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

solving the patient zero inverse problem by using generalized simulated annealing abstract identifying patient zero the initially infected source of a given outbreak is an important step in epidemiological investigations of both existing and emerging infectious diseases here the use of the generalized simulated annealing algorithm gsa to solve the inverse problem of finding the source of an outbreak is studied the classical disease natural histories susceptible infected si susceptible infected susceptible sis susceptible infected recovered sir and susceptible infected recovered susceptible sirs in a regular lattice are addressed both the position of patient zero and its time of infection are considered unknown the algorithm performance with respect to the generalization parameter q v and the fraction of infected nodes for whom infection was ascertained is assessed numerical experiments show the algorithm is able to retrieve the epidemic source with good accuracy even when is small but present no evidence to support that gsa performs better than its classical version our results suggest that simulated annealing could be a helpful tool for identifying patient zero in an outbreak where not all cases can be ascertained

stable enrichment and local preconditioning in the particle partition of unity method this paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods in particular we focus on the particle partition of unity method ppum yet the presented results hold for any partition of unity based enrichment scheme the goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h version independent of the regularity of the solution hence we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme to this end we enrich our ppum function space in an enrichment zone hierarchically near the singularities of the solution this initial enrichment however can lead to a severe ill conditioning and can compromise the stability of the discretization to overcome the ill conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well conditioned basis independent of the employed initial enrichment the construction of this preconditioner is of linear complexity with respect to the number of discretization points we obtain optimal error bounds for an enriched ppum discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super convergence the results of our numerical experiments clearly show that our enriched ppum with local preconditioning recovers the optimal convergence rate of o h p of the uniform h version globally for the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of o h p with delta geq frac 1 2 for p 1 the convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

cracking node method for dynamic fracture with finite elements a new method for modeling discrete cracks based on the extended finite element method is described in the method the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements the method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path a set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled the effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching copyright 2008 john wiley sons ltd

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

hp generalized fem and crack surface representation for non planar 3 d cracks a high order generalized finite element method gfem for non planar three dimensional crack surfaces is presented discontinuous p hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts the gfem is able to model a crack arbitrarily located within a finite element fe mesh and thus the proposed method allows fully automated fracture analysis using an existing fe discretization without cracks we also propose a crack surface representation that is independent of the underlying gfem discretization and controlled only by the physics of the problem the representation preserves continuity of the crack surface while being able to represent non planar non smooth crack surfaces inside of elements of any size the proposed representation also provides support for the implementation of accurate robust and computationally efficient numerical integration of the weak form over elements cut by the crack surface numerical simulations using the proposed gfem show high convergence rates of extracted stress intensity factors along non planar curved crack fronts and the robustness of the method copyright 2008 john wiley sons ltd

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

mesh independent p orthotropic enrichment using the generalized finite element method this paper is aimed at presenting a simple yet effective procedure to implement a mesh independent p orthotropic enrichment in the generalized finite element method the procedure is based on the observation that shape functions used in the gfem can be constructed from polynomials defined in any co ordinate system regardless of the underlying mesh or type of element used numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p orthotropic enrichment copyright 2002 john wiley sons ltd

an efficient augmented finite element method for arbitrary cracking and crack interaction in solids summary this paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary multiple intra elemental discontinuities in 2d solids the augmented finite element method a fem employs four internal nodes to account for the crack displacements due to an intra elemental weak or strong discontinuity and it permits repeated elemental augmentation to include multiple interactive cracks it thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity and to multiple interactive cohesive cracks all within a single bilinear element that employs standard external nodal dofs only a novel elemental condensation procedure has been developed to solve the internal nodal dofs as functions of the external nodal dofs for any irreversible piece wise linear cohesive laws it leads to a fully condensed elemental equilibrium equation with mathematical exactness eliminating the need for nonlinear equilibrium iterations at elemental level the new a fem s high fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous and heterogeneous solids have been demonstrated through several challenging numerical examples it is shown that the proposed a fem empowered by the new elemental condensation procedure is numerically very efficient accurate and robust copyright 2014 john wiley sons ltd

