一种新型裂尖加强函数的显式动态扩展有限元法

基于扩展有限元方法提出了一种新的裂尖加强函数,与传统三角函数基表征的加强函数相比,该裂尖加强函数通过组合传统的函数基,继承了传统附加函数的特性,同时使得结点的奇异附加自由度减少为2个,减少了总体劲度矩阵的规模,提高了计算效率。通过集中质量矩阵考虑结构的惯性效应,使用显式时间积分方法计算了含裂纹结构的瞬间受载问题,并应用相互作用积分得到裂尖端点处的动态应力强度因子。通过相关算例的对比分析,验证了所提出的裂尖加强函数的有效性,同时表明采用显式时间积分方法进行结构动态响应分析的可行性及准确性。     

Improved implementation and robustness study of the X‐FEM for stress analysis around cracks

Numerical crack propagation schemes were augmented in an elegant manner by the X‐FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X‐FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a ‘topological’ enrichment (only the elements touching the front are enriched). We suggest a ‘geometrical’ enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X‐FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that ‘off the shelf’ iterative solver packages can be used and perform as well on X‐FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X‐FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness. Copyright © 2005 John Wiley & Sons, Ltd.     

Cohesive and non-cohesive fracture by higher-order enrichment of XFEM

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.     

Higher order tip enrichment of eXtended Finite Element Method in thermoelasticity

An eXtended Finite Element Method (XFEM) is presented that can accurately predict the stress intensity factors (SIFs) for thermoelastic cracks. The method uses higher order terms of the thermoelastic asymptotic crack tip fields to enrich the approximation space of the temperature and displacement fields in the vicinity of crack tips—away from the crack tip the step function is used. It is shown that improved accuracy is obtained by using the higher order crack tip enrichments and that the benefit of including such terms is greater for thermoelastic problems than for either purely elastic or steady state heat transfer problems. The computation of SIFs directly from the XFEM degrees of freedom and using the interaction integral is studied. Directly computed SIFs are shown to be significantly less accurate than those computed using the interaction integral. Furthermore, the numerical examples suggest that the directly computed SIFs do not converge to the exact SIFs values, but converge roughly to values near the exact result. Numerical simulations of straight cracks show that with the higher order enrichment scheme, the energy norm converges monotonically with increasing number of asymptotic enrichment terms and with decreasing element size. For curved crack there is no further increase in accuracy when more than four asymptotic enrichment terms are used and the numerical simulations indicate that the SIFs obtained directly from the XFEM degrees of freedom are inaccurate, while those obtained using the interaction integral remain accurate for small integration domains. It is recommended in general that at least four higher order terms of the asymptotic solution be used to enrich the temperature and displacement fields near the crack tips and that the J- or interaction integral should always be used to compute the SIFs.     

Modeling crack discontinuities without element‐partitioning in the extended finite element method

In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

XFEM fracture analysis of shells: The effect of crack tip enrichments

In this study the effect of crack tip enrichment functions in the extended finite element analysis of shells is investigated. Utilization of crack tip enrichments leads to reduction of the required number of elements, mesh independency and increased accuracy in computation of fracture mechanics parameters such as the stress intensity factor, the crack tip opening displacement and the crack tip opening angle. The procedure is verified by modeling various shell and plate problems and available benchmark tests. Also, effects of enrichments of in-plane, out-of-plane and rotational degrees of freedom and high order out-of-plane enrichments on different fracture modes are studied. Moreover, reduction of the dependency of crack tip opening angle on the element size in crack propagation problems is discussed.     

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near‐tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack‐tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed‐mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M‐integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors. Copyright © 2006 John Wiley & Sons, Ltd.     

An extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials

We propose a formulation for linear elastic fracture mechanics in which the stress intensity factors are found directly from the solution vector of an extended boundary element method formulation. The enrichment is embedded in the boundary element method formulation, rather than adding new degrees of freedom for each enriched node. Therefore, a very limited number of new degrees of freedom is added to the problem, which contributes to preserving the conditioning of the linear system of equations. The Stroh formalism is used to provide boundary element method fundamental solutions for any degree of anisotropy, and these are used for both conventional and enriched degrees of freedom. Several numerical examples are shown with benchmark solutions to validate the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Stable 3D XFEM/vector level sets for non‐planar 3D crack propagation and comparison of enrichment schemes

We present a three‐dimensional vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting discontinuous linear enrichment functions in place of the asymptotic near‐tip functions. Through the vector level set method, level set values for propagating cracks are obtained via simple geometrical operations, eliminating the need for solution of differential evolution equations. The first XFEM variant ensures optimal convergence rates by means of geometrical enrichment, ie, the use of enriched elements in a fixed volume around the crack front, without giving rise to conditioning problems. The linear enrichment approach, significantly simplifies implementation and reduces the computational cost associated with numerical integration, while providing nonoptimal convergence rates similar to standard finite elements. The 2 dicretization schemes are tested for different benchmark problems, and their combination to the vector level set method is verified for nonplanar crack propagation problems.     

Modeling arbitrary crack propagation in coupled shell/solid structures with X‐FEM

In this paper, the extended finite element method (X‐FEM) formulation for the modeling of arbitrary crack propagation in coupled shell/solid structures is developed based on the large deformation continuum‐based (CB) shell theory. The main features of the new method are as follows: (1) different kinematic equations are derived for different fibers in CB shell elements, including the fibers enriched by shifted jump function or crack tip functions and the fibers cut into two segments by the crack surface or connecting with solid elements. So the crack tip can locate inside the element, and the crack surface is not necessarily perpendicular to the middle surface. (2) The enhanced CB shell element is developed to realize the seamless transition of crack propagation between shell and solid structures. (3) A revised interaction integral is used to calculate the stress intensity factor (SIF) for shells, which avoids that the auxiliary fields for cracks in Mindlin–Reissner plates cannot satisfy exactly the equilibrium equations. Several numerical examples, including the calculation of SIF for the cracked plate under uniform bending and crack propagation between solid and shell structures are presented to demonstrate the performance of the developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

二维压电材料动强度因子的扩展有限元计算

采用扩展有限元求解二维弹性压电材料动断裂问题。扩展有限元的网格独立于裂纹,因此网格生成可大大地简化,且裂纹扩展时不需重构网格。采用相互作用积分技术计算动强度因子。比较了标准的力裂尖加强函数和力-电裂尖加强函数对动强度因子的影响,结果表明标准的力裂尖加强函数能有效地分析压电材料动断裂问题。分析了极化方向对动强度因子的影响。数值分析表明采用扩展有限元获得的动强度因子与其他数值方法解吻合得很好。     

The extended finite element method with new crack‐tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X‐FEM) with new crack‐tip enrichment functions. In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and crack‐tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ? class and κ class, two classes of crack‐tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J‐integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.     

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

An enriched element‐failure method (REFM) for delamination analysis of composite structures

This paper develops an enriched element‐failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element‐failure concepts in the element‐failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near‐tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack‐tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the ‘failed elements’ that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re‐assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross‐ply laminated plate, which is observed in the experiment. Copyright © 2009 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.     

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.     

Time dependent crack tip enrichment for dynamic crack propagation

We study several enrichment strategies for dynamic crack propagation in the context of the extended finite element method and the effect of different directional criteria on the crack path. A new enrichment method with a time dependent enrichment function is proposed. In contrast to previous approaches, it entails only one crack tip enrichment function. Results for stress intensity factors and crack paths for different enrichments and direction criteria are given.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.