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

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

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

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

multilevel preconditioning for partition of unity methods some analytic concepts this paper is concerned with the construction and analysis of multilevel schwarz preconditioners for partition of unity methods applied to elliptic problems we show under which conditions on a given multilevel partition of unity hierarchy mpum one even obtains uniformly bounded condition numbers and how to realize such requirements the main anlytical tools are certain norm equivalences based on two level splits providing frames that are stable under taking subsets

efficient modeling of localized material failure by means of a variationally consistent embedded strong discontinuity approach this paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite strains focus is on localized plastic deformation particularly relevant for slip bands in contrast to already existing models the proposed implementation allows to consider several interacting discontinuities in each finite element based on a proper re formulation of the kinematics an efficient parameterization of the deformation gradient is derived it permits to compute the strains explicitly that improves the performance significantly however the most important novel contribution of the present paper is the advocated variational constitutive update within this framework every aspect is naturally driven by energy minimization i e all unknown variables are jointly computed by minimizing the stress power the proposed update relies strongly on an extended principle of maximum dissipation this framework provides enough flexibility for different failure types and for a broad class of non associative evolution equations by discretizing the aforementioned continuous variational principle an efficient numerical implementation is obtained it shows in addition to its physical and mathematical elegance several practical advantages for instance the physical minimization principle itself specifies automatically and naturally the set of active strong discontinuities copyright 2011 john wiley sons ltd

shape optimization under uncertainty a stochastic programming perspective we present an algorithm for shape optimization under stochastic loading and representative numerical results our strategy builds upon a combination of techniques from two stage stochastic programming and level set based shape optimization in particular usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces which often is much smaller than the number of different realizations of the stochastic forces numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions

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

a particle partition of unity method part iii a multilevel solver in this sequel to part i siam j sci comput 22 2000 pp 853 890 and part ii siam j sci comput 23 2002 pp 1655 1682 we focus on the efficient solution of the linear block systems arising from a galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method pum we present a cheap multilevel solver for partition of unity pu discretizations of any order the shape functions of a pum are products of piecewise rational pu functions varphi i with supp varphi i omega i and higher order local approximation functions psi i n usually a local polynomial of degree leq p i furthermore they are noninterpolatory in a multilevel approach we have to cope with not only noninterpolatory basis functions but also with a sequence of nonnested spaces due to the meshfree construction hence injection or interpolatory interlevel transfer operators are not available for our multilevel pum therefore the remaining natural choice for the prolongation operators are l2 projections here we exploit the pum construction of the function spaces and a hierarchical construction of the pu itself to localize the corresponding projection problem this significantly reduces the computational costs associated with the setup and the application of the interlevel transfer operators the second main ingredient of our multilevel solver is the use of a block smoother to treat the local approximation functions psi i n for all n simultaneously the results of our numerical experiments in two and three dimensions show that the convergence rate of the proposed multilevel solver is independent of the number of patches card omega i the convergence rate is slightly dependent on the local approximation orders pi

on the enrichment zone size for optimal convergence rate of the generalized extended finite element method singular enrichment functions are broadly used in generalized or extended finite element methods gfem xfem for linear elastic fracture mechanics problems these functions are used at finite element nodes within an enrichment zone around the crack tip front in 2 and 3 d problems respectively small zones lead to suboptimal convergence rate of the method while large ones lead to ill conditioning of the system of equations and to a large number of degrees of freedom this paper presents an a priori estimate for the minimum size of the enrichment zone required for optimal convergence rate of the gfem xfem the estimate shows that the minimum size of the enrichment zone for optimal convergence rate depends on the element size and polynomial order of the gfem xfem shape functions detailed numerical verification of these findings is also presented

a quasi static discontinuous galerkin configurational force crack propagation method for brittle materials summary this paper presents a framework for r adaptive quasi static configurational force cf brittle crack propagation cast within a discontinuous galerkin dg symmetric interior penalty sipg finite element scheme cracks are propagated in discrete steps with a staggered algorithm along element interfaces which align themselves with the predicted crack propagation direction the key novelty of the work is the exploitation of the dg face stiffness terms existing along element interfaces to propagate a crack in a mesh independent r adaptive quasi static fashion driven by the cf at the crack tip this adds no new degrees of freedom to the data structure additionally as dg methods have element specific degrees of freedom a geometry driven p adaptive algorithm is also easily included allowing for more accurate solutions of the cf on a moving crack front further for non determinant systems we introduce an average boundary condition that restrains rigid body motion leading to a determinant system to the authors knowledge this is first time such a boundary condition has been described the proposed formulation is validated against single and multiple crack problems with single and mixed mode cracks demonstrating the predictive capabilities of the method this article is protected by copyright all rights reserved