stochastic lattice gas model describing the dynamics of the sirs epidemic process abstract we study a stochastic process describing the onset of spreading dynamics of an epidemic in a population composed of individuals of three classes susceptible s infected i and recovered r the stochastic process is defined by local rules and involves the following cyclic process s i r s sirs the open process s i r sir is studied as a particular case of the sirs process the epidemic process is analyzed at different levels of description by a stochastic lattice gas model and by a birth and death process by means of monte carlo simulations and dynamical mean field approximations we show that the sirs stochastic lattice gas model exhibit a line of critical points separating the two phases an absorbing phase where the lattice is completely full of s individuals and an active phase where s i and r individuals coexist which may or may not present population cycles the critical line that corresponds to the onset of epidemic spreading is shown to belong in the directed percolation universality class by considering the birth and death process we analyze the role of noise in stabilizing the oscillations

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

coded modulation with mismatched csit over mimo block fading channels reliable communication over delay constrained multiple input multiple output mimo block fading channels with discrete inputs and mismatched imperfect channel state information at the transmitter csit is studied the csit mismatch is modeled as gaussian random variables whose variances decay as a power of the signal to noise ratio snr a special focus is placed on the large snr decay of the error and outage probabilities when power control with long term power constraints is used without explicitly characterizing the corresponding power allocation algorithms we derive the outage exponent as a function of the system parameters including the csit noise variance exponent and the exponent of the peak power constraint it is shown that csit even if noisy is always beneficial and leads to important gains in terms of exponents

the mimo arq channel diversity multiplexing delay tradeoff in this paper the fundamental performance tradeoff of the delay limited multiple input multiple output mimo automatic retransmission request arq channel is explored in particular we extend the diversity multiplexing tradeoff investigated by zheng and tse in standard delay limited mimo channels with coherent detection to the arq scenario we establish the three dimensional tradeoff between reliability i e diversity throughput i e multiplexing gain and delay i e maximum number of retransmissions this tradeoff quantifies the arq diversity gain obtained by leveraging the retransmission delay to enhance the reliability for a given multiplexing gain interestingly arq diversity appears even in long term static channels where all the retransmissions take place in the same channel state furthermore by relaxing the input power constraint allowing variable power levels in different retransmissions we show that power control can be used to dramatically increase the diversity advantage our analysis reveals some important insights on the benefits of arq in slow fading mimo channels in particular we show that 1 allowing for a sufficiently large retransmission delay results in an almost flat diversity multiplexing tradeoff and hence renders operating at high multiplexing gain more advantageous 2 mimo arq channels quickly approach the ergodic limit when power control is employed finally we complement our information theoretic analysis with an incremental redundancy lattice space time ir last coding scheme which is shown through a random coding argument to achieve the optimal tradeoff s an integral component of the optimal ir last coding scheme is a list decoder based on the minimum mean square error mmse lattice decoding principle for joint error detection and correction throughout the paper our theoretical claims are validated by numerical results

multilevel particle partition of unity method this paper is concerned with the particle partition of unity method a meshfree generalization of the finite element method we present the fundamental construction principles and abstract approximation properties of the resulting function spaces v pu moreover we discuss the construction of optimal approximation spaces for a reference application in linear elastic fracture mechanics in particular the presented construction not only yields optimal convergence rates globally independently of the regularity of the solution our method shows a super convergence near the singular points of the solution

analysis of multiple antenna systems with finite rate feedback using high resolution quantization theory this paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple antenna systems with finite rate feedback inspired by the results of classical high resolution quantization theory the problem of finite rate quantized communication system is formulated as a general fixed rate vector quantization problem with side information available at the encoder or the quantizer but unavailable at the decoder the framework of the quantization problem is sufficiently general to include quantization schemes with general non mean squared distortion functions and constrained source vectors asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the bennett s integral specifically tight lower and upper bounds of the average asymptotic distortion are proposed sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed rate quantization problems the proposed general methodology provides a powerful analytical tool to study a wide range of finite rate feedback systems to illustrate the utility of the framework we consider the analysis of a finite rate feedback multiple input single output miso beamforming system over independent and identically distributed i i d rayleigh flat fading channels numerical and simulation results are presented that further confirm the accuracy of the analytical results

locally adapted tetrahedral meshes using bisection we present an algorithm for the construction of locally adapted conformal tetrahedral meshes the algorithm is based on bisection of tetrahedra a new data structure is introduced which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected we prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity and we bound the amount of additional refinement that is needed to achieve conformity numerical examples of the effectiveness of the algorithm are presented

