Showing papers on "Extended finite element method published in 1988"

Neumann Expansion for Stochastic Finite Element Analysis

Abstract: With the aid of the finite element method, the present paper deals with the problem of structural response variability resulting from the spatial variability of material properties of structures, when they are subjected to static loads of a deterministic nature. The spatial variabilities are modeled as two‐dimensional stochastic fields. The finite element discretization is performed in such a way that the size of each element is sufficiently small. Then, the present paper takes advantage of the Neumann expansion technique in deriving the finite element solution for the response variability within the framework of the Monte Carlo method. The Neumann expansion technique permits more detailed comparison between the perturbation and Monte Carlo solutions for accuracy, convergence, and computational efficiency. The result from such a Monte Carlo method is also compared with that based on the commonly used perturbation method. The comparison shows that the validity of the perturbation method is limited to the c...

A finite element with embedded localization zones

Abstract: A method is developed by which localization zones can be embedded in four-node quadrilaterals and related elements. This is accomplished by modifying the strain field within the framework of a three-field variational statement. The jumps in strain associated with the localization band are obtained by imposing traction continuity and compatibility within the element; the latter follows naturally from the variational statement. Several one- and two-dimensional applications are shown.

Deformation from Inflation of a Dipping Finite Prolate Spheroid in an Elastic Half‐Space as a Model for Volcanic Stressing

Abstract: Exact analytic expressions are given for the deformation field resulting from inflation of a finite prolate spheroidal cavity in an infinite elastic medium. The field is equivalent to that generated by a parabolic distribution of double forces and centers of dilatation along the spheroid generator. Approximate, but quite accurate, solutions for a dipping spheroid in an elastic half-space are found using the half-space double force and center of dilatation solutions. We compare results of the surface deformation field with those generated by the point source ellipsoidal model of Davis (1986). In the far field both models give identical results. In the near field the finite model must be used to calculate displacements and stresses within the medium. We also test the limits of applicability of the finite model as it approaches the surface by comparing the surface displacement field from a vertical spheroid with that calculated from the finite element method. We find the model gives a satisfactory approximation to the finite element results when the minimum radius of curvature of the upper surface is less than or equal to its depth beneath the free surface. Comparison of surface displacements generated by the point and finite element models gives good agreement, provided this criterion is satisfied. We have used the finite model to invert deformation data from Kilauea volcano, Hawaii. The results, which compare favorably with those obtained from the point ellipsoid model, can be used to estimate the distribution of stresses within the volcano in the near field of the source.

Response Variability of Stochastic Finite Element Systems

Abstract: The response variability of finite element systems arising from the spatial randomness of the material properties is examined. The system is subjected to static loads of a deterministic nature. The problem is analyzed using the finite element method along with a first-order Neumann expansion of the stiffness matrix of the system. The covariance matrix of the response displacement vector is calculated analytically as a function of the number of finite elements. The finite element size necessary to obtain sufficiently accurate values of the stochastic response parameters is examined thoroughly. Various conclusions are drawn concerning the convergence of the coefficient of variation of the response strain to its final value as a function of the number of finite elements. The main advantage of the method is its computational efficiency, since all the response statistics are computed analytically and not through a Monte Carlo simulation.

Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua

Abstract: The fundamental arbitrary Lagrangian-Eulerian (ALE) mechanics and its finite element formulation are given. The tangential stiffness matrix, which is shown to be composed of the linarized material response matrix, the geometrical stiffness matrix, and the ALE transport matrix are derived from a consistent linearization procedure. Various numerical methods for the ALE finite element equations are then presented, and several examples are analyzed to examine some features of the method.

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Abstract: The Lagrange-Galerkin method is a numerical technique for solving convection -- dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation The method is shown to be nonlinearly stable

A quadrilateral finite element including vertex rotations for plane elasticity analysis

Abstract: A finite element of arbitrary quadrilateral shape is presented for plane elasticity analysis. The element is derived from triangular fields of compatible quadratic displacements with vertex connectors which include rotations. All rigid body movements and constant strain states are recovered exactly. Results from several test problems demonstrate that very good numerical accuracy is obtained for both displacements and stresses, even for quite coarse finite element meshes.

A quadratically convergent procedure for the calculation of stability points in finite element analysis

Abstract: In this paper a new finite element formulation is given for the analysis of nonlinear stability problems. The introduction of extended systems opens the possibility to compute limit and bifurcation points directly. Here, the use of the directional derivative yields a quadratically convergent iteration scheme. The combination with arc-length and branch-switching procedures leads to a global algorithm for path-following.

