Multicriteria Design Optimization: Procedures and Applications
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Multicriteria Design Optimization Procedures and Applications
With l7l Figures
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong
Kong Barcelona
Prof. Dr.-Ing. Hans Eschenauer University of Siegen Research
Laboratory for Structural Optimization at the Institute of
Mechanics and Control Engineering 0-5900 Siegen Germany
Assoc. Prof. Dr. Eng. luhani Koski Dept. of Mechanical Engineering
Tampere University of Technology P.O. Box 5Il S F-33 101 Tampere
Finland
Assoc. Prof. Dr. hab. inz. Andrzej Osyczka Technical University of
Cracow Institute of Machine Technology PL-31-155 Cracow
Poland
ISBN 978-3-642-48699-9 ISBN 978-3-642-48697-5 (eBook)
DOl 10.1007/978-3-642-48697-5
Library of Congress Cataloging-in-Publication Data Multicriteria
design optimization: procedures and applications [edited by] Hans
Eschenauer, Juhani Koski, Andrzej Osyczka. Includes bibliographical
references and indexes. ISBN 978-3-642-48699-9 I. Engineering
design-Mathematical models. 2. Mathematical optimization. I.
Eschenauer, Hans. II. Koski,Juhani.1I1. Osyczka, Andrzej. TA174.M85
1990 620'.00425-dc20 90-9931
This work is subject to copyright. All rights are reserved, whether
the whole or part of the material is concerned, specifically the
rights of translation, reprinting, re-use of illustrations,
recitation, broad casting, reproduction on microfilms or in other
ways, and storage in data banks. Duplication of this publication or
parts thereof is only permitted under the provisions ofthe German
Copyright Law of September9, 1965, in its current version and a
copyright fee must always be paid. Violations fall under the
prosecution act of the German Copyright Law.
© Springer-Verlag Berlin, Heidelberg 1990 Softcover reprint of the
hardcover 18t edition 1990
The use of registered names, trademarks,etc. in this publication
does not imply,even in the absence of a specific statement, that
such names are exempt from the relevant protective laws and
regulations and therefore free for general use.
216113020-543210 - Printed on acid-free paper
To Gerda
To Anu
To Laura
The modern era of design optimization began about twenty
years
ago with the recognition of the usefulness of mathematical
programming
techniques. Methods based on mathematical programming were
first
adapted to single-criterion optimum design problems. Now, more
atten
tion is given to multicriteria modelling, as in many engineering
appli
cations often several conflicting criteria have to be considered by
the
designer.
Even though multicriteria optimization goes back as far as V.
Pareto's
study in 1898, a greater interest in such fields as optimization
theory,
operations research, and control theory was not aroused until the
late
1960s. Since that time numerous studies on this topic have
been
published. Most of them deal with the theory of decision making
from
a general point of view whereas only a relatively small number
of
publications can be found in the field of engineering design. Thus,
the
aim of this book is to fill this gap and to provide the designer
with a
new tool for solving the optimization problems in which several
con
flicting and noncommensurable criteria are to be satisfied. In
order to
get a representative survey of the current works, the editors asked
for
contributions from some leading researchers so that a broad range
of
applications could be gathered in a coherent volume.
In order to introduce the subject to the readers, Chapter 1
outlines
the background of multicriteria optimization, broadly describes
the
relevant mathematical procedures, and also shows some
real-life
examples which motivate the designer to apply multicriteria
techniques.
The first part of this volume (Chapters 2-4) deals with
multicriteria
optimization procedures. Chapter 2 presents the optimization
procedure
SAPOP which provides the designer with a general tool for
solving
structural optimum design problems. Most activities in
multicriteria
design optimization concentrate on the application of interactive
proce
dures. Chapter 3 outlines these procedures irrespective of their
role in
the design process and also describes two software packages
which
facilitate the interactive processes for optimum design.
Knowledge-based
systems recently aroused great interest. Their use in multicriteria
design
optimization is described in Chapter 4.
VIII Preface
The second part of this volume is devoted to the application of
multi
criteria techniques to different design problems which are divided
into
subject groups. The first group deals with mechanisms and
dynamic
systems. Here, Chapters 5.1 and 5.2 are devoted to the problem
of
optimal balancing of robot arms using counterweights and spring
mecha
nisms. For the optimum design of spring balancing mechanisms,
a
general method for dealing with computationally expensive
objective
functions has been proposed. Optimization of automotive drive
train
and multibody systems are discussed in Chapters 5.3 and 5.4.
Chapter
5.5 shows a special method for finding a relationship between
FEM
analysis and optimization procedures using regression models.
The
second subject group explores aircraft and space technology
problems.
In Chapter 6.1 multicriteria optimal layouts of aircraft and
spacecraft
structures are discussed whereas Chapter 6.2 presents poblems of
space
craft structures with emphasis on mass and stiffness.
Multicriteria
optimization of machine tool systems is the subject of the thit"d
group.
In Chapter 7.1 design problems of machine tool structures are
presented,
and in Chapter 7.2 the optimum design of machine tool spindle
systems
using a decomposition method is discussed. The fourth subject
group
deals with metal forming and cast metal technology. In Chapter 8.1,
a
multicriteria optimal control approach is applied to die designs
for
symmetric strip drawing. Optimal layouts of heterogeneous
thick-walled,
chilled cast-iron rollers are presented in Chapter 8.2, and a
metal
forming process is optimized and simulated in Chapter 8.3.
Problems
of civil and architectural engineering are considered in two
chapters.
Chapter 9.1 presents the multicriteria optimization of concrete
beams,
trusses, and cable structures, and in Chapter 9.2 multicriteria
opti
mization techniques are applied to architectural planning. Finally,
the
optimization of structures made of advanced materials is
discussed.
Chapters 10.1, 10.2, and 10.4 deal with fibre-reinforced plate and
shell
structures and ceramic components, respectively. In Chapter 10.3
multi
criteria optimization and advanced materials in telescope design
are
presented.
The editors wish to express their appreciation to all authors for
their
contributions and their cooperation in revising the chapters. We
are
especially grateful to Ms Ursula Schmitz (Stud.Ass.) who has
per
formed the type-setting of the book with great skill and
efficiency.
She has also assisted as a translation editor for all chapters and
tried
to meet the editors' requirements with much care and patience.
In
Preface IX
preparing and organizing the publishing process, she did a splendid
job.
We would also like to express our sincere thanks to Ms Birgit
Holl
stein and Mr Michael Wengenroth for supervising the work on
the
book in its final phase. Thanks are also due to Ms Petra Franke,
Ms
Regina Knepper and Ms Henrike StrohbUcker who have done the
drawing
of figures.
The editors wish to express their special thanks and appreciation
to
Dr. R.D. Pat'bery (University of Newcastle/ Australia) for
carefully proof
reading the typescl"ipt. On this occasion, Dr. Parbery would like
to
thank the German Research Community <DFG) for sponsoring his
stay
as a visiting professor at the University of Siegen.
The DFG is also owed a debt of gratitude for its sponsorship
of
Professor A. Osyczka's eight-month-stay at the University of
Siegen
where he did the main part of his work on the book.
Finally, we are indebted to Ms. E. Raufelder and Mr. A. von Hagen
of
Springer Publishing Company, Heidelberg for the excellent
cooperation.
