical applications the stiffness matrix of the 1984 allman triangle which is originally computed from the linear strain triangular element the 6 nodes quadratic triangle can be obtained instead in a cheaper way from that of the constant strain triangular element the constraint of the rotational terms during essential boundary treatment which remains equivocal and ambiguous is understood to be mandatory the partition of unity notion enables a straightforward extension of the allman s rotational dof to meshfree approximations in numerical examples we discuss suppression of spurious zero energy modes and patch tests standard benchmarks are carried out to assess performance of the newly formulated triangle and a meshfree approximation with the rotational dofs copyright 2006 john wiley sons ltd

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

numerical manifold space of hermitian form and application to kirchhoff s thin plate problems summary for second order problems where the behavior is described by second order partial differential equations the numerical manifold method nmm has gained great success because of difficulties in the construction of the h 2 regular lagrangian partition of unity subordinate to the finite element cover however few applications of the nmm have been found to fourth order problems such as kirchhoff s thin plate problems parallel to the finite element methods this study constructs the numerical manifold space of the hermitian form to solve fourth order problems from the minimum potential principle meanwhile the mixed primal formulation and the penalized formulation fitted to the nmm for kirchhoff s thin plate problems are derived the typical examples indicate that by the proposed procedures even those earliest developed elements in the finite element history such as zienkiewicz s plate element regain their vigor copyright 2013 john wiley sons ltd

reproducing polynomial singularity particle methods and adaptive meshless methods for two dimensional elliptic boundary value problems abstract oh et al h s oh j g kim j w jeong the closed form reproducing polynomial particle shape functions for meshfree particle methods comput methods appl mech engrg 196 2007 3435 3461 introduced the reproducing polynomial particle rpp shape functions that are piecewise polynomial and satisfy the kronecker delta property in this paper we introduce reproducing polynomial particle methods rppm that is the galerkin approximation method associated with the use of the rpp approximation space planting particles in the computation domain in a patchwise uniform manner we also introduce patchwise rppm furthermore constructing partition of unity functions with flat top adaptively we introduce adaptive patchwise rppm and compare it with reproducing singularity particle methods rspm that is the galerkin approximation method associated with the use of reproducing singularity particle rsp shape functions

quadrature rules for triangular and tetrahedral elements with generalized functions quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains these quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries specifically the integrands considered here consist of a quadratic polynomial multiplied by a heaviside or dirac delta function operating on a linear polynomial this form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values compared with methods involving explicit integration over subdomains the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical jacobian information describing the dependence of the integrals on the nodal level set values the accuracy of the method is demonstrated for a simple conduction problem with the neumann and robin type boundary conditions copyright 2007 john wiley sons ltd

multifractality stickiness and recurrence time statistics we identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics rts which is based on the concept of poincare recurrences the projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near integrable system the annular billiard these microstructures interfere in the effective transport of a particle in the phase space which can be observed through rts this technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea

an adaptive hp version of the multilevel particle partition of unity method abstract this paper is concerned with the hp adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle partition of unity method the proposed refinement scheme automatically constructs new discretization points or particles the meshfree analogue of an adaptive h refinement and local approximation spaces with better local resolution a p refinement the refinement process is steered with the help of an a posteriori subdomain type error indicator we present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme

general dyson schwinger equations and systems we classify combinatorial dyson schwinger equations giving a hopf subalgebra of the hopf algebra of feynman graphs of the considered quantum field theory we first treat single equations with an arbitrary eventually infinite number of insertion operators we distinguish two cases in the first one the hopf subalgebra generated by the solution is isomorphic to the faa di bruno hopf algebra or to the hopf algebra of symmetric functions in the second case we obtain the dual of the enveloping algebra of a particular associative algebra seen as a lie algebra we also treat systems with an arbitrary finite number of equations with an arbitrary number of insertion operators with at least one of degree 1 in each equation

a hierarchical finite element method based on the partition of unity in this paper we consider the application of hierarchical functions to base approximations which are a partition of unity the particular hierarchical functions used are added to base finite element interpolations which for c0 approximations are a particular case of the partition of unity we also show how the functions may be constructed to preserve the interpolation property of the base finite element functions an application to linear elasticity is used to illustrate the properties and stability of the approximation

a coupled interpolating meshfree method for computing sound radiation in infinite domain summary in this paper the coupling of the improved interpolating element free galerkin iiefg method and the variable order infinite acoustic wave envelope element wee method is studied a coupled iiefg wee method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages the coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the iiefg and wee domains in the iiefg domain the improved interpolating moving least squares iimls method is employed to form the shape functions satisfying the kronecker delta condition while nonsingular weight functions can be used the impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated the numerical results show that the coupled iiefg wee method can take full advantage of both the iiefg and wee methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the wee the experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain this article is protected by copyright all rights reserved

investigation of linear dependence problem of three dimensional partition of unity based finite element methods abstract a known problem of partition of unity based generalized finite element methods is the linear dependence of the approximation space which leads to singular stiffness matrix up to now the linear dependence problem has not been fully understood and an efficient way to alleviate it is not available in our previous paper prediction of rank deficiency in partition of unity based methods with plane triangular or quadrilateral meshes comput methods appl mech engrg 200 2011 665 674 the origin of the linear dependence problem was first dissected and then a method was proposed to reliably predict the rank deficiency of the linearly dependent global approximations of two dimensional partition of unity based generalized finite element methods this paper extends the previous work to three dimensional cases the linear dependence problem is first investigated at an element level and then extended to the whole mesh derivation of general formulations in a three dimensional setting is undoubtedly more challenging than a two dimensional setting because of the complicated element topology the principle of the increase of rank deficiency is once more applied the methodology of summing up the added rank deficiency of each element as that of the whole mesh is further proved to be valid in three dimensional cases this work together with the previous work is regarded as the essential step to successfully and completely solve the linear dependence problem in partition of unity based finite element methods

an augmented lagrangian approach to essential boundary conditions in meshless methods the problem of boundary conditions enforcement in meshless methods has been solved in the literature by several approaches in the present paper the moving least squares mls approximation is introduced in the total potential energy functional for the elastic solid problem and an augmented lagrangian term is added to satisfy essential boundary conditions the method can be easily extended to any kind of constraint for the approximation variables the solution is found by iterating alternatively on approximation variables and on lagrange multipliers the advantages of the proposed formulation are a the ability to deal with the same approach with any constraint type b the number of the variables is not increased by the lagrange multipliers c the hessian of the functional w r t the approximation variables is banded well conditioned and strictly positive definite and d the cost of the augmented lagrangian iteration is a very small fraction of the global computing time therefore the augmented lagrangian element free alef approach represents an attractive and very efficient numerical tool not only for the boundary conditions enforcement but also for the solution of interface and non linear problems copyright 2001 john wiley sons ltd