## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

A common approach for dynamic analysis in current practice is based on a discrete time-integration scheme. This approach can be largely attributed to the absence of a true variational framework for initial value problems. To resolve this problem, a new stationary variational principle was recently established for single-degree-of-freedom oscillating systems using mixed variables, fractional derivatives and convolutions of convolutions. In this mixed convolved action, all the governing differential equations and initial conditions are recovered from the stationarity of a single functional action. Thus, the entire description of linear elastic dynamical systems is encapsulated. For its practical application to structural dynamics, this variational formalism is systemically extended to linear elastic multidegree- of-freedom systems in this study, and a corresponding weak form is numerically implemented via a quadratic temporal finite element method. The developed numerical method is symplectic and unconditionally stable with respect to a time step for the underlying conservative system. For the forced-damped vibration, a three-story shear building is used as an example to investigate the performance of the developed numerical method, which provides accurate results with good convergence characteristics.

To read the full-text of this research,

you can request a copy directly from the authors.

A well-posed time weak form for linear structural dynamics is used to construct a Galerkin time quadrature element formulation. Radau quadrature rule and the generalized differential quadrature analog are used to turn the well-posed weak form into a set of linear equations. The stability and accuracy properties of the formulation are discussed. Numerical examples are given to show the high computational efficiency of the well-posed weak form time quadrature element formulation, as compared with a time finite element solution based on the same weak form using third-order Hermite interpolations.

Purpose
Various time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.
Design/methodology/approach
A variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.
Findings
The weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.
Originality/value
The weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

Although a complete unified theory for elasticity, plasticity, and damage does not yet exist, an approach on the basis of thermomechanical principles may be able to serve as the foundation for such a theory. With this in mind, as a first step, a mixed formulation is developed for fully coupled, spatially discretized linear thermoelasticity under the Lagrangian formalism by using Hamilton's principle. A variational integration scheme is then proposed for the temporal discretization of the resulting Euler-Lagrange equations. With this discrete numerical time-step solution, it becomes possible, for proper choices of state variables, to restate the problem in the form of an optimization. Ultimately, this allows the formulation of a principle of minimum generalized complementary potential energy for the discrete-time thermoelastic system. DOI: 10.1061/(ASCE)EM.1943-7889.0000346. (C) 2012 American Society of Civil Engineers.

The primary objective of the present work is to make further connections between variational methods on the one hand and reversible and irreversible thermodynamics on the other. This begins with the development of a new stationary principle, involving mixed field variables, for continuum problems in infinitesimal dynamic thermoelasticity. By defining Lagrangian and dissipation functions in terms of physically-relevant contributions and invoking the Rayleigh formalism for damped systems, we are able to recover the governing equations of thermoelasticity as the Euler–Lagrange equations. This includes the balance laws of linear momentum and entropy-energy, the constitutive models for elastic response and heat conduction, and the natural boundary conditions. By including energy contributions associated with second sound phenomena, one eliminates the paradox of infinite thermal propagation speeds and the resulting set of governing equations has an elegant symmetry, which is most easily seen in the Fourier wave number domain. A related formulation for dynamic poroelasticity yields two new stationary mixed variational principles. Depending upon the selection of primary field variables, these governing equations can also exhibit an elegant structure, which can deepen our understanding of the underlying phenomena and the thermoelastic–poroelastic analogy. In addition to the theoretical significance, the variational formulations developed here can provide the basis for a class of optimization-based methods for computational mechanics.

We comment on the method of Dreisigmeyer and Young J. Phys. A: Math. Gen.368297 to model nonconservative systems with fractional derivatives. It was previously hoped that using fractional derivatives in an action would allow us to derive a single retarded equation of motion using a variational principle. It is proven that, under certain reasonable assumptions, the method of Dreisigmeyer and Young fails.