Hans A. Eschenauer Juhani Koski Andrzej Osyczka
Siegen, Tampet'e, Cracow March 1990
CONTENTS
H.A. Eschenauer, J. Koski, A. Osyczka
1.1 Introduction .......................... .
Techniques ........................ .
the Design Process. . . . . . . . . . . . . . . . . . . . 3
Design Process. . . . . . . . . . . . . . . . . . . . 4
Optimization . . . . . . . . . . . . . . . . . . . 6
1.3 Components and Plants with their Objectives. .20
1.3.1 Optimum Design of Highly Accurate Parabolic
Antennas . . . . . . . . . . . . . . . . . . . . . . . 20
Collector. . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.4 Optimal Shape Design. . . . . . . . . . . . . . . 25
1.3.5 Optimal Layout of Tube-Flange Structures. .26
1.4 Conclusion .28
2 Optlrn.ization Procedure S.A.POP-.A. General Tool for ~ul ticri
teria Structural Designs 35
M. Bremicker, H.A. Eschenauer, P. U. Post
2.1 Demands on an Optimization Procedure.
2.2 Structure of the Optimization Procedure.
2.2.1 Definitions ............. .
2.3.1 Problem Formulation and Input Data.
2.3.2 SAPOP Main Module ........... .
2.4 Optimization Modelling.
2.4.1 Design Models . . ..
2.4.2 Evaluation Models .
Reduced Line-Search Technique (QPRLT)' . . . .. .55
2.6 Comparison with other Structural Optimization Soft-
ware Systems .....
.58
.58
.65
.66
3.1 Introduction ................. . .71
3.2.2 Approach by Fandel .74
3.2.3 STEP-Method ...... .78
3.3.1 Basic Structure ......... . . . . . . . . . . 86
3.4 Software Package CAM OS. . . . . . . . . . . . . . . .
101
3.4.1 Optimization Algorithms Used in CAMOS . 102
3.4.2 Multicriteria Strategy Approaches . . . . . . . 104
3.4.3 Description of CAMOS . . . . . . . . . . . . . . 104
3.4.4 Interactive MO-Layouts of a Machine Tool
Spindle . 107
M. Balachandran, J.S. Gero
4.3 Role of a Knowledge-Based Approach in Multicriteria
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . ..
.119
4.4.1 Description and Representation of Optimization
Problems. . . . . . . . . . . . . . . . . . . . . . . . . 122
Problems ............................. 127
4.5.1 Recognition of Optimization Formulation . . . . . 133
4.5.2 Optimization Algorithm Selection . . . . . . . .. . 135
4.5.3 Knowledge-Based Control in Pareto-Optimal
Set Generation. . . . . . . 137
Sy & te:r.n.& ................... .... . · .. 151
5.1 Optimal Counterwelght Balancing of Robot Arms Using
Multicriteria Approach . ......................... 151
J. Koski, A. Osyczka
5.1.3 Formulation of the Optimization Problem
5.1.4 Solution Method .....
5.1.5 Pareto-Optimal Designs
A. Osyczka, J. Zajac
of Industrial Robots ....................... 174
Balancing Mechanism . . . . . . . . . . . . . . . . . 181
· 183
5.3 On the Optimal Synthesis of an Automotive Drive Train . ..
184
F. Pfeiffer
5.3.2 The Mechanical Model . . . . . .... .
5.3.3 The Optimization Model . . . . . . . . .
5.3.4 The Solution Procedure
References ... 192
5.4 Modelling of Multibody Systems by Means of Optimiz- ation
Procedures ... 193
H.H. MUller-Sian),
5.4.2 Adaptation of the Model as Optimization Procedure . 195
5.4.3 Example: Adapted Model for the Reflector of a
Parabolic Antenna . 199
5.4.4 Conclusion . 203
D.H. van Campen, R. Nagtegaai, A.J.G. Schoofs
5.5.1 Introduction ................... .
Response Variable .. . ............. .
Responses ...................... .
5.5.5 Computer Program for Experimental Design
5.5.6 Application
5.5.7 Conclusion
6.1 Multicriteria Optimal Layouts of Aircraft and Spacecraft
Structures
G. Kneppe
6.1.1 Introduction
.205
.207
.213
.217
.218
.218
.226
.227
. .229
.. 229
.229
.230
.232
Satellite Structure
6.1.5 Conclusion
References . . . . .
.238
.242
.243
6.2 Multicriteria Design of Spacecraft Structures with Special
Emphasis on Mass and Stiffness . . ..... 244
H. Baier
6.2.4 Solution Strategies ............. .250
6.2.6 Conclusion .258
7.1. Application of Multicriteria Optimization Methods to Machine
Tool Structural Design
M. Yoshimura
7.1.2 Competitive and Cooperative Relationships between
Evaluative Factors . . . . . . . . . . . . . . 266
7.1.4 Application Examples. .274
its Applicatlon to Machine Tool Spindle Design . . . . . . . . .
282
J. Montusiewicz, A. Osyczka, J. Zamorski
7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. 282
7.2.3 Application of the Strategy to the Design of a
Machine Tool Spindle System with Hydrostatic
Bearings ................................ 291
8.1 Optimal Die Design for Symmetric Strip Drawing. .303
W. Stadler
8.1.4 Conclusion
M. Bremicker, H. Eschenauer, H.- W. Wodtke
8.2.1 Introduction .................. .
8.2.3 Structural Analysis ........... .
8.2.5 Optimization Results
References .....
.. 303
. .304
.308
.317
.317
. .... 319
.319
.321
.325
.331
.331
.335
.337
.337
8.3 A Mechanical Model for the Optimization and Simulation of the
Metal Farming Process in Roller Levelling of Sheets 339
L. Henrich, K. Schiffner
.340
.344
References . . . . . . . . . . . . . . . . . . . . ..........
352
XVIII Contents
9.1 Multicriteria Optimization of Concrete Beams, Trusses,
... 355
9.1.3 Optimization of Isostatic Trusses ........... .360
9.1.4 Optimization of Cable Structures. . . . . . . . . . . . .
364
9.1.5 Further Applications of Muticriteria Optimization. .371
References . . . . . . . . . . . . . . . . . . . . . . . . . .. .
372
Programming ................................ 376
9.2.2 Dynamic Programming ........... .
.376
.378
.381
9.2.5 Conclusion .............................. 394
10.1.1 Introduction ............................ 397
10.1.3 Objective Functions - Design Variables - Constraints
405
10.1.4 Examples of Application 407
10.1.5 Conclusion 414
S. Ada/i, K.j. Duffy
10.2.3 Multicriteria Design Problem
K.-H. Stenvers
10.3.2 Program System for Structural Analysis and
Optimization ................... .
References .. . . . . . . . . . . . . . . . . .
.418
.420
.421
.424
.427
.427
J. Koski, R. Si/vennoinen
Components . . . . . . . . . . . . . . . . . . . . .448
10.4.5 Design of a Ceramic Piston Crown. .456
10.4.6 Conclusion .461
Su.bJect Index. .469
LIST OF CONTRIBUTORS
S. Adali Faculty of Engineering, Dept. of Mechanical Engineering,
King George V Ave., Durban 4001, SOUTH AFRICA H. Baier
Dornier-System GmbH, Dept. MEB, Postfach 1360, 07990 Friedrichsha
fen 1, FRG M. Balachandran Dept. of Architectural Science,
University of Sydney, NSW 2006, AUSTRALIA M. Bremicker University
of Michigan, 2212 G.G. Brown Building, Ann Arbor, Mi 48109, USA
D.H. van Campen Eindhoven University of Technology, Building
WHoog-01.144, P.O. Box 513, 5600 MB Eindhoven. NETHERLANDS H.A.
