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

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

a p adaptive scheme for overcoming volumetric locking during plastic flow a feature of many classical plasticity yield functions is that they impose a volumetric constraint on plastic deformation for low order finite elements this results in kinematic restraints on possible deformation modes of an element leading to an overly stiff response the most robust procedure for overcoming volumetric locking is to use higher order finite elements however low order elements are popular for their efficiency and simplicity to remedy volumetric locking in low order elements a p adaptive scheme is proposed where through the partition of unity property of finite element shape functions the underlying basis of low order finite elements undergoing plastic flow is enhanced with higher order polynomial functions the computational result is extra degrees of freedom at existing nodes of elements undergoing plastic flow while the rest of the nodes remain unaltered the performance of the scheme is illustrated through two and three dimensional numerical examples

computational modeling of localized failure in solids xfem vs pf czm abstract this work addressesnumerical comparison between the extended finite element method xfem and the phase field regularized cohesive zone model pf czm wu 2017 2018a for the modeling of cohesive fracture induced localized failure in solids a novel implementation of the pf czm is proposed similarly to the xfem in which the added degrees of freedom characterizing the crack behavior are associated only with those nodes within a small sub domain representative benchmark examples show that both methods are independent of the mesh discretization for problems with the crack path known a priori both methods give almost coincident numerical results though the xfem is advantageous due to its high coarse mesh resolution for complex problems with arbitrary crack propagation even though some discrepancies are observed the numerical results given by both methods are also quantitatively similar if the predicted crack trajectories are close to each other despite the higher computational cost in this case the pf czm is much more favored due to its intrinsic capability of modeling complex crack configurations e g nucleation propagation merging and branching etc with no need of any extra strategy the above facts validate our previous theoretical analysis that the pf czm approaches asymptotically to a discontinuous czm for a vanishing length scale as it avoids most challenging issues exhibited by other approaches the pf czm is not only theoretically elegant but also numerically appealing for the modeling of localized failure in solids

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

an analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids abstract this paper presents an analysis of strong discontinuities in inelastic solids at finite strains solutions exhibiting this type of discontinuities characterized by a discontinuous displacement field are shown to make physical and mathematical sense in a classical multiplicative plasticity continuum model if the softening modulus is reinterpreted as a singular distribution physically the strainsoftening is localized along the discontinuity conditions for the appearance of strong discontinuities in the geometrically nonlinear range are characterized as it is the response of the material during localization in addition these analytical results are exploited in the design of a new class of finite element methods the proposed methods fall within the class of enhanced strain methods and lead to solutions independent of the mesh size and insensitive to mesh alignment without requiring any regularization of the solutions by numerical parameters like a characteristic length

an improved stable xfem is xfem with a novel enrichment function for the computational modeling of cohesive cracks abstract owing to the solution space enriched with a priori known knowledge through the partition of unity the extended or generalized finite element method xfem gfem has been widely applied to the modeling of localized failure in solids as far as cohesive cracks are concerned in this work on the one hand the xfem enriched with heaviside function or the shifted one with respect to nodal values is of optimal accuracy but is prone to the issue of ill conditioned or even singular system matrix for arbitrary crack propagation on the other hand the stable xfem s xfem circumvents the ill conditioning issues by introducing the residual heaviside function with respect to its linear interpolant at the cost of losing accuracy e g spurious stress locking for discontinuities with non uniform displacement jumps aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix this work addresses a simple yet effective improved stable xfem is xfem with a novel enrichment function both the xfem and s xfem are recovered as its particular cases with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale in the proposed is xfem as the enrichment basis functions are not contained in the standard polynomial space conditioning of the resulting system matrix is nearly insensitive to the mesh discontinuity configuration even if the crack path gets close to element nodes furthermore the is xfem with a small stabilization parameter is almost of the same accuracy as the xfem particularly pathological numerical results polluted by spurious stress locking exhibited in the s xfem are not observed across a fully softened discontinuity with non uniform displacement jumps as only the enrichment function is modified the proposed is xfem together with appropriate crack propagation criterion and tracking algorithm can be implemented easily in the existing xfem codes several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed is xfem regarding both accuracy of the solution and conditioning of the system matrix