A general consistent thermodynamic framework for small strain thermoviscoplastic deformations of face-centered cubic (FCC) metals is presented in this study. An appropriate and consistent Helmholtz free energy definition is incorporated, after considering the strain rate effect imbedded through the hardening definition, in deriving the proposed three-dimensional kinematical model. Microstructural physically based thermal and athermal yield function definitions (von Mises type) are utilized in this work for dynamic and static deformations of FCC metals. A length scale parameter introduced implicitly through the viscosity parameter is related to the waiting time of dislocations at an obstacle. The role of material dependence in setting the character of the governing equations is illustrated in the context of a simple uniaxial tensile problem in order to check the effectiveness and the performance of the proposed framework and its finite-element implementation. Results obtained for OFHC copper at low and high strain rates and temperatures show, generally, good comparisons with experimental results.

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton's principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.

Fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems. In this note we address multidimensional fractional actionlike problems of the calculus of variations.

Cited By (since 1996): 42, Export Date: 10 August 2012, Source: Scopus, Art. No.: 053521, doi: 10.1063/1.2929662, Language of Original Document: English, Correspondence Address: El-Nabulsi, R. A.; Department of Nuclear and Energy Engineering, Faculty of Mechanical, Energy and Production Engineering, Cheju National University, Ara-dong 1, Jeju 690-756, South Korea; email: nabulsiahmadrami@yahoo.fr, References: Agrawal, O.P., (2002) Formulation of Euler-Lagrange Equations for Fractional Variational Problems, 272, p. 368. , J. Math. Anal. Appl.;

A new algorithm for nonlinear dynamic simulation of structures is presented. The algorithm is based on a mixed Lagrangian approach described by Sivaselvan and Reinhorn (J. Eng. Mech. (ASCE) 2006; 132(8):795–805). The algorithm developed in this paper is for the simulation of large-scale structural systems. The algorithm is applicable to a wide class of structural systems whose constituent material or component behavior can be derived from a stored energy function and a dissipation potential. The algorithm is based on the fact that for such systems, when using a certain class of time-discretization schemes to numerically compute the system response, the incremental problem of computing the system state at the next sample time knowing the current state and the input is one of convex minimization. As a result, the algorithm possesses excellent convergence characteristics. It is also applicable to geometric nonlinear problems. The implementation of the algorithm is described, and its applicability to the collapse analysis of large-scale structures is demonstrated through numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.

The paper shows the key place of the choice of the bilinear form in order to give a variational formulation to a given problem.
In particular it is shown how the use of a convolution bilinear form makes possible a variational formulation for linear initial
value problems.
A critical survey of the three main methods that was devised in the past to solve the same problem is done.
La nota mette in evidenza il ruolo fondamentale della scelta di una forma bilineare al fine di dare formulazione variazionale
ad un dato problema. In particolare è mostrato come l'uso di una forma bilineare di convoluzione renda possibile la formulazione
variazionale dei problemi ai valori iniziali.
Si fa un esame critico dei tre principali metodi che sono stati escogitati nel passato per risolvere lo stesso problema.

Space-time finite element methods are presented to accurately solve elastodynamics problems that include sharp gradients due to propagating waves. The new methodology involves finite element discretization of the time domain as well as the usual finite element discretization of the spatial domain. Linear stabilizing mechanisms are included which do not degrade the accuracy of the space-time finite element formulation. Nonlinear discontinuity-capturing operators are used which result in more accurate capturing of steep fronts in transient solutions while maintaining the high-order accuracy of the underlying linear algorithm in smooth regions. The space-time finite element method possesses a firm mathematical foundation in that stability and convergence of the method have been proved. In addition, the formulation has been extended to structural dynamics problems and can be extended to higher-order hyperbolic systems.

A series of stationary principles are developed for dynamical systems by
formulating the concept of mixed convolved action, which is written in terms of
mixed variables, using temporal convolutions and fractional derivatives.
Dynamical systems with discrete and continuous spatial representations are
considered as initial applications. In each case, a single scalar functional
provides the governing differential equations, along with all the pertinent
initial and boundary conditions, as the Euler-Lagrange equations emanating from
the stationarity of this mixed convolved action. Both conservative and
non-conservative processes can be considered within a common framework, thus
resolving a long-standing limitation of variational approaches for dynamical
systems. Several results in fractional calculus also are developed.

