Table Of ContentNORTH-HOLLAND SERIES IN
APPLIED MATHEMATICS
AND MECHANICS
EDITORS:
J. D. ACHENBACH
Norlh western University
B. BUDIANSKY
Harvard University
W.T. KOITER
University of Technology, Delft
H. A. LAUWERIER
University of Amsterdam
p. G. SAFFMAN
California Institute of Technology
L. VAN WIJNGAARDEN
Twente University of Technology
J. R.WILLIS
University of Bath
VOLUME 34
m
NORTH-HOLLAND - AMSTERDAM · NEW YORK · OXFORD ·ΤΟΚΥΟ
OPTIMIZATION IN MECHANICS:
PROBLEMS AND METHODS
Pierre BROUSSE
LJniversite Pierre et Marie Curie, Paris
Ecole Centrale de Arts et Manufactures, Paris
1988
NORTH-HOLLAND - AMSTERDAM · NEW YORK · OXFORD ·ΤΟΚΥΟ
^ELSEVIER SCIENCE PUBLISHERS B.V., 1988
All rights reserved. No part of this publication may he reproduced,
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ISBN: 0 444 70494 9
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PRINTED IN THE NETHERLANDS
INTRODUCTION
In this book our purpose is to show and develop several aspects of
Optimization in Mechanics. The study starts from the problems as they
appear in Mechanics. Their different characteristics induce us to
consider various methods of optimization which we introduce as rigorously
as necessary and which allow their solution. We take into account the
recently achieved progress in this fast expanding science which is called
Optimization in Mechanics. The book is intended to encourage thinking
over and to foster the birth of new ideas. Thus we would like it to
be a preparation for research and, at the same time, to be a book useful
for teaching, which provides an overall view of the subject.
The subjects under study are as varied as minimization of masses,
stresses or displacements, maximization of loads, vibration frequencies
or critical speeds of rotating shafts. No prerequisite in optimization
theory is needed. We only suppose that the reader has sufficient know
ledge of Mechanics and Applied Mathematics. This book is written for
students, engineers, scientists and even self-taught individuals. We
hope that they will afterwards be able to read the specialized works
and the numerous proceedings of symposia and congresses, and that they
will even be in a position to begin by themselves research in Optimiz
ation in Mechanics. In order to help them we give many references
throughout the development of this book. Several chapters have formed
the subjects of courses of "Troisieme Cycle" at the University of Paris
VI, at the "Ecole Centrale" of Paris, and of courses in adult education
organized for engineers and professionals.
Here is now a survey of the topics studied in the different chapters.
Chapter 1 deals only with examples. We show very briefly the use
fulness of some optimizations like those of a reinforced shell, a robot,
a booster. We also show how some optimizations of structures discret-
ized by finite element techniques arise. We then evaluate some quant
ities to be minimized or maximized for elastic vibrating structures
and plastic frames and plates, and we completely solve a strength maxim
ization of a structure under stability constraints. The examples show
the necessity of a presentation of preliminary mathematical concepts
which are required in the introduction and further justification of
the methods and algorithms of optimization. These concepts are presented
in Chapter 2. They are illustrated by some examples, certain aspects
vi Introduction
of which are new. These examples are related to structures subject
to several loadings and to fundamental vibration frequencies. Chapter 3
is devoted to the Kuhn Tucker theorem and to duality, with proofs. We
think that the knowledge of these proofs may be a help to a better and
more efficient use of these methods. In chapter 4 we systematically
study pairs of problems which we call associated problems and we draw
new theorems and practical conclusions while giving illustrative examples.
Obviously, we must present the basis of classical numerical methods
of mathematical programming. This is the subject of chapter 5. The
gradient and conjugate gradient methods, the Newton and the so-called
quasi-Newton methods, the linearization, penalty and projection methods
are then analysed starting from their foundations. In due place, we
give the corresponding algorithms which are chosen among the most typical
ones. Approximization methods become necessary for large scale problems.
