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.     

Error controlled adaptive multiscale XFEM simulation of cracks

The accurate and efficient prediction of the interaction of microcracks with macrocracks has been a challenge for many years. In this paper a discretization error controlled adaptive multiscale technique for the accurate simulation of microstructural effects within a macroscopic component is presented. The simulation of cracks is achieved using the corrected XFEM. The error estimation procedure is based on the well known Zienkiewicz and Zhu method extended to the XFEM for cracks such that physically meaningful stress irregularities and non-smoothnesses are accurately reflected. The incorporation of microstructural features such as microcracks is achieved by means of the multiscale projection method. In this context an error controlled adaptive mesh refinement is performed on the fine scale where microstructural effects may lead to highly complex mechanical behavior. The presented method is applied to a few examples showing its validity and applicability to arbitrary problems within fracture mechanics.     

Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods

In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patch‐based version of the DG formulation is developed, which decomposes the domain into enriched and unenriched subdomains. Continuity between patches is enforced with an internal penalty method. An element‐based form is also developed, where each element is considered a patch. The patch‐based DG is shown to have similar accuracy to the element‐based DG for a given discretization but requires significantly fewer degrees of freedom. The method is applied to material interfaces, cracks and dislocation problems. For the dislocations, a contour integral form of the internal forces that only requires integration over the patch boundaries is developed. A previously developed assumed strain (AS) method is also developed further and compared with the DG method for weak discontinuities and linear elastic cracks. The DG method is shown to be significantly more accurate than the standard XFEM for a given element size and to converge optimally, even where the standard XFEM does not. The accuracy of the DG method is similar to that of the AS method but requires less application‐specific coding. Copyright © 2007 John Wiley & Sons, Ltd.     

Crack face contact for a hexahedral-based XFEM formulation

Taking into account arbitrary crack geometries, crack closure generally occurs independently of the load case. As the standard eXtended Finite Element Method (XFEM) does not prevent unphysical crack face penetration in this case, a formulation allowing for crack face contact is proposed in terms of a penalty formulation for normal contact. The discretization is developed for non-planar cracks intersecting hexahedral elements in an arbitrary manner. Typical problems of many crack face contact implementations within the XFEM, like locking or the introduction of additional degrees of freedom, are avoided by projecting the contact contribution onto the hexahedral element nodes. The method is tested by means of suitable numerical examples, finally presenting an application in form of a multiscale setup with arbitrarily arranged micro cracks in the vicinity of a macro crack front.     

Extended finite element method coupled with face‐based strain smoothing technique for three‐dimensional fracture problems

In this work, the extended finite element method (XFEM) is for the first time coupled with face‐based strain‐smoothing technique to solve three‐dimensional fracture problems. This proposed method, which is called face‐based smoothed XFEM here, is expected to combine both the advantages of XFEM and strain‐smoothing technique. In XFEM, arbitrary crack geometry can be modeled and crack advance can be simulated without remeshing. Strain‐smoothing technique can eliminate the integration of singular term over the volume around the crack front, thanks to the transformation of volume integration into area integration. Special smoothing scheme is implemented in the crack front smoothing domain. Three examples are presented to test the accuracy, efficiency, and convergence rate of the face‐based smoothed XFEM. From the results, it is clear that smoothing technique can improve the performance of XFEM for three‐dimensional fracture problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Complete Tangent Stiffness for eXtended Finite Element Method by including crack growth parameters

The eXtended Finite Element Method (XFEM) is a useful tool for modeling the growth of discrete cracks in structures made of concrete and other quasi‐brittle and brittle materials. However, in a standard application of XFEM, the tangent stiffness is not complete. This is a result of not including the crack geometry parameters, such as the crack length and the crack direction directly in the virtual work formulation. For efficiency, it is essential to obtain a complete tangent stiffness. A new method in this work is presented to include an incremental form the crack growth parameters on equal terms with the degrees of freedom in the FEM‐equations. The complete tangential stiffness matrix is based on the virtual work together with the constitutive conditions at the crack tip. Introducing the crack growth parameters as direct unknowns, both equilibrium equations and the crack tip criterion can be handled within the same standard nonlinear iterations. This new solution strategy is believed to provide the modeling capabilities to deal with simultaneous growth of several cracks. A cohesive crack modeling is used. The method is applied to a partly cracked XFEM element of linear strain triangle type with the crack length as the unknown crack growth parameter. In this paper, two examples are given. The first example verifies the theory and the implementation. The second example is the benchmark test three point bending test, where the efficiency of the complete tangential behavior is shown. Copyright © 2013 John Wiley & Sons, Ltd.     

