Showing papers on "Extended finite element method published in 2008"

A comparative study on finite element methods for dynamic fracture

Abstract: The performance of finite element methods for dynamic crack propagation in brittle materials is studied. Three methods are considered: the extended finite element method (XFEM), element deletion method and interelement crack method. The extended finite element method is a method for arbitrary crack propagation without remeshing. In element deletion methods, elements that meet a fracture criterion are deleted. In interelement crack methods, the crack is limited to element edges; the separation of these edges is governed by a cohesive law. We show that XFEM and interelement method show similar crack speeds and crack paths. However, both fail to predict a benchmark experiment without adjustment of the energy release rate. The element deletion method performs very poorly for the refinements studied, and is unable to predict crack branching.

Extended finite element method for fracture analysis of structures

Abstract: Dedication. Preface . Nomenclature . Chapter 1 Introduction. 1.1 ANALYSIS OF STRUCTURES. 1.2 ANALYSIS OF DISCONTINUITIES. 1.3 FRACTURE MECHANICS. 1.4 CRACK MODELLING. 1.4.1 Local and non-local models. 1.4.2 Smeared crack model. 1.4.3 Discrete inter-element crack. 1.4.4 Discrete cracked element. 1.4.5 Singular elements. 1.4.6 Enriched elements. 1.5 ALTERNATIVE TECHNIQUES. 1.6 A REVIEW OF XFEM APPLICATIONS. 1.6.1 General aspects of XFEM. 1.6.2 Localisation and fracture. 1.6.3 Composites. 1.6.4 Contact. 1.6.5 Dynamics. 1.6.6 Large deformation/shells. 1.6.7 Multiscale. 1.6.8 Multiphase/solidification. 1.7 SCOPE OF THE BOOK. Chapter 2 Fracture Mechanics, a Review. 2.1 INTRODUCTION. 2.2 BASICS OF ELASTICITY. 2.2.1 Stress-strain relations. 2.2.2 Airy stress function. 2.2.3 Complex stress functions. 2.3 BASICS OF LEFM. 2.3.1 Fracture mechanics. 2.3.2 Circular hole. 2.3.3 Elliptical hole. 2.3.4 Westergaard analysis of a sharp crack. 2.4 STRESS INTENSITY FACTOR, K . 2.4.1 Definition of the stress intensity factor. 2.4.2 Examples of stress intensity factors for LEFM. 2.4.3 Griffith theories of strength and energy. 2.4.4 Brittle material. 2.4.5 Quasi-brittle material. 2.4.6 Crack stability. 2.4.7 Fixed grip versus fixed load. 2.4.8 Mixed mode crack propagation. 2.5 SOLUTION PROCEDURES FOR K AND G . 2.5.1 Displacement extrapolation/correlation method. 2.5.2 Mode I energy release rate. 2.5.3 Mode I stiffness derivative/virtual crack model. 2.5.4 Two virtual crack extensions for mixed mode cases. 2.5.5 Single virtual crack extension based on displacement decomposition. 2.5.6 Quarter point singular elements. 2.6 ELASTOPLASTIC FRACTURE MECHANICS (EPFM). 2.6.1 Plastic zone. 2.6.2 Crack tip opening displacements (CTOD). 2.6.3 J integral. 2.6.4 Plastic crack tip fields. 2.6.5 Generalisation of J . 2.7 NUMERICAL METHODS BASED ON THE J INTEGRAL. 2.7.1 Nodal solution. 2.7.2 General finite element solution. 2.7.3 Equivalent domain integral (EDI) method. 2.7.4 Interaction integral method. Chapter 3 Extended Finite Element Method for Isotropic Problems. 3.1 INTRODUCTION. 3.2 A REVIEW OF XFEM DEVELOPMENT. 3.3 BASICS OF FEM. 3.3.1 Isoparametric finite elements, a short review. 3.3.2 Finite element solutions for fracture mechanics. 3.4 PARTITION OF UNITY. 3.5 ENRICHMENT. 3.5.1 Intrinsic enrichment. 3.5.2 Extrinsic enrichment. 3.5.3 Partition of unity finite element method. 3.5.4 Generalised finite element method. 3.5.5 Extended finite element method. 3.5.6 Hp-clouds enrichment. 3.5.7 Generalisation of the PU enrichment. 3.5.8 Transition from standard to enriched approximation. 3.6 ISOTROPIC XFEM. 3.6.1 Basic XFEM approximation. 3.6.2 Signed distance function. 3.6.3 Modelling strong discontinuous fields. 3.6.4 Modelling weak discontinuous fields. 