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

cesaro averaging operators we define a family of cesaro operators re 0 and consider the question of their boundedness on h p spaces we also consider discrete versions of these operators acting on sequence spaces

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

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

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

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

domain integral formulation for stress intensity factor computation along curved three dimensional interface cracks new domain integrals for extracting mixed mode stress intensity factors along curved three dimensional bimaterial interface cracks are derived in the derivation the asymptotic auxiliary fields for the plane problem of a bimaterial interface crack are imposed along a curved crack front the general crack tip interaction integral a contour integral surrounding a point on the crack front which is evaluated in the limit as the contour is shrunk onto the crack tip is then expressed in domain form which is more suitable for numerical computation a consequence of imposing the auxiliary fields along a curved crack front is that the auxiliary stress fields do not satisfy equilibrium and the auxiliary strain fields do not satisfy compatibility the terms which arise due to the lack of equilibrium and compatibility are not sufficiently singular to contribute to the crack tip interaction integral or to affect its path independence however these terms become important when domain integral representations are introduced because they involve fields which are not asymptotically close to the crack tip in order to compute the pointwise stress intensity factors along the crack front the domain integrals are evaluated as a post processing step in the finite element method in the numerical results it is demonstrated that it is crucial to incorporate the terms that arise due to lack of equilibrium and compatibility of the auxiliary fields especially in regions where the local crack front curvature is high in the paper we present two numerical examples as a benchmark we first consider the problem of a penny shaped interface crack embedded in a cylinder the results for the complex stress intensity factor and phase angle are found to be in excellent agreement with the analytical solution the problem of an elliptical crack embedded between two dissimilar isotropic materials is also considered and the results are discussed

cesaro averaging operators on hardy spaces it is shown that the cesaro averaging operator 1 satisfies an inequality which immediately implies that it is bounded on certain hardy spaces including h p 0 p h p 0 p 2 for 0 the operator which is conjugate to on h 2 is also shown to be bounded on h p for 1 p p and 0

a stable generalized finite element method sgfem of degree two for interface problems abstract generalized or extended finite element methods gfem xfem of degree 1 for interface problems have been reported in the literature they i yield optimal order of convergence i e o h h is a discretization parameter ii are stable in a sense that conditioning is not worse than that of the standard fem and iii are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements however higher order gfem xfem with the properties i iii have not been successfully addressed yet various enrichment schemes for gfem xfem based on d or d p k d is a distance function or the absolute value of level set function and p k is the polynomial basis of degree k have been reported recently to obtain higher order convergence but they are not stable or robust in general in fact they even may not yield the optimal orders of convergence in this paper we propose a stable gfem xfem of degree 2 sgfem2 for the interface problems where we used the enrichment scheme based on d 1 x y instead of d or d 1 x y x 2 x y y 2 in the literature we prove that the sgfem2 yields the optimal order of convergence i e o h 2 for the interface problems with curved smooth interfaces a local principal component analysis technique has been proposed which ensures that the sgfem2 is stable and robust numerical experiments for straight and curved interfaces have been presented to illuminate these properties

stable generalized finite element method and associated iterative schemes application to interface problems abstract the generalized finite element method gfem is an extension of the finite element method fem where the standard finite element space is augmented with a space of non polynomial functions called the enrichment space the functions in the enrichment space mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully applied to a wide range of problems however it often suffers from bad conditioning i e its conditioning may not be robust with respect to the mesh and in fact the conditioning could be much worse than that of the standard fem in this paper we present a numerical study that shows that if the angle between the finite element space and the enrichment space is bounded away from 0 uniformly with respect to the mesh then the gfem is stable i e the conditioning of gfem is not worse than that of the standard fem a gfem with this property is called a stable gfem sgfem the last part of the paper is devoted to the derivation of a robust iterative solver exploiting this angle condition it is shown that the required wall clock time is greatly reduced compared to popular gfems used in the literature