extended isogeometric analysis for simulation of stationary and propagating cracks summary a novel approach based on a combination of isogeometric analysis iga and extended fem is presented for fracture analysis of structures the extended isogeometric analysis is capable of an efficient analysis of general crack problems using nonuniform rational b splines as basis functions for both the solution field approximation and the geometric description and it can reproduce crack tip singular fields and discontinuity across a crack iga has attracted a lot of interest for solving different types of engineering problems and is now further extended for the analysis of crack stability and propagation in two dimensional isotropic media concepts of the extended fem are used in iga to avoid the necessity of remeshing in crack propagation problems and to increase the solution accuracy around the crack tip crack discontinuity is represented by the heaviside function and isotropic analytical displacement fields near a crack tip are reproduced by means of the crack tip enrichment functions also the lagrange multiplier method is used to impose essential boundary conditions moreover the subtriangles technique is utilized for improving the accuracy of integration by the gauss quadrature rule several two dimensional static and quasi static crack propagation problems are solved to demonstrate the efficiency of the proposed method and the results of mixed mode stress intensity factors are compared with analytical and extended fem results copyright 2011 john wiley sons ltd

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

singularity enrichment for complete sliding contact using the partition of unity finite element method in this paper the numerical modelling of complete sliding contact and its associated singularity is carried out using the partition of unity finite element method sliding interfaces in engineering components lead to crack nucleation and growth in the vicinity of the contact zone to accurately capture the singular stress field at the contact corner we use the partition of unity framework to enrich the standard displacement based finite element approximation by additional enriched functions these enriched functions are derived from the analytical expression of the asymptotic displacement field in the vicinity of the contact corner to characterize the intensity of the singularity a domain integral formulation is adopted to compute the generalized stress intensity factor gsif numerical results on benchmark problems are presented to demonstrate the improved accuracy and benefits of this technique we conduct an investigation on issues pertaining to the extent of enrichment accurate numerical integration of weak form integrals and the rate of convergence in energy the use of partition of unity enrichment leads to accurate estimations of the gsifs on relatively coarse meshes which is particularly beneficial for modelling non linear sliding contacts copyright 2008 john wiley sons ltd

on a finite element method with variable element topology a new finite element like approximation method for problems in solid mechanics here called the variable element topology finite element method vetfem is presented the displacement based variational basis of the conventional finite element method fem is retained in the vetfem as is the discretization of the problem into elements however vetfem elements are not subject to any of the geometric or topological restrictions of conventional elements they may contain any number of nodes in any arrangement the polynomial vetfem shape functions emerge from a constrained minimization process on each element instead of from an isoparametric transformation from a parent element as in the conventional fem all the powerful features normally associated with the conventional fem are exhibited by the new method in addition because of the absence of geometric and or topological restrictions on the elements automatic mesh generation is enormously simplified for this reason the vetfem is thought to be particularly useful for problems involving very complex geometry adaptive remeshing or crack extension

the extended finite element method for rigid particles in stokes flow a new method for the simulation of particulate flows based on the extended finite element method x fem is described in this method the particle surfaces need not conform to the finite element boundaries so that moving particles can be simulated without remeshing the near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh we present a weak form of the equations of motion useful for the simulation of freely moving particles and solve example problems for particles with prescribed and unknown velocities copyright 2001 john wiley sons ltd

