Showing papers on "Extended finite element method published in 2009"
TL;DR: An implementation of the extended finite element method for fracture problems within the finite element software ABAQUSTM is introduced and several numerical examples in fracture mechanics are presented to demonstrate the benefits of the proposed implementation.
322 citations
TL;DR: This paper investigates the approximation properties of the k-method with the theory of Kolmogorov n-widths and conducts a numerical study in which the n- width and sup–inf are computed for a number of one-dimensional cases.
265 citations
TL;DR: In this article, the authors used the Extended Finite element method (X-FEM) for the solution of hydraulic fracture problems, where the presence of an internal pressure inside the crack is taken into account.
Abstract: In this paper, the extended finite element method (X-FEM) is investigated for the solution of hydraulic fracture problems. The presence of an internal pressure inside the crack is taken into account. Special tip functions encapsulating tip asymptotics typically encountered in hydraulic fractures are introduced. We are especially interested in the two limiting tip behaviour for the impermeable case: the classical LEFM square root asymptote in fracture width for the toughness-dominated regime of propagation and the so-called ⅔ asymptote in fracture width for the viscosity-dominated regime. Different variants of the X-FEM are tested for the case of a plane-strain hydraulic fracture propagation in both the toughness and the viscosity dominated regimes. Fracture opening and fluid pressure are compared at each nodes with analytical solutions available in the literature. The results demonstrate the importance of correcting for the loss of partition of unity in the transition zone between the enriched part and the rest of the mesh. A point-wise matching scheme appears sufficient to obtain accurate results. Proper integration of the singular terms introduced by the enrichment functions is also critical for good performance. Copyright © 2008 John Wiley & Sons, Ltd.
239 citations
TL;DR: An analysis is given that shows that the method to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface has optimal order of convergence both in the H^1- and in the L^2-norm.
Abstract: In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface. We give an analysis that shows that the method has optimal order of convergence both in the $H^1$- and in the $L^2$-norm. Results of numerical experiments are included that confirm this optimality.
221 citations
TL;DR: In this paper, a 3D finite element model representing 20 storey buildings was first built to perform the progressive collapse analysis, which accurately displayed the overall behavior of the buildings under the sudden loss of columns, which provided important information for the additional design guidance on progressive collapse.
219 citations
TL;DR: Two issues in the extended finite element method (XFEM) are addressed: efficient numerical integration of the weak form when the enrichment function is self‐equilibrating and blending of the enrichment.
Abstract: Two issues in the extended finite element method (XFEM) are addressed: efficient numerical integration of the weak form when the enrichment function is self-equilibrating and blending of the enrichment. The integration is based on transforming the domain integrals in the weak form into equivalent contour integrals. It is shown that the contour form is computationally more efficient than the domain form, especially when the enrichment function is singular and/or discontinuous. A method for alleviating the errors in the blending elements is also studied. In this method, the enrichment function is pre-multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains. A method for blending step function enrichment with singular enrichments is described. It is also shown that if the enrichment is not shifted properly, the weighted enrichment is equivalent to the standard enrichment. An edge dislocation and a crack problem are used to benchmark the technique; the influence of the variables that parameterize the weight function is analyzed. The resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing XFEM codes
193 citations
TL;DR: In this article, a finite element method for incompressible (or compressible) elasticity problems with discontinuous modulus of elasticity was proposed, where the problem is written on mixed form using P1-continuous displacements and elementwise P0 pressures.
189 citations
TL;DR: In this article, the extended finite element method (XFEM) enables the accurate approximation of solutions with jumps or kinks within elements and achieves high-order convergence for arbitrary curved interfaces.
