Showing papers on "Extended finite element method published in 1984"

An Introduction to the Finite Element Method

Abstract: 1 Introduction 2 Mathematical Preliminaries, Integral Formulations, and Variational Methods 3 Second-order Differential Equations in One Dimension: Finite Element Models 4 Second-order Differential Equations in One Dimension: Applications 5 Beams and Frames 6 Eigenvalue and Time-Dependent Problems 7 Computer Implementation 8 Single-Variable Problems in Two Dimensions 9 Interpolation Functions, Numerical Integration, and Modeling Considerations 10 Flows of Viscous Incompressible Fluids 11 Plane Elasticity 12 Bending of Elastic Plates 13 Computer Implementation of Two-Dimensional Problems 14 Prelude to Advanced Topics

A stable finite element for the stokes equations

Abstract: We present in this paper a new velocity-pressure finite element for the computation of Stokes flow. We discretize the velocity field with continuous piecewise linear functions enriched by bubble functions, and the pressure by piecewise linear functions. We show that this element satisfies the usual inf-sup condition and converges with first order for both velocities and pressure. Finally we relate this element to families of higer order elements and to the popular Taylor-Hood element.

Rational approach for assumed stress finite elements

Abstract: A new method for the formulation of hybrid elements by the Hellinger-Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co-ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4-node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.

PEERS: A new mixed finite element for plane elasticity

Abstract: A mixed finite element procedure for plane elasticity is introduced and analyzed. The symmetry of the stress tensor is enforced through the introduction of a Lagrange multiplier. An additional Lagrange multiplier is introduced to simplify the linear algebraic system. Applications are made to incompressible elastic problems and to plasticity problems.

The finite element method displayed

Abstract: Simplifies the teaching of the finite element method. Topics covered include: the approximation of continuous functions over sub-domains in terms of nodal values; interpolation functions for classical elements in one, two, and three dimensions; fundamental element vectors and matrices and assembly techniques; numerical methods of integration; matrix Eigenvalue and Eigenvector problems; and Fortran programming techniques. Contains tables of formulas and constants for constructing codes.

Finite element solution of diffusion problems with irregular data

Abstract: Diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break-down of the solution's regularity. This may drastically reduce the accuracy of discretization schemes over the whole interval of integration, unless certain precautions are taken. The diagonal Pade schemes of order 2μ, combined with a standard finite element discretization, usually require an unnatural step size restriction in order to achieve even locally optimal accuracy. It is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and, in the case μ=1, does not increase the costs.

Penalty Function Improvement of Waveguide Solution by Finite Elements

Abstract: The finite element method is a well-established method for the solution of a wide range of guided wave problems. One drawback associated with the powerful vector formulation is the appearance of spurious or nonphysical solutions. A penalty function method has been introduced to the finite element formulation, to reduce or eliminate spurious solutions. It also improves the quality of the physical field solutions. The method has been applied for the solution of metallic homogeneous and inhomogeneous guides, and integrated optics guides.

Analysis of mixed finite elements methods for the Stokes problem: a unified approach

Abstract: We develop a method for the analysis of mixed finite element methods for the Stokes problem in the velocity-pressure formulation. A technical "macroelement condition", which is sufficient for the classical Babuska-Brezzi inequality to be valid, is introduced. Using this condition,we are able to verify the stability, and optimal order of convergence, of several known mixed finite element methods.

The self-equilibration of residuals and complementarya posteriori error estimates in the finite element method

Abstract: It is demonstrated that the residual in a compatible (displacement) finite element solution can be partitioned into local self-equilibrating systems on each element. An a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved.

A Dissipative Galerkin Scheme for Open‐Channel Flow

Abstract: The finite element method based on the classical Galerkin formulation produces very poor results when applied to discontinuous channel flow. A variance of the Galerkin method is proposed that exhibits highly selective damping characteristics. The dissipation affects only the numerically-generated high-frequency parasitic waves, while maintaining remarkable accuracy in the approximation to the true solution of the problem. In fact, it is shown that the phase error of the finite element simulation is improved by the introduction of dissipation. The resulting model is second-order accurate with respect to the time step and produces a clean, sharp jump structure that agrees favorably with the exact solution of some test problems. The method is based on discontinuous weighting functions that produce “upwind” effects but at the same time maintain the accuracy of a central difference scheme. The dissipation level is selected by analytical investigations, so that the numerical error is minimized. No second-order pseudo viscosity terms are required, which relaxes the inter-element continuity conditions and results in a very simple and inexpensive scheme.

The discrete wavenumber/finite element method for synthetic seismograms

Abstract: Summary. A new method is presented for computing the complete elastic response of a vertically heterogeneous half-space. The method utilizes a discrete wavenumber decomposition for the horizontal dependence of the wave motion in terms of a Fourier-Bessel series. The series representation is exact if summed to infinity and consequently eliminates the need to integrate a continuous Bessel transform numerically. In practice, a band-limited solution is obtained by truncating the series at large wavenumbers. The vertical and time dependence of the wave motion is obtained as the solution to a system of partial differential equations. These equations are solved numerically by a combination of finite element and finite difference methods which accommodate arbitrary vertical heterogeneities. By using a reciprocity relation, the wave motion is computed simultaneously for all source-observer combinations of interest so that the differential equations need only be solved once. A comparison is made, for layered media, between the solutions obtained by discrete wavenumber/finite element, wavenumber integration, axisymmetric finite element, and generalized rays.