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

orbital enriched flat top partition of unity method for the schrodinger eigenproblem abstract quantum mechanical calculations require the repeated solution of a schrodinger equation for the wavefunctions of the system from which materials properties follow recent work has shown the effectiveness of enriched finite element type galerkin methods at significantly reducing the degrees of freedom required to obtain accurate solutions however time to solution has been adversely affected by the need to solve a generalized rather than standard eigenvalue problem and the ill conditioning of associated system matrices in this work we address both issues by proposing a stable and efficient orbital enriched partition of unity method to solve the schrodinger boundary value problem in a parallelepiped unit cell subject to bloch periodic boundary conditions in the proposed partition of unity method the three dimensional domain is covered by overlapping patches with a compactly supported weight function associated with each patch a key ingredient in our approach is the use of non negative weight functions that possess the flat top property i e each weight function is identically equal to unity over some finite subset of its support this flat top property provides a pathway to devise a stable approximation over the whole domain on each patch we use p th degree orthogonal legendre polynomials that ensure p th order completeness and in addition include eigenfunctions of the radial schrodinger equation furthermore we adopt a variational lumping approach to construct a block diagonal overlap matrix that yields a standard eigenvalue problem for which there exist efficient eigensolvers the accuracy stability and efficiency of the proposed method is demonstrated for the schrodinger equation with a harmonic potential as well as a localized gaussian potential we show that the proposed approach delivers optimal rates of convergence in the energy and the use of orbital enrichment significantly reduces the number of degrees of freedom for a given desired accuracy in the energy eigenvalues while the stability of the enriched approach is fully maintained

error estimates in weighted sobolev norms for finite element immersed interface methods abstract when solving elliptic partial differential equations in a region containing immersed interfaces possibly evolving in time it is often desirable to approximate the problem using an independent background discretisation not aligned with the interface itself optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface at the cost of a higher complexity in the implementation a much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions as done in immersed boundary methods these methods are known to have inferior convergence properties depending on the global regularity of the solution across the interface when compared to enriched methods in this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon restricted to a small neighbourhood of the interface in particular we show that optimal approximations can be constructed in a natural and inexpensive way simply by reformulating the problem in a distributionally consistent way and by resorting to weighted norms when computing the global error of the approximation

interface solutions of partial differential equations with point singularity abstract partial differential equations pdes such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips existing numerical methods for such pdes suffer from the constraint of this low regularity and fail to deliver an optimal high order rate of convergence we approach this problem by i choosing an artificial interface to enclose the center of low regularity and ii representing the solution in the interior of artificial interface as unknown linear combination of known low regularity modes of the solution this results in an interface formulation of the original pde and the linear combination is represented in the interface conditions by enforcing the smooth component of the numerical solution in the interior domain to be approximately zero a least squares problem is obtained for the unknown coefficients the solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the pde in the exterior domain the potential of our interface formulation is favorably demonstrated by numerical experiments on one and two dimensional poisson equations with low regularity solutions high order numerical solutions of unknown coefficients and pdes are obtained this proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems we address the relations between our interface formulation and domain decomposition methods as well as a regularization strategy for the poisson boltzmann equation with singular charge density

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

error probability for coherent and differential psk over arbitrary rician fading channels with multiple cochannel interferers this paper discusses the performance of communication systems using binary coherent and differential phase shift keyed psk modulation in correlated rician fading channels with diversity reception the presence of multiple rician faded cochannel users which may have arbitrary and nonidentical parameters is modeled exactly exact bit error probability bep expressions are derived via the moment generating functions mgfs of the relevant decision statistics which are obtained through coherent detection with maximum ratio combining for coherent psk modulation and differential detection with equal gain combining egc for differential modulation evaluating the exact expressions requires a complexity that is exponential in the number of interferers to avoid this potentially time consuming operation we derive two low complexity approximate methods each for coherent and differential modulation formats which are more accurate than the traditional gaussian approximation approach two new and interesting results of this analysis are 1 unlike in the case of rayleigh fading channels increasing correlation between diversity branches may lead to better performance in rician fading channels and 2 the phase distribution of the line of sight or static fading components of the desired user has a significant influence on the bep performance in correlated diversity channels