In chapter 6 we analyse the so-called optimality criteria: the techniques
of fully-stressed design, the classical optimality criteria, then the
generalized optimality criteria and the mixed methods of C. Fleury which
seem very promising to us. Finally, the ideas, methods and techniques
offered above are combined in order to solve some optimizations of dis
crete or continuous structures subject to dynamical effects. Mass minim
ization and fundamental eigenvalue problems as well as problems of minim
ization of some dynamical responses are studied, certain of them with
new ideas. Computation methods, optimality conditions, results which
bring confidence to the techniques are illustrated in several examples.
The approximation method of N. Olhoff is presented in some detail. It
proves to be efficient for vibrating beams.
Although we try to be rigorous in the presentations and developments,
we omit the too long proofs and those which are not of any real help
in basic understanding. Numerical results are given since they are
obviously important. However, we remark that the use of computers
often benefits from being preceded by a serious study which then allows
simplification of the problem. We do not give computer routines and
we do not introduce complicated structures, thinking that they would
need many other chapters.
We should like to express our grateful thanks to all colleagues who
helped and encouraged us, in particular Prof. M. Dikmen from Istanbul,
Prof. S. Eskinazi from Syracuse University, Dr. C. Fleury from Liege, Prof.
C. Lamoureux from Paris and Prof. E. Masur from the University of Illinois.
Introduction vii
We also thank Dr. J.C. Harvey for the typing and improvements to the
text. We thank especially Prof. W. Koiter and Drs. A. Sevenster, Math
ematics Editors, who accepted this book for publication, and the North-
Holland Company for its usual perfection in the printing and the public
ation of this book.
Pierre Brousse
Chapter 1
EXAMPLES
The aim of this chapter is to give examples of optimization problems
in Mechanics and to consider their application. The problems are of
an industrial nature, even though they are somewhat specialized. They
are all contemporary in character.
First of all we give some short descriptions of structures: a stif
fened shell, a robot, a booster, and without going into details we show
how we are led to ask questions about optimization problems. Later, more
elaborate terms are introduced in the examples denoted by A, B, C, D,
where we can see how the quantities coming with the terms of these prob
lems can be obtained.
Stiffened shells
It is common to use stiffened cylindrical shells in Engineering. Sev
eral types of such shells exist. They may be reinforced by longerons
as shown in Fig. (1.1), or by transverse stiffeners, or again by both
longerons and transverse stiffeners as shown in Fig. (1.2). They may
be used under various conditions. Often they are connected with engines
and machines and may therefore be subjected to effects originating from
other mechanical parts, from fast gas flows, from large variations of
temperature, and so on.
Π
Fig. 1.1 Stiffened Fig. 1.2 Stiffened shell
cylindrical shell
A short look at the figures shows that many parameters remain to
2 1 Examples
be determined after the materials have been specified, even when the
inner diameter and the length are specified. Among these parameters
we have the thickness of the skin, the sizes of the cross-sections of
the longerons and of the transverse stiffeners, and the numbers and
locations of these stiffeners. The choice of these quantities is left
to the designer. They are called design variables, but they cannot
be taken arbitrarily. Indeed, the working conditions require the struc
ture to be able to function satisfactorily without damage and to perform
its assigned role. Moreover, manufacturing limitations require, for
example, that the thickness of the skin or the transverse sizes of the
stiffeners be not too small. Such conditions imposed in advance on
the design variables are called constraints.
It is natural to take advantage of the freedom to select the design
variables to reduce or to increase some quantity which is considered
to be of basic significance. For example we may try to choose values
of the design variables with a view toward making the total mass as
small as possible, or else toward making a certain dynamical effect
as large as possible. The problems thus posed are, respectively, a
mass minimization problem and a dynamical optimization problem.
The reader can look up Refs. [B II, Μ 16, Μ 20, S 3, S 5, S 10, S 17,
S 20] for the discussion of some examples regarding this subject.
