Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, 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 modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

几何非线性扩展有限元法及其断裂力学应用

该文将扩展有限元方法应用到几何非线性及断裂力学问题中,并研制开发了扩展有限元Fortran程序。扩展有限元法其计算网格与不连续面相互独立,因此模拟移动的不连续面时无需对网格进行重新剖分。该文推导了几何非线性扩展有限元法的公式,在常规有限元位移模式中,基于单位分解的思想加进一个阶跃函数和二维渐近裂尖位移场,反映裂纹处位移的不连续性,并用2个水平集函数表示裂纹;采用拉格朗日描述方程建立了有限变形几何非线性扩展有限元方程;采用多点位移外推法计算裂纹应力强度因子并通过最小二乘法拟合得到更精确的结果。最后给出的大变形算例表明该文提出的几何非线性的断裂力学扩展有限元方法和相应的计算机程序是合理可行的,而且对于含裂纹及裂纹扩展的问题,扩展有限元法优于传统的有限元法。     

eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces

An eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial differential equations defined on random domains. This method is based on a marriage between the eXtended Finite Element Method and spectral stochastic methods. In this article, we propose an extension of this method for the numerical simulation of random multi‐phased materials. The random geometry of material interfaces is described implicitly by using random level set functions. A fixed deterministic finite element mesh, which is not conforming to the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

New anisotropic crack-tip enrichment functions for the extended finite element method

In this paper, the extended finite element method (X-FEM) is implemented to analyze fracture mechanics problems in elastic materials that exhibit general anisotropy. In the X-FEM, crack modeling is addressed by adding discontinuous enrichment functions to the standard FE polynomial approximation within the framework of partition of unity. In particular, the crack interior is represented by the Heaviside function, whereas the crack-tip is modeled by the so-called crack-tip enrichment functions. These functions have previously been obtained in the literature for isotropic, orthotropic, piezoelectric and magnetoelectroelastic materials. In the present work, the crack-tip functions are determined by means of the Stroh’s formalism for fully anisotropic materials, thus providing a new set of enrichment functions in a concise and compact form. The proposed formulation is validated by comparing the obtained results with other analytical and numerical solutions. Convergence rates for both topological and geometrical enrichments are presented. Performance of the newly derived enrichment functions is studied, and comparisons are made to the well-known classical crack-tip functions for isotropic materials.     

Solving thermal and phase change problems with the eXtended finite element method

 The application of the eXtended finite element method (X-FEM) to thermal problems with moving heat sources and phase boundaries is presented. Of particular interest is the ability of the method to capture the highly localized, transient solution in the vicinity of a heat source or material interface. This is effected through the use of a time-dependent basis formed from the union of traditional shape functions with a set of evolving enrichment functions. The enrichment is constructed through the partition of unity framework, so that the system of equations remains sparse and the resulting approximation is conforming. In this manner, local solutions and arbitrary discontinuities that cannot be represented by the standard shape functions are captured with the enrichment functions. A standard time-projection algorithm is employed to account for the time-dependence of the enrichment, and an iterative strategy is adopted to satisfy local interface conditions. The separation of the approximation into classical shape functions that remain fixed in time and the evolving enrichment leads to a very efficient solution strategy. The robustness and utility of the method is demonstrated with several benchmark problems involving moving heat sources and phase transformations. Received 20 May 2001 / Accepted 19 December 2001     

Enriched finite elements and level sets for damage tolerance assessment of complex structures

The extended finite element method (X-FEM) has recently emerged as an alternative to meshing/remeshing crack surfaces in computational fracture mechanics thanks to the concept of discontinuous and asymptotic partition of unity enrichment (PUM) of the standard finite element approximation spaces. Level set methods have been recently coupled with X-FEM to help track the crack geometry as it grows. However, little attention has been devoted to employing the X-FEM in real-world cases. This paper describes how X-FEM coupled with level set methods can be used to solve complex three-dimensional industrial fracture mechanics problems through combination of an object-oriented (C++) research code and a commercial solid modeling/finite element package (EDS-PLM/I-DEAS^®). The paper briefly describes how object-oriented programming shows its advantages to efficiently implement the proposed methodology. Due to enrichment, the latter method allows for multiple crack growth scenarios to be analyzed with a minimal amount of remeshing. Additionally, the whole component contributes to the stiffness during the whole crack growth simulation. The use of level set methods permits the seamless merging of cracks with boundaries. To show the flexibility of the method, the latter is applied to damage tolerance analysis of a complex aircraft component.     

X-FEM implementation of VAMUCH: Application to active structural fiber multi-functional composite materials

In multiscale analysis of composite materials, there is usually a need to solve microstructures problems with complex geometries. The variational asymptotic method for unit cell homogenization (VAMUCH) is a recently developed variant of the asymptotic homogenization approach. In contrast to conventional asymptotic methods, VAMUCH carries out an asymptotic analysis of the variational statement, synthesizing the merits of both variational methods and asymptotic methods. This work gives an outline of the Extended Finite Element Method (X-FEM) implementation of VAMUCH for complex multi-material structures. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing an enrichment strategy. The X-FEM/VAMUCH approach is applied successfully to many examples reported in the VAMUCH literature. Numerical experiments on the periodic homogenization of complex unit cells demonstrate the accuracy and simplicity of the X-FEM/VAMUCH approach.     

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.     

Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method

A numerical technique for modeling fatigue crack propagation of multiple coplanar cracks is presented. The proposed method couples the extended finite element method (X-FEM) [Int. J. Numer. Meth. Engng. 48 (11) (2000) 1549] to the fast marching method (FMM) [Level Set Methods & Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, UK, 1999]. The entire crack geometry, including one or more cracks, is represented by a single signed distance (level set) function. Merging of distinct cracks is handled naturally by the FMM with no collision detection or mesh reconstruction required. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. In the X-FEM, 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 [Comput. Meth. Appl. Mech. Engng. 139 (1996) 289]. This enables the domain to be modeled by a single fixed finite element mesh with no explicit meshing of the crack surfaces. In an earlier study [Engng. Fract. Mech. 70 (1) (2003) 29], the methodology, algorithm, and implementation for three-dimensional crack propagation of single cracks was introduced. In this paper, simulations for multiple planar cracks are presented, with crack merging and fatigue growth carried out without any user-intervention or remeshing.     

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.     

A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique

In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of \(u{-}p\) formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.     

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.     

The generalized finite element method: an example of its implementation and illustration of its performance

The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright © 2000 John Wiley & Sons, Ltd.     

X-FEM simulation for two-unequal-collinear cracks in 2-D finite piezoelectric specimen

A study for two-unequal-collinear cracks in a 2-D finite piezoelectric specimen is carried out using a new set of six crack-tip enrichment functions proposed here for piezoelectric media in the X-FEM framework. The intensity factors and energy release rate are calculated using interaction integral in conjugation with the near tip behavior given by the Stroh formalism. Effect of finite size of the specimen is analyzed with respect to offset distances of the cracks from the specimen boundaries. ERR variations are investigated with respect to inter- crack space, crack lengths and electrical/mechanical loadings. Hence, two-unequal-collinear cracks in an infinite domain problem is simulated, analyzed and validated using X-FEM. Further, ERR at the crack tips for the asymmetric case of two collinear equal and unequal cracks, is also computed. It is concluded through this investigation that the proposed enrichment functions could be used to handle the problems of fracture mechanics in 2-D piezoelectric media within a good accuracy.     

Inclusion problem of a two-dimensional finite domain: The shape effect of matrix

With the advance in composite mechanics and micromechanics, there are increasing demands for analytical solutions of inclusion problems in a bounded domain. To echo this need, this study is focused on establishing explicit expressions of elastic fields for a 2D elastic domain containing a circular inclusion at center. Unlike the configuration in the classical Eshelby formulation, the elastic domain in this study is bounded and has shapes other than a circle. To circumvent the mathematical difficulty in solving Green’s function in a finite domain, an approach powered by complex potential method, which has been successfully employed to formulate the elastic fields for inclusion problems where matrix is unbounded or bounded by a circle, is extended to finite domains displaying complicated shapes, particularly, a Pascal’s limaçon and a curved square (an approximation of perfect square) in this study. In order to take advantage of the mathematical simplicity inherent in expressing a circular geometry, conformal mapping is used to transform the complex geometry of the finite domain of interest to a unit circle. The governing complex potentials, which capture the discontinuity on the inclusion–matrix interface due to the uniform eigenstrain within the inclusion, are formulated with the aid of Cauchy integral and then explicitly identified by satisfying the prescribed boundary conditions. In this study, the displacement fields for finite domains bounded by a Pascal’s limaçon and a curved square are obtained based on Dirichlet (displacement) boundary conditions imposed by the far field strain. In addition to asymptotical behaviors, firm agreement is also achieved when the analytical solutions based on complex potentials are compared with the FEM results. Furthermore, inverse of the conformal mapping is discussed here in order to get the explicit expression for elastic fields.     

A three‐dimensional computational procedure for reproducing meshless methods and the finite element method

A flexible computational procedure for solving 3D linear elastic structural mechanics problems is presented that currently uses three forms of approximation function (natural neighbour, moving least squares—using a new nearest neighbour weight function—and Lagrange polynomial) and three types of integration grids to reproduce the natural element method and the finite element method. The addition of more approximation functions, which is not difficult given the structure of the code, will allow reproduction of other popular meshless methods. Results are presented that demonstrate the ability of the first‐order meshless approximations to capture solutions more accurately than first‐order finite elements. Also, the quality of integration for the three types of integration grids is compared. The concept of a region is introduced, which allows the splitting of a domain into different sections, each with its own type of approximation function and spatial integration scheme. Copyright © 2004 John Wiley & Sons, Ltd.     

Variability response functions for apparent material properties

The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions.     

Image-based computational homogenization and localization: comparison between X-FEM/levelset and voxel-based approaches

In material science, images are increasingly used as input data for computational models. In most of the published papers, voxel-based finite element models are employed using a mesh that is automatically built by converting each voxel into a finite element. We have recently proposed (Legrain et al., Int J Numer Methods Eng 86(7): 915–934, 2011) another computational approach for incorporating images in models, based on the extended finite element method (X-FEM) and levelsets. Its main advantages are that the mesh does not need to conform to the geometry and that a smooth representation of physical surfaces is obtained. The aim of this paper is to compare the two approaches in the framework of computational homogenization in elasticity, starting from material microstructural images. Attention will be paid to geometrical approximations, macroscopic properties and local quantities (e.g. stress oscillations, local error etc.). It is shown that the X-FEM/levelset approach is more efficient than voxel-based FEM.