symmetric weak form integral equation method for three dimensional fracture analysis a symmetric galerkin boundary element method is developed for the analysis of linearly elastic isotropic three dimensional solids containing fractures the formulation is based upon a weak form displacement integral equation and a weak form traction integral equation recently developed by li and mear 1997 these integral equations are only weakly singular and their validity requires only that the boundary displacement data be continuous hence allowing standard co elements to be employed as part of the numerical implementation a special crack tip element is developed which has a novel feature in that there exist degrees of freedom associated with the nodes at the crack front as a result a higher degree of approximation is achieved for the relevant displacement data on the crack and further the stress intensity factors are obtained directly in terms of the crack front nodal data various examples are treated for cracks in unbounded domains and for cracks in finite domains including both embedded and surface breaking cracks and it is demonstrated that highly accurate results can be achieved using relatively coarse meshes

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

cesaro operators on hardy spaces in the unit ball this article establishes the boundedness of the generalized cesaro operator on holomorphic hardy spaces in the unit ball the approach consists in writing the generalized cesaro operator as a composition of certain integral operators

enrichment of enhanced assumed strain approximations for representing strong discontinuities addressing volumetric incompressibility and the discontinuous patch test summary we present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary intra finite element discontinuities in the deformation map special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit given an underlying mesh and an arbitrary failure surface we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method x fem with regard to the richer space of functions spanned by the gradient of the enriched approximation we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain in particular in those elements intersected by the failure surface we construct enhanced strain approximations which are orthogonal to piecewise constant stress fields contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields importantly the present formulation gives rise to a symmetric tangent stiffness matrix even in localized elements the present modification also allows for the satisfaction of a discontinuous patch test wherein two different constant stress fields on each side of the failure surface lie in the solution space we demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface particularly for nearly incompressible material response additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics copyright c 2000 john wiley sons ltd

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

on a new integral type operator from the bloch space to bloch type spaces on the unit ball abstract we introduce the following integral type operator on the space h b of all holomorphic functions on the unit ball b c n p g f z 0 1 f t z g t z d t t z b where g h b g 0 0 and is a holomorphic self map of b the boundedness and compactness of the operator from the bloch space b or the little bloch space b 0 to the bloch type space b or the little bloch type space b 0 are characterized in the main results we calculate the essential norm of the operators p g b or b 0 b or b 0 in an elegant way

robustness in stable generalized finite element methods sgfem applied to poisson problems with crack singularities abstract in this paper we study the performance of the generalized finite element method gfem applied to the poisson problem with crack singularities recently a gfem with modified heaviside enrichments was proposed in gupta et al 2013 to approximate the solution of a 2d elasticity problem with a crack it was shown that the gfem is indeed a stable gfem sgfem i e it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method fem however the robustness of the conditioning of the gfem with respect to the position of the mesh relative to the crack was not addressed in gupta et al 2013 in this paper we observed that the conditioning of the gfem with the enrichments used in gupta et al 2013 is not robust when applied to approximate the solution of the poisson problem moreover the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces we proposed using additional singular enrichments at the nodes close to the crack near the crack tip we proved that the gfem with enrichments proposed in this paper yields optimal order of convergence irrespective of the position of the mesh moreover with a local orthogonalization procedure we have shown through numerical experiments that the conditioning of this gfem is not worse than that of the standard fem and the conditioning is robust with respect to the position of the mesh thus the gfem with the enrichments suggested in this paper is indeed an sgfem when applied to a poisson problem with the crack singularities

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

an interaction integral method for 3d curved cracks in nonhomogeneous materials with complex interfaces this work derives an interaction integral for the computation of mixed mode stress intensity factors sifs in three dimensional 3d nonhomogeneous materials with continuous or discontinuous properties the present method is based on a two state integral by the superposition of actual and auxiliary fields in 3d domain formulation of the interaction integral derived here the integrand does not involve any derivatives of material properties furthermore the formulation can be proved to be still valid even when the integral domain contains material interfaces therefore it is not necessary to limit the material properties to be continuous for the present formulation on account of these advantages the application range of the interaction integral can be greatly enlarged this method in conjunction with the finite element method fem is employed to solve several representative fracture problems according to the comparison between the results and those from the published lectures good agreement demonstrates the validation of the interaction integral the results show that the present interaction integral is domain independent for nonhomogeneous materials with interfaces

extended cesaro operators from generally weighted bloch spaces to zygmund space abstract let g be a holomorphic function of the unit ball b in the n dimensional complex space and denote by t g the extended cesaro operator with symbol g starting with a brief introduction to well known results about cesaro operator we investigate the boundedness and compactness of t g from generally weighted bloch spaces b log 0 to zygmund space z in the unit ball and also present some necessary and sufficient conditions