Incremental‐secant modulus iteration scheme and stress recovery for simulating cracking process in quasi‐brittle materials using XFEM

In this paper, an incremental‐secant modulus iteration scheme using the extended/generalized finite element method (XFEM) is proposed for the simulation of cracking process in quasi‐brittle materials described by cohesive crack models whose softening law is composed of linear segments. The leading term of the displacement asymptotic field at the tip of a cohesive crack (which ensures a displacement discontinuity normal to the cohesive crack face) is used as the enrichment function in the XFEM. The opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation. A statically admissible stress recovery (SAR) technique is extended to cohesive cracks with special treatment of non‐homogeneous boundary tractions. The application of locally normalized co‐ordinates to eliminate possible ill‐conditioning of SAR, and the influence of different weight functions on SAR are also studied. Several mode I cracking problems in quasi‐brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme, as well as the characteristics of global responses and local fields obtained numerically by the XFEM. Copyright © 2006 John Wiley & Sons, Ltd.     

Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

This paper presents a new stress recovery technique for the generalized/extended finite element method (G/XFEM) and for the stable generalized FEM (SGFEM). The recovery procedure is based on a locally weighted L² projection of raw stresses over element patches; the set of elements sharing a node. Such projection leads to a block-diagonal system of equations for the recovered stresses. The recovery procedure can be used with GFEM and SGFEM approximations based on any choice of elements and enrichment functions. Here, the focus is on low-order 2D approximations for linear elastic fracture problems. A procedure for computing recovered stresses at re-entrant corners of any internal angle is also presented. The proposed stress recovery technique is used to define a Zienkiewicz-Zhu (ZZ) a posteriori error estimator for the G/XFEM and the SGFEM. The accuracy, computational cost, and convergence rate of recovered stresses together with the quality of the ZZ estimator, including its effectivity index, are demonstrated in problems with smooth and singular solutions.     

求解双材料界面裂纹应力强度因子的扩展有限元法

基于双材料界面裂纹尖端的基本解,构造扩展有限元法(eXtended Finite Element Methods, XFEM)裂尖单元结点的改进函数。有限元网格剖分不遵从材料界面,考虑3种类型的结点改进函数：弱不连续改进函数、Heaviside改进函数和裂尖改进函数,建立XFEM的位移模式,给出计算双材料界面裂纹应力强度因子(Stress Intensity Factors, SIFs)的相互作用积分方法。数值结果表明：XFEM无需遵从材料界面剖分网格,该文的方法能够准确评价双材料界面裂纹尖端的SIFs。     

Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions

Adaptive local refinement is one of the main issues for isogeometric analysis (IGA). In this paper, an adaptive extended IGA (XIGA) approach based on polynomial splines over hierarchical T‐meshes (PHT‐splines) for modeling crack propagation is presented. The PHT‐splines overcome certain limitations of nonuniform rational B‐splines–based formulations; in particular, they make local refinements feasible. To drive the adaptive mesh refinement, we present a recovery‐based error estimator for the proposed method. The method is based on the XIGA method, in which discontinuous enrichment functions are added to the IGA approximation and this method does not require remeshing as the cracks grow. In addition, crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors under linear elastic fracture mechanics. The proposed method has been used to analyze numerical examples, and the stress intensity factors results were compared with reference results. The findings demonstrate the accuracy and efficiency of the proposed method.     

Evaluation of T‐stresses in multiple crack problems of finite plate

This paper provides a solution for T‐stresses for multiple cracks in a finite plate. The results for stress intensity factors (SIFs) are also presented. The case of two cracks in a rectangular plate is taken as an example. In the problem, the crack faces are applied by some loadings, and tractions are free along edges of a rectangular plate. The whole stress field is considered as a superposition of three particular stress fields. The first and second stress fields are initiated by loadings on the first and second crack faces in an infinite plate. The third field is chosen in a polynomial form of complex potentials. After discretization, the loadings on two cracks and the undetermined coefficients in the complex potentials become the unknowns. The relevant algebraic equations are formulated. The solution of algebraic equations will lead to the results of SIFs and T‐stresses at the crack tips. Several numerical examples are presented, which were not reported previously.     

A constitutive relation error estimator based on traction‐free recovery of the equilibrated stress

A new methodology for recovering equilibrated stress fields is presented, which is based on traction‐free subdomains' computations. It allows a rather simple implementation in a standard finite element code compared with the standard technique for recovering equilibrated tractions. These equilibrated stresses are used to compute a constitutive relation error estimator for a finite element model in 2D linear elasticity. A lower bound and an upper bound for the discretization error are derived from the error in the constitutive relation. These bounds in the discretization error are used to build lower and upper bounds for local quantities of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

An explicit residual‐type error estimator for ‐quadrilateral extended finite element method in two‐dimensional linear elastic fracture mechanics

This paper deals with explicit residual a posteriori error estimation analysis for ‐quadrilateral extended finite element method (XFEM) discretizations applied to the two‐dimensional problem of linear elastic fracture mechanics. The result is twofold. First, to enable estimation procedures with application to XFEM, a specific quasi‐interpolation operator of averaging type is constructed. The main challenge here arises from the different types of enrichments implemented, and hence, to impose the constant‐preserving property of the interpolation operator on an element, we use the idea of an extension operator. An upper bound on the discretization error measured in the energy norm and associated local error indicators are then constructed and analyzed. The second result follows from the error analysis and concerns an alternative choice of branch functions used in XFEM applications. In particular, the branch functions have to be chosen to fulfill the divergence‐free conditions within the crack tip element and traction‐free boundary conditions on the crack faces. Then, the corresponding XFEM solution gains a better accuracy with less degrees of freedom. Finally, numerical examples are provided with comparative results. Copyright © 2012 John Wiley & Sons, Ltd.     