3.6.5 Plastic enrichment. 3.6.6 Selection of nodes for discontinuity enrichment. 3.6.7 Modelling the crack. 3.7 DISCRETIZATION AND INTEGRATION. 3.7.1 Governing equation. 3.7.2 XFEM discretization. 3.7.3 Element partitioning and numerical integration. 3.7.4 Crack intersection. 3.8 TRACKING MOVING BOUNDARIES. 3.8.1 Level set method. 3.8.2 Fast marching method. 3.8.3 Ordered upwind method. 3.9 NUMERICAL SIMULATIONS. 3.9.1 A tensile plate with a central crack. 3.9.2 Double edge cracks. 3.9.3 Double internal collinear cracks. 3.9.4 A central crack in an infinite plate. 3.9.5 An edge crack in a finite plate. Chapter 4 XFEM for Orthotropic Problems. 4.1 INTRODUCTION. 4.2 ANISOTROPIC ELASTICITY. 4.2.1 Elasticity solution. 4.2.2 Anisotropic stress functions. 4.2.3 Orthotropic mixed mode problems. 4.2.4 Energy release rate and stress intensity factor for anisotropic. materials. 4.2.5 Anisotropic singular elements. 4.3 ANALYTICAL SOLUTIONS FOR NEAR CRACK TIP. 4.3.1 Near crack tip displacement field (class I). 4.3.2 Near crack tip displacement field (class II). 4.3.3 Unified near crack tip displacement field (both classes). 4.4 ANISOTROPIC XFEM. 4.4.1 Governing equation. 4.4.2 XFEM discretization. 4.4.3 SIF calculations. 4.5 NUMERICAL SIMULATIONS. 4.5.1 Plate with a crack parallel to material axis of orthotropy. 4.5.2 Edge crack with several orientations of the axes of orthotropy. 4.5.3 Single edge notched tensile specimen with crack inclination. 4.5.4 Central slanted crack. 4.5.5 An inclined centre crack in a disk subjected to point loads. 4.5.6 A crack between orthotropic and isotropic materials subjected to. tensile tractions. Chapter 5 XFEM for Cohesive Cracks. 5.1 INTRODUCTION. 5.2 COHESIVE CRACKS. 5.2.1 Cohesive crack models. 5.2.2 Numerical models for cohesive cracks. 5.2.3 Crack propagation criteria. 5.2.4 Snap-back behaviour. 5.2.5 Griffith criterion for cohesive crack. 5.2.6 Cohesive crack model. 5.3 XFEM FOR COHESIVE CRACKS. 5.3.1 Enrichment functions. 5.3.2 Governing equations. 5.3.3 XFEM discretization. 5.4 NUMERICAL SIMULATIONS. 5.4.1 Mixed mode bending beam. 5.4.2 Four point bending beam. 5.4.3 Double cantilever beam. Chapter 6 New Frontiers. 6.1 INTRODUCTION. 6.2 INTERFACE CRACKS. 6.2.1 Elasticity solution for isotropic bimaterial interface. 6.2.2 Stability of interface cracks. 6.2.3 XFEM approximation for interface cracks. 6.3 CONTACT. 6.3.1 Numerical models for a contact problem. 6.3.2 XFEM modelling of a contact problem. 6.4 DYNAMIC FRACTURE. 6.4.1 Dynamic crack propagation by XFEM. 6.4.2 Dynamic LEFM. 6.4.3 Dynamic orthotropic LEFM. 6.4.4 Basic formulation of dynamic XFEM. 6.4.5 XFEM discretization. 6.4.6 Time integration. 6.4.7 Time finite element method. 6.4.8 Time extended finite element method. 6.5 MULTISCALE XFEM. 6.5.1 Basic formulation. 6.5.2 The zoom technique. 6.5.3 Homogenisation based techniques. 6.5.4 XFEM discretization. 6.6 MULTIPHASE XFEM. 6.6.1 Basic formulation. 6.6.2 XFEM approximation. 6.6.3 Two-phase fluid flow. 6.6.4 XFEM approximation. Chapter 7 XFEM Flow. 7.1 INTRODUCTION. 7.2 AVAILABLE OPEN-SOURCE XFEM. 7.3. FINITE ELEMENT ANALYSIS. 7.3.1 Defining the model. 7.3.2 Creating the finite element mesh. 7.3.3 Linear elastic analysis. 7.3.4 Large deformation. 7.3.5 Nonlinear (elastoplastic) analysis. 7.3.6 Material constitutive matrix. 7.4 XFEM. 7.4.1 Front tracking. 7.4.2 Enrichment detection. 7.4.3 Enrichment functions. 7.4.4 Ramp (transition) functions. 7.4.5 Evaluation of the B matrix. 7.5 NUMERICAL INTEGRATION. 7.5.1 Sub-quads. 7.5.2 Sub-triangles. 7.6 SOLVER. 7.6.1 XFEM degrees of freedom. 7.6.2 Time integration. 7.6.3 Simultaneous equations solver. 7.6.4 Crack length control. 7.7 POST-PROCESSING. 7.7.1 Stress intensity factor. 7.7.2 Crack growth. 7.7.3 Other applications. 7.8 CONFIGURATION UPDATE. References . Index