higher order extended finite elements with harmonic enrichment functions for complex crack problems summary in this paper we analyze complex crack problems in elastic media using harmonic enrichment functions in a higher order extended finite element implementation the numerically computed enrichment function of a crack is the solution of the laplace equation with discontinuous dirichlet boundary condition along the crack and its interaction with branches or other cracks is realized by imposing vanishing neumann boundary conditions along those cracks the classical finite element displacement approximation is enriched by adding the enrichment function of a crack through the framework of partition of unity a nested subgrid mesh is used in the laplace solve with a rasterized approximation of a crack which simplifies the numerical integration no partitioning of finite elements is required harmonic enrichment functions readily permit the extension to handle multiple interacting and branched cracks without any special treatment around the junction points several numerical examples are presented that arm the accuracy and eectiveness of the method when applied to complex crack

development of a quadratic finite element formulation based on the xfem and nurbs the fe simulation of inhomogeneous structures such as composite materials biological tissues or foams requires the generation of respective finite element meshes with increasing complexity of the inner architecture of such structures this becomes a time consuming and laborious task additionally the risk of forming bad shaped elements that may lead to ill conditioned numerical problems grows significantly a solution to this problem provides the extended finite element method xfem thereby the interface between different materials is represented by a local enrichment of the displacement approximation as a consequence of this the element boundary need not be aligned to the interface in order to improve the accuracy of the interface approximation the development of a plane element based on the xfem and quadratic shape functions will be presented this element allows for the description of curved material interfaces the computation of the element stiffness matrix requires a numerical integration process that accounts for discontinuous fields regarding a linear element formulation this can be achieved by an adapted triangulation of the element domain however in the case of a curved interface this solution is not applicable hence non uniform rational b spline nurbs surfaces are used to evaluate the integrals numerically finally the results of different examples will show the general properties such as the accuracy of the numerical integration procedure and the convergence behavior of this element formulation copyright 2011 john wiley sons ltd

cohesive and non cohesive fracture by higher order enrichment of xfem summary a comprehensive study is performed on the use of higher order terms of the crack tip asymptotic fields as enriching functions for the extended fem xfem for both cohesive and traction free cracks for traction free cracks the williams asymptotic field is used to obtain highly accurate stress intensity factors sifs directly from the enriched degrees of freedom without any post processing the low accuracy of the results of the original research on this subject by liu et al int j numer meth engng 2004 59 1103 1118 is remedied here by appropriate modifications of the enrichment scheme the modifications are simple and can be easily included into an xfem computer code for cohesive cracks the relevant asymptotic field is used and two widely used criteria including the sifs criterion and the stress criterion are examined for the crack growth simulation both linear and nonlinear cohesive laws are used for the stress criterion averaging is avoided due to the highly accurate crack tip approximation because of the higher order enrichment then a modified stress criterion is proposed which is shown to be applicable to a wider class of problems several numerical examples including straight and curved cracks stationary and growing cracks single and multiple cracks and traction free and cohesive cracks are studied to investigate in detail the robustness and efficiency of the proposed enrichment scheme copyright 2012 john wiley sons ltd

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 finite element method with mesh separation based approximation technique and its application in modeling crack propagation with adaptive mesh refinement summary this paper presents a fem with mesh separation based approximation technique that separates a standard element into three geometrically independent elements a dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation the novel technique makes it very easy for mesh generation of problems with complex or solution dependent varying geometry it offers a flexible way to construct displacement approximations and provides a unified framework for the fem to enjoy some of the key advantages of the hansbo and hansbo method the meshfree methods the semi analytical fems and the smoothed fem for problems with evolving discontinuities the method enables the devising of an efficient crack tip adaptive mesh refinement strategy to improve the accuracy of crack tip fields both the discontinuities due to intra element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level the effectiveness and robustness of the present method are benchmarked with several numerical examples the numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test and it is essential to apply local adaptive mesh refinement for low fracture energy problems copyright 2014 john wiley sons ltd