Abstract: The extended finite element method (XFEM) enables the accurate approximation of solutions with jumps or kinks within elements. Optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces. Higher-order convergence for arbitrary curved interfaces relies on two major issues: (i) an accurate quadrature of the Galerkin weak form for the cut elements and (ii) a careful formulation of the enrichment, which should preclude any problems in the blending elements. For (i), we employ a strategy of subdividing the elements into subcells with only one curved side. Reference elements that are higher-order on only one side are then used to map the integration points to the real element. For (ii), we find that enrichments for strong discontinuities are easily extended to higher-order accuracy. In contrast, problems in blending elements may hinder optimal convergence for weak discontinuities. Different formulations are investigated, including the corrected XFEM. Numerical results for several test cases involving strong or weak curved discontinuities are presented. Quadratic and cubic approximations are investigated. Optimal convergence rates are achieved using the standard XFEM for the case of a strong discontinuity. Close-to-optimal convergence rates for the case of a weak discontinuity are achieved using the corrected XFEM. Copyright © 2009 John Wiley & Sons, Ltd.
172 citations
TL;DR: This paper introduces a new algorithm to define a stable Lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method and shows that a uniform inf–sup condition is satisfied.
Abstract: This paper introduces a new algorithm to define a stable Lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method. In contrast to earlier approaches. we do not work with an interior penalty formulation as, e.g. for Nitsche techniques, but impose the constraints weakly in terms of Lagrange multipliers. Roughly speaking a stable and optimal discrete Lagrange multiplier space has to satisfy two criteria: a best approximation property and a uniform inf-sup condition. Owing to the fact that the interface does not match the edges of the mesh, the choice of a good discrete Lagrange Multiplier space is not trivial. Here we propose a new algorithm for the local construction of the Lagrange Multiplier space and show that a uniform inf-sup condition is satisfied. A counterexample is also presented, i.e. the inf-sup constant depends on the mesh-size and degenerates as it tends to zero. Numerical results in two-dimensional confirm the theoretical ones. Copyright
161 citations
TL;DR: The developed monotone method does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation and the second-order convergence rate is verified with numerical experiments.
152 citations
TL;DR: A review of two-dimensional and three-dimensional finite element grinding models after 1995 and categorizes them by the scale of the modeling approach—either macro- or micro-scale.
Abstract: This paper presents a review of two-dimensional (2D) and three-dimensional (3D) finite element grinding models after 1995 and categorizes them by the scale of the modeling approach—either macro- or micro-scale. Macro-scale models consider the overall wheel–workpiece interaction while micro-scale models focus on the individual grain–workpiece interactions. Each model is discussed and the relevant boundary conditions, material constitutive treatments, and load inputs are compared. Future directions for finite element grinding modeling are then recommended and, based on the results of this review, synthesized current state-of-the-art macro- and micro-scale modeling approaches are presented.
TL;DR: In this paper, a method for treating fracture in composite material by the extended finite element method with meshes that are independent of matrix/fiber interfaces and crack morphology is described. But the results clearly demonstrate that interface enrichment is sufficient to model the correct mechanics of an interface crack.
Abstract: Methods for treating fracture in composite material by the extended finite element method with meshes that are independent of matrix/fiber interfaces and crack morphology are described. All discontinuities and near-tip enrichments are modeled using the framework of local partition of unity. Level sets are used to describe the geometry of the interfaces and cracks so that no explicit representation of either the cracks or the material interfaces are needed. Both full 12 function enrichments and approximate enrichments for bimaterial crack tips are employed. A technique to correct the approximation in blending elements is used to improve the accuracy. Several numerical results for both two-dimensional and three-dimensional examples illustrate the versatility of the technique. The results clearly demonstrate that interface enrichment is sufficient to model the correct mechanics of an interface crack. Copyright © 2008 John Wiley & Sons, Ltd.
TL;DR: In this paper, the scaled boundary finite element method is extended to solve problems of structural dynamics and the dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution.
