# Showing papers in "International Journal for Numerical Methods in Engineering in 1975"

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TL;DR: In this paper, it was shown that a singularity occurs in isoparametric finite elements if the mid-side nodes are moved sufficiently from their normal position to obtain a more accurate solution to the problem of determining the stress intensity at the tip of a crack.

Abstract: It is shown that a singularity occurs in isoparametric finite elements if the mid-side nodes are moved sufficiently from their normal position. By choosing the mid-side node positions on standard isoparametric elements so that the singularity occurs exactly at the corner of an element it is possible to obtain quite accurate solutions to the problem of determining the stress intensity at the tip of a crack. The solutions compare favourably with those obtained using some types of special crack tip elements, but are not as accurate as those given by a crack tip element based on the hybrid principle. However, the hybrid elements are more difficult to use.

1,057 citations

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TL;DR: In this paper, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived starting from continuum mechanics principles, and a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure.

Abstract: SUMMARY Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.

789 citations

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642 citations

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286 citations

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TL;DR: In this article, the numerical stability of simple marching schemes used in elasto/visco-plasticity is investigated and a theoretical stability criterion based on the identification of the numerical process with the integration of a nonlinear, first order, system of ordinary differential equations is derived.

Abstract: The numerical stability of simple marching schemes used in elasto/visco‐plasticity is investigated. An assumption of convexity renders possible the derivation of a theoretical stability criterion based on the identification of the numerical process with the integration of a non‐linear, first order, system of ordinary differential equations. Explicit stability criteria are obtained for commonly used visco‐plastic laws. Selected examples illustrate the necessity and effectiveness of the proposed stability criteria in actual computations. Copyright © 1975 John Wiley & Sons, Ltd

270 citations

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TL;DR: In this article, the authors investigated the achievable accuracy of various time-stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level, and concluded that the Crank-Nicolson scheme with a simple averaging process is superior to the other methods investigated.

Abstract: This paper investigates the phenomenon of ‘noise’ which is common in most time-dependent problems. The emphasis is on the achievable accuracy that is obtained with various time-stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level. A series of experiments are made where the space domain is discretized using the finite element method and the variation with time is approximated by several finite difference methods. The conclusion is reached that the Crank–Nicolson scheme with a simple averaging process is superior to the other methods investigated.

130 citations

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TL;DR: In this paper, a cutting plane algorithm for the solution of generalized geometric programs with bounded variables is described and illustrated by the detailed solution of a small numerical example, and convergence of this algorithm to a Kuhn-Tucker point of the program is assured if an initial feasible solution is available to initiate the algorithm.

Abstract: A cutting plane algorithm for the solution of generalized geometric programs with bounded variables is described and then illustrated by the detailed solution of a small numerical example. Convergence of this algorithm to a Kuhn-Tucker point of the program is assured if an initial feasible solution is available to initiate the algorithm. An algorithm for determining a feasible solution to a set of generalized posynomial inequalities which may be used to find a global minimum to the program as well as test for consistency of the constraint set, is also presented. Finally an application in optimal engineering design with seven variables and fourteen nonlinear inequality constraints is formulated and solved.

110 citations

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94 citations

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TL;DR: In this paper, a class of approximations to the matrix linear differential equation is presented, ranging in accuracy from the simplest forward difference scheme to the exact solution, using the infinite series defining the exponential matrix.

Abstract: A class of approximations to the matrix linear differential equation is presented. The approximations range, in accuracy, from the simplest forward difference scheme to the exact solution. The infinite series defining the exponential matrix is used to ascertain the accuracy of the various approximations. A clear distinction is made between approximations to the system equations and the forcing function, with the forcing term being represented by a piecewise linear function. Special application is given to the equations arising in structural dynamics of the form
For these structural dynamic equations, the measure of the energy of the system is used to analyse the stability of the numerical approximations.

63 citations

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58 citations

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TL;DR: In this paper, a semi-analytical approach for the vibration of conical and cylindrical shells has been proposed based on mass matrices, and good agreement has been found between theory and experiment for thin-walled circular cylinders and cones, a conecylinder combination, and a cooling tower model.

Abstract: Elemental mass matrices have been produced for the vibration of conical and cylindrical shells, based on a semi-analytical approach. Frequencies and modes of vibration have been compared with existing solutions and also with experimental results obtained from other sources. Good agreement has been found between theory and experiment for thin-walled circular cylinders and cones, a cone-cylinder combination, and a cooling tower model.
A theoretical investigation was also made on the vibration of a circular cylinder when subjected to uniform pressure.

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TL;DR: A study of the constraint of incompressibility in the finite element method for plane strain through the use of a Lagrange multiplier and its two approaches from the point of view of rate of convergence and computer time is presented.

Abstract: The constraint of incompressibility is incorporated into the finite element method for plane strain through the use of a Lagrange multiplier. Depending on the approximating function chosen for this multiplier, the constraint condition can be satisfied everywhere within the element or only in an average sense for the entire element. A study of these two approaches from the point of view of rate of convergence and computer time is presented.

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TL;DR: A linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory, is used, and it is obtained convergence in nearly every case, and quadratic convergence in most cases.