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

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions 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 generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

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

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

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

optimal convergence analysis for the extended finite element method we establish some optimal a priori error estimate on some variants of the extended finite element method xfem namely the xfem with a cut off function and the standard xfem with a fixed enrichment area the results are established for the lame system homogeneous isotropic elasticity and the laplace problem the convergence of the numerical stress intensity factors is also investigated we show some numerical experiments which corroborate the theoretical results

xfem based fictitious domain method for linear elasticity model with crack reduction of computational cost of solutions is a key issue for crack identification or crack propagation problems one of the solutions is to avoid remeshing the domain when the crack position changes or when the crack extends to avoid remeshing we propose a new finite element approach for the numerical simulation of discontinuities of displacements generated by cracks inside elastic media the approach is based on a fictitious domain method originally developed for dirichlet conditions for the poisson problem and for the stokes problem which is adapted to the neumann boundary conditions of crack problems the crack is represented by level set functions numerical tests are made with a mixed formulation to emphasize the accuracy of the method as well as its robustness with respect to the geometry enforced by a stabilization technique in particular an inf sup condition is theoretically proven for the latter a realistic simulation with a uniformly pressurized fracture inside a volcano is given for i

strongly stable generalized finite element method ssgfem for a non smooth interface problem ii a simplified algorithm abstract the solution of interface problems with non smooth interfaces having corners connected by straight lines exhibit weak singularity along the interface and may have singularities at the corners in an earlier paper zhang et al 2019 a strongly stable generalized finite element method ssgfem was proposed to approximate solution of such problems the method required two separate enrichment functions at o h 1 number of nodes h is the mesh parameter and a local orthogonalization technique requiring additional operations was used to ensure its stability and robustness in this paper we present a simplified ssgfem that does not require two enrichment functions at any of the nodes as a result we obtain an optimally converging gfem that is stable i e its scaled condition number is of the same order that of a standard fem and robust i e the conditioning is independent of the position of meshes relative to the interface these properties are illuminated by well designed experiments the simplified ssgfem does not require lot and requires less degrees of freedom and is easier to implement in contrast to the ssgfem proposed in zhang et al 2019 we also proved the optimal convergence for the simplified ssgfem in this paper that required a more involved analysis based on a smart observation on approximation of weak singularities near the corner

theory and numerics for finite deformation fracture modelling using strong discontinuities a general finite element approach for the modelling of fracture is presented for the geometrically non linear case the kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements thus the deformation map is defined in terms of one continuous and one discontinuous portion considered as mutually independent giving rise to a weak formulation of the equilibrium consisting of two coupled equations in addition two different fracture criteria are considered firstly a principle stress criterion in terms of the material mandel stress in conjunction with a material cohesive zone law relating the cohesive mandel traction to a material displacement jump associated with the direct discontinuity secondly a criterion of griffith type is formulated in terms of the material crack driving force mcdf with the crack propagation direction determined by the direction of the force corresponding to the direction of maximum energy release apart from the material modelling the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria copyright 2005 john wiley sons ltd

mass lumping strategies for x fem explicit dynamics application to crack propagation this paper deals with the numerical modelling of cracks in the dynamic case using the extended finite element method more precisely we are interested in explicit algorithms we prove that by using a specific lumping technique the critical time step is exactly the same as if no crack were present this somewhat improves a previous result for which the critical time step was reduced by a factor of square root of 2 from the case with no crack the new lumping technique is obtained by using a lumping strategy initially developed to handle elements containing voids to be precise the results obtained are valid only when the crack is modelled by the heaviside enrichment note also that the resulting lumped matrix is block diagonal blocks of size 2 2 for constant strain elements linear simplex elements the critical time step is not modified when the element is cut thanks to the lumped mass matrix the critical time step never tends to zero moreover the lumping techniques conserve kinetic energy for rigid motions in addition tensile stress waves do not propagate through the discontinuity hence the lumping techniques create neither error on kinetic energy conservation for rigid motions nor wave propagation through the crack both these techniques will be used in a numerical experiment copyright 2007 john wiley sons ltd

