Showing papers on "Extended finite element method published in 2005"

High-order extended finite element method for cracked domains

Abstract: The aim of the paper is to study the capabilities of the Extended Finite Element Method (XFEM) to achieve accurate computations in non smooth situations such as crack problems. Although the XFEM method ensures a weaker error than classical finite element methods, the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity. The difficulty can be overcome by modifying the enrichment of the finite element basis with the asymptotic crack tip displacement solutions as well as with the Heaviside function. Numerical simulations show that the modified XFEM method achieves an optimal rate of convergence (i.e. like in a standard finite element method for a smooth problem)

Adaptive finite element methods for differential equations.

Abstract: Gegenstand des Buches ist die Dual Weighted Residual method (DWR), ein sehr effizientes numerisches Verfahren zur Behandlung einer großen Klasse von variationell formulierten Differentialgleichungen. Das numerische Verfahren ist adaptiv, d.h. es konstruiert eigenständig eine Folge von Approximationen für eine gegebene Fragestellung. Typische Fragestellungen sind die Bestimmung gewichteter Mittelwerte der Lösung oder ihrer Ableitungen, die Bestimmung von Randintegralen über Lösungskomponenten (relevant z.B. für die Berechnung von strömungsmechanischen Kenngrößen) oder die Bestimmung von Spannungsintensitätsfaktoren (z.B. in der Bruchmechanik). Das Verfahren basiert auf Projektionsmethoden wie z.B. der Finiten Elemente Methode (FEM). Dort wird die Approximationsgüte durch die Wahl der Gitter gesteuert. Der Kern jeder adaptiven FEM ist deshalb die Art, wie die Gitter gewählt werden. Typischerweise geschieht dies in einer adaptiven Schleife, in der in mehreren Durchgängen schrittweise das Gitter verbessert wird, bis eine gewünschte Genauigkeit erreicht ist. Bei der DWR wird in jedem Schleifendurchgang ein lineares Hilfsproblem—das sog. duale Problem, welches von der vorliegenden Fragestellung abhängt—(näherungsweise) gelöst. Weiterhin wird eine Approximation der Differentialgleichung bestimmt. Aus diesen nun vorliegenden Daten wird dann herausdestilliert, wo das Gitter verfeinert werden sollte bzw. vergröbert werden kann, um eine genauere Lösung zu erhalten. Ziel eines adaptiven Algorithmus ist, das gewünschte Ergebnis möglichst effizient zu bestimmen, d.h. mit möglichst geringem Bedarf an Resourcen (Rechenzeit, Speicherbedarf etc.). Mit zahlreichen Beispielen belegt das Buch, daß die DWR dieses Ziel erreicht. Es sei hier besonders hervorgehoben, daß eine Kosten-Nutzen-Betrachtung für die DWR besonders bei nichtlinearen Problemen günstig ausfällt, da die Kosten für die Lösung des linearen Hilfsproblems vergleichbar mit denen eines Newtonschrittes sind und somit nur einen kleinen Teil der Gesamtkosten ausmachen. Das Buch gibt einen sehr guten Überblick über die Technik und die Möglichkeiten der DWR. In einleitenden Kapiteln wird die DWR an gewöhnlichen Differentialgleichungen und dann an einfachen linearen, elliptischen partiellen Differentialgleichungen sehr klar und verständlich vorgeführt. Anschließend wird die DWR in einem abstrakten funktionalanalytischen Rahmen vorgestellt. Der Rest des Buches illustriert auf eindrucksvolle Weise die Leistungsfähigkeit und Breite der Anwendungsfähigkeit des Konzeptes an Hand von Fallbeispielen: Es werden Eigenwertprobleme, Optimierungsaufgaben mit Zwangsbedingungen, die durch eine partielle Differentialgleichung gegeben sind, Strukturmechanikprobleme (lineare Elastizität, Plastizität), Strömungsmechanik (hydrodynamische Stabilitätsanalyse, Berechnung von Strömungskennwerten) behandelt. Auch zeitabhängige Probleme wie die Lösung der Wellengleichung werden mit der DWR erfolgreich bearbeitet. Insgesamt wird klar ersichtlich, daß die DWR eine sehr flexible und vielseitig anwendbare Technik ist. Die ausgewählten numerischen Beispiele, die vor allem aus umfangreichen numerischen Untersuchungen der Gruppe von Rolf Rannacher aus den letzten 10 Jahren ausgewählt wurden, sind sehr illustrativ. Die Erläuterungen zu den Beispielen sind auch deshalb interessant, weil eine Menge zusätzlicher Informationen über die numerische Behandlung des vorliegenden Problems quasi nebenbei einfließen. Das Buch entstand aus einer fortgeschrittenen Spezialvorlesung, die an der ETH Zürich gehalten wurde. Einen Lehrbuchcharakter erhält das Buch dadurch, daß Übungsaufgaben (mit detailierten Lösungen im Anhang) jedes Kapitel abschließen. Die Aufgaben enthal-

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

Abstract: An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi-static analysis of three-dimensional crack propagation in brittle and quasi-brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity-regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack-band’ version of XFEM. The evolving discontinuity surface is discretized through a C0 surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large-scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.