Abstract: The scaled boundary finite element method is extended to solve problems of structural dynamics. The dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution for the scaled boundary finite element equation. The inertial effect at high frequencies is modeled by high-order terms of the continued fraction without introducing an internal mesh. By using this solution and introducing auxiliary variables, the equation of motion of the bounded domain is expressed in high-order static stiffness and mass matrices. Standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain. Numerical examples for modal and direct time-domain analyses are presented. Rapid convergence is observed as the order of continued fraction increases. A guideline for selecting the order of continued fraction is proposed and validated. High computational efficiency is demonstrated for problems with stress singularity.
TL;DR: In this paper, the authors present a review of the extended finite element method X-FEM for computational fracture mechanics, and discuss the basic ideas and formulation for the newly developed XFEM method.
TL;DR: In this paper, a measurement technique is proposed for estimating full displacement fields by complementing digital image correlation with an additional penalization on the distance between the estimated displacement field and its projection onto the space of elastic solutions.
Abstract: To reduce the measurement uncertainty, a measurement technique is proposed for es- timating full displacement fields by complementing digital image correlation with an additional penalization on the distance between the estimated displacement field and its projection onto the space of elastic solutions. The extended finite element method is used for inserting disconti- nuities independently of the underlying mesh. An application to the brittle fracture of a silicon carbide specimen is used to illustrate the application. To complete the analysis, the crack tip location and the stress intensity factors are estimated. This allows for a characterization of the measurement and identification procedure in terms of uncertainty.
TL;DR: In this paper, an interaction energy contour integral is derived for the computation of mixed-mode stress intensity factors (SIFs) in nonhomogeneous materials with continuous or discontinuous properties.
TL;DR: In this article, a new method for modeling discrete cracks based on the extended finite element method is described, where the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements.
Abstract: A new method for modeling discrete cracks based on the extended finite element method is described. In the method, the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements. The method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path. A set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled. The effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching. Copyright © 2008 John Wiley & Sons, Ltd.
TL;DR: In this article, a high-order generalized finite element method (GFEM) for non-planar 3D crack surfaces is presented, which can model a crack arbitrarily located within a finite element (FE) mesh and thus allows fully automated fracture analysis using an existing FE discretization without cracks.
Abstract: A high-order generalized finite element method (GFEM) for non-planar three-dimensional crack surfaces is presented. Discontinuous p-hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non-planar, non-smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non-planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.
TL;DR: A new fictitious domain approach inspired by the extended finite element method introduced by Moes, Dolbow, and Belytschko is presented and an optimal method is obtained thanks to an additional stabilization technique.
Abstract: The purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by Moes, Dolbow, and Belytschko in [Internat. J. Numer. Methods Engrg., 46 (1999), pp. 131-150]. An optimal method is obtained thanks to an additional stabilization technique. Some a priori estimates are established and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward.
TL;DR: This paper presents a locally conservative finite element method based on enriching the approximation space of the continuous Galerkin method with elementwise constant functions to establish optimal convergence of numerical solutions.
Abstract: This paper presents a locally conservative finite element method based on enriching the approximation space of the continuous Galerkin method with elementwise constant functions. The proposed method has a smaller number of degrees of freedom than the discontinuous Galerkin method. Numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method. We also present a theoretical analysis of the method and establish optimal convergence of numerical solutions.
TL;DR: In this paper, a finite element based method, known as nonlocal finite element method (NL-FEM), is numerically implemented for solving 2D nonlocal elastic problems.
TL;DR: It was found that the stiffness matrix required to describe the effect of axial load is proportional to the mass matrix, which makes the extension to existing software trivial.
TL;DR: In this article, an edge-based smoothed finite element method (ES-FEM) using triangular elements was proposed to improve the accuracy and convergence rate of the existing standard FEM for the elastic solid mechanics problems.
Abstract: An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper, the ES-FEM is extended to more complicated visco-elastoplastic analyses using the von-Mises yield function and the Prandtl–Reuss flow rule. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic and linear kinematic hardening. The formulation shows that the bandwidth of stiffness matrix of the ES-FEM is larger than that of the FEM, and hence the computational cost of the ES-FEM in numerical examples is larger than that of the FEM for the same mesh. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the ES-FEM is more efficient than the FEM.