So far, it is not well known how to deal with dissipative systems. There are
many paths of investigation in the literature and none of them present a
systematic and general procedure to tackle the problem. On the other hand, it
is well known that the fractional formalism is a powerful alternative when
treating dissipative problems. In this paper we propose a detailed way of
attacking the issue using fractional calculus to construct an extension of the
Dirac brackets in order to carry out the quantization of nonconservative
theories through the standard canonical way. We believe that using the extended
Dirac bracket definition it will be possible to analyze more deeply gauge
theories starting with second-class systems.

We generalize the fractional variational problem by allowing the possibility
that the lower bound in the fractional derivative does not coincide with the
lower bound of the integral that is minimized. Also, for the standard case when
these two bounds coincide, we derive a new form of Euler-Lagrange equations. We
use approximations for fractional derivatives in the Lagrangian and obtain the
Euler-Lagrange equations which approximate the initial Euler-Lagrange equations
in a weak sense.

We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the Lagrangian framework by treating the action as a Volterra series. It is then possible to derive two equations of motion, one of these is an advanced equation and the other is retarded.

Recently, an extension of the simplest fractional problem and the fractional variational problem of Lagrange was obtained by Agrawal. The first part of this study presents the fractional Lagrangian formulation of mechanical systems and introduce the Levy path integral. The second part is an extension to Agrawal’s approach to classical fields with fractional derivatives. The classical fields with fractional derivatives are investigated by using the Lagrangian formulation. The case of the fractional Schro¨dinger equation is presented.

This is a textbook written for use in a graduate-level course for students of mechanics and engineering science. It is designed to cover the essential features of modern variational methods and to demonstrate how a number of basic mathematical concepts can be used to produce a unified theory of variational mechanics. As prerequisite to using this text, we assume that the student is equipped with an introductory course in functional analysis at a level roughly equal to that covered, for example, in Kolmogorov and Fomin (Functional Analysis, Vol. I, Graylock, Rochester, 1957) and possibly a graduate-level course in continuum mechanics. Numerous references to supplementary material are listed throughout the book. We are indebted to Professor Jim Douglas of the University of Chicago, who read an earlier version of the manuscript and whose detailed suggestions were extremely helpful in preparing the final draft. He also gratefully acknowledge that much of our own research work on variational theory was supported by the U.S. Air Force Office of Scientific Research. He are indebted to Mr. Ming-Goei Sheu for help in proofreading. Finally, we wish to express thanks to Mrs. Marilyn Gude for her excellent and pains taking job of typing the manuscript. J. T. ODEN J. N. REDDY Table of Contents PREFACE 1. INTRODUCTION 1.1 The Role of Variational Theory in Mechanics. 1 1.2 Some Historical Comments ......... . 2 1.3 Plan of Study ............... . 5 7 2. MATHEMATICAL FOUNDATIONS OF CLASSICAL VARIATIONAL THEORY 7 2.1 Introduction . . . . . . . .