A New Finite Element Gradient Recovery Method: Superconvergence Property

Abstract: This is the first in a series of papers in which a new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz--Zhu patch recovery method. In addition, for uniform triangular meshes, the method is superconvergent for the linear element under the chevron pattern, and ultraconvergent at element edge centers for the quadratic element under the regular pattern. Applications of this new gradient recovery technique will be discussed in forthcoming papers.

An energy‐conserving scheme for dynamic crack growth using the eXtended finite element method

Abstract: This paper proposes a generalization of the eXtended finite element method (X-FEM) to model dynamic fracture and time-dependent problems from a more general point of view, and gives a proof of the stability of the numerical scheme in the linear case. First, we study the stability conditions of Newmark-type schemes for problems with evolving discretizations. We prove that the proposed enrichment strategy satisfies these conditions and also ensures energy conservation. Using this approach, as the crack propagates, the enrichment can evolve with no occurrence of instability or uncontrolled energy transfer. Then, we present a technique based on Lagrangian conservation for the estimation of dynamic stress intensity factors for arbitrary 2D cracks. The results presented for several applications are accurate for stationary or moving cracks. Copyright © 2005 John Wiley & Sons, Ltd.

A Multiscale Finite Element Method for Numerical Homogenization

Abstract: This paper is concerned with a multiscale finite element method for numerically solving second-order scalar elliptic boundary value problems with highly oscillating coefficients. In the spirit of previous other works, our method is based on the coupling of a coarse global mesh and a fine local mesh, the latter being used for computing independently an adapted finite element basis for the coarse mesh. The main idea is the introduction of a composition rule, or change of variables, for the construction of this finite element basis. In particular, this allows for a simple treatment of high-order finite element methods. We provide optimal error estimates in the case of periodically oscillating coefficients. We illustrate our method in various examples.

Finite element formulation for a digital image correlation method

Abstract: A finite element formulation for a digital image correlation method is presented that will determine directly the complete, two-dimensional displacement field during the image correlation process on digital images. The entire interested image area is discretized into finite elements that are involved in the common image correlation process by use of our algorithms. This image correlation method with finite element formulation has an advantage over subset-based image correlation methods because it satisfies the requirements of displacement continuity and derivative continuity among elements on images. Numerical studies and a real experiment are used to verify the proposed formulation. Results have shown that the image correlation with the finite element formulation is computationally efficient, accurate, and robust.

Finite element methods inab initioelectronic structure calculations

Abstract: We review the application of the finite element (FE) method to ab initio electronic structure calculations in solids. The FE method is a general approach for the solution of differential and integral equations which uses a strictly local, piecewise-polynomial basis. Because the basis is composed of polynomials, the method is completely general and its accuracy is systematically improvable. Because the basis is strictly local in real space, the method allows for variable resolution in real space; produces sparse, structured matrices, enabling the effective use of iterative solution methods; and is well suited for parallel implementation. The method thus combines significant advantages of both real-space-grid and basis-oriented approaches, and so is well suited for large, accurate ab initio calculations. We review the construction and properties of the required FE bases and their use in the self-consistent solution of the Kohn?Sham equations of density functional theory.

The Finite Element Method: Basic Concepts and Applications

Abstract: Preface 1.Introduction 2.The Method of Weighted Residuals and Galerkin Approximations 3.The Finite Element Method in One Dimension 4.The Two-Dimensional Triangular Element 5.The Two-Dimensional Quadrilateral ELement 6.Isoparametric Two-dimensional Elements 7.The Three-Dimensional Element 8.Additional Applications References Appendices Index

Non-linear analysis of shells with arbitrary evolving cracks using XFEM

Abstract: A new formulation and numerical procedures are developed for the analysis of arbitrary crack propagation in shells using the extended finite element method. The method is valid for completely non-linear problems. Through-the-thickness cracks in sandwich shells are considered. An exact shell kinematics is presented, and a new enrichment of the rotation field is proposed which satisfies the director inextensibility condition. To avoid locking, an enhanced strain formulation is proposed for the 4-node cracked shell element. A finite strain plane stress constitutive model based on the logarithmic corotational rate is employed. A cohesive zone model is introduced which embodies the special characteristics of the shell kinematics. Stress intensity factors are calculated for selected problems and crack propagation problems are solved. Copyright © 2004 John Wiley & Sons, Ltd.

Analysis of a Discontinuous Finite Element Method for the Coupled Stokes and Darcy Problems

Abstract: The coupled Stokes and Darcy flows problem is solved by the locally conservative discontinuous Galerkin method. Optimal error estimates for the fluid velocity and pressure are derived.

Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations

Abstract: There are three important steps in the computational modelling of any physical process: (i) problem definition, (ii) mathematical model, and (iii) computer simulation

Finite element methods and their applications

Abstract: Elementary Finite Elements.- Nonconforming Finite Elements.- Mixed Finite Elements.- Discontinuous Finite Elements.- Characteristic Finite Elements.- Adaptive Finite Elements.- Solid Mechanics.- Fluid Mechanics.- Fluid Flow in Porous Media.- Semiconductor Modeling.

Trefftz Finite Element Method and Its Applications

Abstract: This paper presents an overview of the Trefftz finite element and its application in various engineering problems. Basic concepts of the Trefftz method are discussed, such as T-complete functions, special purpose elements, modified variational functionals, rank conditions, intraelement fields, and frame fields. The hybrid-Trefftz finite element formulation and numerical solutions of potential flow problems, plane elasticity, linear thin and thick plate bending, transient heat conduction, and geometrically nonlinear plate bending are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions. In the case of geometrically nonlinear plate bending, exact solutions of the Lame-Navier equations are used for the in-plane intraelement displacement field, and an incremental form of the basic equations is adopted. Generation of elemental stiffness equations from the modified variational principle is also discussed. Some typical numerical results are presented to show the application of the finite element approach. Finally, a brief summary of the approach is provided and future trends in this field are identified. There are 151 references cited in this revised article. DOI: 10.1115/1.1995716

Displacement, Flexibility, and Mixed Beam – Column Finite Element Formulations for Distributed Plasticity Analysis

Abstract: This paper presents several beam–column finite element formulations for full nonlinear distributed plasticity analysis of planar frame structures. The fundamental steps within the derivation of displacement-based, flexibility-based, and mixed elements are summarized. These formulations are presented using a total Lagrangian corotational approach. In this context, the element displacements are separated into rigid-body and deformational (or natural) degrees of freedom. The element rigid-body motion is handled separately within the mapping from the corotational to global element frames. This paper focuses on the similarities and differences in the element formulations associated with the element natural degrees of freedom within the corotational frame. The paper focuses specifically on two-dimensional elements based on Euler–Bernoulli kinematics; however, the concepts are also applicable to general beam–column elements for three-dimensional analysis. The equations for the consistent tangent stiffness matric...

A Mixed Finite Element Method for Elasticity in Three Dimensions

Abstract: We describe a stable mixed finite element method for linear elasticity in three dimensions.

Parallel Octree-Based Finite Element Method for Large-Scale Earthquake Ground Motion Simulation

Abstract: We present a parallel octree-based finite element method for large-scale earthquake ground motion simulation in realistic basins. The octree representation combines the low memory per node and good cache performance of finite difference methods with the spatial adaptivity to local seismic wavelengths characteristic of unstructured finite element methods. Several tests are provided to verify the numerical performance of the method against Green’s function solutions for homogeneous and piecewise homogeneous media, both with and without anelastic attenuation. A comparison is also provided against a finite difference code and an unstructured tetrahedral finite element code for a simulation of the 1994 Northridge Earthquake. The numerical tests all show very good agreement with analytical solutions and other codes. Finally, performance evaluation indicates excellent single-processor performance and parallel scalability over a range of 1 to 2048 processors for Northridge simulations with up to 300 million degrees of freedom. keyword: Earthquake ground motion modeling, octree, parallel computing, finite element method, elastic wave propagation

Fundamental Finite Element Analysis and Applications: with Mathematica and Matlab Computations

