bluuebunny/crossref_metadata_2025_split huggingface dataset:
Finite element method for piezoelectric vibration
Authors: Allik Henno , Hughes Thomas J. R. 
Published: August 2005

AbstractA finite element formulation which includes the piezoelectric or electroelastic effect is given. A strong analogy is exhibited between electric and elastic variables, and a ‘stiffness’ finite element method is deduced. The dynamical matrix equation of electroelasticity is formulated and found to be reducible in form to the well‐known equation of structural dynamics, A tetrahedral finite element is presented, implementing the theorem for application to problems of three‐dimensional electroelasticity.
On the numerical solution of a class of problems in a linear first strain–gradient theory of elasticity
Authors: Oden J. T. , Rigsby D. M. , Cornett D. 
Published: August 2005

AbstractThis paper is concerned with the development of general discrete models for the analysis of boundary‐value problems in the first strain–gradient theory of elasticity. Extensions of the finite element method are constructed for this purpose, and general equations of motion are derived for finite elements of a class of micro‐polar materials which are characterized by strain energy functions involving strains and second gradients of strains or displacements. The notion of generalized nodal doublets is introduced. The problem of a composite consisting of a strain–gradient sensitive microlayer embedded between semi‐infinite bodies is examined as an example problem. Some of the results are compared with available exact solutions.
A multidata method of approximate Laplace transform inversion
Authors: Cost Thomas L. , Becker Eric B. 
Published: August 2005

AbstractA newly developed numerical Laplace transform inversion technique is described. A derivation of the method, termed the ‘multidata method’, is presented along with a description of a similar collocation method. Similarities and differences between the two numerical methods are described and discussed. Results of parameter studies of both methods are presented which demonstrate the sensitivity to error displayed by the collocation method and the magnitude of the improvement in accuracy obtainable with the multidata method as compared with the collocation method, especially when errors exist in the function to be inverted.
An application of the finite element technique to boundary value problems of potential flow
Authors: Doctors L. J. 
Published: August 2005

AbstractThis report summarizes an experimental application of the finite element approach to two‐dimensional inviscid fluid flow.The method results in a matrix equation relating the vector of velocities at the nodal points with the vector of singularities. The singularities, which are concentrated at the nodes, consist of a source and a vortex.On solution of this equation (defining the flow in the region), boundary conditions must te stipulated. This usually involves setting the internal singularities to zero and making certain modifications ro those lying on the boundary of the region.Some results of the application to a particular problem are included in the paper.
A conforming triangular finite element plate bending solution
Authors: Anderheggen Edoardo 
Published: August 2005

AbstractA conforming plate bending solution using simple polynomial deflection functions of third‐degree inside each triangular element is presented. In order to avoid normal slope discontinuities along the sides of the elements, the plate displacement parameters are subjected to ‘slope continuity conditions’ acting as constraints to the minimum potential energy problem. This is then solved by the classical method of Lagrange introducing multipliers as new auxiliary variables. If a special variational formulation of the problem is used, it can be shown that the Lagrangean multipliers are generalized stress parameters. The suggested solution is therefore basically a ‘mixed’ solution, the unknown variables of the problem being both displacement and stress parameters. Several numerical results are presented.
Calculations of plate frequencies from complementary energy formulation
Authors: Sakaguchi R. L. , Tabarrok B. 
Published: August 2005

AbstractThe complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments. It is shown that in this formulation, the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self‐equilibrating bending moments. In applying the Rayleigh Ritz procedure for the non‐zero frequency modes of the plate, it is shown that it the assumed functions are orthogonal to only a finite number of zero frequency modes, then one may obtain frequencies which are lower than the true frequencies of the plate. An iliustrative example is given in the paper.
The numerical solution of non‐linear consolidation problems
Authors: Poskitt T. J. 
Published: August 2005

AbstractIt has long been recognized that permeability, compressibility and viscosity vary during the consolidation of soil. However, the introduction of these variations into consolidation theory leads to formidable difficulties due to their rendering the equations highly non‐linear. Numerical methods are given herein which enable the governing equation to be solved for a variety of different non‐linear laws.
Non‐axial‐symmetric thermal stresses in circular discs or cylinders
Authors: Kettleborough C. F. 
Published: August 2005

AbstractA solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single‐valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non‐zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.
Note on dynamic relaxation
Authors: Wood Winifred L. 
Published: August 2005

AbstractThe Dynamic Relaxation (DR) method of solving a set of simultaneous linear equations requires an estimate of the spectral radius of the matrix. Dividing each equation by the corresponding row sum of moduli of the elements of the matrix gives a convenient upper bound of unity to this. This note shows that the DR method then gives a faster asymptotic rate of the convergence than the degenerate Chebyshev method which it closely resembles.
Use of optimization methods to solve heat convection problems with unknown wall boundary conditions
Authors: Gray Michael , Smith Russell A. 
Published: August 2005