The former Essay contained a general method for reducing all the most important problems of dynamics to the study of one characteristic function, one central or radical relation. It was remarked at the close of that Essay, that many eliminations required by this method in its first conception, might be avoided by a general transformation, introducing the time explicitly into a part S of the whole characteristic function V; and it is now proposed to fix the attention chiefly on this part S, and to call it the Principal Function . The properties of this part or function S, which were noticed briefly in the former Essay, are now more fully set forth; and especially its uses in questions of perturbation, in which it dispenses with many laborious and circuitous processes, and enables us to express accurately the disturbed configuration of a system by the rules of undisturbed motion, if only the initial components of velocities be changed in a suitable manner. Another manner of extending rigorously to disturbed motion the rules of undisturbed, by the gradual variation of elements, in number double the number of the coordinates or other marks of position of the system, which was first invented by Lagrange, and was afterwards improved by Poisson, is considered in this Second Essay under a form perhaps a little more general; and the general method of calculation which has already been applied to other analogous questions in optics and in dynamics by the author of the present Essay, is now applied to the integration of the equations which determine these elements. This general method is founded chiefly on a combination of the principles of variations with those of partial differentials, and may furnish, when it shall be matured by the labours of other analysts, a separate branch of algebra, which may be called perhaps the Calculus of Principal Functions ; because, in all the chief applications of algebra to physics, and in a very extensive class of purely mathematical questions, it reduces the determination of many mutually connected functions to the search and study of one principal or central relation. When applied to the integration of the equations of varying elements, it suggests, as is now shown, the consideration of a certain Function of Elements , which may be variously chosen, and may either be rigorously determined, or at least approached to, with an indefinite accuracy, by a corollary of the general method. And to illustrate all these new general processes, but especially those which are connected with problems of perturbation, they are applied in this Essay to a very simple example, suggested by the motions of projectiles, the parabolic path being treated as the undisturbed. As a more important example, the problem of determining the motions of a ternary or multiple system, with any laws of attraction or repulsion, and with one predominant mass, which was touched upon in the former Essay, is here resumed in a new way, by forming and integrating the differential equations of a new set of varying elements, entirely distinct in theory (though little differing in practice) from the elements conceived by Lagrange, and having this advantage, that the differentials of all the new elements for both the disturbed and disturbing masses may be expressed by the coefficients of one disturbing function.

The theoretical development of the laws of motion of bodies is a problem of such interest and importance, that it has engaged the attention of all the most eminent mathematicians, since the invention of dynamics as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton. Among the successors of those illustrious men, Lagrange has perhaps done more than any other analyst, to give extent and harmony to such deductive researches, by showing that the most varied consequences respecting the motions of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem. But the science of force, or of power acting by law in space and time, has undergone already another revolution, and has become already more dynamic, by having almost dismissed the conceptions of solidity and cohesion, and those other material ties, or geometrically imaginable conditions, which Lagrange so happily reasoned on, and by tending more and more to resolve all connexions and actions of bodies into attractions and repulsions of points: and while the science is advancing thus in one direction by the improvement of physical views, it may advance in another direction also by the invention of mathematical methods. And the method proposed in the present essay, for the deductive study of the motions of attracting or repelling systems, will perhaps be received with indulgence, as an attempt to assist in carrying forward so high an inquiry. In the methods commonly employed, the determination of the motion of a free point in space, under the influence of accelerating forces, depends on the integration of three equations in ordinary differentials of the second order; and the determination of the motions of a system of free points, attracting or repelling one another, depends on the integration of a system of such equations, in number threefold the number of the attracting or repelling points, unless we previously diminish by unity this latter number, by considering only relative motions. Thus, in the solar system, when we consider only the mutual attractions of the sun and of the ten known planets, the determination of the motions of the latter about the former is reduced, by the usual methods, to the integration of a system of thirty ordinary differential equations of the second order, between the coordinates and the time; or, by a transformation of Lagrange, to the integration of a system of sixty ordinary differential equations of the first order, between the time and the elliptic elements: by which integrations, the thirty varying coordinates, or the sixty varying elements, are to be found as functions of the time. In the method of the present essay, this problem is reduced to the search and differentiation of a single function, which satisfies two partial differential equations of the first order and of the second degree: and every other dynamical problem, respecting the motions of any system, however numerous, of attracting or repelling points, (even if we suppose those points restricted by any conditions of connexion consistent with the law of living force,) is reduced, in like manner, to the study of one central function, of which the form marks out and characterizes the properties of the moving system, and is to be determined by a pair of partial differential equations of the first order, combined with some simple considerations. The difficulty is therefore at least transferred from the integration of many equations of one class to the integration of two of another: and even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from the reduction of the most complex and, probably., of all researches respecting the forces and motions of body, to the study of one characteristic function, the unfolding of one central relation.