Eschenauer Research Laboratory for Applied Structural Optimization
at the Institute of Mechanics and Control Engineering, University
of Siegen, Paul-Bo natz-Str .. 05900 Siegen, FRG W. Fuchs
Lemmel'Z- Werke KGaA, Dept. TEF, Postfach 1120, 05330 Konigswinter
1, FRG J.S. Gero Dept. of Al'chitectural Science, University of
Sydney, NSW 2006, AUSTRALIA L. Henrich Institute of Mechanics and
Control Engineering, University of Siegen, Paul-Bonatz-Str., 05900
Siegen, FRG S. Jendo Polish Academy of Sciences, Institute of
Fundamental Technological Research, 00-049 Warszawa, Swietokrzyska
21, POLAND G. Kneppe MBB Hubschrauber und Flugzeuge,
CAE/Informationssysteme LWD 132, Postfach 801160, 08000 MUnchen 80,
FRG J. Koski Tampere Univel'sity of Technology, P.O. Box 527, 33101
Tampere, FINLAND J. Montusiewicz Technical University of Lublin,
20-618 Lublin, ul. Nadbystrzycka 38, POLAND
XXII List of Contributors
B.B. MWler-Slany Research Laboratory for Applied Structural
Optimization, University of Siegen, Paul-Bonatz-Str., D 5900
Siegen, FRG R. Nagtegaal Eindhoven University of Technology,
Building WHoog-0t.144, P.O. Box 513, 5600 Eindhoven, NETHERLANDS
A.Osyczka Technical University of Cracow, 31-155 Cracow, ul.
Warszawska 24, POLAND F. pfeiffer Institute of Mechanics, TU
MUnchen, Postfach, D 8000 MUnchen 2, FRG P.u. Post Festo KG, Dept.
EF-BE, Ruiterstr. 82, D 7300 Esslingen 1, FRG M.A. Rosenman Dept.
of Architectural Science, Univel"sity of Sydney, NSW 2006,
AUSTRALIA R. Silvennoinen Tampere University of Technology, P.O.
Box 527, 33101 Tampere, FINLAND E. Schiifer Research Laboratory for
Applied Structural Optimization, University of Siegen,
Paul-Bonatz-Str., D 5900 Siegen, FRG K. Schiffner Institute of
Mechanics and Control Engineering, University of Siegen,
Paul-Bonatz-Str., D 5900 Siegen, FRG AJ.G. Schoofs Eindhoven
University of Technology, Building W-Hoog-0t.144, P.O. Box 513,
5600 MB Eindhoven, NETHERLANDS W. Stadler San Francisco State
University, Division of Engineering, 1600 Holloway Ave., San
Francisco, California 94132, USA K.B. Stenvers Krupp
Industrietechnik, GmbH, Dept. Systems Technology, Postfach 141960,
D 4100 DUisburg 14, FRG B.W. Wodtke Research Laboratory for Applied
Structural Optimization, University of Siegen, Paui-Bonatz-Str., D
5900 Siegen. FRG M. Yoshimura Dept. of Precision Engineering, Kyoto
University, Sakyo-ku, Kyoto 606, JAPAN J. Zamorsk1 Technical
University of Cracow, 31-155 Cracow, ul. Warszawska 24, POLAND J.
Zajac Technical University of Cracow, 31-155 Cracow, ul. Warszawska
24, POLAND
LIST OF SYMBOLS
Note: The following list is restricted to the most important
subscripts,
notations and letters in the book. Further terms are
explained
in the text.
Scalar quantities are printed in roman letters, vectors in
boldface,
tensors or matrices in capital letters and in boldface.
1. Subscripts and Notations
Nabla operator with respect to a vector x
difference
quantity a is valid under the assumptions b,c
all elements a have the attribute A
quantity a equals quantity b
A is a sufficient condition for B
x is an element of set M
x is not an element of set M
set M is a subset of set N
union of set M and set N
intersection of set M and set N
Euclidean norm of a vector x
absolute value of a scalar x
substitute problem
sensitivity matrix
concrete area
1) pay-off table matrix
systematic departure (bias) from the true physical relation
ship
moisture concentration
damping matrix
1) expected value
2) Young's modulus
i-th unit vector
objective function, objective function vector, j-th
coordinate
function of the vector f ( j=l, ... , m )
criterion map
reduced gain, gain of an ideal parabolic antenna
vector of inequality ·constraints, j-th inequality constraint
( j=1, ... , p )
( i=l, ... , q)
criterion
2) stiffness matrix reduced to plane stresses
vector of degrees of freedom
set of all real numbers
n-dimensional Euclidean vector space
3) excitation vector
transformation matrix
1) state variable vector
2) displacement vector
2) weight
coefficient or factor
complete Pareto-optimal solution set
vector of IR n , design variable vector, i-th design variable
1) vector of input variables
2) vector of coded or standardized input quantities
compromise (suitable) solution
vector
slack variables
subspace
1) analysis variables from linear transformation
2) vector of output or response variables
point of the functional-efficient boundary, minimal point
vector of the demand level, j-th demand level
set of feasible solutions
3. Greek Letters
vector of unknown coefficients
curvature of the middle surface
coefficients of the thermal stress tensor
coefficients of the hygrothermal stress tensor
r
(3
8
coefficients of the shear strain tensor
direction of minimization step
physical stresses
stress tensor
modal matrix
MOTIVATION
1.1 Introduction 1.1.1 On the Historical Development of
Optimization Techniques
A well-known statement of the energy principles says that
among
all possible displacements the actual displacements make the
total
potential energy an absolute minimum. This means that the
application
of the principle of minimum potential energy leads to the
fundamental
equations of the boundary value problem in the theory of
elasticity.
The principles of mechanics go back to the 17th century. They
allowed
the formulation of classical problems in numerous fields of the
natu
ral and engineering sciences by means of the calculus of
variation
[1,2]. G.W. Leibniz (1646-1716) and L. Euler (1707-1783) found a
suitable
mathematical tool for finding the extreme values of given
functions
by introdUcing the infinitesimal calculus, with which it is
possible to
carry out an integrated treatment of energy principles in all
fields of
mechanics with application to dynamics of rigid bodies, general
elastic
ity theory, analysis of load supporting structures (frames,
trusses,
plates, shells), the theory of buckling, the theory of vibrations,
etc.
Some very interesting examples from the field of classical
mechanics
are the "curve of the shortest falling time" ("brachistochrone")
and the
isoperimetric problem investigated by J. Bernoulli (1655-1705) and
D.
Bernoulli (\700-1782). Another important problem is that of the
"smallest
resistance of a body of revolution" solved by Sir I. Newton
(\643-1727).
With the principle of least action and the integral principles J.L.
de
Lagrange (\736-1813) and W.R. Hamilton (1805-1865) contributed to
the
perfection of the calculus of variation which still serves as the
basis
for various optimization methods. A further important application
of
minimal principles is to introduce special approach functions.
