Error estimation for crack simulations using the XFEM

The extended finite element method (XFEM) is by now well‐established for crack calculations in linear elastic fracture mechanics. An advantage of this method is its discretization independence for crack simulations. Nevertheless, discretization errors occur when using the XFEM. In this paper, a simple recovery based error estimator for the discretization error in XFEM‐calculations for cracks is presented. The method is based on the Zienkiewicz and Zhu error estimator. Enhanced smoothed stresses incorporating the discontinuities and singularities because of the cracks are recovered to enable the error estimation for arbitrary distributed cracks. This approach also allows the consideration of materials with generally inelastic behaviour. The enhanced stresses are computed by means of a least square fit problem. To assess the quality of the error estimator, global norms and the effectivity index for the global energy norm for examples with known analytical solutions are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

An XFEM model for cracked porous media: effects of fluid flow and heat transfer

In this work, a numerical model is developed to investigate the influence of fluid flow and heat transfer on the thermo-mechanical response of a cracked porous media. The fluid flow, governed by the Darcy’s law, is discretized with the nonconforming finite element method. Time splitting is used with the energy conservation equation to solve the fluid and the solid phases separately. A combination of Discontinuous Galerkin (DG) and multi-point flux approximation methods is used to solve the advection-diffusion heat transfer equation in the fluid phase. While the conductive heat transfers equation in the solid phase is solved using the eXtended finite element method (XFEM) to better handle the temperature discontinuities and singularities caused by the cracks. Further, the resulted temperature is used as body force to solve the thermo-mechanical problem using the XFEM. In the post processing stage, the thermal stress intensity factor is computed using the interaction integral technique at each time step and used to validate the obtained results. A good agreement was found when the results were compared with the existing ones in the literature.     

The extended/generalized finite element method: An overview of the method and its applications

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non‐smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non‐smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi‐field problems. Copyright © 2010 John Wiley & Sons, Ltd.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.     

Fatigue life prediction of cracked attachment lugs using XFEM

In the present paper, a three dimensional finite element method (FEM) is used to compute the stress intensity factor (SIF) in straight lugs of Aluminum 7075-T6. Extended finite element method (XFEM) capability available in ABAQUS is used to calculate the stress intensity factor. Crack growth and fatigue life of single through-thickness and single quarter elliptical corner cracks in attachment lug are estimated and then compared with the available experimental data for two different load ratios equal to 0.1 and 0.5. The SIF calculated from XFEM shows that the introduction of different loading boundary conditions significantly affect the estimated fatigue life.     

Numerical simulation of bi-material interfacial cracks using EFGM and XFEM

In this paper, bi-material interfacial cracks have been simulated using element free Galerkin method (EFGM) and extended finite element method (XFEM) under mode-I and mixed mode loading conditions. Few crack interaction problems of dissimilar layered materials are also simulated using extrinsic partition of unity enriched approach. Material discontinuity has been modeled by a signed distance function whereas strong discontinuity has been modeled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The results obtained by EFGM and XFEM for bi-material edge and center cracks are compared with those available in literature. In order to check the validity of simulations, the results have been obtained for two different ratio of Young’s modulus.     

基于XFEM的强震区砼重力坝开裂与配筋抗震措施研究

已有震害表明,混凝土坝遭遇强烈地震将不可避免地产生开裂。扩展有限元法（XFEM）通过在相关节点的影响域上富集非连续位移模式,使得对非连续位移场的表征独立于单元边界,可以有效描述混凝土中的裂纹扩展。基于扩展有限元模型,采用合理的地震波动模型对国内某混凝土重力坝强震下的动力破坏过程进行了分析;针对大坝破坏情况,应用嵌入式滑移模型模拟了混凝土重力坝配筋前后的地震响应和破坏状况,据此评价局部配筋的抗震效果。研究表明,局部配置抗震钢筋虽无法防止裂缝的发生,但可有效限制坝体裂缝的开裂扩展范围及深度,减少裂缝的开度,有效改善坝体的抗震性能。     

Fatigue Investigations on Steel Pipeline Containing 3D Coplanar and Non-Coplanar Cracks

Fluctuated loadings from currents, waves and sea ground motions are observed on offshore steel pipelines, and they will result in small cracks to propagate continuously and cause unexpected damage to offshore/geotechnical infrastructures. In spite of the availability of efficient techniques and high-power computers for solving crack problems, investigations on the fatigue life of offshore pipelines with 3D interacting cracks are still rarely found in open literature. In the current study, systematic numerical investigations are performed on fatigue crack growth behaviours of offshore pipelines containing coplanar and non-coplanar cracks. Extended finite element method (XFEM) is adopted to simulate the fatigue crack growth. The qualitative validations of numerical results are made for certain cases with available experimental results. Parametric studies are conducted to investigate the influences of various important parameters on fatigue crack growth. The results will be helpful to assess the fatigue behaviours of steel pipeline with 3D interacting cracks.     

