Unification of Approximate Methods
It seems that the approximate analysis of physical problems started with the calculus of variations, was raised with the finite differences and matured with the finite element method, depending upon the nature of a problem; however, each has its own merit
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LECTURES - No. 221
HAYRETTIN KARDESTUNCER UNIVERSITY OF CONNECTICUT
DISCRETE MECHANICS A UNIFIED APPROACH
SPRINGER-VERLAG WIEN GMBH
This work is subject to copyright.
AII rights are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
©
1975 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1975
ISBN 978-3-211-81379-9 DOI 10.1007/978-3-7091-4350-6
ISBN 978-3-7091-4350-6 (eBook)
To
Vinicio TUREILO
for his continuous effort in providing a truly academic atmosphere for great masters and students in mechanics of all nations. Such an effort will undoubtedly benefit the scientific world.
PREFACE
Rightfully or not, most researchers in mechanics today have divorced themselves (or tend to) from searching closed-form solutions for their problems. The complicated nature of these problems and the availability of the powerful number-crunching equipment are the main causes of this development. Shortcomings seem to arise, however, if, while observing these rapid changes one disassociates himself from the step stones of the past. The text introduces very briefly the three most popular approximate methods in their sequential order: calculus of variations, finite difference and finite elements. It then advocates the combination of all three in one physical problem. Each of these three has its own beauty and advantage over the other two yet no one alone is satisfactory for some problems. The initial and the boundary value combinations, particularly, such as path and time dependent problems force us to unify these methods. The text is intended to be a pioneering attempt in this direction. I am very grateful to Professor
w.
Olszak, Rector of CISM, for providing me with the
opportunity to present this work to the participants of the UNESCO series of lectures at the Center.
H. Kardestuncer
Udine July 1975
Chapter 1 INTRODUCfION
Ogni azione latta
e
dalla natura latta nel pi,} breve modo. Leonardo da Vinci
Nature always tends to minimize its effort in achieving its goal. Whether it is the maximum area of Queen Dido of Carthage, the minimum resistance of Newton or the shortest time of Bernouilli, most physical problems fall, theoretically at least, within the jurisdiction of variational calculus. Although the basic concept (minimum, maximum) of variational calculus was known to ancient Egyptians (reallocation of land after the Nile £loodings), and Greeks (maximum area or minimum distance), it could not be considered a mathematical discipline until the period of Newton, Leibnitz and the Bernouillis. Much of the formulation of this mathematics was developed by Euler [1] (1707-83) who was a pupil of Jacob Bernouilli. The earliest accounted problem in this discipline, the brachistochrone problem, was proposed by Johann Bernouilli in 1696 and simultaneously solved by his brother Jacob, Sir Isaac Newton and
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