AbstractIn convection heat transfer problems, it is convenient to solve the governing differential equations by using straightforward marching techniques for numerical integration. However, the boundary layer equations in these heat transfer problems are not of the initial value type even though they are parabolic equations. Consequently marching techniques are not successful until correct initial values are known. A means of determining these initial values is presented herein. Selected optimization methods are utilized in conjunction with marching integration techniques to solve third‐ and fourth‐order ordinary differential equations which have three unknown initial boundary conditions. Two optimization methods, one deterministic and the other non‐deterministic in nature, are used both independently and in combination as determined by the particular circumstances. Two applications are used to demonstrate the technique and comparison is made with existing solutions.
A simple algorithm for the stiffness matrix of triangular plate bending elements
Authors: Kratochvíl J. , Ženíšek A. , Zlámal M. 
Published: August 2005

AbstractFor a hierarchy of polynomials on the triangle there is derived an algorithm for computing the stiffness matrix of the plate bending element. The algorithm is easy to program and means a considerable saving of the computing time. The same approach can be used for any elliptic equation with constant coefficients.
Global‐local finite element
Authors: Mote C. D. 
Published: August 2005

AbstractFinite element procedures usually require more degrees of freedom for a specified accuracy than does a classical Ritz procedure if suitable coordinate functions are available. This paper develops a combined global and local dependent variable representation which couples the conventional and finite element Ritz methods. This hybrid method preserves much of the flexibility of the finite element method while increasing the solution accuracy for a specified system order. The method is illustrated by examination of a beam and a plate vibration problem.
On completeness of shape functions for finite element analysis
Authors: Taylor Robert L. 
Published: August 2005

AbstractSome elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n‐dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three‐dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third‐degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher‐order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.
Finite element analysis of plates with curved edges
Authors: Chernuka M. W. , Cowper G. R. , Lindberg G. M. , Olson M. D. 
Published: August 2005

AbstractThe application of the high precision triangular plate bending element to problems with curved boundaries is considered. Appropriate edge conditions for nodal points on these boundaries are derived. The error inherent in representing the shape of a curved boundary by a series of straight segments is found to be the limiting factor on accuracy, while the effect of approximations in the actual boundary conditions is minor. To overcome the first type of error, the high precision element is modified to include one curved edge. Substantial improvements in accuracy are obtained, as demonstrated in example calculations for circular and elliptical plates.
On the stability of explicit methods for the numerical integration of the equations of motion in finite element methods
Authors: Fu C. C. 
Published: August 2005

AbstractThis paper deals with the stability of the numerical solutions of a dynamic finite element analysis. The solutions are obtained through a stepwise integration of the equations of motion. Upper bounds on the steplength of the integration are obtained from a stability analysis of using a simple finite difference approximation for the equations of motion, and are shown to depend strongly on the particular element is use and on how the mass of the element is distributed at its nodes.As an example, the two‐dimensional wave propagation in a semi‐infinite plate subjected to a suddenly applied moment along its edge is studies. Through the example, we show that the bound on the steplength, obtained from the simple analysis, can provide a useful guide on choosing the steplength in other higher order integration methods. In particular, we show that, for stability considerations, the upper bound on the steplength should also hold for a fourth order explicit method. In order to achieve an acceptable accuracy of the solution, we show that the steplength should be approximately one half of the bound for the higher order explicit method as well as a higher order implicit method.Solution of the example has been compared with that of the Timoshenko theory.
Viscoelastic stress analysis
Authors: Carpenter William Calvin 
Published: August 2005

AbstractThis paper formulates the viscoelastic problem so as to allow the use of higher‐order differential equation solution techniques. Viscoelastic models describing the functional relationships of stress and strain for the material are used. However, the procedure outline is valid for more general relationships. The advantages of using Runge–Kutta integration formulae are indicated and an example demonstrates the excellent results obtainable. The computational efficiency of the method is emphasized and error prediction is discussed.
Solution of eigenvalue problems by sturm sequence method
Authors: Gupta K. K. 
Published: August 2005

AbstractA generalized eigenvalue algorithm is presented herein along with the complete listing of the associated computer program, which may be conveniently utilized for the efficient solution of certain broad classes of eigenvalue problems. Extensive applications of the procedure are envisaged in the analysis of many important engineering problems, such as stability and natural frequency analysis of practical discrete structural systems, idealized by the finite element technique. The procedure based on the Sturm sequence method is accurate and fast, possessing several significant advantages over other known methods of such analysis. Numerical results are also presented for two representative structural engineering problems.
Tetrahedral polynomial finite elements for the Helmholtz equation
Authors: Silvester P. 
Published: August 2005

