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

extended finite element method for three dimensional large plasticity deformations on arbitrary interfaces in this paper the extended finite element method is presented for large elasto plastic deformation in 3d solid mechanics problems the x fem computational algorithm is presented in the framework of lagrangian description in order to model the arbitrary interfaces in large deformations in x fem the material interface is represented independently of element boundaries and the process is accomplished by partitioning the domain with several tetrahedral sub elements whose gauss points are used for integration of the domain of elements the interface between two bodies is modeled by using the x fem technique and applying a modified level set enrichment function in order to simulate the nonlinear behavior of materials the lagrangian plasticity formulation is coupled with the x fem technique finally several numerical examples are analyzed to demonstrate the efficiency of the x fem technique in large plasticity deformations

a general procedure for improving substructures representation in dynamic synthesis in the general substructure synthesis method for the dynamic analysis of complex flexible structures developed earlier by the authors the motion of each substructure is represented by a given number of substructure admissible functions and the otherwise disjoint substructures are connected together to form a whole structure by imposing approximate geometric compatibility conditions by means of weighted residuals in this paper a general procedure for improving the admissible functions representing each substructure in the synthesis is developed the procedure is an iterative one and it leads to an improved eigensolution for the intermediate structure i e a fictitious structure defined by the approximate compatibility conditions where the improved eigensolution is obtained while always using the same number of degrees of freedom to represent a given substructure in any iteration the computations associated with each substructure are independent of those for all other substructures and they can be performed in parallel by increasing the number of iterations the eigensolution for the intermediate structure is approached the procedure is applicable to assembled structures that are either positive definite or positive semidefinite moreover the procedure represents an extension of the concept of subspace iteration to structures composed of substructures

fracture mechanics analysis using the wavelet galerkin method and extended finite element method summary this paper presents fracture mechanics analysis using the wavelet galerkin method and extended finite element method the wavelet galerkin method is a new methodology to solve partial differential equations where scaling wavelet functions are used as basis functions in solid structural analyses the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain to improve accuracy wavelet functions are superposed on the scaling functions within a region having a high stress concentration such as near a hole or notch thus the method can be considered a refinement technique in fixed grid approaches however because the basis functions are assumed to be continuous in applications of the wavelet galerkin method there are difficulties in treating displacement discontinuities across the crack surface in the present research we introduce enrichment functions in the wavelet galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method this paper presents the mathematical formulation and numerical implementation of the proposed technique as numerical examples stress intensity factor evaluations and crack propagation analyses for two dimensional cracks are presented copyright 2012 john wiley sons ltd

eigenvalue convergence in the finite element method in static force deflection applications of the finite element method convergence rates for the p version in which the polynomial degree of element interpolation functions is increased while the mesh remains fixed are superior to those for the h version in which the element degree remains fixed while the mesh is refined so that element size approaches zero in structural dynamics applications one does not seek to approximate a single solution as in static applications but seeks estimates for a number of the lower system eigenvalues this paper identifies factors responsible for poorer accuracy in higher computed eigenvalues in addition it explains why the p version of the finite element method can be expected to exhibit significantly better eigenvalue convergence than the h version numerical examples demonstrate the superiority of the p version over the h version they also show the effects of various mechanisms limiting eigenvalue convergence

crack growth analysis using mesh superposition technique and x fem in this paper a new approach to crack propagation analysis for large scale or complicated geometry structures is presented that is the connection of mesh superposition technique and the extended finite element method the former is a technique that increases the accuracy of analysis locally by superimposing additional mesh of higher resolution onto the mesh that represents the rough deformation of the structure in this technique the boundaries and nodes of both meshes do not have to coincide with each other in the latter technique the discontinuity across the crack segment and singularity around the crack tips are represented in the approximation by enriching the nodal degrees using partition of unity condition this technique does not require meshes to conform to the crack geometry and this enables crack propagation procedure with no re meshing process by connecting both techniques it becomes possible to analyse crack growth models of large scale and complicated shape with highly flexible modelling and no remeshing process by using this approach some numerical examples are analysed and appropriate results are obtained copyright 2007 john wiley sons ltd

combined extended and superimposed finite element method for cracks a combination of the extended finite element method xfem and the mesh superposition method s version fem for modelling of stationary and growing cracks is presented the near tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity the two displacement fields are matched through a transition region the method can robustly deal with stationary crack and crack growth it simplifies the numerical integration of the weak form in the galerkin method as compared to the s version fem numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method copyright 2004 john wiley sons ltd