Instead
of solving the governing differential equations together with the
boun-
2 Multicriteria Optimization - Fundamentals and Motivation
dary conditions, often a mathematically difficult task, the problem
can
be interpreted as finding functions which satisfy some or all
boundary
conditions and minimize the potential energy and the
complementary
energy, respectively. Useful approximation methods based on the
vari
ational principles of mechanics were devised by Lord Rayleigh
(1842-1919),
W. Ritz (1878-1909), B.G. Galerkin (1871-1945), and others.
In a first application on "Optimum Structural Design"
variational
methods have been treated by J.L. de Lagrange, T. Clausen
(1801-1885),
and B. de Saint-Venant (1797-1886). The investigations of finding
out
optimal designs of one-dimensional structures under various
loadings
should be mentioned here. Typical examples are the bar subjected
to
buckling loads or the cantilever beam under single load or dead
weight,
for which optimal cross-sectional shapes were found by means of
the
calculus of variation. For this purpose, optimality criteria are
derived
in terms of necessary conditions, e.g. Euler's equations in the
case of unconstrained problems. If constraints are considered,
Lagrange's
multiplier method is applied. This corresponds to the solution of
an
isoperimetric problem [3].
Fig. 111 shows an interesting comparison between the natural
shape
of a tree branch and the optimal shape design of a cantilever
beam
under dead weight. W. Stadler treated this problem using
structural
control and multicriteria optimization techniques [71 Nowadays,
some
al
a) A tree branch as a natural shape
b) A cantilever beam under dead weight
1.1 Introduction 3
researchers in the field of applied structural optimization use
vari
ational principles for their investigations [4]. In developing
optimization
procedures together with mathematical p,"ogramming methods,
the
"optimality criteria"- method can be effectively integrated into
the prob
lem system.
Process
procedures into the practical design phase:
1) Increasing the quality and quantity of products and plants and
at
the same time reducing costs and thereby being competitive.
2) Fulfilling the permanently increasing specification demands as
well
as considering reliability and safety, observing severe
pollution
regulations and saving energy and raw materials.
3) Introducing inevitable rationalization measures in development
and
design offices <CAD, CAE) in order to save more time for the
staff
to work creatively.
solutions" can only be met if
- sufficient time is available for the development of
alternative
solutions,
- mathematical optimization methods are available and
applicable.
Although there are numerous activities in establishing
CAD-work
stations, the two first points are only partially fulfilled. In
addition,
many branches of industry have reservations concerning structural
opti
mization. This is due to the yet unrealized demand of practitioners
for
reliable, efficient and robust methods which are simple in
application
and furthermore proven to be problem-independent. It is a major
con
cern of research in the field of design optimization to meet
these
practical demands.
The rapid development of efficient computer systems allows the
inte
gration of an optimization procedure into the process of
engineering de
sign [401 This requires detailed mathematical-physical modelling
for
any structural analysis of a technical design problem and coupling
it
4 1 Multicriteria Optimization - Fundamentals and Motivation
with a suitable optimization algorithm as well. Both are combined
via
the problem-programs of optimization modelling for tasks in
engi
neering design (see Chapter 2).
For all these tasks of design optimization mentioned above, it
is
equally essential to note that the application of optimization
theories
in the design process depends upon the fundamental aspects of
the
technical problems.
1.1.3 Multicriteria Optimization 88 a Strategy in the Design
Process
Nowadays, a multitude of new constructions and their
corresponding
designs require much closer attention because more than one
main
criterion corresponding to the aspects 1) and 2) in Section 1.1.2
is given
respectively. Such optimization problems for multipJe criteria
are
called either Vector or Multicriteria Optimization ProbJems.
With
reference to V. Pareto (1848-1923), the French-Italian economist
and
sociologist, who established an optimality concept in the field of
econ
omics based on a multitude of objectives, i.e. on the permanent
conflict
of interests and antagonisms in social life, this special field of
opti
mization is also called Pareto optimization [5] .
The application of multicriteria optimization (MO) to problems
in
structural mechanics or technology in general took quite a long
time.
It was W. Stadler who referred to the scientific application of
Pareto's
optimality concept in the 1970's for the first time and who
published
several papers especially on natural shapes [6,7]. Since the end of
the
seventies, vector optimization has been more and more integrated
into
problems of optimal designs in the papers of a number of
scientists
(e.g. 18-16]).
Within the scope of a design process a designer and an
analyst
have to figure out which dimensions and shapes will fulfill
certain
main criteria of a structure in the best possible way. From
common
experience it is known, however, that all these demands can rarely
be
met simultaneously. Competing objectives require an estimation
of
the importance of the individual objectives by weighing them
corres
pondingly. The application of vector optimization as a strategy in
a
design process is particularly suitable for decision making
during
single phases of development of new constructions or
components.
Fig. 1/2 shows the appropriate phases of such a process. In order
to
System specifications - Loads - Objecti yes - Constraints - Design
Variables - Bounds
Il Feasibility stud)' - Topo log) - Geomef.r~
- Versions , ( Decision
final s~ stem - Specification for
a selected version , ( Decision
, ( Decision
t
1.1 Introduction S
MO -Tool
Global behaviour of the entire system. Compliance with the most
important demands
I
Globa l behav iour of the ent.ire system and the main
assemblies
I
Global behaviour of the main s;.,s t e m . Loca l beha ,'iour of
the subsys t ems (assemblies'
I Local behav iour of t he subsystems , Compliance with demands of
subsys t ems (si ngle components!. Interacth<e MO
I Global and local beha\ iou r of the final ... ersion inclu- ding
all assemblies and components
Fig. 1/2. Use of multic riteria techniques during the several
phases of
a design process
obtain statements concerning the global behaviour of different
design
versions , it is advantageous to find a number of Pareto-optimal
or
functional-efficient solutions (see Section 1.3), The particular
local
behaviour of a st.·uctural component or a group of structural
compo
nents (assemblies) can efficiently be found via an interactive
design
process (s ee Chapter 3),
6 Multicriteria Optimization - Fundamentals and Motivation
1.2 Mathematical Fundamentals
1.2.1 General Definitions and Notations in Scalar
Optimization
The objective of design optimization is to select the values of
the
design variables xi (j=I .. ... n) under consideration of various
constraints
in such a way that an objective function f=f(x) attains an extreme
value.
This can be expressed in the abbreviated form
min {f(x): h(x) = O. g(x) sO} xE IR n
(1-0
with IRn the set of real numbers. f an objective function. x E IRn
a vector
of n design variables. g a vector of p inequality constraints. h a
vector
of q equality constraints (e.g .. system equations for the
determination
of stresses and deformations). and X: = {x E IR n : h(x) = O. g(x)
sO}
the "feasible" domain where s has to be interpreted for each
individual
component.
An additional problem in design optimization is that the
objective
function(s) and the constraints are generally nonlinear functions
of the
design variable vector x E IRn. The continuity of the functionals
as
well as of their derivatives is usually assumed (Fig.1I3).
Before treating multicriteria optimization problems some
relevant
definitions and conditions from scalar optimization will be
considered.
X2
1.2 Mathematical Fundamentals 7
The subset XC IRn will be given as the domain of definition. U (x*)
E
describes the s-neighbourhood of the point x*; i.e. the number of
all
points x whose distance from x* is smaller than s > O. The
distance
is given by the Euclidean metric.
Definition 1.1 Global and local minima
i. A point x* E X is a global minimum. if and only if
(!-2a)
:Ii ii. A point x E X is called a local minimum point of f on X, if
and
only if for some s
f (x*) :> f( x) (t-2b)
The value f*= f (x*) is accordingly called a local (relative)
minimum.