Multiscale aggregating discontinuities: A method for circumventing loss of material stability

Abstract: New methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described. These methods, called multiscale aggregating discontinuities, are based on the concept of ‘perforated’ unit cells, which exclude subdomains that are unstable, i.e. exhibit loss of material stability. Using this concept, it is possible to compute an equivalent discontinuity at the coarser scale, including both the direction of the discontinuity and the magnitude of the jump. These variables are then passed to the coarse-scale model along with the stress in the unit cell. The discontinuity is injected at the coarser scale by the extended finite element method. Analysis of the procedure shows that the method is consistent in power and yields a bulk stress–strain response that is stable. Applications of this procedure to crack growth in heterogeneous materials are given. Copyright © 2007 John Wiley & Sons, Ltd.

A new crack tip element for the phantom-node method with arbitrary cohesive cracks

Abstract: We have developed a new crack tip element for the phantom node method. In this method, a crack tip can be placed inside an element. Therefore cracks can propagate almost independent of the finite element mesh. We developed two different formulations for the three-node triangular element and four-node quadrilateral element, respectively. Although this method is well suited for the one-point quadrature scheme, it can be used with other general quadrature schemes. We provide some numerical examples for some static and dynamic problems.

Extended digital image correlation with crack shape optimization

Abstract: The methodology of eXtended finite element method is applied to the measurement of displacements through digital image correlation. An algorithm, initially based on a finite element decomposition of displacement fields, is extended to benefit from discontinuity and singular enrichments over a suited subset of elements. This allows one to measure irregular displacements encountered, say, in cracked solids, as demonstrated both in artificial examples and experimental case studies. Moreover, an optimization strategy for the support of the discontinuity enables one to adjust the crack path configuration to reduce the residual mismatch, and hence to be tailored automatically to a wavy or irregular crack path. Copyright © 2007 John Wiley & Sons, Ltd.

NURBS-enhanced finite element method (NEFEM)

Abstract: The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques.

The extended finite element method in thermoelastic fracture mechanics

Abstract: The extended finite element method (XFEM) is applied to the simulation of thermally stressed, cracked solids. Both thermal and mechanical fields are enriched in the XFEM way in order to represent discontinuous temperature, heat flux, displacement, and traction across the crack surface, as well as singular heat flux and stress at the crack front. Consequently, the cracked thermomechanical problem may be solved on a mesh that is independent of the crack. Either adiabatic or isothermal condition is considered on the crack surface. In the second case, the temperature field is enriched such that it is continuous across the crack but with a discontinuous derivative and the temperature is enforced to the prescribed value by a penalty method. The stress intensity factors are extracted from the XFEM solution by an interaction integral in domain form with no crack face integration. The method is illustrated on several numerical examples (including a curvilinear crack, a propagating crack, and a three-dimensional crack) and is compared with existing solutions. Copyright © 2007 John Wiley & Sons, Ltd.

