Showing papers on "Finite element method published in 1969"
TL;DR: In this paper, a numerical method for the dynamic analysis of infinite continuous systems is developed, applicable to systems for which all exciting forces and geometrical irregularities are confined to a limited region and is applicable to both transient and steady state problems.
Abstract: A numerical method for the dynamic analysis of infinite continuous systems is developed. The method is applicable to systems for which all exciting forces and geometrical irregularities are confined to a limited region and is applicable to both transient and steady state problems. The infinite system is replaced by a system consisting of a finite region subjected to a boundary condition which simulates an energy absorbing boundary. The resulting systems may be analyzed by the finite element method. Examples applying the method to foundation vibration problems are presented. Good agreement with existing solutions is found and new results for embedded footings are presented.
2,172 citations
TL;DR: In this article, a review and reinterpretation of the existing finite element methods and other alternative schemes are presented, and a comparison of the relative merits of the various methods is made.
Abstract: Finite element methods can be formulated from the variational principles in solid mechanics by relaxing the continuity requirements along the interelement boundaries. The combination of different variational principles and different boundary continuity conditions yields numerous types of approximate methods. This paper reviews and reinterprets the existing finite element methods and indicates other alternative schemes. Plate bending problems are used to compare the relative merits of the various methods.
417 citations
TL;DR: In this article, a variational principle is used in conjunction with the finite element method to solve the initial boundary value problem of flow in a saturated porous elastic medium, which results in a powerful solution technique for the determination of stress and displacement history, both for the solid and the liquid phases, for arbitrary boundary conditions and within complex geometrical configurations.
Abstract: A variational principle is used in conjunction with the finite-element method to solve the initial boundary value problem of flow in a saturated porous elastic medium. This results in a powerful solution technique for the determination of stress and displacement history, both for the solid and the liquid phases, for arbitrary boundary conditions and within complex geometrical configurations. Direct application is to problems of consolidation and drainage of saturated soils under load. Linear theory of the coupled fields is treated but extension to nonlinear problems is possible through use of incremental procedures.
278 citations
218 citations
TL;DR: In this paper, an analytic derivation for high-accuracy triangular finite elements useful for numerical solution of field problems involving Laplace's, Poisson's, Helmholtz's, or related elliptic partial differential equations in two dimensions is given.
194 citations
TL;DR: In this paper, the Kirchhoff theory of shells is refined to include a transverse shear deformation and the Lagrange equations are introduced to derive the equations of the discrete system.
186 citations
TL;DR: In this paper, the idea of finite element discretisation is applied to time dependent dynamic phenomena, where the time is discretised into a set of finite elements which are taken to be the same for all structural elements.
186 citations
TL;DR: In this article, the essential shakedown theory and the basis of relevant solution procedures are presented in compact form. And for systems with associated flow-laws, the second shakedown theorem (Koiter's) is extended in order to allow variable dislocations (e.g. temperature cycles).
Abstract: The matrix description of the mechanical behaviour resting on finite element discretization and piecewise linearization of yield surfaces, is adopted instead of the traditional continuous field description. By the use of linear programming concepts, the essential of a general shakedown theory and the basis of relevant solution procedures are presented in compact form. For systems with associated flow-laws, the second shakedown theorem (Koiter's) is extended in order to allow for variable dislocations (e. g. temperature cycles). For systems with nonassociated flow-laws two theorems are given which supply lower and upper bounds to the safety factor.
185 citations
178 citations
TL;DR: In this paper, the topological properties of finite element models of functions defined on spaces of finite dimension were examined, and a number of applications of the general theory were presented, such as the generation of finite-element models in the time domain and certain problems in wave propagation, kinetic theory of gases, non-linear partial differential equations, nonlinear continuum mechanics, and fluid dynamics.
Abstract: SUMMARY In Part I of the this paper. topological properties of finite element models of functions defined on spaces of finite dimension were examined. In this part, a number of applications of the general theory are presented. These include the generation of finite element models in the time domain and certain problems in wave propagation, kinetic theory of gases, non-linear partial differential equations, non-linear continuum mechanics, and fluid dynamics.