Lagrangian and Hamiltonian mechanics can be formulated to include derivatives of fractional order [F. Riewe, Phys. Rev. 53, 1890 (1996)]. Lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction. The present work continues the development of fractional-derivative mechanics by deriving a modified Hamilton's principle, introducing two types of canonical transformations, and deriving the Hamilton-Jacobi equation using generalized mechanics with fractional and higher-order derivatives. The method is illustrated with a frictional force proportional to velocity. In contrast to conventional mechanics with integer-order derivatives, quantization of a fractional-derivative Hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives. Instead, a quantum-mechanical wave equation is proposed that follows from the Hamilton-Jacobi equation by application of the correspondence principle.

The principle of mixed convolved action provides a new rigorous weak variational formalism for a broad range of initial value problems in mathematical physics and mechanics. Here, the focus is initially on classical single-degree-of-freedom oscillators incorporating either Kelvin-Voigt or Maxwell dissipative elements and then, subsequently, on systems that utilize fractional-derivative constitutive models. In each case, an appropriate mixed convolved action is formulated, and a corresponding weak form is discretized in time using temporal shape functions to produce an algorithm suitable for numerical solution. Several examples are considered to validate the mixed convolved action principles and to investigate the performance of the numerical algorithms. For undamped systems, the algorithm is found to be symplectic and unconditionally stable with respect to the time step. In the case of dissipative systems, the approach is shown to be robust and to be accurate with good convergence characteristics for both classical and fractional-derivative based models. As part of the derivations, some interesting results in the calculus of Caputo fractional derivatives also are presented.

The dynamic analysis of progressive collapse faces a great number of obstacles that often lead to the collapse of the analysis prior to the actual analysis of collapse. Hence, the Mixed Lagrangian Formulation that has been shown to be very robust was adopted as a framework to accommodate such analysis. By modifying the loading function and the numerical scheme, the capabilities of this framework were extended to account for strength degradation and fracture, while some insight to its behavior is introduced as well. The examples presented show a very robust and stable behavior of the numerical scheme in terms of the time step size required, even in cases where a sudden fracture takes place. Copyright © 2009 John Wiley & Sons, Ltd.

A systematic procedure is presented for the stability and accuracy analysis of direct integration methods in structural dynamics. Amplitude decay and period elongation are used as the basic parameters in order to compare various integration methods. The specific methods studied are the Newmark generalized acceleration scheme, the Houbolt method and the Wilson θ-method. The advantages of each of these methods are discussed. In addition, it is shown how the direct integration of the equations of motion is related to the mode superposition analysis.

This paper presents a general finite element formulation for a class of Fractional Variational Problems (FVPs). The fractional derivative is defined in the Riemann–Liouville sense. For FVPs the Euler–Lagrange and the transversality conditions are developed. In the Fractional Finite Element Formulation (FFEF) presented here, the domain of the equations is divided into several elements, and the functional is approximated in terms of nodal variables. Minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme. Three examples are considered to show the performance of the algorithm. Results show that as the number of discretization is increased, the numerical solutions approach the analytical solutions, and as the order of the derivative approaches an integer value, the solution for the integer order system is recovered. For unspecified boundary conditions, the numerical solutions satisfy the transversality conditions. This indicates that for the class of problems considered, the numerical solutions can be obtained directly from the functional, and there is no need to solve the fractional Euler–Lagrange equations. Thus, the formulation extends the traditional finite element approach to FVPs.

New results on the patterns of linearly independent rows and columns among the blocks of a symplectic matrix are presented. These results are combined with the block structure of the inverse of a symplectic matrix, together with some properties of Schur complements, to give a new and elementary proof that the determinant of any symplectic matrix is +1. The new proof is valid for any field. Information on the zero patterns compatible with the symplectic structure is also presented.

Traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction. A method is proposed that uses a Lagrangian containing derivatives of fractional order. A direct calculation gives an Euler-Lagrange equation of motion for nonconservative forces. Conjugate momenta are defined and Hamilton's equations are derived using generalized classical mechanics with fractional and higher-order derivatives. The method is applied to the case of a classical frictional force proportional to velocity.

The Theory of Sound, Macmillan and co

- J W S Rayleigh

J. W. S. Rayleigh, The Theory of Sound, Macmillan and co.,
London (1877).