mixed stabilized finite element methods in nonlinear solid mechanics part ii strain localization this paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining a standard local isotropic rankine damage model with strain softening is used as exemplary constitutive model both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated from this analysis the issue of local discretization error in the pre localization regime is deemed as the main difficulty to be overcome in the discrete problem focus is placed on low order finite elements with continuous strain and displacement fields triangular p1p1 and quadrilateral q1q1 although the presented approach is very general numerical examples show that the resulting procedure is remarkably robust it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh bias dependence

new simple closed form ber expressions for mrc diversity bpsk in correlated rayleigh fading and cci exact closed form expressions are derived for the bit error rate of binary phase shift keying in rayleigh fading channels with maximal ratio combining diversity in the presence of cochannel interference and additive white gaussian noise the desired signal and the interferer signals are subject to correlated rayleigh fading and binary phase shift keying modulation is used by all users

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

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

stable generalized finite element method sgfem for parabolic interface problems abstract we study a stable generalized finite element method sgfem for parabolic interface problems with smooth interfaces the standard finite element fe space based on a simple mesh is enriched by a one side distance function to simulate gradient jumps of exact solutions on the interfaces the mesh is quasi uniform and independent of the interfaces so that mesh matching or refinement along the interfaces is not needed the main idea to derive the sgfem is to modify the one side distance function by subtracting its fe interpolant the sgfem approximates the parabolic interface problem with an optimal convergence rate and is stable in that its scaled condition number is of the same order as the fem o h 2 h is a mesh size parameter and does not depend on the relative positions of the mesh and interfaces the proposed sgfem is a conforming method and is free from any penalty parameters or stability schemes moreover the components of the approximation functions of the sgfem are independent when extended to vector valued interface problems the optimal convergence rates of sgfem for curved interfaces are proven theoretically for the backward euler and crank nicolson formulations the convergence and conditioning are tested by numerical experiments

fracture analysis in plane structures with the two scale g xfem method abstract generalized or extended finite element method g xfem uses enrichment functions that holds a priori knowledge about the problem solution these enrichment functions are mostly limited to two dimensional problems a well established solution for problems without any specific types of analytically derived enrichment functions is to use numerically build functions in which they are called global local enrichment functions these functions are extracted from the solution of boundary value problems defined around the region of interest discretized by a fine mesh such solution is used to enrich the global solution space through the partition of unity framework of the g xfem here it is presented a two scale global local g xfem approach to model crack propagation in plane stress strain and reissner mindlin plate problems discontinuous functions along with the asymptotic crack tip displacement fields are used to represent the crack without explicitly represent its geometries under the linear elastic fracture mechanics approach the stress intensity factor obtained from a domain based interaction energy integral can be used to either determine the crack propagation direction or propagation status i e the crack can start to propagate or not the proposed approach is presented in detail and validated by solving several linear elastic fracture mechanics problems for both plane stress strain and reissner mindlin plate cases to demonstrate its the robustness and accuracy

strongly stable generalized finite element method ssgfem for a non smooth interface problem abstract in this paper we propose a strongly stable generalized finite element method ssgfem for a non smooth interface problem where the interface has a corner the ssgfem employs enrichments of 2 types the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions both types of enrichments have been modified by a simple local procedure of subtracting the interpolant a simple local orthogonalization technique lot also has been used at the nodes enriched with both enrichment functions we prove that the ssgfem yields the optimal order of convergence the numerical experiments presented in this paper indicate that the conditioning of the ssgfem is not worse than that of the standard finite element method and the conditioning is robust with respect to the position of the mesh relative to the interface