174 citations
TL;DR: In this paper, a variational principle is formulated as the foundation of the finite element method proposed by Pian, and an extension of Pian's original proposal is made and the convergence of the approximate solution to the exact one is proved.
TL;DR: Finite element solution of incompressible lubrication problem by minimum principle for transient incompressibility Reynolds equation with boundary conditions as discussed by the authors, where the minimum principle is used to solve the problem.
Abstract: Finite element solution of incompressible lubrication problem by minimum principle for transient incompressible Reynolds equation with boundary conditions
TL;DR: In this paper, a general computer program for determining sets of propagating modes and cutoff frequencies of arbitrarily shaped waveguides is described, which uses a new method of analysis based on approximate extremization of a functional whose Euler equation is the scalar Helmholtz equation, subject to homogeneous boundary conditions.
Abstract: A very general computer program for determining sets of propagating modes and cutoff frequencies of arbitrarily shaped waveguides is described. The program uses a new method of analysis based on approximate extremization of a functional whose Euler equation is the scalar Helmholtz equation, subject to homogeneous boundary conditions. Subdividing the guide cross section into triangular regions and assuming the solution to be representable by a polynomial in each region, the variational problem is approximated by a matrix eigenvalue problem, which is solved by Householder tridiagonalization and Sturm sequences. For reasonably simple convex polygonal guide shapes, the dominant eigenfrequencies are obtained to 5-6 significant figures; for nonconvex or complicated shapes, the accuracy may fall to 3 significant figures. Use of the program is illustrated by calculating the propagating modes of a class of degenerate mode guides of current interest, for which experimental data are available. Numerical studies of convergence rate and discretization error are also described. It is believed that the new program produces waveguide analyses of higher accuracy than any general program previously available.
TL;DR: In this article, a review of variational methods for the solution of electromagnetic field problems is presented, including the Rayleigh-Ritz approach for determining the minimizing sequence, and a brief description of the finite element method.
Abstract: This paper reviews some of the more useful, current and newly developing methods for the solution of electromagnetic fields. It begins with an introduction to numerical methods in general, including specific references to the mathematical tools required for field analysis, e.g., solution of systems of simultaneous linear equations by direct and iterative means, the matrix eigenvalue problem, finite difference differentiation and integration, error estimates, and common types of boundary conditions. This is followed by a description of finite difference solution of boundary and initial value problems. The paper reviews the mathematical principles behind variational methods, from the Hilbert space point of view, for both eigenvalue and deterministic problems. The significance of natural boundary conditions is pointed out. The Rayleigh-Ritz approach for determining the minimizing sequence is explained, followed by a brief description of the finite element method. The paper concludes with an introduction to the techniques and importance of hybrid computation.
TL;DR: In this paper, a numerical approach to transient fluid flow in multilayered aquifers has been developed using the finite element method, in which an initial boundary value problem is converted to a variational problem and applied to a descretized system of elements.
Abstract: A new approach to transient fluid flow in multilayered aquifers has been developed using the finite element method. This is a numeric technique in which an initial boundary value problem is converted to a variational problem and applied to a descretized system of elements. Details for the case of isotropic systems have been presented elsewhere. This paper extends the treatment to anisotropic systems. The method has been used to investigate potential distributions in multilayered aquifers of finite radial extent being pumped at constant rate using completely penetrating wells. An analysis of two-layer systems with permeability contrasts of up to 100:1 indicates that at early time the drawdown behavior can vary significantly from the Theis solution, depending on where observations are made. As time increases, however, the results eventually converge on the Theis solution regardless of the permeability contrast. A 13-layer aquifer containing either isotropic or anisotropic layers has been examined, and the results are in general agreement with the behavior for two-layer systems.