Definition 1. 2 Conve.,ity
i. A subset X of IRn is convex if and only if
(t-3a)
for each xl'x2 E X and for each real number 0:> [l:> 1.
ii. A real-valued function f on the convex subset X is convex on X
if
and only if
(1-3b)
for each x1,x 2 E X and for each 0 :> [l :> 1.
Theorem 1.1 Optimality conditions for unconstrained minimum
problems
i. Necessa/:1 Condition:
(1-4b)
is positive definite, f(x) has a local minimum at x*.
8 Multicriteria Optimization - Fundamentals and Motivation
Theorem 1.2 Conditions for constrained minimum problems
For the dete,-mination of optimality conditions, the Lag,-angian
function
is now introduced [23]
q p
L(x,«,(3) = f (x) + L cx. h.(x) + L (3. g.(x) , i= 1 I I j= 1 J
J
(1-5)
ments yield the following conditions:
i. Necessary conditions for a local minimum
The Kuhn-Tucker conditions [24] are applied to test local
optimality
at a point x*
q p V L(x*) = Vf(x*) + L cx* V h .(x*) + ~ (3. *V g.(x*) = 0
i=1 i I j =1 J J
and h.(x*) = 0 = 1, ... q (1-6) I
,
~(x*) :>; 0 j = 1....,p
(3~g. (x*) = 0 (3~ ~ 0 = 1, ... ,p J J J
ii. Sufficient conditions
Fo,- problems for which f(x) is convex, the equality constraints
are
linear, and the inequality constraints are convex functions, i.e.
for
so-called convex p,-oblems, the Kuhn-Tucker conditions are also
suffi
cient conditions (see [23]).
Fig. 1I-t. shows a geometric interp,-etation in the presence of
three
inequality constraints. According to the constraints (1-6), the
points A
and B in Fig. 114 satisfy the following conditions:
1. At point A -> 'i7f(x*)= (3*'i7g (x*) + (3*'i7g (x*). 1 1 3
3
(1-7a)
The gradient does not lie in the cone «(31 < 0) set by the
gradients of
the constraint functions; A is not a minimum point because the
function
value can be reduced within the feasible domain.
2. At point B (t-7b)
The considered point B is a local optimum because there is no
direction
within the feasible domain in which the function value can be
reduced.
1.2 Mathematical Fundamentals 9
consideration of three inequality constraints
1.2.2 The Multicriteria Programming Problem
In problems with mUltiple criteria one deals with a design
variable
vector x which fulfills all constraints and renders the m
components
of an objective function vector f(x) as small as possible . A
completion
of the problem (1-1) yields the vector optimization problem:
min (f(x) : h(x) = 0, g(x) s; 0 l XEIR"
(1-8)
A characteristic feature of such optimization problems with
multiple
criteria is the appearance of an objectil'e conflict, i.e. none of
the
feasible solutions allows the simultaneous minimization of all
objectives ,
or the i ndi v idual sol u tions of each sing Ie objecti ve fu
nction differ.
Consequently, the subject of multicriteria optimization deals with
all
kinds of conflicting problems.
Definition 1.3 Convexity of MO
A multicriteria optimization problem on IR m is convex if and only
if
(a) the components of the objective function vector f(x) are
convex,
(b) the components of the vector of the inequality constraints
g(x)
are convex, and
(c) the components of the vector of the equality constraints
hare
affine-linear functions of x .
10 Multicriteria Optimization - Fundamentals and Motivation
Definition 1.4 Functional-efficiency or Pareto-optimality
([6,25,30])
A vector x* E X is Pareto-optimal resp. p-efficient or
functional-efficient
for the problem (1-8), if and only if there is no vector X E X with
the
characteris tics
and (1-9)
f.(x) < f. (x*) for at least one j E {1, ... ,m} . J J
For all non-Pareto-optimal vectors, the value of at least one
objective
function fj can be reduced without increasing the functional values
of
the other components. Fig. 115 shows a mapping of the
two-dimen
sional design space X C IR2 into the criterion space Y C IR2 where
the
Pareto-optimal solutions lie on the curved section AB. Solutions
of
nonlinear vector optimization problems can be found in different
ways.
By defining so-called substitute problems these are normally
reduced
to scalar optimization problems.
The problem
min p[f(x)] (1-1Oa) xEX
is a substitute problem if there exists x E X* sLich that
p[f(x)] = min p[f(x)] . xEX
(1-1Ob)
The function p is called a preference function or a substitute
objective
function OJ" a criterion of control effectiveness (the last term is
mainly
used in control engineering) [8,1O-12,15.26J. It is obviously
important to study whether the solutions X of the
substitute problems are Pareto-optimal with respect to X and to
the
set of objective functions ft, ... ,fm' i.e. that a point y=f(x)
actually
lies on the efficient boundary ay * [6,11].
A number of publications have dealt with various methods fOJ"
trans
forming vector optimization problems into substitute problems
[11-16,
32,38]. In the following these transformation ru les will be
denoted
"strategy" when referring to the optimization procedure. Since the
prob
lem-dependence of the various strategies may be highly relevant,
it
is of interest to test their efficiency and thus their preference
behav
iour on typical structures [16].
1.2 Mathematical Fundamentals 11
Design Space X Criterion Space Y
Fig. 115. Mapping of a feasible set into the criterion space
Some of the strategies used at"e described below:
a) Method of Objective Weighting
Objective weighting is obviously one of the most usual
substitute
models for vector optimization problems. It permits a
preference
formulation that is independent of the individual minima for
positive
weights; it also guarantees that all points will lie on the
efficient
boundary for convex problems. The pt"eference function or utility
func
tion is here determined by the linear combination of the criteria
fl' ... ,f m
togethet" with the cOl"responding weighting factors wt, ... ,wm
:
In
p[f( x)] := 2 [w. f. (x)] = w T f (x) , j=l J J
XEX. 0-11)
m 2 w. = 1 .
J j=l
It is possible to genet"ate Pat"eto-optima for the odginal problem
(1-8)
by vat"ying the weights Wj in the preference function. In
engineering and
in economics this approach has been applied for quite some time
[9,27,
28]. The deficiency of this stategy in structural optimization has
been
discussed for example in [39].
12 1 Multicriteria Optimization - Fundamentals and Motivation
bJ Method of Distance Functions
Distance functions are frequently applied and also lead to a
scalarization
of the vector optimization problem. A decision maker specifies a
so
called demand level vector 1 = (Y, ... ,Y )T with the objective
function t m
value to be achieved in the best possible way. In design
optimization
this corresponds to a set of assumed specification values or
demands
for the single objective functions. The respective substitute
problem
then reads
j=1 J 1 " r " 00, X EX, (1-12)
where the variation of I' meets various interpretations of the
"distance"
between the demand levels 1 and the functional-efficient solutions.
The
following distance functions are most frequently used:
r = 1: p[f(xl] = ~ ! f.(xl-Y! ' j = t J J
r = 2: p[f(xl] = [ ~ (f.(xl _ y.)2]V2 j = 1 J J
r - 00: p[f(x) ]
(H3cl
The choice of a demand level may cause problems. Therefore, Fig.
1/6
qualitatively gives the solutions of the substitute problem for
various
demand levels. It shows that the choice of 11 yields a solution x
of
the substitute pt-oblem for which y 1 = f (x) E c)Y * is efficient
with
respect to Y. The choice of 1 2 , howevet-, yields an y 2 E c) Y *
not lying
on the efficient boundary, and with the choice of the inner point 1
3 ,
the respective solution y 3 is not an efficient point of the
boundary of Y.