on a new iterative approach of component mode synthesis a wholly new concept of intermediate structure is developed in this paper based on this a new component mode iteration cmi or subdomain subspace iteration method is proposed in the new cmi method the ordinarily required step of ensuring interface compatibility is eliminated an outstanding achievement of this method is that it retains the merit of the standard subspace iteration ssi technique concerning the number of vectors required in computation in the present method iteraton is carried out at subdomain levels on machines capable of parallel processing this work can be performed simultaneously numerical examples clearly demonstrate the excellence of this new method

a general dynamic synthesis for structures with discrete substructures abstract this paper presents a substructure synthesis method for the dynamic simulation of complex structures where the structures consist of an assemblage of discrete substructures an analogy between distributed and discrete structures is extensively invoked to stimulate the motion of discrete substructures the concept of admissible vectors is introduced where admissible vectors represent the discrete counterpart of admissible functions for distributed substructures the individual substructures are forced to act as a whole structure by imposing certain geometric compatibility on internal boundaries shared by any two substructures a numerical example illustrating the method is presented

a compactness property for solutions of the ricci flow on orbifolds in this paper we consider ricci flow on orbifolds with at most isolated singularities we generalize two results of r hamilton to orbifolds the first result is the compactness property of ricci flow and the second is an injectivity radius estimate for ricci flow in the proof of the compactness property we give an orbifold version of the injectivity radius estimate of j cheeger m gromov and m taylor also along the way we prove a compactness property of riemannian orbifolds

the rs method for material failure simulations a new method for propagating arbitrary failure modes is presented arbitrary failure modes are resolved on a refined local patch of elements and then embedded into the coarse grid using partition of unity method strong discontinuities are propagated by means of element erosion in the superimposed patch of elements only the method coined as the rs version of the finite element method or reduced order s method has been integrated in abaqus and verified on several test problems copyright 2007 john wiley sons ltd

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

eigensolution of augmented graph products using shifted inverse iteration method two important matrices associated with graphs are adjacency and laplacian matrices in this paper efficient methods are presented for eigensolution of graph products augmented by other graphs for augmentations that do not destroy the symmetry of the graph products a method is proposed for decomposition of matrices resulting in considerable simplification of their eigensolution for graphs composed of two non overlapping graph products joined through a small number of link members a method based on shifted inverse iteration is proposed which utilizes all eigenvalues and eigenvectors of each individual graph products owing to the availability of fast methods for eigensolution of graph products this method simplifies the eigensolution of a variety of graph models and proves to be very efficient in determining the few smallest eigenpairs of these models with high levels of accuracy

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

higher order xfem for curved strong and weak discontinuities the extended finite element method xfem enables the accurate approximation of solutions with jumps or kinks within elements optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces higher order convergence for arbitrary curved interfaces relies on two major issues i an accurate quadrature of the galerkin weak form for the cut elements and ii a careful formulation of the enrichment which should preclude any problems in the blending elements for i we employ a strategy of subdividing the elements into subcells with only one curved side reference elements that are higher order on only one side are then used to map the integration points to the real element for ii we find that enrichments for strong discontinuities are easily extended to higher order accuracy in contrast problems in blending elements may hinder optimal convergence for weak discontinuities different formulations are investigated including the corrected xfem numerical results for several test cases involving strong or weak curved discontinuities are presented quadratic and cubic approximations are investigated optimal convergence rates are achieved using the standard xfem for the case of a strong discontinuity close to optimal convergence rates for the case of a weak discontinuity are achieved using the corrected xfem copyright 2009 john wiley sons ltd

efficient algorithm for edge cracked geometries the stress field in a finite edge cracked specimen under load is computed using algorithms based on two slightly different integral equations of the second kind these integral equations are obtained through left regularizations of a first kind integral equation in numerical experiments it is demonstrated that the stress field can be accurately computed highly accurate stress intensity factors and t stresses are presented for several setups and extensive comparisons with results from the literature are made for simple geometries the algorithms presented here achieve relative errors of less than 10 10 it is also shown that the present algorithms can accurately handle both geometries with arbitrarily shaped edge cracks and geometries containing several hundred edge cracks all computations were performed on an ordinary workstation copyright c 2005 john wiley sons ltd less