TL;DR: In this article, the procedure for extending the finite element displacement method to include geometric nonlinearity is outlined and a consistent formulation applicable to arbitrary plate and shallow shell elements is derived, and examples illustrating the application of a rectangular element to a plate, a shallow cylindrical shell, and a shallow hyperbolic paraboloid are presented and the results are compared with other theoretical and experimental results.
Abstract: The procedure for extending the finite-element displacement method to include geometric nonlinearity is outlined and a consistent formulation applicable to arbitrary plate and shallow shell elements is derived. Examples illustrating the application of a rectangular element to a plate, a shallow cylindrical shell, and a shallow hyperbolic paraboloid are presented and the results are compared with other theoretical and experimental results. Close agreement is obtained even for fairly coarse mesh spacing.
TL;DR: In this article, a specialized form of Reissner's variational principle is developed which is suitable for anisotropic incompressible and nearly-incompressible thermoelasticity.
TL;DR: In this paper, a finite element formulation for the analysis of thin elastic plates subjected to transverse loading and including nonlinear geometric effects associated with large deflections is presented, which utilizes local co-ordinate axes which translate and rotate with the individual elements so that the small deflection Kirchhoff formulation remains valid, with respect to these axes, and may be applied to determine element stiffness and resisting forces.
Abstract: A finite element formulation for the analysis of thin elastic plates subjected to transverse loading and including nonlinear geometric effects associated with large deflections is presented. The analysis utilizes local co-ordinate axes which translate and rotate with the individual elements so that the small deflection Kirchhoff formulation remains valid, with respect to these axes, and may be applied to determine element stiffness and resisting forces. Nonlinear terms in the strain displacement equations are included in the coordinate transformation and change of plate configuration is included in the equilibrium equations. The formulation is applied, using an iterative technique and an approximate incremental stiffness, to obtain solutions to a number of typical large deflection plate problems. Results include an inextensional plate problem, cylindrical bending, and a square plate with simple edge condition. Deflection and stress results are compared with solutions available in the literature.
TL;DR: In this paper, the large deflection behavior of a shallow circular arch subjected to a vertical point load is studied analytically using the Rayleigh-Ritz finite element method.
TL;DR: In this paper, a variational principle based on assumed stress hybrid method suitable for incompressible or near-incompressible solid is formulated for finite element analysis, and one example to illustrate the use of the method is given.
TL;DR: In this article, a modified version of the finite element method is used to solve a series of real world problems associated with GEOLOGICAL STRUCTURES, such as the formation of mullions.
Abstract: A NUMERICAL APPROACH TO THE STUDY OF FINITE, QUASI-STATIC PLANE DEFORMATIONS OF VISCOUS SOLIDS IS INTRODUCED. THE PROBLEM IS FORMULATED IN TERMS OF A SERIES OF INCREMENTAL PROBLEMS. A MODIFIED VERSION OF THE FINITE ELEMENT METHOD IS EMPLOYED IN THE SOLUTION OF THIS SERIES OF PROBLEMS. IT IS HOPED THAT THIS APPROACH MAY PROVE CONVENIENT FOR THE STUDY OF CERTAIN PROBLEMS ASSOCIATED WITH GEOLOGICAL STRUCTURES. THE EXAMPLE OF THE FINITE COMPRESSION OF A BODY WITH AN INITIALLY IMPERFECT FREE SURFACE MIGHT SHED SOME LIGHT ON THE FORMATION OF MULLIONS. /AUTHOR/
TL;DR: In this article, two commonly suggested forms of the equation linking head loss and velocity for flow of water through coarse granular media are the Forchheimer and exponential relations combined with the continuity expression, these relations give the differential equations applicable, within the limits of validity of the parent relations, to actual regions of flow.
Abstract: Two commonly suggested forms of the equation linking head loss and velocity for flow of water through coarse granular media are the Forchheimer and exponential relations. Combined with the continuity expression, these relations give the differential equations applicable, within the limits of validity of the parent relations, to actual regions of flow. The resultant nonlinear partial differential equations are amenable to solution by the numerical technique known as the method of finite elements. This technique has advantages when dealing with complex boundary shapes. Solutions have been obtained for some examples of unconfined flow with boundary conditions similar to those likely to be encountered in practical applications. Experimental work in an open flume has shown that agreement between observed and calculated values of discharge and piezometric head can be obtained when the coefficients in the head loss equations are accurately known.
