Showing papers on "Extended finite element method published in 1974"

On a Finite Element Method for Solving the Neutron Transport Equation

Abstract: Publisher Summary This chapter discusses the finite element method for solving the neutron transport equation and spatial discretization. It also discusses the numerical approximation of a problem by a finite element method using triangular or quadrilateral elements and other methods for solving the neutron transport equation. The discrete Galerkin method is equivalent to some implicit Runge-Kutta method. The existence and uniqueness of the approximate solution and an algorithm for computing the approximate solution is given. In the chapter, the finite element method is studied which provides an effective way for numerically solving such problems.

Local and global smoothing of discontinuous finite element functions using a least squares method

Abstract: The concepts and potential advantages of local and global least squares smoothing of discontinuous finite element functions are introduced. The relationship between local smoothing and the ‘reduced’ integration' technique is established. Examples are presented to illustrate the application of the two smoothing techniques to the finite element stresses from several structural analysis problems. The paper concludes with some practical recommendations for discontinuous finite element function smoothing.

Finite Element and Finite Difference Methods for Hyperbolic Partial Differential Equations

Abstract: Publisher Summary This chapter defines finite element and finite difference methods for hyperbolic partial differential equations. The advantage of the finite element method is that the resulting procedures are automatically stable and there is extreme flexibility in choosing the basic functions. Therefore, in very complicated domains or for problems with complicated interfaces, the method is the only feasible one. For hyperbolic partial differential equations it is essential to control the dispersion, dissipation, and the propagation of discontinuities. This is easily done by using suitable difference approximations. The main disadvantage of finite difference methods is that it may be difficult to handle boundaries properly.

Representation of singularities with isoparametric finite elements

Abstract: The development of a generalized quadrilateral finite element that includes a singular point at a corner node is presented. Inter-element conformability is maintained so that monotone convergence is preserved. The global-local concept of finite elements is used to formulate the complete set of equations. Examples of crack tip singularities are given.

Transient two‐dimensional heat conduction problems solved by the finite element method

Abstract: A finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.

A second order finite element method for the one‐dimensional Stefan problem

Abstract: We describe a finite element method for the one-dimensional Stefan problem The elements are quadrilaterals of the space-time plane which are determined at each time-step in relation with the position of the free boundary The method appears as a generalization of the classical Crank-Nicolson scheme, since it is identical to this scheme in the case of rectangular elements; it has the advantage of providing a simple and accurate determination of the free boundary Numerical experiments show that the order of accuracy is equal to 2

The Relative Efficiency of Finite Difference and Finite Element Methods. I: Hyperbolic Problems and Splines

Abstract: The finite element method, using smooth splines as basis functions, applied to the model problem $u_t = cu_x $ with periodic data generates a differential-difference equation whose phase error is closely estimated and compared with the phase error of both explicit and high order implicit centered differencing. We also compute and compare the minimum work required to obtain a fixed error for several fully discrete schemes.

Steady state flow in rigid networks of fractures

Abstract: The finite element has been used to develop two numerical methods of calculating the flow characteristics of rigid networks of planar fractures. One method uses triangular elements to investigate details of laminar flow in fractures of irregular cross section combined with that of a permeable rock matrix. The other method uses line elements and is designed only for flow in networks of planar fractures in an impermeable matrix. As an example of the application of the finite element approach the line element method was used to develop a series of dimensionless graphs that characterize seepage in idealized fracture systems beneath dams. These methods treat two-dimensional flow in the laminar regime for networks of fractures of arbitrary orientation and aperture distribution.

On the accuracy of finite element solutions to the transient heat-conduction equation

Abstract: This paper discusses three numerical integration schemes connected with a finite element solution to the transient heat-conduction equation. The main emphasis is on the achievable accuracy. A simple recurrence relation derived from the Galerkin process is suggested for the treatment of fast-varying boundary conditions as being significantly better than the usual Crank-Nicholson scheme in matters of short-time accuracy.

Integrations of the primitive equations on a sphere using the finite element method

Abstract: Integrations of the shallow water equations on the sphere using the finite element method are performed and compared with published integrations of Doron et al. (1974). Better results are obtained with the finite element method than with a second order finite difference method using four times the number of grid points, in particular the breakdown of a Rossby wave of zonal wavenumber 8 is correctly predicted.

