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

three dimensional non planar crack growth by a coupled extended finite element and fast marching method a numerical technique for non planar three dimensional linear elastic crack growth simulations is proposed this technique couples the extended finite element method x fem and the fast marching method fmm in crack modeling using x fem the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two dimensional asymptotic crack tip displacement fields the initial crack geometry is represented by two level set functions and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation crack modeling is performed without the need to mesh the crack and crack propagation is simulated without remeshing crack growth is conducted using fmm unlike a level set formulation for interface capturing no iterations nor any time step restrictions are imposed in the fmm planar and non planar quasi static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique copyright 2008 john wiley sons ltd

a nurbs based generalized finite element scheme for 3d simulation of heterogeneous materials a 3d nurbs based interface enriched generalized finite element method nigfem is introduced to solve problems with complex discontinuous gradient fields observed in the analysis of heterogeneous materials the method utilizes simple structured meshes of hexahedral elements that do not necessarily conform to the material interfaces in heterogeneous materials by avoiding the creation of conforming meshes used in conventional fem the nigfem leads to significant simplification of the mesh generation process to achieve an accurate solution in elements that are crossed by material interfaces the nigfem utilizes non uniform rational b splines nurbs to enrich the solution field locally the accuracy and convergence of the nigfem are tested by solving a benchmark problem we observe that the nigfem preserves an optimal rate of convergence and provides additional advantages including the accurate capture of the solution fields in the vicinity of material interfaces and the built in capability for hierarchical mesh refinement finally the use of the nigfem in the computational analysis of heterogeneous materials is discussed development of generalized finite element method for 3d mesoscale modeling of materials with complex microstructures novel nurbs based enrichment functions built in capability to perform hierarchical local mesh refinement method is applicable to engineering problems characterized by discontinuous gradient and complex internal geometry

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

algebraic equations determining quantum dimensions we show that a quantum dimension dq lambda for a representation rho of uq g a quantized universal enveloping algebra of a compact and simple lie group g is computed from the algebraic equations which we found recently in studying 2 1 dimensional chern simons theory we solve the equations explicitly for the typical examples of all compact and simple lie groups this method can be applied to super lie groups such as su m n and osp m n

a multiscale extended finite element method for crack propagation in this paper we propose a multiscale strategy for crack propagation which enables one to use a refined mesh only in the crack s vicinity where it is required two techniques are used in synergy a multiscale strategy based on a domain decomposition method to account for the crack s global and local effects efficiently and a local enrichment technique the x fem to describe the geometry of the crack independently of the mesh the focus of this study is the avoidance of meshing difficulties and the choice of an appropriate scale separation to make the strategy efficient we show that the introduction of the crack s discontinuity both on the microscale and on the macroscale is essential for the numerical scalability of the domain decomposition method to remain unaffected by the presence of a crack thus the convergence rate of the iterative solver is the same throughout the crack s propagation

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

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

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

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

finite elements with embedded displacement discontinuity a generalized variable formulation this paper is intended to bring a contribution towards a satisfactory simulation of those fracture phenomena which result in the appearance and development of discrete cracks to this purpose a general mixed finite element formulation is proposed based on the concept of generalized variables in prager s sense the displacement field inside an element is modelled by the sum of two contributions a regular continuous part which is governed by standard shape functions and a possibly discontinuous one which is introduced soon after a suitable criterion is satisfied the formulation is first specialized to a one dimensional case then a triangular element for two dimensional problems is described in detail analytical and numerical examples are presented in order to clarify the formulation and to point out the essential role of inter element conformity copyright 2000 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

finite element analysis on implicitly defined domains an accurate representation based on arbitrary parametric surfaces in this paper we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis the novel contributions of this paper are threefold 1 describe and validate a method to represent arbitrary parametric surfaces implicitly 2 represent arbitrary solids implicitly including sharp features using level sets and boolean operations 3 impose arbitrary dirichlet and neumann boundary conditions on the resulting implicitly defined boundaries the methods proposed do not require local refinement of the finite element mesh in regions of high curvature ensure the independence of the domain s volume on the mesh do not rely on boundary regularization and are well suited to methods based on fixed grids such as the extended finite element method xfem numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries this approach is one step in the desired direction of tying numerical simulations to computer aided design cad similarly to the isogeometric analysis paradigm

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

determining hysteresis thresholds for edge detection by combining the advantages and disadvantages of thresholding methods hysteresis is an important technique for edge detection but the unsupervised determination of its parameters is not an easy problem in this paper we propose a method for unsupervised determination of hysteresis thresholds using the advantages and disadvantages of two thresholding methods the basic idea of our method is to look for the best hysteresis thresholds in a set of candidates first the method finds a subset and a overset of the unknown edge points set then it determines the best edge map with the measure 2 compared with a general method to determine the parameters of an edge detector our method performs well and is less computationally complex the basic idea of our method can be generalized to other pattern recognition problems

a combined extended finite element and level set method for biofilm growth this paper presents a computational technique based on the extended finite element method xfem and the level set method for the growth of biofilms the discontinuous derivative enrichment of the standard finite element approximation eliminates the need for the finite element mesh to coincide with the biofilm fluid interface and also permits the introduction of the discontinuity in the normal derivative of the substrate concentration field at the biofilm fluid interface the xfem is coupled with a comprehensive level set update scheme with velocity extensions which makes updating the biofilm interface fast and accurate without need for remeshing the kinetics of biofilms are briefly given and the non linear strong and weak forms are presented the non linear system of equations is solved using a newton raphson scheme example problems including 1d and 2d biofilm growth are presented to illustrate the accuracy and utility of the method the 1d results we obtain are in excellent agreement with previous 1d results obtained using finite difference methods our 2d results that simulate finger formation and finger tip splitting in biofilms illustrate the robustness of the present computational technique copyright 2007 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

accurate recovery based upper error bounds for the extended finite element framework this paper introduces a recovery type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method xfem the paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a fem framework the upper bound property requires the recovered solution to be equilibrated and continuous the proposed technique consists of using a recovery technique especially adapted to the xfem framework that yields equilibrium at a local level patch by patch then a postprocess based on the partition of unity concept is used to obtain continuity the result is a very accurate but only nearly statically admissible recovered stress field with small equilibrium defaults introduced by the postprocess sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults as demonstrated by the numerical tests

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 mixed augmented lagrangian extended finite element method for modelling elastic plastic fatigue crack growth with unilateral contact the complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods a new technique for the finite element modelling of elastic plastic fatigue crack growth with unilateral contact on the crack faces is presented the extended finite element method x fem is used to discretize the equations allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh this paper presents an augmented lagrangian formulation in the x fem framework that is able to deal with elastic plastic crack growth with treatment of contact an original formulation which takes advantages of two powerful numerical methods is presented next the numerical issues such as contact treatment and numerical integration are addressed and finally numerical examples are shown to validate the method copyright 2007 john wiley sons ltd

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

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