Showing papers on "Finite element method published in 1973"

The finite element method with Lagrangian multipliers

Abstract: The Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essential (stable) boundary conditions. The implementation is based on the application of Lagrangian multiplier. The rate of convergence is proved.

Conforming and nonconforming finite element methods for solving the stationary Stokes equations I

Abstract: — The paper is devoted to a gênerai finite element approximation ofthe solution of the Stokes équations for an incompressible viscous fluid, Both conforming and nonconforming finite element methods are studied and various examples of simplicial éléments well suitedfor the numerical treatment of the incompressibility condition are given. Optimal error estimâtes are derived in the energy norm and in the L-norm.

Finite Element Method Accuracy for Wave Propagation Problems

Abstract: The purpose of this paper is presentation of the results of a study concerning the accuracy of displacements caused by a single, harmonic, one-dimensional elastic wave propagating through a finite element mesh. Results are presented for the steady-state response of a finite model of the semi-infinite elastic constrained rod; both the homogeneous and two material cases were analyzed.

Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic-plastic materials

Abstract: The plane strain elastic-plastic state at a crack tip is determined for compact tension, bend, double edge-cracked and centre-cracked specimens using a finite element method with triangular constant-strain elements. The solutions are found to differ by 10 to 30 per cent at the ASTM-limit as regards fracture surface displacement, normal stress and plastic zone size. In order to bring the boundary layer solution for the crack problem into agreement with the solution for a specific specimen one has to modify this solution. The modification consists of an addition to the boundary tractions for the boundary layer problem of tractions corresponding to the non-singular, constant second term in a series expansion of the normal stress parallel to the crack plane.

Construction of curvilinear co-ordinate systems and applications to mesh generation

Abstract: Computer-oriented mesh generators, which serve as pre-processors to finite element programs, have recently been developed by several investigators to alleviate the frustration and to reduce the amount of time involved in the tedious manual subdividing of a complex structure into finite elements. Our purpose here is to describe how the techniques of bivariate ‘blending-function’ interpolation, which were originally developed for, and applied to, geometric problems of computer-aided design and numerically controlled machining of free-form surfaces such as automobile exterior panels, can be adapted and applied to the problems of mesh generation for finite element analyses. We concentrate attention on the problem of curvilinearly co-ordinating simply connected planar domains ℛ by constructing invertible maps of the unit square [0, 1] × [0, 1] onto ℛ. Extensions of the methods described herein to shells in 3-space is straightforward and is illustrated by a practical example taken from the automobile industry. Analogous mesh generators for three-dimensional solids can be developed on the basis of the trivariate ‘blending-function’ formulae found at the end of the second section.

Incompatible Displacement Models

Abstract: Publisher Summary This chapter introduces incompatible displacement modes at the element level in order to improve element accuracy. One of the main causes of inaccuracies in lower-order finite elements is their inability to represent certain simple stress gradients. The same basic method of introducing incompatible displacement modes in order to improve the bending properties can be used in three dimensions. The first eight are the standard compatible interpolation functions. The last three are incompatible and are associated with linear shear and normal strains. The nine incompatible modes are eliminated at the element stiffness level by static condensation. As the three-dimensional element degenerates to the same approximation as in the two-dimensional element, the same improvement in accuracy is obtained. This element has been found to be extremely effective in the analysis of massive three-dimensional structures subjected to bending. One element in the thickness direction of arch dams or thick pipe joints has been found to be adequate.

Transfinite element methods: Blending-function interpolation over arbitrary curved element domains

Abstract: In order to better conform to curved boundaries and material interfaces, curved finite elements have been widely applied in recent years by practicing engineering analysts. The most well known of such elements are the "isoparametric elements". As Zienkiewicz points out in [18, p. 132] there has been a certain parallel between the development of "element types" as used in finite element analyses and the independent development of methods for the mathematical description of general free-form surfaces. One of the purposes of this paper is to show that the relationship between these two areas of recent mathematical activity is indeed quite intimate. In order to establish this relationship, we introduce the notion of a "transfinite element" which, in brief, is an invertible mapping $$\vec T$$ from a square parameter domainJ onto a closed, bounded and simply connected region? in thexy-plane together with a "transfinite" blending-function type interpolant to the dependent variablef defined over?. The "subparametric", "isoparametric" and "superparametric" element types discussed by Zienkiewicz in [18, pp. 137---138] can all be shown to be special cases obtainable by various discretizations of transfinite elements Actual error bounds are derived for a wide class of semi-discretized transfinite elements (with the nature of the mapping $$\vec T$$ :J?? remaining unspecified) as applied to two types of boundary value problems. These bounds for semi-discretized elements are then specialized to obtain bounds for the familiar isoparametric elements. While we consider only two dimensional elements, extensions to higher dimensions is straightforward.