Inclusion of Transverse Shear Deformation in Finite Element Displacement Formulations

Abstract: A stiffness matrix is derived for a beam element with transverse shear deformation. It is shown that straightforward energy minimization yields the correct stiffness matrix in displacement formulations when transverse shear effects are considered. Since the TIM4 beam element does not represent the geometric boundary conditions for a cantilever beam the rotation of the normal must be retained as a grid point degree of freedom.

Finite elements in the solution of electromagnetic induction problems

Abstract: This paper describes a method of solving electromagnetic induction problems by means of the finite element technique. A variational equation associated with the differential equation for the vector potential is formulated and solved via the concept of finite elements. Sinusoidal driving currents and linear, isotropic, but inhomogeneous media are assumed. A typical example is included in order to illustrate the suggested solution technique.

An application of finite element techniques to post-yield analysis of proposed standard three-point bend fracture test pieces

Abstract: Making use of an Elastic-Plastic Finite Element Program, two recommended fracture testing geometries are analysed Special attention is given to the singular environment at a crack tip, characterised by Rice's contour integral, and its relationship with the COD concept suggested by Wells

Finite Element Methods for Parabolic Equations

Abstract: The initial-boundary value problem for a linear parabolic equation with the Dirichlet boundary condition is solved approximately by applying the finite element discreti- zation in the space dimension and three types of finite-difference discretizations in time: the backward, the Crank-Nicolson and the Calahan discretization. New error bounds are derived. 1. Introduction. A number of years ago, engineers applied the finite element method to the solution of the heat conduction problem. We mention the papers by Visser (7) and by Wilson and Nickell (8). Their idea is that in the space dimension a finite element discretization is used whereas in time a finite-difference method is applied. Recently, there appeared papers in mathematical journals where these methods were analyzed as well as new methods proposed, some of them of higher order of accuracy, and where error bounds of a different kind were derived. We mention the papers by Douglas and Dupont (3), Hlavacek (5) and Bramble and

Solution of Problems with Interfaces and Singularities

Abstract: Publisher Summary This chapter discusses the solution of problems with interfaces and singularities. There has been considerable progress made in theoretical investigations of the finite element method. There is an area in the finite element method where neither theoretical nor practical progress has been made, which is comparable with the progress in the general treatment of this method. This area is connected with handling the singularities of the solution arising especially because of interfaces and unsmoothness of the domain. There are two kinds of problems that have to be investigated separately: the global problems and local problems. The global solution is used for the one dimensional problem. The local problem has a quite different character from global problems. The difficulties lie in the problem of determining how much the convergence of the finite element method will be slowed down when the solution of the problem has some singularities.

Galerkin approximation of the time derivative in the finite element analysis of groundwater flow

Abstract: Whereas considerable effort has been expended in generating approximations to the spatial derivatives encountered in porous media flow, the time derivative has received relatively little attention. In spite of the fact that sophisticated finite element formulations have been developed for the spatial derivatives, finite difference methods are generally applied to the time derivative. The Galerkin method of approximation permits a high-order approximation in time as well as in space. The resulting approximate equations have been successfully solved by using a prismatic element with triangular cross section. The time axis runs the length of the prism and is subdivided into elements that may be linear, quadratic, or cubic. Because this formulation requires in general the solution for several time levels simultaneously, there is a resulting increase in computer time required to solve the larger matrix. Numerical experiments indicate that the selection of an optimum numerical scheme is dependent not only on the particular problem considered but also on the sequence of time steps used.

Finite Element Formulation and Solution of Contact-Impact Problems in Continuum Mechanics

Abstract: This report presents a general theory of contract-impact problems cast in a variational theorem suitable for implementation with the finite element method. The numerical scheme is described as is the structural analysis computer code in which it is contained.

Least squares finite element analysis of laminar boundary layer flows

Abstract: Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non-linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.

Nonlinear Thermo-Elastic-Plastic and Creep Analysis by the Finite-Element Method

Abstract: Consistent with a Lagrangian displacement formulation, an incremental constitutive relation for uncoupled thermoelastic-plastic and creep deformations is presented. The nonisothermal von Mises yield function and its associated flow rule are utilized, together with both isotropic and kinematic hardening rules. Steady-state creep deformations are considered using Norton-Odqvist's power law. This development is particularly applicable to the nonlinear finite element analysis of three-dimensional structures with timeand temperature-dependent material properties. Using a nonlinear general-purpose computer program which has been developed on the basis of this formulation, a number of numerical examples are solved and the results compared with the closed-form solutions.