Abstract: Preface. 1. Finite Element Method: The Big Picture. 1.1 Discretization and Element Equations. 1.1.1 Plane Truss Element. 1.1.2 Triangular Element for Two Dimensional Heat Flow. 1.1.3 General Remarks on Finite Element Discretization. 1.1.4 Triangular Element for Two Dimensional Stress Analysis. 1.2 Assembly of Element Equations. 1.3 Boundary Conditions and Nodal Solution. 1.3.1 Essential Boundary Conditions by Re-arranging Equations. 1.3.2 Essential Boundary Conditions by Modifying Equations. 1.3.3 Approximate Treatment of Essential Boundary Conditions. 1.3.4 Computation of Reactions to Verify Overall Equilibrium. 1.4 Element Solutions and Model Validity. 1.4.1 Plane Truss Element. 1.4.2 Triangular Element for Two Dimensional Heat Flow. 1.4.3 Triangular Element for Two Dimensional Stress Analysis. 1.5 Solution of Linear Equations. 1.5.1 Solution Using Choleski Decomposition. 1.5.2 Conjugate Gradient Method. 1.6 Multipoint Constraints. 1.6.1 Solution Using Lagrange multipliers. 1.6.2 Solution Using Penalty function. 1.7 Units. 2. Mathematical Foundation of the Finite Element Method. 2.1 Axial Deformation of Bars. 2.1.1 Differential equation for axial deformations. 2.1.2 Exact solutions of some axial deformation problems. 2.2 Axial Deformation of Bars Using Galerkin Method. 2.2.1 Weak form for axial deformations. 2.2.2 Uniform bar subjected to linearly varying axial load. 2.2.3 Tapered bar subjected to linearly varying axial load. 2.3 One Dimensional BVP Using Galerkin Method. 2.3.1 Overall solution procedure using Galerkin method. 2.3.2 Higher-Order Boundary Value Problems. 2.4 Rayleigh-Ritz Method. 2.4.1 Potential Energy for Axial Deformation of Bars. 2.4.2 Overall solution procedure using the Rayleigh-Ritz method. 2.4.3 Uniform bar subjected to linearly varying axial load. 2.4.4 Tapered bar subjected to linearly varying axial load. 2.5 Comments on the Galerkin & the Rayleigh-Ritz Methods. 2.5.1 Admissible assumed solution. 2.5.2 Solution convergence - the completeness requirement. 2.5.3 Galerkin versus Rayleigh-Ritz. 2.6 Finite Element Form of Assumed Solutions. 2.6.1 Linear interpolation functions for second-order problems. 2.6.2 Lagrange interpolation. 2.6.3 Galerkin weighting functions in the finite element form. 2.6.4 Hermite interpolation for fourth-order problems. 2.7 Finite Element Solution of Axial Deformation Problems. 2.7.1 Two Node Uniform Bar Element for Axial Deformations. 2.7.2 Numerical examples. 3. One Dimensional Boundary Value Problem. 3.1 Selected Applications of 1D BVP. 3.1.1 Steady state heat conduction. 3.1.2 Heat flow through thin fins. 3.1.3 Viscous fluid flow between parallel plates - Lubrication problem. 3.1.4 Slider bearing. 3.1.5 Axial deformation of bars. 3.1.6 Elastic buckling of long slender bars. 3.2 Finite Element Formulation for Second Order 1D BVP 3.2.1 Complete Solution Procedure. 3.3 Steady State Heat Conduction. 3.4 Steady State Heat Conduction and Convection. 3.5 Viscous Fluid Flow Between Parallel Plates. 3.6 Elastic Buckling of Bars. 3.7 Solution of Second Order 1D BVP. 3.8 A Closer Look at the Inter-Element Derivative Terms. 4. Trusses, Beams, and Frames. 4.1 Plane Trusses. 4.2 Space Trusses. 4.3 Temperature Changes and Initial Strains in Trusses. 4.4 Spring Elements. 4.5 Transverse Deformation of Beams. 4.5.1 Differential equation for beam bending. 4.5.2 Boundary conditions for beams. 4.5.3 Shear stresses beams. 4.5.4 Potential energy for beam bending. 4.5.5 Transverse deformation of a uniform beam. 4.5.6 Transverse deformation of a tapered beam fixed at both ends. 4.6 Two Node Beam Element. 4.6.1 Cubic assumed solution. 4.6.2 Element equations using Rayleigh-Ritz method. 4.7 Uniform Beams Subjected to Distributed Loads. 4.8 Plane Frames. Contents 4.9 Space Frames. 4.9.1 Element equations in local coordinate system. 4.9.2 Local to global transformation. 4.9.3 Element Solution. 4.10 Frames in Multistory Buildings. 5. Two Dimensional Elements. 5.1 Selected Applications of the 2D BVP. 5.1.1 Two dimensional potential flow. 5.1.2 Steady-state heat flow. 5.1.3 Bars subjected to torsion. 5.1.4 Waveguides in Electromagnetics. 5.2 Integration by Parts in Higher Dimensions. 5.3 Finite Element Equations Using the Galerkin Method. 5.4 Rectangular Finite Elements. 5.4.1 Four node rectangular element. 5.4.2 Eight node rectangular element. 5.4.3 Lagrange interpolation for rectangular elements. 5.5 Triangular Finite Elements. 5.5.1 Three node triangular element. 5.5.2 Higher-order triangular elements. 6. Mapped Elements. 6.1 Integration Using Change of Variables. 6.1.1 One dimensional integrals. 6.1.2 Two dimensional area integrals. 6.1.3 Three dimensional volume integrals. 6.2 Mapping Quadrilaterals Using Interpolation Functions. 6.2.1 Mapping lines. 6.2.2 Mapping quadrilateral areas. 6.2.3 Mapped mesh generation. 6.3 Numerical Integration Using Gauss Quadrature. 6.3.1 Gauss quadrature for one dimensional integrals. 6.3.2 Gauss quadrature for area integrals. 6.3.3 Gauss quadrature for volume integrals. 6.4 Finite Element Computations Involving Mapped Elements. 6.4.1 Assumed solution. 6.4.2 Derivatives of the assumed solution. 6.4.3 Evaluation of area integrals. 6.4.4 Evaluation of boundary integrals. Fundamental Finite Element Theory and Applications. 6.5 Complete Mathematica and Matlab Based Solutions of 2DBVP Involving Mapped. Elements. 6.6 Triangular Elements by Collapsing Quadrilaterals. 6.7 Infinite Elements. 6.7.1 One dimensional BVP. 6.7.2 Two dimensional BVP. 7. Analysis of Elastic Solids. 7.1 Fundamental Concepts in Elasticity. 7.1.1 Stresses. 7.1.2 Stress failure criteria. 7.1.3 Strains. 7.1.4 Constitutive equations. 7.1.5 Temperature effects and initial strains. 7.2 Governing Differential Equations. 7.2.1 Stress equilibrium equations. 7.2.2 Governing differential equations in terms of displacements. 7.3 General Form of Finite Element Equations. 7.3.1 Potential energy functional. 7.3.2 Weak form. 7.3.3 Finite Element Equations. 7.3.4 Finite Element Equations in the Presence of Initial Strains. 7.4 Plane Stress and Plane Strain. 7.4.1 Plane stress problem. 7.4.2 Plane strain problem. 7.4.3 Finite element equations. 7.4.4 Three node triangular element. 7.4.5 Mapped quadrilateral elements. 7.5 Planar Finite Element Models. 7.5.1 Pressure Vessels. 7.5.2 Rotating Disks and Flywheels. 7.5.3 Residual Stresses due to Welding. 7.5.4 Crack-Tip Singularity. 8. Transient Problems. 8.1 Transient Field Problems. 8.1.1 Finite element equations. 8.1.2 Triangular element. 8.1.3 Transient heat flow. 8.2 Elastic Solids Subjected to Dynamic Loads. 8.2.1 Finite Element Equations. 8.2.2 Mass matrices for common structural elements. Contents 8.2.3 Free Vibration Analysis. 8.2.4 Transient Response Examples. 9. p-Formulation. 9.1 p-Formpulation for Second-Order 1D BVP. 9.2 p-Formpulation for Second-Order 2D BVP. Appendix A: Use of Commercial FEA Software. A.1 Ansys Applications. A.1.1 General Steps. A.1.2 Truss Analysis. A.1.3 Steady-State Heat Flow. A.1.4 Plane Stress Analysis. A.2 Optimizing Design Using Ansys. A.2.1 General Steps. A.2.2 Heat Flow example. A.3 Abaqus Applications. A.3.1 Execution procedure. A.3.2 Truss Analysis. A.3.3 Steady State Heat Flow. A.3.4 Plane Stress Analysis. Appendix B: Variational Form for Boundary Value Problems. B.1 Basic concept of variation of a function. B.2 Derivation of Equivalent Variational Form. B.3 Boundary Value Problem Corresponding to a Given Functional. Bibliography. Index.

Stress analysis around crack tips in finite strain problems using the eXtended Finite Element Method

Abstract: Fracture of rubber-like materials is still an open problem. Indeed, it deals with modelling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X-FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works.

Continuum–discontinuum modelling of shear bands

Abstract: A methodology to model shear bands as strong discontinuities within a continuum mechanics context is presented. The loss of hyperbolicity of the IBVP is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction–separation model acting at the discontinuity surface. The extended finite element method (XFEM) is employed for the spatial discretization of the governing equations. This enables the shear bands to be arbitrarily positioned within the mesh. Examples that study the shear band progression within a rate-independent material are presented. Copyright © 2005 John Wiley & Sons, Ltd.