Showing papers on "Smoothed finite element method published in 1973"
TL;DR: In this paper, two methods of finite element discretisation are presented, and a comparison of the effeciency of the methods associated with the solution of particular problems is made.
1,202 citations
574 citations
463 citations
TL;DR: A penalty method approach is used for achieving convergence of a finite element method using nonconforming elements and error estimates are given.
Abstract: A penalty method approach is used for achieving convergence of a finite element method using nonconforming elements. Error estimates are given.
248 citations
TL;DR: For problems with singularities, the convergence rate for the finite element method is often controlled by the nature of the solution near the points of singularity as discussed by the authors. And unless the singularities are properly handled, the regular so-called high-accuracy element will not be able to improve the rate of convergence.
164 citations
TL;DR: In this article, a finite element method based on a stationary variational principle (the Reissner principle) was proposed for bi-harmonic boundary value problems, and error estimates and the existence of finite element solution were proved.
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.
139 citations
TL;DR: The finite element method as discussed by the authors is the most powerful technique for solving a large class of partial differential equations, and it was proposed by Courant in a lecture to the American Mathematical Society.
Abstract: Thirty years ago, Courant gave a remarkable lecture to this Society. My talk today is more or less a progress report on an idea which he described near the end of that lecture. There are a lot of people in this city, and a few in this room, who worked very closely with Courant—but the idea I am talking about came to fruition in a different and more unexpected way. To begin with, his idea was forgotten. Perhaps you have forgotten it too ; it had to do with approximation by piecewise polynomials, and I will try to explain it properly in a moment. Ten years later Pólya made a very similar suggestion [3], [4], without reference to Courant's lecture. At the same time, and independently, Synge did exactly the same thing [10]. Meanwhile Schoenberg had written the paper [5] which gave birth to the theory of splines—again proposing that, for approximation and interpolation, the most convenient functions were piecewise polynomials. Certainly there was an idea whose time was coming. When it finally came, fifteen years after Courant's lecture, it developed into what is now the most powerful technique for solving a large class of partial differential equations—the finite element, method. The only sad part is that virtually the whole development took place as if Courant had never existed. It is like the story of Romulus and Remus (I think); in this case, the wolves who eventually took care of the orphan happened to be structural engineers. They needed a much better technique for the solution of complicated elliptic systems, and in numerical analysis the algorithms which survive and mature are those which are needed. We want to describe this finite element method, and then at the end to propose an open problem; its interest may be more algebraic-combinatorial than practical, but it is directly suggested by the construction of finite elements. Prior to Courant, the usual approximating functions were sines and cosines, or Bessel functions, or Legendre polynomials. For a simple problem on a regular domain, these are still completely adequate; their approximation properties are well known, and integrations are simplified
92 citations
69 citations
TL;DR: The application of the finite element method to problems in neutron diffusion in space, energy, and time is studied in this paper, where the use of piecewise polynomials with a variational form of the diffusion equation leads to algebraic systems of equations with characteristics similar to the usual finite difference equations.
Abstract: The application of the finite element method to problems in neutron diffusion in space, energy, and time is studied. The use of piecewise polynomials with a variational form of the diffusion equation leads to algebraic systems of equations with characteristics similar to the usual finite difference equations. In Part I, a theoretical analysis of the finite element method, with Hermite polynomials, is presented and rigorous error bounds for the approximate solution are developed. In Part II, numerical studies are presented for problems in space and time. The results confirm the theoretical analysis and indicate the power of the method for diffusion problems.
TL;DR: In this paper, the accuracy and stability of the techniques presented are discussed in relation to the well-known "numerical diffusion and dispersion" phenomena prevalent in popular finite difference methods, and it is suggested that previous variational (Rayleigh-Ritz) formulations of finite elements for some problems are misleading and are of limited application when compared with Galerkin's method.
Abstract: Finite element methods are presented for the solution of certain two-dimensional partial differential equations of interest in water resource problems. Earlier work using Galerkin's method for one-dimensional problems is shown to be a prototype finite element technique. It is suggested that previous variational (Rayleigh-Ritz) formulations of finite elements for some problems are misleading and are of limited application when compared with Galerkin's method. The accuracy and stability of the techniques presented are discussed in relation to the well-known ‘numerical diffusion and dispersion’ phenomena prevalent in popular finite difference methods.
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01 Jan 1973
TL;DR: In this article, the basic equations of linear elasticity are summarized and two fundamental variational principles of solid mechanics, the principles of virtual force, are developed for the analysis and solution of complex structural problems by computer methods.
Abstract: Using the formulations of matrix algebra, the mathematically powerful methods of finite elements are readily applicable to the analysis and solution of complex structural problems by computer methods. The book starts by summarizing the basic equations of linear elasticity and then develops the two fundamental variational principles of solid mechanics, the principles of virtual force. Chapter 2 treats the analysis of member systems and provides a link between the conventional matrix structural analysis approach and finite element methods. Chapters 3-7 contain a general development of the finite displacement method and its application to plane stress, three-dimensional bodies, plate bending and shell structures. The fundamentals of thin-shell theory are developed and current shell finite element models are discussed.
17 Dec 1973
TL;DR: In this paper, the one-dimensional diffusion-convection equation with the finite element representation employing the Galerkin approach was formulated with a linear shape function and two-dimensional triangular and rectangular elements in space and time.
Abstract: The one-dimensional diffusion-convection equation is formulated with the finite element representation employing the Galerkin approach. A linear shape function and two-dimensional triangular and rectangular elements in space and time were used in solving the problem. The results are compared with finite difference solutions as well as the exact solution. As another example, the convective term is set equal to zero and these techniques are applied to the resulting heat equation and similar comparisons are made.
TL;DR: A formulation and some computational details are presented dealing with a general elastic/visco-plastic material where nonlinear elasticity is admissible and the flow rule and yield condition need not be associated.
Abstract: In this paper, authors present a formulation and some computational details dealing with a general elastic/visco-plastic material where nonlinear elasticity is admissible and the flow rule and yield condition need not be associated
17 Dec 1973
TL;DR: In this paper, the authors applied the finite element method to the analysis of the wind-driven circulation of variable-depth, shallow, homogenous lakes, focusing upon higher-order description of the flow phenomena within the individual elements and upon the use of these higher order functions in the definition of curved element boundaries (isoparametric elements).
TL;DR: In this article, the convergence of finite element approximations for static problems in elasticity is studied and various types of incremental loading techniques for finite elastic deformations are examined, and a number of representative solved problems in finite elasticity are given.
01 Jan 1973
TL;DR: In this article, the effect of using plane quadrilateral membrane elements for modeling nonplanar structures was investigated by analyzing a simplified finite element model with the aid of the structure network analysis program.
Abstract: Investigation of the effect of using plane quadrilateral membrane elements for modeling nonplanar structures. The effect is assessed by analyzing a simplified finite element model with the aid of the structure network analysis program. The results obtained indicate that the use of planar quadrilateral membrane elements for modeling bending problems can lead to large errors if the four points that define the quadrilateral are not in the same plane.