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

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

critical behavior of the susceptible infected recovered model on a square lattice by means of numerical simulations and epidemic analysis the transition point of the stochastic asynchronous susceptible infected recovered model on a square lattice is found to be c0 0 176 500 510 where c is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor this point corresponds to an infection recovery rate of c 1 c0 c0 4 665 713 and a net transmissibility of 1 c0 1 3 c0 0 538 4102 which falls between the rigorous bounds of the site and bond thresholds the critical behavior of the model is consistent with the two dimensional percolation universality class but local growth probabilities differ from those of dynamic percolation cluster growth as is demonstrated explicitly

joint source channel codes for mimo block fading channels we consider transmission of a continuous amplitude source over an l block rayleigh fading mt x mr multiple input multiple output mimo channel when the channel state information is only available at the receiver since the channel is not ergodic shannon s source channel separation theorem becomes obsolete and the optimal performance requires a joint source channel approach our goal is to minimize the expected end to end distortion particularly in the high signal to noise ratio snr regime the figure of merit is the distortion exponent defined as the exponential decay rate of the expected distortion with increasing snr we provide an upper bound and lower bounds for the distortion exponent with respect to the bandwidth ratio among the channel and source bandwidths for the lower bounds we analyze three different strategies based on layered source coding concatenated with progressive superposition or hybrid digital analog transmission in each case by adjusting the system parameters we optimize the distortion exponent as a function of the bandwidth ratio we prove that the distortion exponent upper bound can be achieved when the channel has only one degree of freedom that is l 1 and min mt mr 1 when we have more degrees of freedom our achievable distortion exponents meet the upper bound for only certain ranges of the bandwidth ratio we demonstrate that our results which were derived for a complex gaussian source can be extended to more general source distributions as well

gradient enhanced damage modeling of cracked bodies by reproducing kernel element method abstract in crack modeling by extended finite element method the enrichment by the asymptotic analytical solution works excellent for linear elastic fracture mechanics however for the more sophisticated condition of a gradient enhanced damage model firstly the analytical solution is not available and secondly higher order continuity of the shape functions is required in this study the extended finite element model of the cracked body is augmented with shape functions of higher order continuity and higher order consistency generated by reproducing kernel element method they are used in a region near the crack tip to have an appropriate interpolation of the damaged zone while for the rest of the solution domain standard shape functions are used to reduce the computational cost no new nodes are needed when defining the higher order shape functions by reproducing kernel element method and thus remeshing is avoided during the crack growth after describing the formulations several numerical examples are implemented to investigate the ability of the proposed method for both the linear elastic fracture mechanics and the gradient enhanced damage models

hybrid digital analog hda joint source channel codes for broadcasting and robust communications we consider the problem of transmitting a band limited gaussian source on an additive band limited gaussian noise channel the well known threshold effect dictates that the more powerful a code is the more sensitive it is to the exact knowledge of the channel noise a code is said to be robust if it is asymptotically optimal for a wide range of channel noise thus robust codes have a graceful degradation characteristic and are free of the threshold effect it is demonstrated that robust codes exist whenever the source and channel bandwidths are equal in the unequal bandwidth case a collection of nearly robust joint source channel codes is constructed using a hybrid digital analog hda coding technique for designing nearly robust codes a matched tandem code whose channel encoder s output is partially fully matched to its input is proposed and the existence of an asymptotically optimal matched tandem code is shown the nearly robust codes achieve the shannon limit theoretically optimum distortion and have a less severe threshold effect finally for the case of two different noise conditions the distortion regions of these codes are determined

high order boundary and finite elements for 3d fracture propagation in brittle materials abstract the quasi static propagation of fracture in brittle materials was studied in several recent publications a variational formulation salvadori 2008 salvadori and carini 2011 salvadori and fantoni 2013 led to three dimensional crack tracking strategies salvadori and fantoni 2014 2 3 salvadori and fantoni 2016 one of the complexities of this new class of algorithms is the evaluation of high order terms of the expansion of the crack opening and sliding in this paper new types of finite and boundary elements are formulated that capture the near crack front asymptotical displacement behavior up to the order 3 2 the use of these elements with the quasi static crack propagation algorithms of the above references is demonstrated for a simple crack configuration