a stable generalized extended p hierarchical fem for three dimensional linear elastic fracture mechanics abstract in this paper the quadratic stable generalized finite element method sgfem proposed in sanchez rivadeneira and duarte 2019 is extended to 3 d fracture problems with non planar crack surfaces and curved crack fronts this sgfem is based on p hierarchical fem enrichments and its approximation space is the same as in its gfem counterpart singular enrichments are modified using a discontinuous finite element interpolant for conditioning control this leads with the proper choice of enrichments to stiffness matrices with a scaled condition number of o h 2 the same order as in the standard fem the robustness of the method with respect to the position of a crack relative to the mesh is demonstrated a scalar implementation of singular enrichments that can exactly reproduce the first term of the mode i ii and iii of the asymptotic expansion of the elasticity solution is presented improved approximations of signed distance functions based on integration sub elements are also presented convergence studies of a fully 3 d fracture problem with a non planar crack surface and a curved crack front are presented they show that the convergence rate of the proposed method on a sequence of uniform meshes is three times higher than quadratic quarter point finite elements

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

analysis of three dimensional crack initiation and propagation using the extended finite element method an erratum has been published for this article in international journal for numerical methods in engineering 2005 63 8 1228 we present a new formulation and a numerical procedure for the quasi static analysis of three dimensional crack propagation in brittle and quasi brittle solids the extended finite element method xfem is combined with linear tetrahedral elements a viscosity regularized continuum damage constitutive model is used and coupled with the xfem formulation resulting in a regularized crack band version of xfem the evolving discontinuity surface is discretized through a c0 surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two the element s properties allow a closed form integration and a particularly efficient implementation allowing large scale 3d problems to be studied several examples of crack propagation are shown illustrating the good results that can be achieved copyright 2005 john wiley sons ltd

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

on the iterated logarithmic bloch space on the unit ball abstract we introduce a new space of analytic functions on the unit ball the so called iterated logarithmic bloch space b log k and study some of its properties among other features we study the boundedness and compactness of weighted composition operators on the unit disk as well as of a recently introduced integral type operator on the unit ball with i domain the iterated logarithmic bloch space b log k or the little iterated logarithmic bloch space b log k 0 and ii range the bloch type space b or the little bloch type space b 0 or the weighted space h or the little weighted space h 0

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

stable generalized finite element method sgfem abstract the generalized finite element method gfem is a partition of unity method pum where the trial space of standard finite element method fem is augmented with non polynomial shape functions with compact support these shape functions which are also known as the enrichments mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully used to solve a variety of problems with complicated features and microstructure however the stiffness matrix of gfem is badly conditioned much worse compared to the standard fem and there could be a severe loss of accuracy in the computed solution of the associated linear system in this paper we address this issue and propose a modification of the gfem referred to as the stable gfem sgfem we show that sgfem retains the excellent convergence properties of gfem does not require a ramp function in the presence of blending elements and the conditioning of the associated stiffness matrix is not worse than that of the standard fem moreover sgfem is very robust with respect to the parameters of the enrichments we show these features of sgfem on several examples

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

higher order stable generalized finite element method the generalized finite element method gfem is a galerkin method where the trial space is obtained by augmenting the trial space of the standard finite element method fem by non polynomial functions called enrichments that mimic the local behavior of the unknown solution of the underlying variational problem the gfem has excellent approximation properties but its conditioning could be much worse than that of the fem however if the enrichments satisfy certain properties then the conditioning of the gfem is not worse than that of the standard fem and the gfem is referred to as the stable gfem sgfem in this paper we address the higher order sgfem that yields higher order convergence and suggest a specific modification of the enrichment function that guarantees the required conditioning yielding a robust implementation of the higher order sgfem

a closed form ber expression for bpsk using mrc in correlated cci and rayleigh fading an exact closed form expression for the average bit error rate ber of coherent bpsk using maximal ratio combining mrc with correlated branches in the presence of cochannel interference cci and noise is derived this expression is valid for an arbitrary correlation structure under the assumption that the branch gains of the desired user signal and interfering signals experience correlated rayleigh fading with the same correlation eigenvector matrix