Efficient linear and nonlinear heat conduction with a quadrilateral element

Abstract: A method is presented for performing efficient and stable finite element calculations of heat conduction with quadrilaterals using one-point quadrature. The stability in space is obtained by using a stabilization matrix which is orthogonal to all linear fields and its magnitude is determined by a stabilization parameter. It is shown that the accuracy is almost independent of the value of the stabilization parameter over a wide range of values; in fact, the values 3, 2 and 1 for the normalized stabilization parameter lead to the 5-point finite difference, 9-point finite difference and fully integrated finite element operators, respectively, for rectangular meshes; numerical experiments reported here show that the three have identical rates of convergence in the L2 norm. Eigenvalues of the element matrices, which are needed for stability limits, are also given. Numerical applications are used to show that the method yields accurate solutions with large increases in efficiency, particularly in nonlinear problems.

A finite element -- Capacitance method for elliptic problems on regions partitioned into subregions

Abstract: A method is given for the solution of linear equations arising in the finite element method applied to a general elliptic problem. This method reduces the original problem to several subproblems (of the same form) considered on subregions, and an auxiliary problem. Very efficient iterative methods with the preconditioning operator and using FFT are developed for the auxiliary problem.

Time Finite Element Discretization of Hamilton's Law of Varying Action

Abstract: Hamilton's Law of Varying Action is used as a variational source for the derivation of finite element discretization procedure in the time domain. Three different versions of the proposed algorithms are presented and verified for accuracy and stability. [The first one is the high-precision, finite time element, analogous to the standard finite elements, with cutoff frequency; the second version is the step by step, one-time element from which the unconditionally stable, with slightly altered accuracy, third algorithm is derived. ] The new operator, connected with the proposed algorithms, bears attractive properties of much greater accuracy than other existing stable methods, and easy computer implementation. Thus, the work herein shows that the reservations expressed against the use of finite elements in time domain seem unjustified.

Finite element mesh generation

Abstract: The capabilities of a geometric modeller are extended towards finite element analysis by a mesh generator which extracts all its geometric and topological information from the model. A coarse mesh is created and subsequently refined to a suitable finite element mesh, which accomodates material properties, loadcase and analysis requirements. The mesh may be optimized by adaptive refinement, ie according to estimates of the discretization errors. A survey of research and development in geometric modelling and finite element analysis is presented, then an implementation of a mesh generator for 3D curvilinear and solid objects is described in detail.

On the implementation of finite strain plasticity equations in a numerical model

Abstract: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

A total lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements. Part I. Two-dimensional problems: Shell and plate structures

Abstract: A total Lagrangian finite element formulation for the geometrically nonlinear analysis (large displacement/large rotations) of shells is presented. Explicit expressions of all relevant finite element matrices are obtained by means of the definition of a local co-ordinate system, based on the shell principal curvature directions, for the evaluation of strains and stresses. A series of examples of nonlinear analysis of shell and plate structures is given.

A shear‐flexible triangular finite element model for laminated composite plates

Abstract: Formulation and numerical evaluation of a shear-flexible triangular laminated composite plate finite element is presented in this paper. The element has three nodes at its vertices, and displacements and rotations along with their first derivatives have been chosen as nodal degrees-of-freedom. Computation of element matrices is highly simplified by employing a shape function subroutine, and an optimal numerical integration scheme has been used to improve the performance. The element has satisfactory rate of convergence and acceptable accuracy with mesh refinement for thick as well as thin plates of both homogeneous isotropic and laminated anisotropic materials. The numerical studies also suggest that reliable prediction of the behaviour of laminated composite plates necessitates the use of higher order shear-flexible finite element models, and the proposed finite element appears to have some advantages over available elements.

Finite element and boundary element methods for extrusion computations

Abstract: The present paper reports some new finite element and boundary element results for the expansion of a plane jet of Maxwell fluid. Finite element results for the axisymmetric problem are also presented. The boundary element method represents a new method of solving this problem. The Maxwell stresses are evaluated by integrating along the streamlines; these are then incorporated in the boundary element formulation. The new results are compared with existing finite element solutions of this problem. Although all sets of results predict the same overall behaviour, there is still little agreement on the exact behaviour

Interelement Continuity in the Boundary Element Method

Abstract: It is well known that homogeneous elliptic field problems may be alternatively posed as infinite systems of boundary integral equations obtained by using a suitable family of kernel functions and integration by parts [1], [2]. A determinate system of non-homogeneous linear algebraic equations is obtained therefrom by discretizing the system, using elements defined after the manner of finite elements and a finite member of Kernel functions [1]. The result is a practical and powerful numerical method for the solution of elliptic field problems which may easily be generalized to rival the finite element method in its range of applicability.

Beam on Elastic Foundation Finite Element

Abstract: A stiffness matrix for a beam on elastic foundation finite element and element load vectors due to concentrated forces, concentrated moments, and linearly distributed forces are developed for plane frame analysis. This element stiffness matrix can be readily adopted for the conventional displacement method. For the force method, an element flexibility matrix and element displacement vectors due to the aforementioned loads are also presented. Whereas most other analyses of a beam on elastic foundation finite element approximate the foundation by discrete springs or by cubic hermitian polynomials, the present stiffness and flexibility matrices are derived from the exact solution of the differential equation. Thus, results of this finite element analysis are exact for Navier and Winkler assumptions. Numerical examples are given to demonstrate the efficiency and simplicity of the element.