AbstractIt is shown that the Helmholtz equation in three dimension leads to finite element approximations on tetrahedral elements that closely resemble the corresponding two‐dimensional treatment on triangle. For each polynomial order, there exist two numeric universal matrices independent to tetrahedron size and shape; the element matrices are always given as linear combinations of row and column permutations of these. Numeric matrices are given up to third‐order, and the permutation schemes are shown in detail. Experimental computer programs using these elements have shown fast matrix assembly times; convergence rates are essentially similar to those obtained with the corresponding triangular elements.
Effect of shear deformation on the elastica with axial strain
Authors: Huddleston John V. 
Published: August 2005

AbstractThe problem of the elastica is generalized to include the effect if shear as well as axial strain. This leads to a non‐linear two‐point boundary‐value problem composed of five first‐order differential equations. At first the problem is linearized to study the way in which the bucking loads are modified by the combination of effects. For a simple column, the eigenvalues are found to be zeros of a third‐order polynomial, and these are evaluated by a straightforward Newton–Raphson technique using a digital computer. The postbucking behaviour is then studied by using the computer and a ‘shooting’ method to solve the non‐linear problem. For the special case of the prismatic column, graphs plotted by the computer are displayed to show the variation of load with both lateral and axial deflection for various combinations of axial compressibility and shear deformability. For one particular case, the deflected center line at various load levels is shown.
On the unity of the constant strain/constant moment finite element methods
Authors: Hellan Kåre 
Published: August 2005

AbstractThe paper is an attempt to unify the finite element methods of ‘first’ order in plate stretching and plate bending, i.e. those assuming in each element constant strains in stretching or constant moments in bending. Reference is made to the stretching–bending analogy between the method of Turner, Clough, Martin and Topp and the method of Elias. By another analogy property a plate stretching method which is the dual to the constant moment method of Herrmann (and Hellan) is derived. The transformation of variables which carries the last pair of dual methods identically into the first pair is finally demonstrated.
A mixed finite element method for bending of plates
Authors: Poceski Apostol 
Published: August 2005

AbstractA 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.
Numerical stability in quasi‐static elasto/visco‐plasticity
Authors: Cormeau I. 
Published: August 2005

AbstractThe 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.
Solution of generalized geometric programs
Authors: Avriel M. , Dembo R. , Passy U. 
Published: August 2005

AbstractA 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 non‐linear inequality constraints is formulated and solved.
Use of the over‐relaxation technique in the simulation of large groundwater basins by the finite element method
Authors: Gambolati Giuseppe 
Published: August 2005

AbstractThe over‐relaxation technique has been used in conjunction with the finite element method in a regional time‐dependent simulation of the subsidence of Venice. One year of computer experiments have shown that in basins with a ratio h/R ⩽ 10−2 between the vertical and the horizontal dimension, the over‐relaxation technique can lead to unsatisfactory results. In single precision and for relatively large time steps, the solution of the final linear system can be inaccurate if the optimum over‐relaxation factor ω is not correctly assessed. In transient analyses, instability can also occur. The latter may be avoided by properly reducing the time step or by switching to double precision. Steady state simulations can also require double precision to provide accurate results even when the best ω is used. Instability and inaccuracy disappear in basins for which h/R ⩾ 10−1. In addition the processor time diminishes significantly as in this case the number of iterations necessary to obtain a good solution is considerably smaller than the order of the matrix.
The method of moments and its optimization
Authors: Leventhal Stephen H. 
Published: August 2005

AbstractThe method of moments is a semidiscrete numerical method for solving partial differential equations. The method approximates the solution of a partial differential equation by a finite sum of products of two functions. One function in the product is an unknown function of a single variable and the other function (moment function) is a prescribed function in the remaining variables. Using variational technique we obtain a finite system of boundary value problems of ordinary differential equations for the unknown functions.The main goal of this paper is the study of the theoretical background and numerical effectiveness of the method of moments for solving linear partial differential equations on rectangular‐like domains.The mathematical formulation of the method together with error estimates and the theory of optimal moment functions are given. If for the one‐dimensional moment functions piecewise polynomials of degree K are used then finite element type error bounds are obtained for the approximate solution in two dimensions.We also consider the numerical implementation of the method through the factorization method and efficient initial value methods. Several numerical examples showing the efficiency of the method are presented.
Finite element formulations for large deformation dynamic analysis
Authors: Bathe Klaus‐Jürgen , Ramm Ekkehard , Wilson Edward L. 
Published: August 2005

AbstractStarting 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.
Mathematical programming in wing theory
Authors: Bera Rajendra K. 
Published: August 2005

AbstractIt is shown that mathematical programming (MP) finds useful application in wing theory. This is illustrated by posing wing theoretical problems as programming problems within the framework of the linearized lifting surface theory. A great merit of programming techniques is their ability to accommodate inequality constraints in a problem. This feature allows a wing theoretician to expand into a domain of practical problems where qualitative constraints can be included into his formulations. A few numerical examples involving the Prandtl lifting line equation with equality and inequality constraints are given to show that the method works. Its extension to complex wing‐body problems poses no conceptual difficulties, but requires further numerical experimentation.