Finite elements for symmetric tensors in three dimensions

Abstract: We construct finite element subspaces of the space of symme- tric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger-Reissner mixed formulation of the elasticty equations, when stan- dard discontinous finite element spaces are used to approximate the displace- ment field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites

Abstract: In a nanostructured material, the interface-to- volume ratio is so high that the interface energy, which is usually negligible with respect to the bulk energy in solid mechanics, can no longer be neglected. The interfaces in a number of nanomaterials can be appropriately characterized by the coherent interface model. According to the latter, the displacement vector field is continuous across an interface in a medium while the traction vector field across the same interface is discontinuous and must satisfy the Laplace–Young equation. The present work aims to elaborate an efficient numerical approach to dealing with the interface effects described by the coherent interface model and to determining the size-dependent effective elastic moduli of nanocomposites. To achieve this twofold objective, a computational technique combining the level set method and the extended finite element method is developed and implemented. The numerical results obtained by the developed computational technique in the two-dimensional (2D) context are compared and discussed with respect to the relevant exact analytical solutions used as benchmarks. The computational technique elaborated in the present work is expected to be an efficient tool for evaluating the overall size-dependent elastic behaviour of nanomaterials and nano-sized structures.

A contact algorithm for frictional crack propagation with the extended finite element method

Abstract: We present an incremental quasi-static contact algorithm for path-dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non-linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.

Smooth finite element methods: Convergence, accuracy and properties

Abstract: A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi-equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one-subcell element with a quasi-equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one-cell smoothed four-noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.

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

Abstract: SUMMARY 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 q 2007 John Wiley & Sons, Ltd.

Polyhedral finite elements using harmonic basis functions

Abstract: Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.

Convergence and optimal complexity of adaptive finite element eigenvalue computations

Abstract: In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper.

A combined extended finite element and level set method for biofilm growth

Abstract: This paper presents a computational technique based on the extended finite element method (XFEM) and the level set method for the growth of biofilms. The discontinuous-derivative enrichment of the standard finite element approximation eliminates the need for the finite element mesh to coincide with the biofilm–fluid interface and also permits the introduction of the discontinuity in the normal derivative of the substrate concentration field at the biofilm–fluid interface. The XFEM is coupled with a comprehensive level set update scheme with velocity extensions, which makes updating the biofilm interface fast and accurate without need for remeshing. The kinetics of biofilms are briefly given and the non-linear strong and weak forms are presented. The non-linear system of equations is solved using a Newton–Raphson scheme. Example problems including 1D and 2D biofilm growth are presented to illustrate the accuracy and utility of the method. The 1D results we obtain are in excellent agreement with previous 1D results obtained using finite difference methods. Our 2D results that simulate finger formation and finger-tip splitting in biofilms illustrate the robustness of the present computational technique. Copyright © 2007 John Wiley & Sons, Ltd.

Approximation capability of a bilinear immersed finite element space

Abstract: This article discusses a bilinear immersed finite element (IFE) space for solving second-order elliptic boundary value problems with discontinuous coefficients (interface problem). This is a nonconforming finite element space and its partition can be independent of the interface. The error estimates for the interpolation of a Sobolev function indicate that this IFE space has the usual approximation capability expected from bilinear polynomials. Numerical examples of the related finite element method are provided. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Free vibration analysis of composite and sandwich plates using an improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory

Abstract: The new improved discrete Kirchhoff quadrilateral element based on the third-order zigzag theory developed earlier by the present authors for the static analysis of composite and sandwich plates is extended for dynamics and assessed for its performance for the free vibration response. The element is free from the shear locking. The finite element formulation is validated by comparing the results for simply supported plates with the analytical Navier solution of the zigzag theory. Comparison of the present results for the natural frequencies with those of a recently developed triangular element based on the zigzag theory, for composite and sandwich plates, establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed for composite and sandwich plates with various boundary conditions and aspect ratio by comparing the finite element results with the 3D elasticity analytical and finite element solutions.

Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method

Abstract: A numerical technique for non-planar three-dimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method (X-FEM) and the fast marching method (FMM). In crack modeling using X-FEM, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields. The initial crack geometry is represented by two level set functions, and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated without remeshing. Crack growth is conducted using FMM; unlike a level set formulation for interface capturing, no iterations nor any time step restrictions are imposed in the FMM. Planar and non-planar quasi-static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique. Copyright © 2008 John Wiley & Sons, Ltd.