Saturated-unsaturated seepage by finite elements

Abstract: A Galerkin-type finite element method is employed to solve the quasilinear partial differential equations of transient seepage in saturated-unsaturated porous media. The resulting computer program is capable of handling nonuniform flow regions having complex boundaries and arbitrary degrees of local anisotropy. Flow can take place in a vertical plane, in a horizontal plane, or in a three-dimensional system with radial symmetry. An arbitrary number of seepage faces can be considered simultaneously, and the positions of the exit points on these boundaries are adjusted automatically during each time step. Two examples, one of seepage through an earth dam with a sloping core and horizontal drainage blanket, and the other of seepage through a layered medium cut by a complex topography, are included. These examples indicate that the classical concept of a free surface is not always applicable when dealing with transient seepage through soils.

Finite Element Analysis of Reinforced Concrete

Abstract: Cracking, crushing, and yielding of reinforced concrete beams is investigated by the finite element technique. Three-dimensional, 20-node isoparametric elements are used. The analysis incorporates the elasto-plastic behavior of concrete and reinforcing steel and the cracking of concrete stressed in tension beyond a limiting tensile stress. Shear retention in the cracking plane is also considered. The results obtained by this procedure compare well with those obtained experimentally and analytically.

Computational fracture mechanics

Abstract: Numerical procedures for accurate determination of elastic stress intensity factors for the general two-dimensional crack problem are reviewed. The elastic perfectly plastic state of crack tip deformation is studied by a finite element procedure. Elastic-plastic fields in the immediate vicinity of a crack tip are determined numerically by finite element procedures based on asymptotic studies of crack tip singularities in plastic materials. The small-scale yielding problem is modeled, and expressions for crack tip opening displacement, shear singularity amplitude, and plastic zone extent are derived. A finite element solution to the large-scale yielding of a circumferentially cracked round tension bar is obtained. The three-dimensional aspects of flawed structures and numerical methods of treating them are studied. Ductile fracture mechanisms, in particular crack tip fracture on the microscale, are discussed.

Calibration of distributed parameter groundwater flow models viewed as a multiple‐objective decision process under uncertainty

Abstract: Owing to the deterministic nature of most groundwater flow models there has been a tendency in the past to overlook the strong element of uncertainty that invariably enters into the problem of parameter identification. It is shown that because of this uncertainty an approach based on the minimization of a single error functional does not in general lead to satisfactory results. A multiple-objective decision process is postulated taking into account all the available information on the aquifer flow system as well as the range of environmental conditions under which the system is expected to operate in the future. According to this new approach a continuous or discrete set of alternative solutions to the identification problem is generated with the aid of mathematical programing techniques, and the decision maker is asked to apply his own value judgment in selecting a particular model structure. The method is illustrated by applying parametric linear programing to a finite element model of steady state flow in a locally anisotropic aquifer. The reliability of each parameter estimate is ascertained with the aid of a postoptimal sensitivity analysis.

Curved Elements in the Finite Element Method. I

Abstract: Curved elements, introduced by the author in [13] and [14], which are suitable for solving boundary value problems of the second order in plane domains with an arbitrary boundary are discussed. An approximation theorem is proved, the Dirichlet problem for a ${\mathop W\limits^{\circ}} _2^{(1)} $-elliptic equation is considered as a model problem and error bounds are derived.

Nonconforming Elements in the Finite Element Method with Penalty

Abstract: A penalty method approach is used for achieving convergence of a finite element method using nonconforming elements. Error estimates are given.

Transient shell response by numerical time integration

Abstract: In using the finite element method to compute a transient response, two choices must be made. First, some form of mass matrix must be decided upon. Either the consistent mass matrix prescribed by the finite element method can be employed or some form of diagonal mass matrix may be introduced. Secondly, some particular time integration procedure must be adopted. The procedures available divide themselves into two classes: the conditionally stable explicit schemes and the unconditionally or conditionally stable implicit schemes. The choices should be guided by both economy and accuracy. Using exact discrete solutions compared to the exact solutions of the differential equations, the results of these choices are displayed. Concrete examples of well-matched methods, as well as ill-matched methods, are identified and demonstrated. In particular, the diagonal mass matrix and the explicit central difference time integration method are shown to be a good combination in terms of accuracy and economy.

Finite Element for Rock Joints and Interfaces

Abstract: The objective of this paper is to present a method for dynamic stress analysis of saturated soil structures subjected to earthquake loading. The soil is idealized as a fluid saturated porous solid. The coupled equations of motion of the finite element discretized system are solved by a step-by-step integration scheme to determine the motion of the constituent materials along with the intergranular stresses and the pore pressure. Finally, some results of an illustrative example of an earth dam-reservoir system are presented and analyzed.

Uni-moment method of solving antenna and scattering problems

Abstract: It has been shown by this investigator and numerous others [6], [7], [8] that exterior boundary value problems involving localized inhomogeneous media are most conveniently solved using finite difference or finite element techniques together with integral equations or harmonic expansions, which satisfy the radiation conditions. The methods result in large matrices that are partly full and partly sparse; and methods to solve them, such as iteration or banded matrix methods are not very satisfactory. The unimoment method alleviates the difficulties by decoupling exterior problems from the interior boundary value problems. This is done by solving the interior problem many times so that N linearly independent solutions are generated. The continuity conditions are then enforced by a linear combination of the N independent solutions, which may be done by solving much smaller matrices. Methods of generating solutions of the interior problems are discussed.