Abstract: We present a program which has given excellent results for uniform approximation of functions by polynomials, rational functions, generalized polynomials, and generalized rational functions. The algorithm is described in detail and several examples are discussed. The approximation is done over a finite point set, which is commonly a set of real numbers or points in the plane (in the latter case we are doing what is often known as surface fitting). Input to and output from the program is in tabular form. The method used is a linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory (quadratically in some situations). In practice, we have obtained convergence in nearly every case, and quadratic convergence in most cases. The program can also be used for simultaneous approximation of several functions.

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TL;DR: In this article, the results of a finite difference analysis are presented for the problem of incompressible laminar flow heat transfer in concentric annuli with simultaneously developing hydrodynamic and thermal boundary layers, the boundary conditions of one wall being isothermal and the other wall adiabatic.

Abstract: The results of a finite difference analysis are presented for the problem of incompressible laminar flow heat transfer in concentric annuli with simultaneously developing hydrodynamic and thermal boundary layers, the boundary conditions of one wall being isothermal and the other wall adiabatic. This corresponds to the fundamental solution of the third kind according to the four fundamental solutions classified by Reynolds, Lundberg and McCuen1.
Firstly, the hydrodynamic entry length problem, based on the boundary layer simplifications of the Navier–Stokes equations, was solved by means of an extension of the linearized finite difference scheme used previously by Bodia and Osterle2 to solve a similar problem between parallel plates. The energy equation is then solved, using the velocity profiles previously obtained, by means of an implicit finite difference technique.
The accuracy of the numerical solution was checked by comparing results for the annulus of radius ratio 0.25 with the avaiable solution of Shumway and McEligot3.

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TL;DR: In this paper, a gradient projection optimal control algorithm incorporating conjugate gradient directions of search is described and applied to several minimum weight panel design problems subject to a flutter speed constraint.

Abstract: A gradient projection optimal control algorithm incorporating conjugate gradient directions of search is described and applied to several minimum weight panel design problems subject to a flutter speed constraint. New numerical solutions are obtained for both simply-supported and clamped homogeneous panels of infinite span for various levels of inplane loading and minimum thickness. The minimum thickness inequality constraint is enforced by a simple transformation of variables.

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TL;DR: In this article, the general stiffness matrix for a beam element is derived from the Bernoulli-Euler differential equation with the inclusion of axial forces, and the terms of this matrix are expanded into a power series as a function of the two variables: the axial force, and; the vibrating frequency.

Abstract: The general stiffness matrix for a beam element is derived from the Bernoulli–Euler differential equation with the inclusion of axial forces. The terms of this matrix are expanded into a power series as a function of the two variables: the axial force, and; the vibrating frequency. It is shown that the first three terms of the resulting series, which are derived in the technical literature from assumed static displacement functions, correspond respectively to the elastic stiffness matrix, the consistent mass matrix, and the geometric matrix. Higher order terms up to the second order terms of the series expansion are obtained explicitly. Also a discussion is presented for establishing the region of convergence of the series expansion for the dynamic stiffness matrix, the stability matrix, and the general stiffness matrix.

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General Motors

^{1}TL;DR: In this article, finite element methods are used to solve hydrodynamic lubrication problems involving compressible lubricants and porous bearing solids, and the particular calculation scheme permits solution at high compressibility numbers (Λ > 100) to be obtained without any numerical difficulty.

Abstract: Finite element methods are used to solve hydrodynamic lubrication problems involving compressible lubricants and porous bearing solids. The particular calculation scheme permits solution at high compressibility numbers (Λ > 100) to be obtained without any numerical difficulty. Finite element and finite difference results for the porous, gas lubricated journal bearing are presented and compared.

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TL;DR: In this paper, a method for solving structural design problems that allows a continuous distribution of material along structural elements is presented, which is an extension of the generalized steepest descent method presented in Reference 1.

Abstract: A method for solving structural design problems that allows a continuous distribution of material along structural elements is presented. The method is an extension of the generalized steepest descent method presented in Reference 1. Inequality constraints on design variables, displacement, natural frequency, and buckling are explicitly treated and a minimum weight cost function is employed. A steepest descent method for boundary-value state equations is developed and a computational algorithm is given. Several example problems in minimum weight structural design are solved and compared with results obtained by discretization techniques.

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TL;DR: In this article, a Lagrangian variational formulation for the analysis of large displacement elastic-plastic and elastic-viscoplastic dynamic analysis of deformable bodies is presented, where linearized incremental equations of motion are solved using direct integration procedures, with added accuracy obtained from application of equilibrium correction at each step.

Abstract: Incremental equations of motion are derived from a Lagrangian variational formulation for the large displacement elastic-plastic and elastic-viscoplastic dynamic analysis of deformable bodies. The material constitutive behaviour is described in terms of the symmetric Piola–Kirchhoff stress and Lagrangian strain tensors. Degenerate isoparametric elements, permitting relaxation of the Kirchhoff–Love hypothesis, are used in a finite element formulation specialized for the analysis of shells of revolution subjected to axisymmetric loading. The linearized incremental equations of motion are solved using direct integration procedures, with added accuracy obtained from application of equilibrium correction at each step. The effectiveness of the numerical techniques is illustrated by the dynamic response analyses carried out on a shallow spherical cap subjected to uniform external step pressure loadings.