A rational approach for choosing stress terms for hybrid finite element formulations

Abstract: A new approach for choosing the stress terms for hybrid stress elements is based on the condition of vanishing of the virtual work along the element boundary due to the stress terms higher than constant and the additional incompatible displacements. Examples using 4-node plane stress elements have shown that when the incompatible displacements also satisfy the constant strain patch test the resulting element will provide the most accurate solutions.

Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra

Abstract: A technique for representing large finite rotations in terms of only three independent parameters, the conformal rotation vector, is described and applied to the finite element formulation of 3-D mechanisms problems. A beam finite element that takes into account large finite rotations and various types of rigid joints have been developed. Some test examples which demonstrate the applicability of the proposed technique are presented.

A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation

Abstract: Adding to the classical Hellinger Reissner formulation another residual form of the equilibrium equation, a new Petrov-Galerkin finite element method is derived. It fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations, including equal-order discontinuous stress and continuous displacement interpolations which are unstable within the Galerkin approach. Error estimates are presented using the Babuska-Brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method.

Applications of the hybrid finite element-boundary element method in electromagnetics

Abstract: The concept of combining the finite-element method with the boundary-element method for electromagnetic problems is introduced. The general equations are derived, and examples are given for a number of two- and three-dimensional cases. These include both static and time-varying problems. >

The p‐version of the finite element method for domains with corners and for infinite domains

Abstract: A special approach to deal with elliptic problems with singularities is introduced. It is shown that this approach, to be called an auxiliary mapping technique, in the framework of the p-version of the finite element method yields an exponential rate of convergence. It is also shown that this technique can deal with elliptic problems on unbounded domains in R2 as well. (AMS(MOS) subject classifications: Primary, 65N30, 65N15.)

3-D finite element method for analyzing magnetic fields in electrical machines excited from voltage sources

Abstract: A method for analyzing 3-D magnetic fields and currents in electrical machines excited from voltage sources has been developed. It was obtained by expanding the 2-D finite-element method into three dimensions, using the A phi method. The basic idea and the finite-element formulation are described. The effectiveness of the method is shown by some application examples. >

The Localized Finite Element Method and Its Application to the Two-Dimensional Sea-Keeping Problem

Abstract: We study the various convergence properties of the localized finite element method, which is devoted to the numerical solution of linear problems set in unbounded domains The problem under consideration is the so-called sea-keeping problem, ie, the determination of the motion of ships undergoing the action of an incident swell, but in fact the method applies to numerous other problems Copyright © 1988 Society for Industrial and Applied Mathematics

Generalized coupled transient thermoelastic plane problems by laplace transform/finite element method

Abstract: A general finite-element model is proposed to deal with dynamic theromelastic problems especially with longer transient period. The method consists of formulating and solving the problem in the Laplace transform domain by the Finite Element Method (FEM) and then numerically inverting the transformed solution to obtain the time-domain response. Therefore, the transient solutions at any time could be evaluated directly. A number of examples are presented which demonstrate the accuracy, efficiency, and versatility of the proposed method, and show the effects of relaxation times, inertia, and thermoelastic coupling terms.

Calculation of three dimensional eddy current by FEM-BEM coupling method

Abstract: A coupled finite element-boundary element method is applied to three-dimensional eddy-current problems. The magnetic vector potential A and scalar potential phi are adopted for the formulation. The finite element method is used in the conducting region, while the boundary element method is used in the exterior vacant region. A symmetric linking method is chosen for the coupling procedure. The calculations show good agreement with analytical results and results obtained by other numerical methods. >

Numerical solution of the Fokker Planck equation using moving finite elements

Abstract: Numerical solution of the Fokker Planck equation for the probability density function of a stochastic process by traditional finite difference or finite element methods produces erroneous oscillations and negative values whenever the drift is large compared to the diffusion Upwinding schemes to eliminate the oscillations introduce false numerical diffusion because it is impossible to make the one step drift large enough to match the original equation without making the one step diffusion too large A variation of the moving finite element method is presented that overcomes these difficulties by using basis functions that satisfy the drift part of the equation by moving along the trajectories of the deterministic dynamical system associated with the stochastic process A Galerkin type method can then be used to find the coefficients in the remaining pure diffusion equation Solutions of two test equations are presented to illustrate the effectiveness of the method