The use of distance functions is subject to the following
disadvan
tages [11]:
1. The selection of "wrong" demand levels 1 will lead to
nonefficient
solutions (Fig. 1/6),
2. The selection of "correct" or "valid" demand levels 1 requires
know
ledge of the individual minima of the m objective functions
f/xl.
j=I, ... ,m which is not easy to achieve with nonconvex
problems.
1.2 Mathematical Fundamentals 13
f 1min
Fig. 1/6. Solution of the substitute problem for various demand
levels
The methods of the distance functions can also be parametrized
to
generate Pareto-optima for the original problem (1-8), For example
in
[38] several possibilities for choosing the parameters and their
relations
to Pareto-optima have been considered in detail.
c) Method of Constraint Oriented Transformation (Trade-off
Method)
Retransformation of the vector optimization problem into a scalar
sub
stitute problem may also be achieved by minimizing only one
objective
function with all others bounded [11,12]):
p [f (x)] = XEX
j=2, ... , m
Thus, f 1 is called the main objective, and f 2" " ,f m are called
secondary
01' side objectives. The given problem can be interpreted in such a
way
that when minimizing fl the other components are not allowed
to
exceed the values y 2 , ... , Y m ' The dependence of the solution
on the
selection of these constraint levels for the two-dimensional case
is
shown in Fig . 117.
Fig. 1/7. Solution of a constraint-oriented transformation
depending
on the constraint level
The main objective function f1 is generally one for which no a
priori
estimation of an upper limit Yt is available.
If the const)'aint levels are taken as equality constraints, and if
other
constraints a)'e not considered, the problem corresponds to the
minimi
zation of the )'espective Lagrangian fu nction [2<1]
m L(x,cx) ;= f1 (x) + L cx.[f.(x)-Y.],
j=2 J J J (1-15)
which is used in this case as a preference function. The
necessary
optimality criteria corresponding to the Kuhn-Tucker conditions
(1-6)
without inequality constraints are
c}L = ~ + f cx . ~~ ~ 0, c}x i c}x i j=2 J i
i = 1, ... , n ,
( 1-16a)
( 1-16b)
They are the basis fo)' calculating the optimal values for xt"",xn
and
those of the adequate Lagrange mUltipliers cx z '"'' cx m' The
introduction
of the abbreviations
~ jj j = 2
m In L<x,a) L rx. f. (x) - L rx.Y· =
j=2 J J j=2 J J
(1-18)
Thus. the expression (1-18) corresponds locally to the substitute
prob
lem with objective weighting if one disregards the normalization of
the
weighting factors and the additive parametel- C. which are
irrelevant
for solving the problem.
Finally, one can state that the considered "Trade-off"-formulation
yields
the Pareto-optimal set of solutions if one critedon is replaced by
a
sequence of inequality constraints. Therefore. this stl-ategy is
sometimes
called "multi-constraint" or "bound formulation" (see [4-,
4-1]).
d) Method of Min-Max Formulation
Besides the preference functions described above. the min-max
formu
lation plays a vel-) important role in solving substitute problems.
It
is based on the minimization of relative deviations of the single
objec
tive functions from the respective individual minimum
[12,32].
For the interpretation of a min-max fOl-mulation three given
objective
functions with the domain of definition xl S X S x2 are
considered
(Fig. 118), If the extrema f. are established separately for each
objective J
function (critedon), the desired solution is that x which results
in the
smallest value of the maximum deviation of all objective
functions.
Therefore, the scalar substitute problem can be defined according
to
the min- max formulation as follows:
p[f(x)] = max [z.(x)] j=l .... ,m J
with
(t-19a)
(I-l9b)
16 Multicriteria Optimization - Fundamentals and Motivation
Fig. VB. Preference function of a min-max formulation in the
one
dimensional case
In [.:/.], a weIl-known modification of equation (1-18) is applied
for
p.'actical computations. It consists of the minimization of a new
variable
xn+t (comparable to a slack variable, e.g. see [31]) while
simultaneously
considering the additional const.'aints:
X E X j = 1, ... , m . <1-20)
Equation 0-20> is especiaIly appropriate for nonlinear
optimization
problems using effectively the inequality constraints in the
optimization
algorithms <e.g. method of sequential linearization) [21].
For the min-max formulation (1-20) a geometric inte"pretation
can
be given on the basis of the hypothesis that all inequality
const.'aints
z.<x) - xn+1 5. 0 (j=I, ... ,m) are active within the min-max
optimum X, J
i.e. Zj<X)- xn+\ = O. Without going into detailed proof here, it
can
be stated that there will be a parameter graph within the
hyperspace
IRm from which one can conclude that the optimal solution point
f(x)
must lie on a line within the criterion space. Therefore, it yields
a
fit'st geometric position for f(i). The second one results from
mini
mizing the distance (r=2, Euclidean norm) between the reference
point
f and any random point of the line in the criterion space. It can
be
shown that this precisely corresponds to the minimization of
xn+l'
The min-max optimum can therefore be interpreted as the
intersection
of a line in the space with the functional-efficient solution set
ay -:lE.
Fig. 1/9 shows this behaviour for two objective functions.
These investigations show that there are certain
interdependencies
between the min-max fOJ'mulation and the method of distance
functions.
1.2 Mathematical Fundamentals 17
J=1.2 fj
~f T 1
Fig. 1/9. Geometric interpretation of a min-max formulation for
two
criteria
Starting from the general distance formulation according to (1-13),
the
min-max formulation for r--;>co (Chebyshev metric) results
in
p[f(x)] = max If.(x)-Yl. j = 1 ..... In J J
XEX ( 1-21)
with the components of the demand level vector}.. If the minima r.
J J
of the individual objective function components are selected as
compo-
nents fo.' the demand level vecto.'. and if every objective
function is
related to the respective r. . then the distance function
formulation is J
t.'ansfOl'med into the min-max formulation in accordance with Eq.
(1-19).
The min-max formulation described above yields the compromise
solution x considering all objective functions with equal priority.
But
if the single objectives have to meet a special order or if the
functional
efficient solution set X" is of great importance for the decision
maker.
the min-max formulations can be modified or extended (see
[16.21,34]):
- Min-Mav Formulation with Objective Weighting The introduction of
dimensionless weighting factors w j ;:: 0 transforms
the substitute problem (1-19) into
p[f(x)] = max [w.z.(x)], j=1 ..... m J J
XEX, (1-22)
where Zj (x) denotes the j-th relative deviation as in (1-19).
The
weighting factors describe the priority of the single objective
functions.
Thus. it is possible to select definite compromise solutions
from
18 Multicriteria Optimization - Fundamentals and Motivation
f-IXI-T] p[f(x)] = max [ w. ~
j=1.2 J fj
consideration of different weighting-factor relations
random fields of functional- efficient sets. Mo,-eover, the
variation of
w. allows the establishment of the ,-epresentative solution set. A
J
similar modification also exists for Eq. (1-20)
j = 1, ... ,m. (1-23)
Fig. 1110 shows the geometric interpretation of Eq. (1-23) for the
two
dimensional case. It is obvious that depending on the ratio Wt/W2
of
the two weighting factors one obtains different compromise
solutions
describing the whole functional-efficient boundary.