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

embedded interfaces by polytope fem in this paper the polytope finite element method is employed to model an embedded interface through the body independent of the background fem mesh the elements that are crossed by the embedded interface are decomposed into new polytope elements which have some nodes on the interface line the interface introduces discontinuity into the primary variable strong or into its derivatives weak both strong and weak discontinuities are studied by the proposed method through different numerical examples including fracture problems with traction free and cohesive cracks and heat conduction problems with dirichlet and dirichlet neumann types of boundary conditions on the embedded interface for traction free cracks which have tip singularity the nodes near the crack tip are enriched with the singular functions through the extended finite element method the concept of natural element coordinates necs is invoked to drive shape functions for the produced polytopes a simple treatment is proposed for concave polytopes produced by a kinked interface and also for locating crack tip inside an element prior to using the singularity enrichment the proposed method pursues some implementational details of extended generalized finite element methods for interfaces but here the additional dofs are constructed on the interface lines in contrast to x g fem which attach enriched dofs to the previously existed nodes copyright 2011 john wiley sons ltd

a discrete formulation for elastic solids with damaging interfaces abstract an elastic solid with embedded interfaces loci of possible displacement discontinuities is considered here decohesion and quasi brittle fracture processes are simulated by making use of softening interface laws which relate tractions to displacements jumps a discrete formulation in terms of interface variables only is obtained the space discretization is carried out by means of a mixed finite element approach in which all interface variables are modelled study of uniqueness of the rate problem formulated in terms of interface variables and of stability of the equilibrium states is presented some examples are shown in order to clarify the formulation

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

harmonic enrichment functions a unified treatment of multiple intersecting and branched cracks in the extended finite element method a unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented within each element that is intersected by a crack the enrichment function for the crack is obtained via the solution of the laplace equation with dirichlet and vanishing neumann boundary conditions a single algorithm emanates for the enrichment field for multiple cracks as well as intersecting and branched cracks without recourse to special cases which provides flexibility over the existing approaches in which each case is treated separately numerical integration is rendered to be simple there is no need for partitioning of the finite elements into conforming subdivisions for the integration of discontinuous or weakly singular kernels stress intensity factor computations for different crack configurations are presented to demonstrate the accuracy and versatility of the proposed technique copyright 2010 john wiley sons ltd

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

element local level set method for three dimensional dynamic crack growth summary an approximate level set method for three dimensional crack propagation is presented in this method the discontinuity surface in each cracked element is defined by element local level sets ellss the local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least square sense a simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements the discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method xfem the ells method and the phantom node technology are combined for the solution of dynamic fracture problems numerical examples for three dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method copyright q 2009 john wiley sons ltd

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

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

the streaming dmt of fading channels we study sequential transmission of a stream of messages over a block fading multi input multi output mimo channel a new message arrives in each coherence block and the decoder is required to output each message after a delay of t coherence blocks we establish the optimal diversity multiplexing tradeoff dmt and show that it can be achieved using a simple interleaving of messages the converse is based on an outage amplification technique which appears to be new we also discuss another coding scheme based on a sequential tree code this coding scheme only requires the knowledge of delay at the decoder and yet realizes the optimal dmt we finally discuss some extensions when multiple messages at uniform intervals arrive within each coherence period

the threshold of coexistence and critical behaviour of a predator prey cellular automaton we study a probabilistic cellular automaton to describe two population biology problems the threshold of species coexistence in a predator prey system and the spreading of an epidemic in a population by carrying out mean field approximations and numerical simulations we obtain the phase boundaries thresholds related to the transition between an active state where prey and predators present a stable coexistence and a prey absorbing state the numerical estimates for the critical exponents show that the transition belongs to the directed percolation universality class in the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed

a higher order discontinuous enrichment method for the solution of high peclet advection diffusion problems on unstructured meshes a higher order discontinuous enrichment method dem with lagrange multipliers is proposed for the efficient finite element solution on unstructured meshes of the advection diffusion equation in the high peclet number regime following the basic dem methodology the usual galerkin polynomial approximation is enriched with free space solutions of the governing homogeneous partial differential equation pde in this case these are exponential functions that exhibit a steep gradient in a specific flow direction exponential lagrange multipliers are introduced at the element interfaces to weakly enforce the continuity of the solution the construction of several higher order dem elements fitting this paradigm is discussed in detail numerical tests performed for several two dimensional benchmark problems demonstrate their computational superiority over stabilized galerkin counterparts especially for high peclet numbers copyright 2009 john wiley sons ltd