Crack identification by ‘arrival time’ using XFEM and a genetic algorithm

A computational framework is developed in which cracks in two‐dimensional structures are identified, in conjunction with non‐destructive testing of specimens. As opposed to a previous study by the authors, which was based on time‐harmonic excitation with a single frequency, here the transient response of the structure to a short‐duration signal is measured along part of the external boundary. Crack detection is performed using the solution of an inverse time‐dependent problem. It is shown that the arrival time of the input signal to the points of measurement is a good criterion for crack identification in the time domain. The inverse problem of identification is solved using a genetic algorithm, while each forward problem is solved by the time‐dependent extended finite element method (XFEM). The XFEM scheme is efficient in that it allows the use of a single regular mesh for a large number of forward time response problems with different crack geometries. Numerical examples involving a crack in a flat membrane are presented. Identification based on ‘arrival time’ is shown to perform better than that based on time‐harmonic response. Copyright © 2008 John Wiley & Sons, Ltd.     

A consistent partly cracked XFEM element for cohesive crack growth

Present extended finite element method (XFEM) elements for cohesive crack growth may often not be able to model equal stresses on both sides of the discontinuity when acting as a crack‐tip element. The authors have developed a new partly cracked XFEM element for cohesive crack growth with extra enrichments to the cracked elements. The extra enrichments are element side local and were developed by superposition of the standard nodal shape functions for the element and standard nodal shape functions for a sub‐triangle of the cracked element. With the extra enrichments, the crack‐tip element becomes capable of modelling variations in the discontinuous displacement field on both sides of the crack and hence also capable of modelling the case where equal stresses are present on each side of the crack. The enrichment was implemented for the 3‐node constant strain triangle (CST) and a standard algorithm was used to solve the non‐linear equations. The performance of the element is illustrated by modelling fracture mechanical benchmark tests. Investigations were carried out on the performance of the element for different crack lengths within one element. The results are compared with previously obtained XFEM results applying fully cracked XFEM elements, with computational results achieved using standard cohesive interface elements in a commercial code, and with experimental results. The suggested element performed well in the tests. Copyright © 2007 John Wiley & Sons, Ltd.     

An improvement of the EDI method in linear elastic fracture mechanics by means of an a posteriori error estimator in G

In this paper, an error estimator that quantifies the effect of the finite element discretization error on the computation of the stress intensity factor in linear elastic fracture mechanics is presented. In order to obtain the proposed estimator, a shape design sensitivity analysis (SDSA) is applied to the fracture mechanics problem. Following this approach, one of the most efficient post‐processing techniques for computing the strain energy release rate G, the well‐known EDI method, may be interpreted as a continuum method of the SDSA. The proposed error estimator is based on the recovery of the gradient fields and its reliability has been checked by means of numerical problems, yielding very good estimations of the true error. The new estimator remarkably improves the results given by a previous error estimator, which is based on a discrete analytical approach of SDSA. As a consequence, the combination of the new error estimator and the result given by the EDI method provides a much more accurate estimation of G. Copyright © 2003 John Wiley & Sons, Ltd.     

模拟三维裂纹问题的自适应多尺度扩展有限元法

为了在大型结构分析中考虑小裂纹或以小的代价提高裂纹附近求解精度,该文建立了分析三维裂纹问题的自适应多尺度扩展有限元法。基于恢复法评估三维扩展有限元后验误差,大于给定误差值的单元进行细化。所有尺度单元采用八结点六面体单元,采用六面体任意结点单元连接不同尺度单元。采用互作用积分法计算三维应力强度因子。三维I 型裂纹和I-II 复合型裂纹算例分析表明了该方法的正确性和有效性。     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

Combined extended and superimposed finite element method for cracks

A combination of the extended finite element method (XFEM) and the mesh superposition method (s‐version FEM) for modelling of stationary and growing cracks is presented. The near‐tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity. The two displacement fields are matched through a transition region. The method can robustly deal with stationary crack and crack growth. It simplifies the numerical integration of the weak form in the Galerkin method as compared to the s‐version FEM. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

3D adaptive finite element modeling of non-planar curved crack growth using the weighted superconvergent patch recovery method

In this paper, an adaptive finite element analysis is presented for 3D modeling of non-planar curved crack growth. The fracture mechanical evaluation is performed based on a general technique for non-planar curved cracks. The Schollmann’s crack kinking criterion is used for the process of crack propagation in 3D problems. The Zienkiewicz-Zhu error estimator is employed in conjunction with a weighted SPR technique at each patch to improve the accuracy of error estimation. Applying the proposed technique to 3D non-planar curved crack growth problems shows significant improvements particularly at the boundaries and near crack tip regions. Several numerical examples are presented to illustrate the robustness of the proposed technique.