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

simulating curvilinear crack propagation in two dimensions with universal meshes summary we formulate a class of delicately controlled problems to model the kink free evolution of quasistatic cracks in brittle isotropic linearly elastic materials in two dimensions the evolving crack satisfies familiar principles griffith s criterion local symmetry and irreversibility a novel feature of the formulation is that in addition to the crack path the loading is also treated as an unknown specifically a scaling factor for prescribed dirichlet and neumann boundary conditions is computed as part of the solution to yield an always propagating and ostensibly kink free crack and a continuous loading history beyond the initial step a dimensionless statement of the problem depends only on the poisson s ratio of a homogeneous material and is in particular independent of its young s modulus and fracture toughness numerical resolution of the formulated problem relies on two new ideas the first is an algorithm to compute triangulations conforming to cracked domains by locally deforming a given background mesh in the vicinity of evolving cracks the algorithm is robust under mild assumptions on the sizes and angles of triangles near the crack and its smoothness hence a subset if not the entire family of cracked domains realized during the course of a simulation can be discretized with the same background mesh we term the latter a universal mesh for such a family of domains universal meshes facilitate adopting a discrete representation for the crack as splines in our examples preclude the need for local splitting retriangulation operations liberate the crack from following directions prescribed by the mesh and enable the adoption of standard fems to compute the elastic fields in the cracked solid second we employ a method specifically designed to approximate the stress intensity factors for curvilinear cracks we examine the performance of the resulting numerical method with detailed examples including comparisons with an exact solution of a crack propagating along a circular arc which we construct and comparisons with experimental fracture paths in all cases we observe convergence of computed paths their derivatives and loading histories with refinement of the universal mesh copyright 2014 john wiley sons ltd

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

modeling quasi static crack growth with the extended finite element method part i computer implementation abstract the extended finite element method x fem is a numerical method for modeling strong displacement as well as weak strain discontinuities within a standard finite element framework in the x fem special functions are added to the finite element approximation using the framework of partition of unity for crack modeling in isotropic linear elasticity a discontinuous function and the two dimensional asymptotic crack tip displacement fields are used to account for the crack this enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces and hence quasi static crack propagation simulations can be carried out without remeshing in this paper we discuss some of the key issues in the x fem and describe its implementation within a general purpose finite element code the finite element program dynaflow is considered in this study and the implementation for modeling 2 d cracks in isotropic and bimaterial media is described in particular the array allocation for enriched degrees of freedom use of geometric based queries for carrying out nodal enrichment and mesh partitioning and the assembly procedure for the discrete equations are presented we place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework

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

the exceptional lie algebra e7 25 multiplets and invariant differential operators in the present paper we continue the project of systematic construction of invariant differential operators on the example of the non compact exceptional algebra e7 25 our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras which we call conformal lie algebras which have very similar properties to the conformal algebras of n dimensional minkowski spacetime this class of algebras is identified and summarized in a table another motivation is related to the ads cft correspondence we give the multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators

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

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 with edge based strain smoothing esm xfem for linear elastic crack growth abstract this paper presents a strain smoothing procedure for the extended finite element method xfem the resulting edge based smoothed extended finite element method esm xfem is tailored to linear elastic fracture mechanics and in this context to outperform the standard xfem in the xfem the displacement based approximation is enriched by the heaviside and asymptotic crack tip functions using the framework of partition of unity this eliminates the need for the mesh alignment with the crack and re meshing as the crack evolves edge based smoothing es relies on a generalized smoothing operation over smoothing domains associated with edges of simplex meshes and produces a softening effect leading to a close to exact stiffness super convergence and ultra accurate solutions the present method takes advantage of both the es fem and the xfem thanks to the use of strain smoothing the subdivision of elements intersected by discontinuities and of integrating the singular derivatives of the approximation functions is suppressed via transforming interior integration into boundary integration numerical examples show that the proposed method improves significantly the accuracy of stress intensity factors and achieves a near optimal convergence rate in the energy norm even without geometrical enrichment or blending correction