epidemic outbreaks on random voronoi delaunay triangulations abstract we study epidemic outbreaks on random delaunay triangulations by applying the asynchronous sir susceptible infected removed dynamics coupled to two dimensional voronoi delaunay triangulations in order to investigate the critical behavior of the model we obtain the cluster size distribution by using newman ziff algorithm allowing to simulate random inhomogeneous lattices and measure any desired observable related to percolation we numerically calculate the order parameter defined as the wrapping cluster density the mean cluster size and binder cumulant ratio defined for percolation in order to estimate the epidemic threshold our findings suggest that the system falls into two dimensional dynamic percolation universality class and the quenched random disorder is irrelevant in agreement with results for classical percolation

on source transmission over mimo channels with limited feedback the problem of source channel coding over a multiple antenna mimo channel with quantized channel state information at the transmitter csit is considered upper bounds on the distortion exponents achieved with partial csit under a long term power constraint are developed it is shown that the distortion exponent with perfect csit grows unbounded as the ratio between the channel and source bandwidth increases while the exponent achieved with any feedback link of fixed finite resolution is bounded above by a polynomial of the product between the number of transmit and number of receive antennas the resolution of the feedback link should grow with the bandwidth ratio to make the distortion exponent scale as fast as that in the perfect csit case we show that in order to achieve the optimal scaling the csit feedback resolution must grow logarithmically with the bandwidth ratio for mimo channels and faster than linear for the single input single output channel the achievable distortion exponent of some hybrid schemes with heavily quantized feedback is also derived the results demonstrate that dramatic performance improvement over the case of no csit can be achieved by combining simple schemes with a very coarse csit feedback

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

two scale approach to predict multi site cracking potential in 3 d structures using the generalized finite element method this research utilized the novel computational framework of the generalized finite element method gfem to predict the potential for crack propagation in concrete slabs a two scale approach using the global local concept within the gfem framework gfemg l is applied to multi site cracking problems msc where different crack geometries are placed simultaneously at different positions in a three dimensional airfield slab loaded by new generation aircraft gears the gfemg l approach efficiently simulated multiple cracks without discretization in the global mesh but only in the local domain the gfemg l enrichment functions allow the displacements of the local problem to be represented in the global domain through enrichment functions from the local problems rather than explicitly modeling each crack discretely in the global domain the main contribution of this work was extension of the gfemg l approach to a class of three dimensional msc problems involving realistic boundaries conditions and existence of multiple cracks spanning different orders of magnitude in size scales within the domain for the linear elastic structure bottom and surface initiated cracks with small dimensions were considered in conjunction with a larger macro crack unlike traditional numerical methods the proposed gfemg l made it possible to tackle this class of problems by avoiding refined crack front meshes in the global domain as well as numerical round off errors furthermore the two scale approach significantly reduces the computational cost for large scale 3 d msc problems

generalized finite element enrichment functions for discontinuous gradient fields a general gfem xfem formulation is presented to solve two dimensional problems characterized by c0 continuity with gradient jumps along discrete lines such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media the new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks we show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines the use of the proposed enrichments in both linear and quadratic approximations is investigated as well as their combination with interface enrichment functions available in the literature through a detailed convergence study we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems in the linear case the effectiveness of correction functions proposed in the literature is also investigated copyright 2009 john wiley sons ltd

design and analysis of mimo spatial multiplexing systems with quantized feedback this paper investigates the problem of transmit beamforming in multiple antenna spatial multiplexing sm systems employing a finite rate feedback channel assuming a fixed number of spatial channels and equal power allocation we propose a new criterion for designing the codebook of beamforming matrices that is based on minimizing an approximation to the capacity loss resulting from the limited rate in the feedback channel using the criterion we develop an iterative design algorithm that converges to a locally optimum codebook under the independent identically distributed channel and high signal to noise ratio snr assumption the effect on channel capacity of the finite bit representation of the beamforming matrix is analyzed central to this analysis is the complex multivariate beta distribution and tractable approximations to the voronoi regions associated with the code points furthermore to compensate for the degradation due to the equal power allocation assumption we propose a multimode sm transmission strategy wherein the number of data streams is determined based on the average snr this approach is shown to allow for effective utilization of the feedback bits resulting in a practical and efficient multiple input multiple output system design