TL;DR: In this article, a node-based smoothed finite element method (NS-FEM) for solving solid mechanics problems using a mesh of general polygonal elements was proposed.
Abstract: A node-based smoothed finite element method (NS-FEM) for solving solid mechanics problems using a mesh of general polygonal elements was recently proposed. In the NS-FEM, the system stiffness matrix is computed using the smoothed strains over the smoothing domains associated with nodes of element mesh, and a number of important properties have been found, such as the upper bound property and free from the volumetric locking. The examination was performed only for two-dimensional (2D) problems. In this paper, we (1) extend the NS-FEM to three-dimensional (3D) problems using tetrahedral elements (NS-FEM-T4), (2) reconfirm the upper bound and free from the volumetric locking properties for 3D problems, and (3) explore further other properties of NS-FEM for both 2D and 3D problems. In addition, our examinations will be thorough and performed fully using the error norms in both energy and displacement. The results in this work revealed that NS-FEM possesses two additional interesting properties that quite simi...
TL;DR: In this article, an application of the extended finite element method (X-FEM) to the analysis of fracture in piezoelectric materials is presented, based on specific asymptotic crack tip solutions.
Abstract: This paper presents an application of the extended finite element method (X-FEM) to the analysis of fracture in piezoelectric materials. These materials are increasingly used in actuators and sensors. New applications can be found as constituents of smart composites for adaptive electromechanical structures. Under in service loading, phenomena of crack initiation and propagation may occur due to high electromechanical field concentrations. In the past few years, the X-FEM has been applied mostly to model cracks in structural materials. The present paper focuses at first on the definition of new enrichment functions suitable for cracks in piezoelectric structures. At second, generalized domain integrals are used for the determination of crack tip parameters. The approach is based on specific asymptotic crack tip solutions, derived for piezoelectric materials. We present convergence results in the energy norm and for the stress intensity factors, in various settings.
TL;DR: This study proposes enhancing the blending elements by adding hierarchical shape functions where appropriate, which permits compensating for the unwanted terms in the interpolation, and shows that the proposed enhancement always results in greater accuracy.
TL;DR: In this paper, a general mass lumping technique for explicit dynamics simulations using the eXtended Finite Element Method with arbitrary enrichment functions is presented, which is a generalization of previously published results for cracks and holes.
TL;DR: In this article, the feasibility of a hybrid scheme using Daubechies wavelet functions and the finite element method to obtain numerical solutions of some problems in structural mechanics is investigated.
TL;DR: It is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps.
Abstract: This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted $L^2$ projection onto $H(\mathrm{div})$ conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps.
TL;DR: The application of the method is seen to be straightforward even in the complicated case of laminated sandwich panels, and accurate predictions of the dispersion curves are found at negligible computational cost.
Abstract: This paper describes a wave finite element method for the numerical prediction of wave characteristics of cylindrical and curved panels. The method combines conventional finite elements and the theory of wave propagation in periodic structures. The mass and stiffness matrices of a small segment of the structure, which is typically modeled using either a single shell element or, especially for laminated structures, a stack of solid elements meshed through the cross-section, are postprocessed using periodicity conditions. The matrices are typically found using a commercial FE package. The solutions of the resulting eigenproblem provide the frequency evolution of the wavenumber and the wave modes. For cylindrical geometries, the circumferential order of the wave can be specified in order to define the phase change that a wave experiences as it propagates across the element in the circumferential direction. The method is described and illustrated by application to cylinders and curved panels of different constructions. These include isotropic, orthotropic, and laminated sandwich constructions. The application of the method is seen to be straightforward even in the complicated case of laminated sandwich panels. Accurate predictions of the dispersion curves are found at negligible computational cost.