Incremental Finite Element Matrices

Abstract: A common technique in geometrically nonlinear finite element analysis is to express the total potential in terms of Lagrangian displacement coordinates, differentiate the potential to obtain the equilibrium equations, and form the differentials of the equilibrium equations to obtain linear incremental equilibrium equations. The geometric nonlinearities in the strain-displacement equations give rise to incremental matrices in the preceding equations. The form of these matrices is not unique in the expression for the total potential. The paper presents expressions for incremental matrices that remain valid in the equilibrium equations and in the linear incremental equilibrium equations. The construction of such matrices is illustrated for truss elements, in-plane bending elements, membrane elements, and plate flexural elements. An examination of some of the recent literature indicates that some investigators have used inappropriate forms of these incremental matrices.

Flow of compressible fluid in porous elastic media

Abstract: A finite element formulation for flow of fluid in a porous elastic media has been derived from a Gurtin type variational principle. Biot's field equations for porous media have been used in which the constitutive relations include the compressibility of the fluid. It has been shown that by proper choice of the form of the compressibility of the fluid as a function of the state variables, this option of the formulation can be used to treat the partially saturated soil. The method is general with respect to geometry, boundary conditions and material properties. Finally, the results of a series of examples have been presented and compared with exact results to demonstrate accuracy and applicability.

Earthquake-Induced Soil Pressures on Structures

Abstract: The earthquake-induced pressures on soil-retaining structures are investigated. The study was motivated by the lack of suitable earthquake design data for relatively rigid structures on firm foundations in situations where the foundation, structure and retained soil remain essentially elastic. Pressures and forces on the walls of a number of idealized wall-soil problems are analyzed. The solutions obtained are evaluated for a range of the important parameters to give results useful for design. In the idealized problems the soil is represented by an elastic layer of finite length bonded to a rigid foundation or rock layer. The wall or structure is represented by a rigid element resting on the rock layer and is permitted to undergo rotational deformation about the base. The mass or moment of inertia of the structure and its rotational stiffness are included as parameters in the idealization. Static and dynamic solutions are obtained using both analytical and finite element methods. Solutions are evaluated for the assumption of perfectly rigid behavior of the wall. The general solution for the deformable wall case was developed by superposition of the solution for the perfectly rigid case and solutions derived for displacement forcing of the wall structure. The assumption of linear elastic behavior of the wall- soil system is likely to be approximately satisfied in situations where a building or other large civil engineering structure is founded on firm soil or rock strata. In contrast to the linearly elastic assumption made in this study, the commonly used Mononobe-Okabe method employs the assumption of sufficiently large wall deformations to induce a fully plastic stress condition in the soil. It was concluded that both the elastic theory and the Mononobe-Okabe method have valid applications in the design of wall structures subjected to earthquake motions, but that because of significant differences in the solutions obtained from each method, care is required in selecting the most appropriate method for a particular situation.

A Galerkin‐finite element simulation of groundwater contamination on Long Island, New York

Abstract: The Galerkin method of approximation in conjunction with the finite element method of analysis may be used to simulate the movement of groundwater contaminants. In solving the groundwater flow and mass transport equations this approach allows a functional representation of the dispersion tensor, transmissivity tensor, and fluid velocity, as well as an accurate representation of boundaries of irregular geometry. A field application of the method to chromium contamination on Long Island, New York, shows that accurate simulations can be obtained by using the Galerkin-finite element approach.

Bending and vibration of multilayer sandwich beams and plates

Abstract: A finite element displacement analysis of multilayer sandwich beams and plates, each with n stiff layers and n−1 weak cores, is presented. Each layer of the sandwich structure may have individual orthotropic properties of its own and the bending rigidities of the stiff layers are taken into account while direct stresses in cores are neglected in the analysis. The condition of common shear angle for all cores, which has been used by several authors is not implied in the formulation. Several examples on bending problems have been solved using lower-order elements and the accuracy of the results has been shown to be excellent. Two higher-order elements have also been developed but have not been found to yield much better results. The free vibration problems of multilayer sandwich structures have also been solved, and good accuracy is demonstrated.

On the convergence of a mixed finite-element method for plate bending problems

Abstract: In this paper we justify a finite element method for biharmonic boundary value problems. The method is based on a stationary variational principle (the Reissner principle), and was introduced by Hellan, Herrmann and Visser. We prove error estimates and the existence of a finite element solution.

Higher-order finite-element analysis of topographic guides supporting elastic surface waves

Abstract: A method of computing the eigenmodes of acoustic waveguides of arbitrary cross section is described. It is a finite‐element technique, based on a variational formulation of the guided‐wave problem. The accuracy of the method is assessed by comparing numerical and experimental results for a few specific structures. In particular we describe an acoustic surface ridge waveguide with low dispersion over a broad band.