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

robust and direct evaluation of j2 in linear elastic fracture mechanics with the x fem numerical simulation of fracture mechanics is of major importance in the design of structures such as planes or dams for such complex structures robustness and accuracy of the simulations are necessary the finite element method is classically considered for the simulation of such problems however the crack must be explicitly meshed in order to incorporate the discontinuity of the displacement field across crack faces this process has to be handled at each step of propagation the main drawback of this approach is the computational and human cost of remeshing the x fem method was proposed as a response to this issue 1 2 it uses the concept of partition of unity 3 in two ways first to take into account the displacement jump across crack faces far away from the crack tip and also to enrich the approximation close to the tip with some analytical asymptotic fields another important issue in fracture mechanics is the prediction of crack growth direction several criteria have been proposed one can cite for example the maximum hoop stress criterion 4 the maximum tangential stress criterion or the symmetry principle 5 the maximum energy release rate criterion 6 also called vectorial j integral criterion 7 is an alternative which was initially proposed by hellen and blackburn 7 the main difference in evaluating this criterion with respect to the classical j integral rely on the difficulties in the evaluation of the second component j2 of the energy release rate vector j it is well known that j1 which is basically the j integral is path independent concerning j2 herrmann and herrmann 8 have shown that it is path independent only in a modified manner i e by adding an extra contribution on the crack faces however this new contribution requires an accurate determination of the mechanical fields near the crack tip it is well known that the quality of these fields near a singularity is not ensured in the finite element context this is why some authors proposed specific strategies to avoid the integration near the tip for an accurate numerical evaluation of j2 9 10 for enriched finite

a stable generalized extended fem with discontinuous interpolants for fracture mechanics abstract this paper presents numerical studies with three classes of quadratic generalized fem gfem approximations and shows that all of them lead to errors that are orders of magnitude smaller than the fem with quarter point elements provided that appropriate enrichments are selected however all of them lead to severely ill conditioned systems of equations with a condition number up to o h 10 in the case of the gfem based on a quadratic partition of unity enrichment modifications able to address the ill conditioning of quadratic gfem approximations while preserving their optimal convergence are proposed a robust enrichment modification strategy based on a discontinuous fe interpolant is proposed to control the conditioning of branch function enrichment the discontinuous fe interpolant is a generalization of the continuous one used with the stable gfem sgfem we show that sgfem spaces based on p hierarchical fem enrichments are the same as their gfem counterparts this guarantees that both gfem and sgfem spaces will lead to the same solution this is not the case for the other classes of second order spaces the robustness of the proposed approximation spaces with respect to the crack location in the mesh is also demonstrated

calculation of fracture mechanics parameters for an arbitrary three dimensional crack by the equivalent domain integral method summary in this paper an equivalent domain integral edi method and the attendant numerical algorithms arc presented for the computation of a near crack tip field parameter the vector je integral and its variation along the front of an arbitrary three dimensional crack in a structural component account is taken of possible non elastic strains present in the structure in this case the near tip je values may be significantly different from the far field values jf especially under non proportional loading

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

three dimensional improved xfem ixfem for static crack problems abstract an improved xfem ixfem for three dimensional linear elastic fracture mechanics lefm problems is developed it utilizes an extra dof free pu approximation to fundamentally overcome the linear dependence and the ill conditioning issues of the standard xfem and the corrected xfem cxfem the introduced pu approximation is based on local least squares fitting with a one point interpolation constraint the resulting pu approximation interpolates and has no difficulties in boundary treatments detailed comparisons are made with the current xfems in 3d in terms of accuracy convergence and conditioning properties or solver efficiency better accuracy and convergence are demonstrated the remarkably better efficiency with a subspace iterative solver the de facto in large scale simulation is observed the number of iterations for the system of linear equations is reduced by orders of magnitude compared with the current xfems

convergence of domain integrals for stress intensity factor extraction in 2 d curved cracks problems with the extended finite element method summary the aim of this study is the analysis of the convergence rates achieved with domain energy integrals for the computation of the stress intensity factors sif when solving 2 d curved crack problems with the extended fem xfem domain integrals specially the j integral and the interaction integral are widely used for sif extraction and provide high accurate estimations with fems the crack description in xfem is usually realized using level sets this allows to define a local basis associated with the crack geometry in this work the effect of the level set local basis definition on the domain integral has been studied the usual definition of the interaction integral involves hypotheses that are not fulfilled in generic curved crack problems and we introduce some modifications to improve the behavior in curved crack analyses despite the good accuracy of domain integrals convergence rates are not always optimal and convergence to the exact solution cannot be assured for curved cracks the lack of convergence is associated with the effect of the curvature on the definition of the auxiliary extraction fields with our modified integral proposal the optimal convergence rate is achieved by controlling the q function and the size of the extraction domain copyright 2013 john wiley sons ltd