- Min-Ma, .... Formulations Presuming a Demand-Level Vector
If the definition of the relative deviations in (1-19b) is not
based
on the individual minima f. but on the given components y. of the J
_ J
demand level vector with the characteristics Yj = f j' one can
get
analogous substitute problems to (1-22) and (1-23). However,
the
problem formulation does not guarantee that all inequality
constraints
become active at the solution point x. In other words that they
can
be regarded as equality constraints. Only if all inequality
constraints
become active, the solution vector X lies on the intersection of
the
line in the space with the functional- efficient solution set
ClY*.
The difference with respect to the previously mentioned
formulation
is illustrated in Fig.1Ill. If the line passing through the point y
and
defined by the relation w t /w 2 intersects the
functional-efficient
boundary, the intersection point is also the compromise solution.
If
1.2 Mathematical Fundamentals 19
J=1.2 J Yj
vector Y
there is no intersection point , the point corresponding to f 1 or
to f 2
is the solution depending on the ratio w/w 2 .
The special selection of a demand-level vector Y =0 along
with
omitting the division by y. within the relative deviation z.(x)
yields a J J
further modification of the min-max formulation a formulation
p[f(x)) = max [w . f.(x)], j=I . .... m J J
XEX (1-24)
1.2.3 The Multicriteria Control Problem [35. 36]
As mentioned in the preceding section. some of the optimum
design
problems can be modelled using an optimal control approach
[35,36].
Let the state X E A C IR n be controlled by means of a control
vector
u( ·): [to,tt] - U c IR r in the state equation
X = g(x,U) (1-25)
with x(to ) E X corresponding to the initial set and x(t1) E X
corre
sponding the terminal set and with xn = t, the independent
variable,
so that gn(x,u) = I. Furthermore,
g( . ): A x U - B (open) C IR n (1-26)
20 Multicriteria Optimization - Fundamentals and Motivation
is the velocity function and U is the control constraint set, the
set of
all possible values of u(·). It is usual to assume that u(')
belongs to a
nonempty set F of admissible controls. A criterion map f(·); F -
IRn is
defined in terms of the component integrals
tl
where
goi (.) ; A x U - C i (open) C IRn. i=I, ... ,m . (1-28)
The state space IRn is augmented with
(1-29)
where y € IRm is the criterion space and go=(gol' .... gom >.
Let u(·) € F.
and x(·) be a corresponding solution of the state equation (1-25),
and
let s(·) be a solution of (1-29) corresponding to the pair (x('),
u(·».
The attainable criteria set is then defined by
(1.30)
Multicriteria control problems can be stated as finding an
"optimal"
control u*(·) € F for f(u(·» subject to u(·) E F.
1.3 Components and Plants with their Objectives
In this chapter several examples illustrate the advantages of
multi
criteria optimization techniques in decision making during the
planning
and the design process of complex components and plants.
1.3.1 Optimum Design of Highly Accurate Parabolic Antennas
[17,18]
A practical application of the optimization strategies and
procedures
is to figure out the layout of the main components for highly
accurate
focusing parabolic antennas. Antennas can be defined as
so-called
wavetype transducers. As transmitting antennas they transform
cable
1.3 Components and Plants with their Objectives 21
gUided high- frequency energy into wave types convenient for an
ex
tension into free space, and as receiving antennas they retransform
the
energy taken from free space into cable guided waves. Moreover, one
tries to achieve a transformation from one condition into the
otheJo
with least possible losses in order to get optimal antenna gain.
The
transmission and reception of waves in the dm- , cm-, and
mm-range
(micro-wave range) are usually realized by means of parabolic
reflectors
based on the laws of geometrical optics.
The rays radiated from the focus of a paraboloid during
transmission
are reflected on its surface and leave the mirror as parallel,
in-phase
rays. This process is reversed for wave reception. ThE' in-phase
condition
of the rays essentially depends on the existence of an accurate
para
bolic surface. As the ray reception is analogous to that of
optical
astronomy, radio astronomists usually call their parabolic
antennas
Fig. 1/12. View of a 30-m-radio telescope for millimeter-wave
range
(Max-Planck Institute for Radio Astronomy, Bonn, FRG)
22 Multicriteria Optimization - Fundamentals and Motivation
"radio telescopes" in contrast to the "mirror telescopes" in
optical
astronomy. Ideally, all incident rays should intersect in the
focus
assuming an ideal surface as exact as possible in any given
position.
It is obVious that due to this demand, the reflector and its
supporting
structure are the most impo)·tant components of a movable
parabolic
antenna. In practice, however, such a highly accurate surface is
hardly
attainable. Fig. 1112 shows the latest radio telescope for the
mm-wave
range (MRT) with 30 m aperture diameter developed, designed
and
manufactured by two German companies (Krupp Industrietechnik,
Duisburg and MAN, Gustavsburg) and ordered by the Max- Planck
Association, Munich (FRG).
The reflector consisting of single adjustable panels (Fig. 1113)
suppor
ted on a rear spatial framework is deformed by dead weight, by
wind,
and by temperature loads . Furthermore, there are manufacturing
tole
rances as well as measuring and adjusting faults during the
positioning
of the reflector surface. Due to these systematic and statistical
diffe
rences, the phases of the individual rays will be different. Part
of the
energy will be diffused and radiated towards other directions.
According
panel surface
~----------d----------~
Fig. 1/13. Design of a parabolic reflector with circular aperture
in
panel surfaces
1.3 Components and Plants with theit· Objectives 23
to Ruze, the reduced gain G can be described by a Gaussian
error
equation [17]
= ? e - (4rrCl/)..) -
The relation GIGo expresses the "efficiency" of an antenna,
and
(1-31 )
(1-32)
is the "gain of an ideal parabolic antenna with d = apertul'e
diameter,
A = wavelength, (j = standard deviation or root mean square
value
(rms-value). 1] = sul'face efficiency.
The rms-value (j is defined as a measure for the surface accuracy.
It
is determined by the method of least squares with a
"bestfit"-surface
being described by a set of n given points of the deformed and
imper
fect reflector surface [tSl As the efficiency of a parabolic
antenna
substantially depends on the surface accuracy, the rms-value plays
the
most important role besides the weight of the design of an
antenna.
Both objectives must be fulfilled in the best possible way. They
are
used as criteria in this bicriteria optimization problem.
1.3.2 Optimal Layout of a Novel Solar Energy Collector [19]
As a further example, a special type of a concentrated solar
energy
collector, the so-called "Rear-Focus Collector", is considered (see
Fig.
1114-), It consists of several frustum-type reflector shells linked
together
by two intersecting ribs. The focus of the rays and accordingly
the
absorber are located behind the collector. The system efficiency 1]
of the
concentrating collectors depends on the geometry, the shape
accuracy
of the reflector. and the tracking en'or. These aspects have a
substantial
influence on the two relevant quantities of 1]. the concentration
factor
and the intercept factor. For this collector, unlike a similarly
designed
one developed by A. Spyridonos in the early seventies, the
optimal
arrangement of the single shells of the collector are determined
by
means of the mathematical programming. Apart from one
objective
function representing the system efficiency, the volume is included
as a
further one.
24 Multicriteria Optinlization - Fundarnentals and Motivation
Fig. 1/14. Front view of the Rear Focus Collector (University
of
Calgary, Canada)
The supporting behaviour of shells can be considerably improved
by
shape optimization without incl'easing the weight, This is
especially
impoJ'tant for constructions in satellite technology because it is
exactly
this field of technology which has enormous demands on light
con
structions with high reliability.
TheJ'efoJ'e, a method fOJ' the optimal layout of the middle sudace
and
the wall thickness distdbution of satellite tanks was developed.