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TL;DR: The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed in this article, and convergence criteria and bounds for the direct flexibility-influence coefficient are examined.

Abstract: Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.

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TL;DR: In this paper, the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi-infinite elastic foundations, where no assumption regarding the contact pressure distribution has been made.

Abstract: In the present work the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi-infinite elastic foundations. No assumption regarding the contact pressure distribution has been made. The equations have been developed in non-dimensional form and also the results have been obtained in non-dimensional form. These results have been compared with the available experimental results and the agreement between them is found to be much better than that of the previous works. The same method with slight modification can be applied for Winkler type foundations and problems of circular plates with varying thickness.

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TL;DR: In this article, a simple mixed finite element method is developed, where the deflections are assumed to be simple four-element polynomials, bending moments, Mx and My with a partially linear distribution, expressed in terms of the node deflections.

Abstract: A simple mixed finite element method is developed. The finite element is a rectangular triangle and rectangle. In the element the deflections are assumed to be simple four-element polynomials, bending moments, Mx and My with a partially linear distribution, and a constant, Mxy, expressed in terms of the node deflections. The element matrix is of the order of 8 × 8. It is derived in a common engineering way. The unknowns are the deflections at the nodes and mid-diagonal, the two moments at the end of the diagonal and the two moments on the cathetus. The results obtained by this method show good convergence and an improvement in the accuracy of the moments as well as in the deflections, compared with results obtained by similar methods, such as those of Herrmann.

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TL;DR: In this paper, a variational formulation of the finite element strategy is used to solve the elliptic boundary value problem in the class of subsonic irrotational flows in a nozzle and around one or several lifting bodies of arbitary shape.

Abstract: Solutions of elliptic boundary value problems are obtained with variational formulations by minimization of functionals.
Conventional tools of the finite element strategy are emphasized; linear or quadratic approximations on triangular elements, curved iso-parametric elements, Ni shape funtions, Li homogeneous co-ordinates. High speed of computation of the final assembled matrices require the use of the Formal–FORMAC calculus. Illusrations of the finite element method are chosen in the class of the 2d–3d compressible subsonic irrotational flows in a nozzle and around one or several lifting bodies of arbitary shape.

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TL;DR: Numerical solutions to three-dimensional elasto-plastic problems illustrating the applicability of isoparametric elements and the order of computation times involved are presented.

Abstract: Advances in technology and interest in limit state design have made the inclusion of nonlinear effects, such as elasto-plastic behavior, desirable in the analysis of many structures. Improvements in solution algorithms coupled with parallel developments in high speed digital computers have now made the practical solution of such problems possible. The paper presents numerical solutions to three-dimensional elasto-plastic problems illustrating the applicability of isoparametric elements and the order of computation times involved.

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Virginia Tech

^{1}TL;DR: In this article, the minimum cross-section of a vibrating beam is controlled by the introduction of an inequality constraint, and a finite element displacement formulation is then used in an iterativve manner to arrive at the optimum fundamental frequency and the corresponding material distribution for the discretized beam models with various boundary conditions.

Abstract: Accounting for shear deformations and rotary inertia effects, necessary condition for optimum fundamental frequency of a vibrating beam of constant volume and with a given distribution of non-structural mass, is obtained through the calculus of variations. Minimum cross-section of the beam is controlled by the introduction of an inequality constraint. A finite element displacement formulation is then used in an iterativve manner to arrive at the optimum fundamental frequency and the corresponding material distribution for the discretized beam models with various boundary conditions. A comparison is then made with the corresponding results of an Euler beam.

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TL;DR: In this paper, the authors present a guide for those who wish to undertake spectral analyses using Discrete Fast Fourier Transforms (DFT) for spectral analysis of a signal with discontinuities along the time axis.

Abstract: The present study is intended as a guide for those who wish to undertake spectral analyses using Discrete Fast Fourier Transforms. Points of particular difficulty in using Fourier Transforms are derived in some detail. Experimental results are offered to illustrate the mathematical derivations. Finally the case of a signal with discontinuities along the time-axis is discussed.

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TL;DR: The main project objective is the development of a mathematical modeling capability for the benefit of the rail transportation industry that will permit design optimization of key structural components such that the probability of fatigue failure can be minimized with respect to a given load environment.

Abstract: This is the first of a series of reports to appear in which the development of a new finite element stress analysis technique will be documented. The work is being conducted at Washington University under a cooperative research program with AMCAR Division of ACF Industries, Inc. Washington University's participation is sponsored by the U.S. Department of Transportation under the Program of University Research and by the Association of American Railroads. The main project objective is the development of a mathematical modeling capability for the benefit of the rail transportation industry that will permit design optimization of key structural components such that the probability of fatigue failure can be minimized with respect to a given load environment. The current finite element technology is not cost-effective in fatigue design applications because a very large number of successive analyses must be executed with progressively refined finite element subdivision in order to establish confidence in the accuracy of solution in those areas where stresses change rapidly.