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

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

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

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

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

finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions a new finite element formulation aimed at the solution of problems involving strain localization is presented the proposed formulation incorporates displacement interpolated embedded localization lines results are shown to converge to an exact solution when the mesh is refined and also to be quite insensitive to mesh distortions

enrichment strategies and convergence properties of the xfem for hydraulic fracture problems abstract in two recent papers gordeliy and peirce 2013 investigating the use of the extended finite element method xfem for modeling hydraulic fractures hf two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful xfem hf algorithms in this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies in addition we derive a novel set of crack tip enrichment functions that enable the xfem to model hf with the full range of power law r behavior of the displacement field and the corresponding r 1 singularity in the stress field for 1 2 1 this novel crack tip enrichment enables the xfem to achieve the optimal convergence rate which is not achieved by existing enrichment functions used for this range of power law the two xfem boundary value problem classes are as follows i a neumann to dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the xfem is used to solve for the corresponding crack width p w and ii a mixed hybrid formulation of the xfem that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it p w the two enrichment schemes considered are i the xfem t scheme with full singular crack tip enrichment and ii a simpler more efficient xfem s scheme in which the singular tip behavior is only imposed in a weak sense if enrichment is applied to all the nodes of tip enriched elements then the resulting xfem stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions which is a situation that also holds for the classic set of square root enrichment functions for the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix numerical experiments indicate that the xfem t scheme with the new tip enrichment achieves the optimal o h 2 convergence rate we expect of the underlying piece wise linear fem discretization which is superior to the enrichment functions currently available in the literature for 1 2 1 the xfem s scheme with only signum enrichment to represent the crack geometry achieves an o h convergence rate it is also demonstrated that the standard p w formulation based on the variational principle of minimum potential energy is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce while the mixed hybrid p w formulation based on the hellinger reissner variational principle does not have this disadvantage

partition of unity enrichment for bimaterial interface cracks partition of unity enrichment techniques are developed for bimaterial interface cracks a discontinuous function and the two dimensional near tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity this enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces the crack tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack the concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation the mixed mode complex stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized copyright 2004 john wiley sons ltd

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

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

a finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation abstract this paper presents a finite element model for strain localization analysis of elastoplastic solids subjected to discontinuous displacement fields based on standard galerkin approximation strain enhancements via jumps in the displacement field are captured and condensed on the material level leading to a formulation that does not require static condensation to be performed on the element level the mathematical formulation revolves around the dual response of a macroscopic point cut by a shear band which requires the satisfaction of the yield condition on the band as the same stress point unloads elastically just outside the band precise conditions for the appearance of slip lines including their initiation and evolution are outlined for a rate independent strain softening drucker prager model and explicit analytical expressions are used to describe the orientation of the slip line in a plane strain setting at post localization the stress point integration algorithm along the band is exact and amenable to consistent linearization numerical examples involving simple shearing of elastoplastic solids with deviatoric plastic flow as well as plane strain compression of dilatant cohesive frictional materials are presented to demonstrate absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions

plane strain finite element analysis of pressure sensitive plasticity with strong discontinuity numerical simulations of localized deformation in solids should capture the structural phenomenon of localization and associated loss of material body strength in a manner independent of spatial discretization many regularization techniques have been proposed to address the ill posedness associated with rate independent softening plasticity that leads to mesh dependent numerical simulations one approach to alleviating mesh dependence is the strong discontinuity approach which represents localized deformation as a slip surface within a plasticity model a strong discontinuity is a discontinuous displacement field this approach is used in this paper to formulate a plane strain pressure sensitive nonassociative plasticity model with strong discontinuity and to implement the model along with an enhanced bilinear quadrilateral element via an assumed enhanced strain method numerical examples demonstrate mesh independence of the method for pressure sensitive materials

a critical displacement approach for predicting structural instability abstract a new technique for predicting structural instability points using the finite element method is presented the approach is based on the estimation of the critical displacement pattern by writing an approximation of the tangent stiffness singularity condition at the instability point the critical load is subsequently computed by using a secant load displacement relationship details of this procedure are given together with explicit forms of the secant stiffness matrix for finite element analysis of solids and trusses the accuracy and effectiveness of the method are clearly shown in a number of examples of two and three dimensional bar structures

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

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

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