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。     

A stabilized discrete shear gap extended finite element for the analysis of cracked Reissner–Mindlin plate vibration problems involving distorted meshes

In this work, a new approach in the framework of the local partition of unity finite element method (XFEM) to significantly improve the accuracy of natural frequency in free vibration analysis of cracked Reissner–Mindlin plates is presented. Different from previous approaches, the present formulation is expected to be more accurate and effective in modeling cracked plates by integrating the stabilized discrete shear gap (DSG) into the XFEM setting. Intensive numerical results at low frequency demonstrated that the novel DSG-based XFEM approach possesses the following desirable properties: (1) the awkwardness of transverse shear-locking phenomenon can be overcome easily; (2) the DSG-based XFEM can be applicable to both moderately thick and thin plates straightforwardly; (3) the representation of cracks is independent of finite element mesh; (4) mesh distortion is insensitive and controllable; (5) the accuracy of natural frequency obtained by the present method is high and (6) the present method uses three-node triangular elements that can be much easily generated automatically for problems even with complicated geometry. These properties of the DSG-based XFEM are confirmed through several numerical examples of cracked plates with different boundary conditions.     

基于X-SBFEM的裂纹体非网格重剖分耦合模型研究

扩展有限元法利用了非网格重剖分技术,但需要基于裂尖解析解构造复杂的插值基函数,计算精度受网格疏密和插值基函数等因素影响。比例边界有限元法则在求解无限域和裂尖奇异性问题优势明显,两者衔接于有限元法理论内,可建立一种结合二者优势的断裂耦合数值模型。该文从虚功原理出发,利用位移协调与力平衡机制,提出了一种断裂计算的新方法X-SBFEM,达到了扩展有限元模拟裂纹主体、比例边界有限元模拟裂尖的目的。在数值算例中,通过边裂纹和混合型裂纹的应力强度因子计算,并与理论解对比,验证了该方法的准确性和有效性。     

On time integration in the XFEM

The extended finite element method (XFEM) is often used in applications that involve moving interfaces. Examples are the propagation of cracks or the movement of interfaces in two‐phase problems. This work focuses on time integration in the XFEM. The performance of the discontinuous Galerkin method in time (space–time finite elements (FEs)) and time‐stepping schemes are analyzed by convergence studies for different model problems. It is shown that space–time FE achieve optimal convergence rates. Special care is required for time stepping in the XFEM due to the time dependence of the enrichment functions. In each time step, the enrichment functions have to be evaluated at different time levels. This has important consequences in the quadrature used for the integration of the weak form. A time‐stepping scheme that leads to optimal or only slightly sub‐optimal convergence rates is systematically constructed in this work. Copyright © 2009 John Wiley & Sons, Ltd.     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

Topology optimization of geometrically nonlinear structures using an evolutionary optimization method

The topology optimization using isolines/isosurfaces and extended finite element method (Iso-XFEM) is an evolutionary optimization method developed in previous studies to enable the generation of high-resolution topology optimized designs suitable for additive manufacture. Conventional approaches for topology optimization require additional post-processing after optimization to generate a manufacturable topology with clearly defined smooth boundaries. Iso-XFEM aims to eliminate this time-consuming post-processing stage by defining the boundaries using isovalues of a structural performance criterion and an extended finite element method (XFEM) scheme. In this article, the Iso-XFEM method is further developed to enable the topology optimization of geometrically nonlinear structures undergoing large deformations. This is achieved by implementing a total Lagrangian finite element formulation and defining a structural performance criterion appropriate for the objective function of the optimization problem. The Iso-XFEM solutions for geometrically nonlinear test cases implementing linear and nonlinear modelling are compared, and the suitability of nonlinear modelling for the topology optimization of geometrically nonlinear structures is investigated.     