a discontinuous enrichment method for variable coefficient advection diffusion at high peclet number a discontinuous galerkin method with lagrange multipliers is presented for the solution of variable coefficient advection diffusion problems at high peclet number in this method the standard finite element polynomial approximation is enriched within each element with free space solutions of a local constant coefficient homogeneous counterpart of the governing partial differential equation hence in the two dimensional case the enrichment functions are exponentials each exhibiting a sharp gradient in a carefully chosen flow direction the continuity of the enriched approximation across the element interfaces is enforced weakly by the aforementioned lagrange multipliers numerical results obtained for two benchmark problems demonstrate that elements based on the proposed discretization method are far more competitive for variable coefficient advection diffusion analysis in the high peclet number regime than their standard galerkin and stabilized finite element comparables copyright 2010 john wiley sons ltd

generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids an extension of the generalized finite element method to the class of mixed finite element methods is presented to tackle heterogeneous systems with nearly incompressible non linear hyperelastic material behavior in particular heterogeneous systems with large modulus mismatch across the material interface undergoing large strains are investigated using two formulations one based on a continuous deformation map the other on a discontinuous one a bimaterial patch test is formulated to assess the ability of the two formulations to reproduce constant stress fields while a mesh convergence study is used to examine the consistency of the formulations finally compression of a model heterogeneous propellant pack is simulated to demonstrate the robustness of the proposed discontinuous deformation map formulation

a unified enrichment approach addressing blending and conditioning issues in enriched finite elements abstract we present a combination of techniques to improve the convergence and conditioning properties of partition of unity pu enriched finite element methods by applying these techniques to different types of enrichment functions namely polynomial discontinuous and singular higher order convergence rates can be obtained while keeping condition number growth rates similar to the ones corresponding to standard finite elements

on the distortion snr exponent of hybrid digital analog space time coding we consider the transmission of a real independent and identically distributed i i d analog source over a quasi static m input n output multiple input multiple output mimo block fading channel the relevant performance criterion is end to end average quadratic distortion d versus channel signal to noise ratio snr for given spectral efficiency eta defined as the ratio of the source bandwidth over the channel bandwidth in the limit of high snr we define the distortion snr exponent a eta as the largest a such that d esdot snr a over all possible source channel coding schemes of spectral efficiency eta we find a simple upper bound on a eta an achievable lower bound asep eta achievable by separated tandem source channel coding and a tighter lower bound ahybrid eta achievable by new hybrid digital analog space time coding schemes as a corollary we have that a eta is completely determined for the scalar case m n 1 and for the bandwidth compression case eta ges 2 min m n expiicit and simple construction of hybrid space time codes achieving ahybrid eta are also given

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

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

mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard fem analysis of 3 d cohesive fractures summary a robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures this class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small sized critical regions the proposed approach is based on the division of a simulation into a suitable number of sub simulations where adaptive mesh refinement is performed only once based on refinement window s around crack front process zone s the initialization of newton raphson nonlinear iterations at the start of each sub simulation is accomplished by solving a linear problem based on a secant stiffness rather than a volume mapping of nonlinear solutions between meshes the secant stiffness is evaluated using material state information stored read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework moreover a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software data transfer between sub simulations is not required in the simplified strategy the computational efficiency accuracy and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3 d copyright 2016 john wiley sons ltd

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

susceptible infected recovered and susceptible exposed infected models two stochastic epidemic lattice models the susceptible infected recovered and the susceptible exposed infected models are studied on a cayley tree of coordination number k the spreading of the disease in the former is found to occur when the infection probability b is larger than bc k 2 k 1 in the latter which is equivalent to a dynamic site percolation model the spreading occurs when the infection probability p is greater than pc 1 k 1 we set up and solve the time evolution equations for both models and determine the final and time dependent properties including the epidemic curve we show that the two models are closely related by revealing that their relevant properties are exactly mapped into each other when p b k k 1 b these include the cluster size distribution and the density of individuals of each type quantities that have been determined in closed forms