TL;DR: The present paper seeks to apply the ideas of discretisation to time dependent phenomena and demonstrates that the general case of a multi-degree of freedoms system can be made to depend on the matrices which describe the unidimensional motion of a mass point.
Abstract: The present paper seeks to apply the ideas of discretisation to time dependent phenomena. As a suitable variational statement we may use Hamilton's principle. In practise this means that the time is discretised into a set of finite elements which are taken to be the same for all structural elements. A finite element in time consists simply of a fixed time interval. In our present discussion we detail in particular the case when at the beginning and end of the time interval the generalised displacements and velocities are given. For dynamic problems this is the minimum of information required, but the technique may easily be extended to account for additional “timewise degrees of freedoms”. Introducing an appropriate interpolation procedure we may obtain the displacement and velocity at any instant of time. It is then possible to carry out in the variational statement the time integration explicitly and to obtain hence a system of linear equations. The method is extremely simple, since the time interpolation of all structural freedoms of an element in space is the same. We also demonstrate that the general case of a multi-degree of freedoms system can be made to depend on the matrices which describe the unidimensional motion of a mass point.
TL;DR: In this paper, the stiffness matrix of a plate-bending element of general polygonal shape having any number of nodes is derived by assuming various numbers of unknown coefficients in the stress distributions.
Abstract: The assumed stress distribution approach is used to derive the stiffness matrix of a plate-bending element of general polygonal shape having any number of nodes. The effect of assuming various numbers of unknown coefficients in the stress distributions is examined and the convergence properties of the resulting elements compared with others derived form assumed displacements.
TL;DR: In this paper, a new element which has rotational degrees of freedom at the node points is proposed for the analysis of simple shear wall problems with a single row of openings.
Abstract: Most previously derived plane stress finite elements cannot be connected with line elements in bending. This difficulty is overcome by devising a new element which has rotational degrees of freedom at the node points. Application to analysis of simple shear wall problems with a single row of openings illustrates the method and comparison of results with other solutions demonstrates its validity. A study of the behavior of the new element in comparison with results for a more conventional element shows: (1) For the same number of elements, the accuracy of estimation of deflection is increased and (2) the ratio of length to breadth of the elements has a less significant effect on the results.
TL;DR: In this article, the stiffness matrix of a rectangular plate element for bending proposed by Greene is employed, and results of numerical examples duly justifies applicability of the present method, and a finite element method and iteration technique are employed.
Abstract: The large deflection problem of a rectangular plate is analysed by using the finite element method and employing the iteration technique. In the present study, the stiffness matrix of a rectangular plate element for bending proposed by Greene is employed, and results of numerical examples duly justifies applicability of the present method.
TL;DR: In this article, a solution of plane elastic problems by piecewise linear approximation is outlined, based upon Galerkin error distribution technique, which leads to simultaneous algebraic equations identical to those associated with the Finite Element Method.
Abstract: Solution of plane elastic problems by piecewise linear approximation is outlined. This method is based upon Galerkin error distribution technique, which leads to simultaneous algebraic equations identical to those associated with the Finite Element Method. In addition, this method permits definition of the discretization error, which can be computed once the displacement components are known. Properties of the interpolation functions are discussed, and a sequence of internally compatible plane elastic elements is defined.
01 Oct 1969
TL;DR: In this article, a discretised finite-element equation for wave propagation in inhomogeneous waveguides is derived for the dominant LSM mode in a dielectric-slab-loaded waveguide.
Abstract: Finite-element equations are derived for wave propagation in inhomogeneous waveguides. A variational expression is used for the wavenumber k0. A discretised finite-element equation is found at a point surrounded by four dielectrics, and special cases deduced. The method is applied to the solution of the dominant LSM mode in a dielectric-slab-loaded waveguide.