Predicting crack propagation with peridynamics: a comparative study

The fidelity of the peridynamic theory in predicting fracture is investigated through a comparative study. Peridynamic predictions for fracture propagation paths and speeds are compared against various experimental observations. Furthermore, these predictions are compared to the previous predictions from extended finite elements (XFEM) and the cohesive zone model (CZM). Three different fracture experiments are modeled using peridynamics: two experimental benchmark dynamic fracture problems and one experimental crack growth study involving the impact of a matrix plate with a stiff embedded inclusion. In all cases, it is found that the peridynamic simulations capture fracture paths, including branching and microbranching that are in agreement with experimental observations. Crack speeds computed from the peridynamic simulation are on the same order as those of XFEM and CZM simulations. It is concluded that the peridynamic theory is a suitable analysis method for dynamic fracture problems involving multiple cracks with complex branching patterns.     

Applications of normal stress dominated cohesive zone models for mixed-mode crack simulation based on extended finite element methods

In conventional cohesive zone models the traction-separation law starts from zero load, so that the model cannot be applied to predict mixed-mode cracking. In the present work the cohesive zone model with a threshold is introduced and applied for simulating different mixed-mode cracks in combining with the extended finite element method. Computational results of cracked specimens show that the crack initiation and propagation under mixed-mode loading conditions can be characterized by the cohesive zone model for normal stress failure. The contribution of the shear stress is negligible. The maximum principal stress predicts crack direction accurately. Computations based on XFEM agree with known experiments very well. The shear stress becomes, however, important for uncracked specimens to catch the correct crack initiation angle. To study mixed-mode cracks one has to introduce a threshold into the cohesive law and to implement the new cohesive zone based on the fracture criterion. In monotonic loading cases it can be easily realized in the extended finite element formulation. For cyclic loading cases convergence of the inelastic computations can be critical.     

Dynamic crack propagation analysis of orthotropic media by the extended finite element method

Dynamic crack propagation of composites is investigated in this paper based on the recent advances and development of orthotropic enrichment functions within the framework of partition of unity and the extended finite element method (XFEM). The method allows for analysis of the whole crack propagation pattern on an unaltered finite element mesh, defined independent of the existence of any predefined crack or its propagation path. A relatively simple, though efficient formulation is implemented, which consists of using a dynamic crack initiation toughness, a crack orientation along the maximum circumferential stress, and a simple equation to presume the crack speed. Dynamic stress intensity factors (DSIFs) are evaluated by means of the domain separation integral method. The governing elastodynamics equation is first transformed into a standard weak formulation and is then discretized into an XFEM system of time dependent equations, to be solved by the unconditionally stable Newmark time integration scheme. A number of benchmark and test problems are simulated and the results are compared with available reference results.     

Extended finite element fracture analysis of a cracked isotropic shell repaired by composite patch

An extended finite element method (XFEM) is developed to study fracture parameters of cracked metal plates and tubes that are repaired on top of the crack with a composite patch. A MATLAB® stand‐alone code is prepared to model such structures with eight‐noded doubly curved shell elements in the XFEM framework. Crack trajectory studies are performed for a diagonally cracked panel under fatigue loading. Verification studies are investigated on different shell type structures such as a cracked spherical shell and cracked cylindrical pipe with different crack orientations. The effects of using patch repairs with different fibre orientations on the reduction of stress intensity factors (SIFs) is also studied which can be useful for design purposes. XFEM is selected as any crack geometry can be embedded in the finite element mesh configuration with no need to coincide the crack geometry with meshed elements and so re‐meshing with fine mesh generation is not needed in the current method.     

摩擦接触裂纹问题的扩展有限元法

扩展有限元法(XFEM)是一种在常规有限元框架内求解强和弱不连续问题的新型数值方法。扩展有限元法分析闭合型裂纹时,必须考虑裂纹面间的接触问题。已有文献均采用迭代法求解裂纹面的接触问题。该文建立了闭合型摩擦裂纹问题的扩展有限元线性互补模型,将裂纹面非线性摩擦接触转化为一个线性互补问题求解,不需要迭代求解。算例分析说明了该方法的正确性和有效性,同时表明扩展有限元法结合线性互补法求解接触问题具有较好的前景。     

The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method

In this paper, a novel geometric method combined with the piecewise linear function method is introduced into the extended finite element method (XFEM) to determine the crack tip element and crack surface element. Then, by combining with the advanced mesh technique, a novel method is proposed to improve the modelling of crack propagation in triangular 2D structure with the XFEM. The numerical tests show that the accuracy, the convergence, and the stability of the XFEM can be improved using the proposed method. Moreover, the applicability of the conventional multiple enrichment scheme is discussed. Compared with the proposed method, the conventional multiple enrichment scheme has deficiency in mixed mode I and II crack. Finally, a comparative study shows that the performance of the XFEM by using the proposed method to model the crack propagation can be greatly improved.