a multiple level set approach to prevent numerical artefacts in complex microstructures with nearby inclusions within xfem in this paper we highlight that when the extended finite element method xfem is employed to model a microstructure in which inclusions are involved and the distance between two inclusions is small enough to be comparable with the mesh size three numerical artefacts are induced significantly affecting the convergence and accuracy of the numerical solution to the problem with such a microstructure these artefacts are a an artificial percolation of nearby inclusions b an artificial distortion of phase domains c an enrichment deficiency we propose to improve the xfem levelset method so as to avoid these artefacts the new technique leading to this improvement uses one level set function for each inclusion and adds additional enrichment in an element whose support is cut by several interfaces a local description of the multiple level sets is provided to avoid the storage of all level set functions a simple integration rule is employed for numerical quadrature in elements cut by several interfaces we show that the artefacts mentioned hereinbefore are circumvented in this framework the performances of the method are demonstrated through benchmarks and examples applied to the homogenization of concrete materials in 2d and 3d cases

explicit space time codes achieving the diversity multiplexing gain tradeoff a recent result of zheng and tse states that over a quasi static channel there exists a fundamental tradeoff referred to as the diversity multiplexing gain d mg tradeoff between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space time st code this tradeoff is precisely known in the case of independent and identically distributed i i d rayleigh fading for tgesnt nr 1 where t is the number of time slots over which coding takes place and nt nr are the number of transmit and receive antennas respectively for t nt case we present two general techniques for building d mg optimal rectangular st codes from their square counterparts a byproduct of our results establishes that the d mg tradeoff for all tgesnt is the same as that previously known to hold for tgesnt n r 1

susceptible infected recovered model with recurrent infection abstract we analyze a stochastic lattice model describing the spreading of a disease among a community composed by susceptible infected and removed individuals a susceptible individual becomes infected catalytically an infected individual may spontaneously either become recovered that is acquire a permanent immunization or become again susceptible the critical properties including the phase diagram is obtained by means of mean field theories as well as numerical simulations the model is found to belong to the universality class of dynamic percolation except when the recovering rate vanishes in which case the model belongs to the directed percolation universality class

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

an algebraic treatment of essential boundary conditions in the particle partition of unity method this paper is concerned with the treatment of essential boundary conditions in meshfree methods in particular we focus on the particle partition of unity method ppum however the proposed technique is applicable to any partition of unity based approach we present an efficient scheme for the automatic construction of a direct splitting of a ppum function space into the degrees of freedom suitable for the approximation of the dirichlet data and the degrees of freedom that remain for the approximation of the pde by simple linear algebra notably our approach requires no restrictions on the distribution of the discretization points nor on the employed local approximation spaces we attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces hence the global splitting can be computed with sub linear complexity due to this direct splitting of the meshfree ppum function space we can implement a conforming local treatment of essential boundary data so that the realization of dirichlet boundary values in the meshfree ppum is straightforward the presented approach yields an optimally convergent scheme which is demonstrated by the presented numerical results

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

parameter free weak imposition of dirichlet boundary conditions and coupling of trimmed and non conforming patches we present a parameter free domain sewing approach for low order as well as high order finite elements its final form contains only primal unknowns that is the approach does not introduce additional unknowns at the interface additionally it does not involve problem dependent parameters which require an estimation the presented approach is symmetry preserving that is the resulting discrete form of an elliptic equation will remain symmetric and positive definite it preserves the order of the underlying discretization and we demonstrate high order accuracy for problems of non matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p we also demonstrate how the method may be used to model material interfaces which may be curved and for which the interface does not coincide with the underlying mesh this novel approach is presented in the context of the p version and b spline version of the finite cell method an embedded domain method of high order and compared with more classical methods such as the penalty method or nitsche s method

approximately universal codes over slow fading channels performance of reliable communication over a coherent slow fading multiple input multiple output mimo channel at high signal to noise ratio snr is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains this paper studies the problem of designing codes that optimally tradeoff the diversity and multiplexing gains the main contribution is a precise characterization of codes that are universally tradeoff optimal i e they optimally tradeoff the diversity and multiplexing gains for every statistical characterization of the fading channel this characterization is referred to as approximate universality the approximation is in the connection between error probability and outage capacity with diversity and multiplexing gains respectively the characterization of approximate universality is then used to construct new coding schemes as well as to show optimality of several schemes proposed in the space time coding literature

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