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

gibbs phenomenon in structural mechanics the concepts of complementary boundary condition and interior condition are discussed guidelines are developed to eliminate the gibbs effect when selecting trial functions for the series expansion

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

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

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

domain decomposition methods for eigenvalue problems abstract this paper proposes several domain decomposition methods to compute the smallest eigenvalue of linear self adjoint partial differential operators let us be given a partial differential operator on a domain which consists of two nonoverlapping subdomains suppose a fast eigenvalue solver is available for each of these subdomains but not for the union of them the first proposed method is a scheme which determines an appropriate boundary condition at the interface separating the two regions this boundary condition can be derived from the zero of a nonlinear operator which plays the same role for the eigenvalue problem as the steklov poincare operator for the linear equation an iterative method can be used to solve this nonlinear equation yielding the exact boundary condition at the interface this enables the determination of the eigenpair on the whole domain the same concept can be applied to a schwarz alternating method for the eigenvalue problem in case the subdomains overlap a nonoverlapping scheme in the spirit of schur complement will also be discussed these ideas are also applicable to a domain imbedding algorithm

component mode iteration for frequency calculations this paper presents component mode iteration a new method for carrying out a form of subspace iteration on the substructure level for structures composed of substructures in component mode iteration substructure displacement is in terms of component modes with the interface displacements as coefficients component modes are chosen such that exact compatibility is assured throughout the structure the reduced structure eigenvalue problem is assembled and solved for lower mode approximations the resulting estimates of lower mode interface motion are used to refine the component modes by approximating the dynamic response of substructures to interface motion with a vector iteration process for this step the substructures are independent and can be processed in parallel reassembly and solution of the reduced structure problem yields improved lower mode estimates further iterations of refinement and reassembly are performed until convergence takes place the initial component modes are generated automatically and at negligible expense and numerical examples show that convergence with these is excellent yielding very accurate results after only one iteration indeed convergence is shown to be significantly faster than in the standard subspace iteration method

a new generalized finite element method for two scale simulations of propagating cohesive fractures in 3 d summary this paper presents a novel numerical framework based on the generalized finite element method with global local enrichments gfemgl for two scale simulations of propagating fractures in three dimensions a non linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters the cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity concept provided by the generalized extended finite element method and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface the proposed method employs fine scale solutions of non linear local boundary value problems extracted from the original global problem in order to not only construct scale bridging enrichment functions but also to identify damaged states in the global problem thus enabling accurate global solutions on coarse meshes this is in contrast with the available gfemgl in which the local solution field contributes only to the kinematic description of global solutions the robustness efficiency and accuracy of this approach are demonstrated by results obtained from representative numerical examples copyright 2015 john wiley sons ltd

frankel conjecture and sasaki geometry abstract we classify simply connected compact sasaki manifolds of dimension 2 n 1 with positive transverse bisectional curvature in particular the kahler cone corresponding to such manifolds must be bi holomorphic to c n 1 0 as an application we recover the theorem of mori and siu yau on the frankel conjecture and extend it to certain orbifold version the main idea is to deform such sasaki manifolds to the standard round sphere in two steps both fixing the complex structure on the kahler cone first we deform the metric along the sasaki ricci flow and obtain a limit sasaki ricci soliton with positive transverse bisectional curvature then by varying the reeb vector field which essentially decreases the volume functional we deform the sasaki ricci soliton to a sasaki einstein metric with positive transverse bisectional curvature i e a round sphere the second deformation is only possible when one treats simultaneously regular and irregular sasaki manifolds even if the manifold one starts with is regular quasi regular i e kahler manifolds orbifolds

ricci flow on kahler einstein manifolds this is the continuation of our earlier article 10 for any kahler einstein surfaces with positive scalar curvature if the initial metric has positive bisectional curvature then we have proved see 10 that the kahler ricci flow converges exponentially to a unique kahler einstein metric in the end this partially answers a long standing problem in ricci flow on a compact kahler einstein manifold does the kahler ricci flow converge to a kahler einstein metric if the initial metric has positive bisectional curvature in this article we give a complete affirmative answer to this problem