Robot
A second example deals with a robot, such as the one shown diagramat-
ically in Fig. (1.3). The robot has to carry heavy castings from a
position P^^ to a position P^ by means of a moving system comprising
a part C which may go up and down, an arm A and a forearm F which can
turn round their axes. These motions are controlled by special internal
motors programmed in advance.
At first sight, several optimal conditions may be essential.
The first consists of choosing the position of the support, the leng
ths of the arm and forearm, and the speeds of translation and rotation,
so that the time for a cycle, i.e. for a turn, is minimum. This time,
which is to be minimized, is called the objective function. The elements
over which the designer may exercise a choice, i.e. the position, lengths
and speeds, are called design variables. They are subject to limitations,
in particular in relation to the powers of motors commercially avail
able.
1 Examples
Fig. 1.3 Robot
Now let us suppose that the geometry of the structure is specified,
as well as the speeds of the moving parts. A second optimization prob
lem consists in reducing as far as possible the operating cost during
each cycle of the motion. Perhaps the nature of the motors is to be
considered again. However, a sure way to reduce the cost is to reduce
the weight of the moving system, by a suitable choice of the so-called
design variables, for example the shapes and the transverse sizes and
the material characteristics. Obviously, the design variables are
again subject to constraints, since the structure must be strong enough
to carry out its mission. The weight to be minimized represents another
objective function.
Finally, some designers may desire to reduce the manufacturing price.
More generally, they may take into consideration a generalized cost or
objective function, including, in specified proportions, the time of
a motion cycle, the weight of the moving system and the manufacturing price.
These are some of the optimization problems which may be considered
regarding the robot under construction.
Booster
A recent example of optimization concerns the European launcher
Ariane. In a future version, many improvements will be made to the
present launcher. Thus four boosters instead of two will be used
in order to increase the thrust. But the more complicated the struct
ure is, the heavier it becomes. However, to have a lightweight launcher
^ 1 Examples
would be important, and to reduce the mass is therefore of fundamental
concern. Indeed, each kilogram saved on the booster allows increasing
the payload by 0.15 kilogram. This is why some parts of the structure
have been submitted to adequate optimization techniques in order to
reduce their masses.
Let us briefly present an optimization problem of the engine mount
structure which is located at the base of the booster. Details con
cerning the methods will be found in a paper by C. Fleury and V. Braib-
ant in Ref. [F 18]. The foundations of similar methods will be dealt
with in Chapt. 6 of this book. However, significant difficulties
have remained. First, they were due to the performance to be obtained:
to reduce again the weight of a structure already studied is not an
easy thing. Then, the problem involves various and numerous constraints,
such as stiffness requirements at joints and many other places, limit
ations on the normal stress flow in rings, limitations on stress under
different loadings, and so on. Moreover, the structure was really
complex, and this complexity came as well from internal geometry as
from manufacturing processes. Thus, efficient techniques have required
finite element models involving several thousands of degrees of freedom
and about one thousand elements.
This example calls attention to the importance of finite elements
in structural analysis. It is therefore essential to examine with
details the main optimization problems concerning discretized structures.
This is the objective of the first subchapter (l.A).
l.A STRUCTURES DISCRETIZED BY FINITE ELEMENT TECHNIQUES
Discretization by finite element techniques represents a general
and commonly used method of analysing elastic structures. It is there
fore important to study how optimization problems relating to these
structures may be approached.
The problem of mass minimization has so far received the most attent
ion. Indeed, it is a fundamental engineering problem, in which the
reduction of the operating cost plays a leading part, as in Aeronautics.
However, other problems may arise, as shown in Sect. (1.6.4).
In a preliminary Sect. (1.1) we briefly recall some fundamentals
of discretization techniques by the displacement method. Then, in
Sect. (1.2), we shall present general classes of structures for which
various types of optimization have become indispensable.
