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Showing papers on "Extended finite element method published in 2004"


MonographDOI
25 Mar 2004
TL;DR: The Finite Element Method: A Review 3. Heat Transfer and other Field Problems in One Dimension 4. Nonlinear Bending of Straight Beams 5. Bending Elastic Plates 7. Flows of Viscous Incompressible Fluids 8. Non-linear Analysis of Time-Dependent Problems 9. Finite Elements Formulations of Solids and Structures 10. Material Nonlinearities and Coupled Problems as mentioned in this paper
Abstract: 1. Introduction 2. The Finite Element Method: A Review 3. Heat Transfer and other Field Problems in One Dimension 4. Nonlinear Bending of Straight Beams 5. Heat Transfer and other Field Problems in Two Dimensions 6. Nonlinear Bending of Elastic Plates 7. Flows of Viscous Incompressible Fluids 8. Nonlinear Analysis of Time-Dependent Problems 9. Finite Element Formulations of Solids and Structures 10. Material Nonlinearities and Coupled Problems A1 Solution Procedures for Nonlinear Equations A2 Banded Symmetric and Unsymmetric Solvers

896 citations


Proceedings Article
17 May 2004
TL;DR: This method extends the warped stiffness finite element approach for linear elasticity and combines it with a strain-state-based plasticity model and produces realistic animations of a wide spectrum of materials at interactive rates that have typically been simulated off-line thus far.
Abstract: In this paper we present a fast and robust approach for simulating elasto-plastic materials and fracture in real time. Our method extends the warped stiffness finite element approach for linear elasticity and combines it with a strain-state-based plasticity model. The internal principal stress components provided by the finite element computation are used to determine fracture locations and orientations. We also present a method to consistently animate and fracture a detailed surface mesh along with the underlying volumetric tetrahedral mesh. This multi-resolution strategy produces realistic animations of a wide spectrum of materials at interactive rates that have typically been simulated off-line thus far.

524 citations



Journal ArticleDOI
TL;DR: The particle finite element method (PFEM) as mentioned in this paper is a general formulation for the analysis of fluid-structure interaction problems using the Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains.
Abstract: We present a general formulation for the analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use of a Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are thus viewed as particles which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral form, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility condition in the fluid is introduced via the finite calculus (FIC) method. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM method to solve a number of fluid-structure interaction problems involving large motions of the free surface and splashing of waves are presented.

461 citations


Journal ArticleDOI
TL;DR: This paper presents a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods and special attention is paid to the mesh- free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of theGalerkin weak form.

426 citations


Journal ArticleDOI
TL;DR: In this article, a new stabilized finite element method for the Stokes problem is presented by modifying the mixed variational equation by using local L 2 polynomial pressure projections, which leads to a stable variational formulation.
Abstract: A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L2 polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal-order approximations for the Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure–velocity mismatch eliminates this inconsistency and leads to a stable variational formulation. Unlike other stabilization methods, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The new method can be implemented at the element level and for affine families of finite elements on simplicial grids it reduces to a simple modification of the weak continuity equation. Numerical results are presented for a variety of equal-order continuous velocity and pressure elements in two and three dimensions. Copyright © 2004 John Wiley & Sons, Ltd.

352 citations


Journal ArticleDOI
TL;DR: A modified mixed multiscale finite element method for solving elliptic problems with rough coefficients arising in porous media flow that incorporates the effect of small-scale heterogeneous structures in the elliptic coefficients into the base functions and produces a detailed velocity field that can be used to solve phase transport equations at a subgrid scale.
Abstract: In this paper we propose a modified mixed multiscale finite element method for solving elliptic problems with rough coefficients arising in, e.g., porous media flow. The method is based on the construction of special base functions which adapt to the local property of the differential operator. In particular, the method incorporates the effect of small-scale heterogeneous structures in the elliptic coefficients into the base functions and produces a detailed velocity field that can be used to solve phase transport equations at a subgrid scale. The method is mass conservative and accounts for radial flow in the near-well region without resorting to complicated well models or near-well upscaling procedures. As such, the method provides a step toward a more accurate and rigorous treatment of advanced well architectures in reservoir simulation. The accuracy of the method is demonstrated through a series of three-dimensional incompressible two-phase flow simulations.

288 citations


Journal ArticleDOI
TL;DR: Partition of unity enrichment techniques are developed for bimaterial interface cracks in this article, where 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.
Abstract: 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.

277 citations


Journal ArticleDOI
TL;DR: An order-N atomic-scale finite element method (AFEM) is proposed, which is as accurate as molecular mechanics simulations, but is much faster than the widely usedOrder-N 2 conjugate gradient method.

265 citations


Journal ArticleDOI
TL;DR: An overview of the main ideas of the GFEM can be found in this paper, where the authors present the basic results, experiences with, and potentials of this method, as well as various forms of meshless methods used in engineering.
Abstract: This paper is an overview of the main ideas of the Generalized Finite Element Method (GFEM). We present the basic results, experiences with, and potentials of this method. GFEM is a generalization of the classical Finite Element Method — in its h, p, and h-p versions — as well as of the various forms of meshless methods used in engineering.

261 citations


Journal ArticleDOI
TL;DR: In this article, a combination of discrete element method (DEM) and finite element method for dynamic analysis of geomechanics problems is presented, which can employ spherical (or cylindrical in 2D) rigid elements and finite elements in the discretization of different parts of the system.

Journal ArticleDOI
TL;DR: Investigation of the influence of the material mapping algorithm on the results predicted by the finite element analysis showed that the choice of the mapping algorithm influences the material distribution, but this did not always propagate into the finiteelement results.

Journal ArticleDOI
TL;DR: The dispersive properties of high order finite element schemes are analyzed in the setting of the Helmholtz equation, and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order.
Abstract: The dispersive properties of high order finite element schemes are analyzed in the setting of the Helmholtz equation, and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order. It is shown that the numerical dispersion displays three different types of behavior, depending on the size of the order of the method relative to the mesh-size and the wave number. Quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes. All estimates are fully explicit and are shown to be sharp. Limits are obtained, where transitions between the different regimes occur, and used to provide guidelines for the selection of the mesh-size and the polynomial order in terms of the wave number so that the dispersion error is controlled.

Journal ArticleDOI
TL;DR: In this article, a method for modeling the growth of multiple cracks in linear elastic media is presented, which uses the extended finite element method for arbitrary discontinuities and does not require remeshing as the cracks grow; the method also treats the junction of cracks.
Abstract: SUMMARY A method for modelling the growth of multiple cracks in linear elastic media is presented. Both homogeneous and inhomogeneous materials are considered. The method uses the extended finite element method for arbitrary discontinuities and does not require remeshing as the cracks grow; the method also treats the junction of cracks. The crack geometries are arbitrary with respect to the mesh and are described by vector level sets. The overall response of the structure is obtained until complete failure. A stability analysis of competitive cracks tips is performed. The method is applied to bodies in plane strain or plane stress and to unit cells with 2‐10 growing cracks (although the method does not limit the number of cracks). It is shown to be efficient and accurate for crack coalescence and percolation problems. Copyright ! 2004 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: This work considers the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, and derives new estimates for approximation byquadrilateral Raviart--Thomas elements (requiring less regularity) and proposes a new quadrilaterally finite element space which provides optimal order approximation in H(div).
Abstract: We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to a space of vector fields on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism of the square onto the element. For affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in L2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, the situation is more complicated and we give a precise characterization of what is needed for optimal order L2-approximation of the function and of its divergence. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of H(div). We also derive new estimates for approximation by quadrilateral Raviart--Thomas elements (requiring less regularity) and propose a new quadrilateral finite element space which provides optimal order approximation in H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of Poisson's equation.

Journal ArticleDOI
TL;DR: In this paper, the extended finite element method (XFEM) is improved to directly evaluate mixed mode stress intensity factors (SIFs) without extra post-processing, for homogeneous materials as well as for bimaterials.
Abstract: The extended finite element method (XFEM) is improved to directly evaluate mixed mode stress intensity factors (SIFs) without extra post-processing, for homogeneous materials as well as for bimaterials. This is achieved by enriching the finite element (FE) approximation of the nodes surrounding the crack tip with not only the first term but also the higher order terms of the crack tip asymptotic field using a partition of unity method (PUM). The crack faces behind the tip(s) are modelled independently of the mesh by displacement jump functions. The additional coefficients corresponding to the enrichments at the nodes of the elements surrounding the crack tip are forced to be equal by a penalty function method, thus ensuring that the displacement approximations reduce to the actual asymptotic fields adjacent to the crack tip. The numerical results so obtained are in excellent agreement with analytical and numerical results available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a least-squares quadratic B-spline finite element method for calculating the numerical solutions of the one-dimensional Burgers-like equations with appropriate boundary and initial conditions is presented.

Journal ArticleDOI
TL;DR: The material point method as discussed by the authors is a variant of the finite element method formulated in an arbitrary Lagrangian-Eulerian description of motion, where the motion of material points, representing subregions of the analysed continuum, is traced against a background of the computational element mesh.

Journal ArticleDOI
TL;DR: In this article, an immersed finite element (IFE) space is introduced for solving a second-order elliptic boundary value problem with discontinuous coefficients (interface problem), where the IFE space is nonconforming and its partition can be independent of the interface.
Abstract: This article discusses an immersed finite element (IFE) space introduced for solving a second-order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body-fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004

Reference EntryDOI
15 Nov 2004
TL;DR: In this paper, the p-version of the finite element method, where the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached.
Abstract: In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached. In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems. We present numerical examples which illustrate the effectiveness of the p-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity.We also discuss roundott error and computational costs associated with the p-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implemen- tation of the p-version.

Journal ArticleDOI
TL;DR: In this article, the authors analyse fracture in multi-phase composite microstructures and a variable stiffness bilinear cohesive model and give a criterion for solution convergence for meshes with uniform, cross-triangle elements.
Abstract: The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack-tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi-phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross-triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al2O3 ceramic and in an Al2O3/TiB2 ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. Copyright © 2004 John Wiley & Sons, Ltd.