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

kron s method for symmetric eigenvalue problems kron s method has been used successfully by engineers for over 30 years to find eigenvalues of large symmetric matrices these matrices arise from domain decomposition of nonoverlapping domains of self adjoint partial differential operators this paper discusses some theoretical aspects of kron s method specifically the poles and zeroes of eigenvalues of the kron matrix and their relationships with the eigenvalues of the original matrix are given

on three modal synthesis variants abstract a unified formulation of fixed free and loaded interface variants of modal synthesis is presented a formal proof of the exactness of all three variants in the absence of modal truncation is established the sensitivity of the truncation error to the different interface conditions is studied

approximate eigensolution of laplacian matrices for locally modified graph products laplacian matrices and their spectrum are of great importance in algebraic graph theory there exist efficient formulations for eigensolutions of the laplacian matrices associated with a special class of graphs called product graphs in this paper the problem of determining a few approximate smallest eigenvalues and eigenvectors of large scale product graphs modified through the addition or deletion of some nodes and or members is investigated the eigenproblem associated with a modified graph model is reduced using the set of master eigenvectors and linear approximated slave eigenvectors from the original model implicitly restarted lanczos method is employed to obtain the required eigenpairs of the reduced problem examples of large scale models are included to demonstrate the efficiency of the proposed method compared to the direct application of the irl method

another look at admissible functions abstract problems associated with using global admissible functions in boundary value problems are discussed it is shown that if care is not taken it is possible to construct sets of trial functions that yield inaccurate results which take too long to converge the concept of complementary boundary conditions is introduced and their existence is established complementary boundary conditions indicate that the solution to the boundary value problem cannot take certain values at the boundaries it is shown that trial functions that yield incorrect or very slowly converging results are those that violate the complementary boundary conditions a revision for the definition of admissible and comparison functions is proposed that takes the complementary boundary conditions into consideration

new development in extended finite element modeling of large elasto plastic deformations this paper presents new achievements in the extended finite element modeling of large elasto plastic deformation in solid problems the computational technique is presented based on the extended finite element method x fem coupled with the lagrangian formulation in order to model arbitrary interfaces in large deformations in x fem the material interfaces are represented independently of element boundaries and the process is accomplished by partitioning the domain with some triangular sub elements whose gauss points are used for integration of the domain of elements the large elasto plastic deformation formulation is employed within the x fem framework to simulate the non linear behavior of materials the interface between two bodies is modeled by using the x fem technique and applying the heaviside and level set based enrichment functions finally several numerical examples are analyzed including arbitrary material interfaces to demonstrate the efficiency of the x fem technique in large plasticity deformations copyright 2008 john wiley sons ltd

a moving superimposed finite element method for structural topology optimization level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures in this paper a dynamic implicit boundary based moving superimposed finite element method s version fem or s fem is developed for structural topology optimization using the level set methods in which the variational interior and exterior boundaries are represented by the zero level set both a global mesh and an overlaying local mesh are integrated into the moving s fem analysis model a relatively coarse fixed eulerian mesh consisting of bilinear rectangular elements is used as a global mesh the local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function in numerical integration using the gauss quadrature rule the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes a double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co ordinate systems an element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency a simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint a physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated pde solving procedure the proposed moving s fem is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems for the classical elasticity problems in the literature the present s fem can achieve numerical results in good agreement with those from the theoretical solutions and or numerical results from the standard fem for the minimum compliance topology optimization problems in structural optimization the present approach significantly outperforms the well recognized ersatz material approach as expected in the accuracy of the strain field numerical stability and representation fidelity at the expense of increased computational time it is also shown that the present approach is able to produce structures near the theoretical optimum it is suggested that the present s fem can be a promising tool for shape and topology optimization using the level set methods copyright 2005 john wiley sons ltd

mesh independent modelling of cracks by using higher order shape functions a mesh independent crack modelling approach based on displacement approximation with higher order shape functions is proposed the heaviside step function based local enrichment method known as extended finite element method is modified by replacing the step function with a higher order shape functions approximation polynomial b spline approximation functions are used in the present paper an advantage of the proposed method is that its implementation only involves integration of the products of original shape functions and their derivatives and does not require modification of the integration domains a volume integral based expression is proposed to calculate the effective surface area of the crack modelled by using an approximate step function it is shown to give the actual crack surface area in the limit of the approximate step function approaching the heaviside function the convergence and accuracy of the method is illustrated in examples of transverse and oblique with respect to loading direction crack problems in rectangular plates uniaxial tension of a unidirectional composite with an open hole is considered hoop stress relaxation due to longitudinal splitting is successfully modelled by the method proposed and compared to direct modelling by using ansys software published in 2002 by john wiley sons ltd

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions