Showing papers on "Extended finite element method published in 1999"
TL;DR: In this article, a displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method.
Abstract: SUMMARY An improvement of a new technique for modelling cracks in the nite element framework is presented. A standard displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Copyright ? 1999 John Wiley & Sons, Ltd.
5,815 citations
Book•
14 Jan 1999
TL;DR: In this paper, the Finite Element Method (FEM) was used to solve saddle point problems in incompressible fluid flow and in the theory of elasticity, respectively.
Abstract: Variational formulation of boundary value problems The Finite Element Method (FEM): definition, basic properties hp- Finite Elements in one dimension hp- Finite Elements in two dimensions Finite Element analysis of saddle point problems, mixed hp-FEM in incompressible fluid flow hp-FEM in the theory of elasticity
725 citations
TL;DR: This paper provides a detailed convergence analysis of the multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale.
Abstract: We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.
573 citations
TL;DR: This paper presents a new method which exhibits the best features of both finite volume and finite element techniques, and is robust and capable of delivering high accuracy.
471 citations
TL;DR: A new method for the solution of the Euler and Navier-Stokes equations is introduced, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact, higher-order accurate and stable solver.
Abstract: We introduce a new method for the solution of the Euler and Navier-Stokes equations, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact, higher-order accurate and stable solver. The method involves a weak imposition of continuity conditions on the state variables and on inviscid and diffusive fluxes across interelement and domain boundaries. Within each element the field variables are approximated using polynomial expansions with local support; therefore, this method is particularly amenable to adaptive refinements and polynomial enrichment. Moreover, the order of spectral approximation on each element can be adaptively controlled according to the regularity of the solution. The particular formulation on which the method is based makes possible a consistent implementation of boundary conditions, and the approximate solutions are locally (elementwise) conservative. The results of numerical experiments for representative benchmarks suggest that the method is robust, capable of delivering high rates of convergence, and well suited to be implemented in parallel computers
317 citations
TL;DR: In this paper, a finite element implementation of the Fleck-Hutchinson phenomenological strain gradient theory is reported, which fits within the Toupin-Mindlin framework and deals with first-order strain gradients and the associated work-conjugate higher-order stresses.
Abstract: A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first-order strain gradients and the associated work-conjugate higher-order stresses. In conventional displacement-based approaches, the interpolation of displacement requires C1-continuity in order to ensure convergence of the finite element procedure for higher-order theories. Mixed-type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C0-continuous shape functions and can achieve the same convergence as C1 elements. These C0 elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.
291 citations
TL;DR: A large deformation, linear, viscoelastic model, suitable for direct use with commercially available finite element software packages such as ABAQUS is constructed and the agreement between the proposed theoretical model and the experiment is good.
277 citations
TL;DR: In this paper, the authors investigated the use of least squares methods to approximate the Helmholtz equation and proved convergence theorems for the method and to some extent, control the conditioning of the resulting linear sy stem.
247 citations
TL;DR: In this article, a general formulation for the analysis of complex Bravais crystals using atomic energy functionals embedded within a finite element framework is presented, which uses atomistic potentials to determine the constitutive response of the system.
Abstract: A general formulation for the analysis of complex Bravais crystals using atomic energy functionals embedded within a finite element framework is presented. The method uses atomistic potentials to determine the constitutive response of the system. Unlike traditional finite element methods, the nonlinear elastic effects, the symmetries of the underlying crystal, and the possibility of uniform structural phase transformations are naturally included in this formulation. Explicit expressions for empirical energy functionals with separable two- and three-body potentials, and semiempirical tight-binding energy functionals with two-center integrals are presented. A simple application to silicon underscores the importance of including internal relaxation in a finite element treatment of a complex crystal. In a forthcoming companion paper, the method presented here is applied to the nanoindentation of silicon.
247 citations
TL;DR: A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges.
Abstract: A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an M-matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems.
237 citations
Journal Article•
TL;DR: In this article, the authors present theoretical results on the Arbitrary Lagrangian Eulerian (ALE) formulation for parabolic evolution problems and its relation with Geometric Conservation Laws.
Abstract: In this paper we present some theoretical results on the Arbitrary Lagrangian Eulerian (ALE) formulation. This formulation may be used when dealing with moving domains and consists in recasting the governing differential equation and the related weak formulation in a frame of reference moving with the domain. The ALE technique is first presented in the whole generality for conservative equations and a result on the regularity of the underlying mapping is proven. In a second part of the work, the stability property of two types of finite element ALE schemes for parabolic evolution problems are analyzed and its relation with the so-called Geometric Conservation Laws is addressed.
Book•
17 Nov 1999
TL;DR: In this article, the authors introduce the Finite Difference Method (DFM) and Galerkin Methods to solve the problem of high-frequency problems with Finite Elements, and the solution of the problem is given.
Abstract: 1. Basic Principles of Electromagnetic Fields 2. Overview of Computational Methods in Electromagnetics 3. The Finite Difference Method 4. Variational and Galerkin Methods 5. Shape Functions 6. The Finite Element Method 7. Integral Equations 8. Open Boundary Problems 9. High- Frequency Problems with Finite Elements 10. Low-Frequency Applications 11. Solution of Equations A. Vector Operators B. Triangle Area in Terms of Vertex Coordinates C. Fourier Transform Mehtod D. Integrals of Area Coordinates E. Integrals of Voluume Coordinates F. Gauss-Legendre Quadrature Formulae, Abscissae and, Weight Coefficients G. Shape Functions for 1D Finite Elements H. Shape Functions for 2D Finite Elements I. Shape Functions for 3D Finite Elements References Index
TL;DR: This method is based on the Runge--Kutta discontinuous Galerkin finite element method for solving conservation laws and has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high-order accuracy with a local, compact stencil, and is suited for efficient parallel implementation.
Abstract: In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton--Jacobi equations. This method is based on the Runge--Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high-order accuracy with a local, compact stencil, and is suited for efficient parallel implementation. One- and two-dimensional numerical examples are given to illustrate the capability of the method. At least kth order of accuracy is observed for smooth problems when kth degree polynomials are used, and derivative singularities are resolved well without oscillations, even without limiters.
TL;DR: In this paper, a model of rubber elasticity, the extended tube-model, is introduced, which considers the topological constraints as well as the limited chain extensibility of network chains in filled rubbers.
Abstract: A novel model of rubber elasticity—the extended tube-model—is introduced. The model considers the topological constraints as well as the limited chain extensibility of network chains in filled rubbers. It is supplied by a formulation suitable for an implementation into a finite element code. Homogeneous states of deformation are evaluated analytically to yield expressions required e.g., for parameter identification algorithms. Finally, large scale finite element computations compare the extended tube-model with experimental investigations and with the phenomenological strain energy function of the Yeoh-model. The extended tube-model can be considered as an interesting approach introducing physical considerations on the molecular scale into the formulation of the strain energy function which is on the other hand the starting point for the numerical realization on the structural level. Thus, the gap between physics and numerics is bridged. Nevertheless, this study reveals the importance of a proper...
TL;DR: In this paper, a mixed finite element formulation using the standard Galerkin weighted residual method is presented and the resulting first-order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme.
Abstract: An equivalent new expression of the triphasic mechano-electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement us, modified electrochemical/chemical potentials ϵw, ϵ+and ϵ− (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton–Raphson iterative procedure is employed to handle the non-linear terms. The resulting first-order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One-dimensional (1-D) linear isoparametric element is developed. The final algebraic equations form a non-symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1-D triphasic stress relaxation problem in the confined compression configuration and a 1-D triphasic free swelling problem. The formulation accuracy and convergence for 1-D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods. Copyright © 1999 John Wiley & Sons, Ltd.
TL;DR: In this article, an improved unusual finite element method is studied for a second-order linear scalar differential equation including a zero order term, which consists in subtracting from the standard Galerkin method a mesh dependent term suggested by static condensation of the bubbles.
TL;DR: The Dirichlet-to-Neumann (DtN) Finite Element Method as discussed by the authors is a general technique for the solution of problems in unbounded domains, which arise in many fields of application.
Abstract: The Dirichlet-to-Neumann (DtN) Finite Element Method is a general technique for the solution of problems in unbounded domains, which arise in many fields of application. Its name comes from the fact that it involves the nonlocal Dirichlet-to-Neumann (DtN) map on an artificial boundary which encloses the computational domain. Originally the method has been developed for the solution of linear elliptic problems, such as wave scattering problems governed by the Helmholtz equation or by the equations of time-harmonic elasticity. Recently, the method has been extended in a number of directions, and further analyzed and improved, by the author's group and by others. This article is a state-of-the-art review of the method. In particular, it concentrates on two major recent advances: (a) the extension of the DtN finite element method tononlinear elliptic and hyperbolic problems; (b) procedures forlocalizing the nonlocal DtN map, which lead to a family of finite element schemes with local artificial boundary conditions. Possible future research directions and additional extensions are also discussed.
TL;DR: A fully developedr-adaptive finite element method can be expected to be ideally suited to complement the currently popularh-pfinite element methods and to provide increased reliability and efficiency for mesh adaptation.
TL;DR: In this paper, the authors developed a new stabilization operator which globally reconstructs the derivatives not present in the local element function space, which is seen to engender a stronger consistency leading to better convergence and improved accuracy.
TL;DR: In this paper, a numerical procedure is developed for the form finding of minimal surfaces and prestressed membrane structures based on an iteratively adapted regularisation of the stiffness matrix which gives the method its name: the updated reference strategy.
Abstract: Starting from a rigorous mechanical formulation, a numerical procedure is developed for the form finding of minimal surfaces and prestressed membrane structures. The method is based on an iteratively adapted regularisation of the stiffness matrix which gives the method its name: the updated reference strategy. The method can be applied to any finite element discretization of cable and membrane structures subjected to pre-stress as well as lateral pressure or other external loading. It is very robust and reliable as is shown by many illustrative examples. Further analysis states that the well known force density method is the special case of applying the updated reference strategy to cable nets.
TL;DR: The goal of this paper is to define a procedure for bounding the error in a conforming finite element method that is fully explicit and can be computed locally.
Abstract: The goal of this paper is to define a procedure for bounding the error in a conforming finite element method. The new point is that this upper bound is fully explicit and can be computed locally. Numerical tests prove the efficiency of the method. It is presented here for the case of the Poisson equation and a first order finite element approximation.
TL;DR: In this article, a stabilized mixed finite element method for finite elasticity is presented, which circumvents the fulfillment of the Ladyzenskaya-Babuska-Brezzi condition by adding meshdependent terms, which are functions of the residuals of the Euler-Lagrange equations, to the usual Galerkin method.
TL;DR: In this article, a finite element algorithm is presented for the simulation of steady incompressible fluid flow with heat transfer using triangular meshes, which is modified by employing the artificial compressibility concept to provide coupling between the pressure and velocity fields of the fluid.
Abstract: A finite element algorithm is presented for the simulation of steady incompressible fluid flow with heat transfer using triangular meshes. The continuity equation is modified by employing the artificial compressibility concept to provide coupling between the pressure and velocity fields of the fluid. A standard Galerkin finite element method is used for spatial discretization and an explicit multistage Runge‐Kutta scheme is used to march in the time domain. The resulting procedure is stabilized using an artificial dissipation technique. To demonstrate the performance of the proposed algorithm a wide range of test cases is solved including applications with and without heat transfer. Both natural and forced convection applications are studied.
TL;DR: A new variant of the discontinuous Galerkin method is presented which is applicable to the numerical solution of diffusion problems, not requiring auxiliary variables such as those used in mixed methods.
Abstract: This paper presents the mathematical analysis of a new variant of the discontinuous Galerkin method which is applicable to the numerical solution of diffusion problems, not requiring auxiliary variables such as those used in mixed methods. The focus of this study is on a class of linear second-order boundary value problems for which we prove stability and a priori error estimates in both the finite- and infinite-dimensional spaces.
TL;DR: In this paper, an immersed finite element space is used to solve the elliptic interface problems by a finite volume element method, and special nodal basis functions are introduced in a triangle whose interior intersects with the interface so that the jump conditions across the interface are satisfied.
TL;DR: In this article, a family of explicit space-time finite element methods for the initial boundary value problem for linear, symmetric hyperbolic systems of equations is described and analyzed.
Abstract: A family of explicit space-time finite element methods for the initial boundary value problem for linear, symmetric hyperbolic systems of equations is described and analyzed. The method generalizes the discontinuous Galerkin method and, as is typical for this method, obtains error estimates of order $O(h^{n+1/2})$ for approximations by polynomials of degree $\le n$.
TL;DR: In this paper, a co-rotational total Lagrangian finite element formulation for the geometrically nonlinear dynamic analysis of spatial Euler beam with large rotations but small strain is presented.
TL;DR: In this paper, the Stokes and Navier-Stokes equations were approximated by a piecewise-constant element for the pressure and a stable element in both cases for which optimal error estimates for the velocity and pressure in L2 can be established.
Abstract: Recently, Douglas et al. [4] introduced a new, low-order, nonconforming rectangular element for scalar elliptic equations. Here, we apply this element in the approximation of each component of the velocity in the stationary Stokes and Navier–Stokes equations, along with a piecewise-constant element for the pressure. We obtain a stable element in both cases for which optimal error estimates for the approximation of both the velocity and pressure in L2 can be established, as well as one in a broken H1-norm for the velocity.
TL;DR: In this paper, a modified Hu-Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2, which is more efficient in numerical simulations, especially for large deformation problems.
Abstract: In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive B functions which avoid numerical inversion of matrices. The B-strain matrix is sparse and has the same structure as the strain matrix B obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a B-strain matrix instead of the standard B-matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed-enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements. Copyright © 1999 John Wiley & Sons, Ltd.
TL;DR: In this paper, the authors presented appropriate finite element concepts accounting for finite elastoplastic strains and isotropic stress response in arbitrary shells and discussed three parameterization strategies for the calculation of thin and thick smooth shells as well as shell intersections.