Journal ArticleDOI
H. Ji1, John E. Dolbow1
TL;DR: In this paper, the authors consider phase transitions in stimulus-responsive hydrogels, where a sharp interface separates swelled and collapsed phases, and show that as the reciprocal interfacial mobility vanishes, it plays the role of a penalty parameter enforcing a pure Dirichlet constraint.
Abstract: We consider a problem stemming from recent models of phase transitions in stimulus-responsive hydrogels, wherein a sharp interface separates swelled and collapsed phases. Extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly 'fit' the interface geometry are emphasized. Attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface. We show that as the reciprocal interfacial mobility vanishes, it plays the role of a penalty parameter enforcing a pure Dirichlet constraint, eventually triggering oscillations in the interfacial velocity. We also examine alternative formulations employing a Lagrange multiplier to enforce this constraint. It is shown that the most convenient choice of basis for the Lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms, suggesting a lack of stability in the discrete formulation. Under such conditions, neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the Lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux. Fortunately, the adaptation and use of local, domain-integral methodologies considerably improves the flux evaluations. Several example problems are presented to compare and contrast the various techniques and methods.

Journal ArticleDOI
TL;DR: This paper explains how the evaluation of integrals and the transfer between arbitrary finite element spaces can be implemented easily and computed efficiently.
Abstract: The basis of mapped finite element methods are reference elements where the components of a local finite element are defined. The local finite element on an arbitrary mesh cell will be given by a map from the reference mesh cell. This paper describes some concepts of the implementation of mapped finite element methods. From the definition of mapped finite elements, only local degrees of freedom are available. These local degrees of freedom have to be assigned to the global degrees of freedom which define the finite element space. We will present an algorithm which computes this assignment. The second part of the paper shows examples of algorithms which are implemented with the help of mapped finite elements. In particular, we explain how the evaluation of integrals and the transfer between arbitrary finite element spaces can be implemented easily and computed efficiently.

Journal ArticleDOI
TL;DR: In this paper, a new three-dimensional quadratic interface finite element is developed to simulate damage by particle fracture and interface decohesion in composites by numerical simulation in three-dimensions of a representative volume element which reproduces the microstructure.

Journal ArticleDOI
Patrick Wu1
TL;DR: In this article, the authors consider the applicability of commercial finite element (FE) codes for geophysical studies involving long wavelength deformation or viscoelasticity and provide in detail how and why the commercial codes have to be modified when incompressibility is assumed.
Abstract: SUMMARY Modifications to commercial finite element (FE) packages must be applied before they can be used for geophysical studies involving long wavelength deformation or viscoelasticity. This paper provides in detail how and why the commercial codes have to be modified when incompressibility is assumed. Both the non-self-gravitating flat earth and self-gravitating spherical earth will be considered. The latter involves an iterative procedure, which converges within 5 iterations. This is demonstrated both analytically and numerically. In addition, implementation of the gravitationally self-consistent sea level equation on a self-gravitating spherical earth is also described. Good agreement between numerical results obtained with this coupled finiteelement method and the conventional spectral method is also demonstrated. In all cases, the interpretation of the outputs of FE models are particularly important in modelling the state of stress.

Journal ArticleDOI
TL;DR: The method involves a simple shift of the integration points to locations away from conventional Gauss or Gauss–Lobatto integration points, which results in fourth-order accuracy with respect to dispersion error (error in wavelength), as opposed to the second- order accuracy resulting from conventional integration.

Journal ArticleDOI
TL;DR: In this article, a finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described, which generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy-Born rule.
Abstract: The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy-Born rule. The constitutive model for a two-dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper-elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin-shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi-million systems illustrate the computational savings which can be achieved.

Journal ArticleDOI
TL;DR: In this article, an extended stochastic formulation of the triangular composite facet shell element TRIC is presented for the case of combined uncertain material (Young's modulus, Poisson's ratio) and geometric (thickness) properties.

Journal ArticleDOI
TL;DR: In this paper, a partial mixed layerwise finite element model for adaptive plate structures is presented by considering a Reissner mixed variational principle, and the mixed functional is formulated using transverse stresses, displacement components and electric and magnetic potentials as primary variables.

Journal ArticleDOI
TL;DR: In this paper, the fundamentals of a vector form intrinsic finite element procedure (VFIFE) are summarized and numerical results are calculated by using an explicit algorithm using a set of deformation coordinates for the description of kinematics.
Abstract: In a series of three articles, fundamentals of a vector form intrinsic finite element procedure (VFIFE) are summarized. The procedure is designed to calculate motions of a system of rigid and deformable bodies. The motion may include large rigid body motions and large geometrical changes. Newton's law, or a work principle, for particle is assumed to derive the governing equations of motion. They are obtained by using a set of deformation coordinates for the description of kinematics. A convected material frame approach is proposed to handle very large deformations. Numerical results are calculated by using an explicit algorithm. In the first article, using the plane frame element as an example, basic procedures are described. In the accompanied articles, plane solid elements, convected material frame procedures and numerical results of patch tests are given.