In
this case the weight of the tank has to be minimized while at
the
same time the volume has to be maximized and the feasible
stJ'esses
have to be fulfilled. In ordeJ' to avoid buckling problems,
negative
stresses should not be permitted. Fig. 1115 shows a view of
such
satellite tanks.
t!M 2 TOf"M !...to OR
Fig. 1/15. View of satellite tanks (MBB-ERNO, Bremen, West
Germany)
1.3.4 Optimal Shape Design of a Conveyer Belt Drum [21,22]
Efficient conveyers are necessary for extensive soil shifting
operations
in open mining. Here, large rubber-belt conveyel"S could prove to
be
successfully used. According to Fig. tlt6a, the conveyer belt drum
, the
track supporting roller, and the belt are essential components of
belt
conveyers. A conveyer belt drum consists of the cylindrical drum
shell
(t) and the bottom (2) (Fig. \/\6bl. Drum bottom and shaft (4)
are
connected by a clamping ring (3) .
Development in this field is characterized by continuously
increasing
demands on conveying capacity, conveying track, and opel"ational
safe
ty which leads to enlarged distances between the axes and the
conveyer
belt width. As the criteria can be sufficiently realized by
essentially
larger tension fOI"ces, the stresses in the belt drums are
inevitably
enlarged, too. It was attempted to reduce these stresses by
extending
the wall thickness and by implementing ribs . But these
measures
often led to an extreme increase in weight so that damage and
failure could not be avoided. The optimal shape design of a
conveyer
belt drum was treated with the dil"ect method of shape
optimization.
26 Multicdteda Optimization - Fundamentals and Motivation
bl Detail A
1 Drum shell 4 2 Drum bottom 3 Clamping ring 4 Shaft
Fig. 1/16. Sketch of the belt conveyer a) complete system
b) conveyer belt cylinder
On the basis of a given midsurface contour dcp) , the optimal wall
thick
ness distribution t(cp) had to be determined in a way that the
criteria
of optimization "minimal weight W " and "minimal reference
stress
Cl ref max" were fulfilled as well. Constraints were specified in
terms
of other limitations.
Steel engineering in various areas of "structural engineering"
(crane
technology, steel engineering, "offshore"-techniques, piping
construc
tions), circular cylindrical shells are often employed as
structural units
connected with other elements (e.g. plates>. The main parts of a
slewing
crane (Fig. 1I17a) are the overhang beam (1) , the tubular column
(2) ,
and the slewing ball bearing (3)' The bottom flange (Fig.1I17b)
which
is welded to the tubular column is connected to the foundation
with
anchor bolts . With the force F at the overhang beam the column
and
also the flange connection are loaded by an axial force and a
bending
moment.
al
bl
t
F
b) bottom flange
, f
ds
In the design phase, particu larl) the shape of the region near the
edges is
important. As there are stress concentrations near the edges the
problem
is to find out suitable dimensions for the flange leading to a
mini
mization of stl'ess concentrations but additional weight
reduction.
These two competing objectives lead to a multicriteria
optimization
problem.
Some results are presented in Fig. 1118. The conflict between
the
criteria f 1(x) ~ W(x) (weight) and f2 (x) ~ o(x) (stress
concentration)
is given . Des igns with low stresses give relatively high weight
values.
The sensitivity of the flange height x1= h is much higher than the
inner
diameter x 2 = d i of the flange. During the variation only h is
changed
by the optimization algorithm. If h=h max is reached, d j varies as
well.
The results discussed here are adequate and very important not
only
for decision making on this particular design problem of the
investi
gated connection but also for all other examples.
28 Multicriteria Optimization - Fundamentals and Motivation
h "' .75 ho
10-2 10- 1 10 0 10 1
Fig. 1/18. Conflict between the two criteria weight and max.
stress
00' W 0 specification values
w 2/w t ratio of weighting factors
1.4 Conclusion
This first chapter is a presentation of the fundamentals of scalar
and
multicriteria optimization and illustrates the necessity of
application
of multicriteria optimization techniques to develop and to
layout
components and stl'uctures by means of some real-life examples.
The
applicaton of MO-techniques is primarily due to the fact that
today
the manufacturing of machines does not only require a minimization
of
costs but also observes objectives such as shape accuracy and
reliability. Such problems are defined as "optimization problems
with
multiple objectives" (multicriteria optimization).
The objectives which are mostly competitive and nonlinear do
not
lead to one solution point for the optimum but rather to a
"functional
efficient" (Pareto-optima/) solution set, i.e. the decision maker
selects
the most efficient compromise solution out of such a set. The use
of
preference fUnctions transforms the multicriteria optimization
problem
References 29
into a scalar substitute problem. This so-called optimization
strategy is
a basic part of optimization modelling (see Chapter 2). For the
trans
formation a number of preference functions such as objective
weighting,
distance functions, constraint-oriented transformation
(trade-off
method) and min-max formulation have been analysed and tested.
It
can be shown that the efficiency of the single preference
functions
depend both on the problem and on the adaptation to certain
opti
mization algorithms.
The examples from industrial practice given in Section 1.3 show
how
important it is for the designer to get a tool for decision making
in
the design pt"ocess, especially when there is more than one
criterion
to be fulfilled. A large number of possible multicriteria
formulations
which go fat" beyond these examples is presented in the second
pat"t
of the book (Applications).
design can be summarized as follows:
(j) Multicriteria modelling very well reflects the design process
in
which usually several conflicting objectives have to be
satisfied.
(ij) The designer has the possibility to explore a broader range
of
altemative solutions than with single criteria models for
which
the solution is immediately fixed after the
problem-formulation.
(ijj) Multicriteria formulation provides a basis for explicit
trade-off
between conflicting objectives or interests.
References
Gattingen, Heidelberg: Springer 1972
Stuttgart: Birkhauset" 1979
Mathematiques et Astronomiques I (1849-1853) 279-294
[4] Bends0e, M.P.; Olhoff, N.; Taylor, J.E.: A Variational
Formulation
for Multicriteria Structural Optimization. Joumal of Struc
tural Mechanics, Vol. 11, No.4, 1983
30 1 Multicriteria Optimization - Fundamentals and Motivation
[5] Pareto, V.: Manual of Political Economy. Translation of the
French
edition (1927) by A.S. Schwier. London-Basingslohe: The
McMillan
Press Ltd., 1971
Optimality. In: Marzolio/Leitmann (edsJ: Multicriterion
Decision Making. CISM Courses and Lectures.Berlin,
Heidelberg,
New York: Springer 1975
(1978) 169-217
Tragwerken insbesondere bei mehrfachen Zielen. Dissertation,
TH Darmstadt, 1978
University of Technology Publication, Tampere 1979
[10] Eschenauel', H.: tiber die Optimierung hochgenauer
TI'agstruk
turen. Karl- Marguerre-Gedachtnisband, Schriftenreihe "THD
Wissenschaft und Technik", (1980) 89-101
[11] Sattler, H.-J.: Ersatzprobleme fUr
Vektoroptimierungaufgaben
und ihre Anwendung in del' Strukturmechanik. Dissertation,
Universi tat-GH-Siegen, 1982
York, Chichester, Brisbane, Toronto: John Wiley, 1984
[13] Radford, A.D.; Gero, J.S.; Roseman, M.A.; Balachandran,
M.:
Pareto Optimization as a Computer-Aided Design Tool. In:
Gero,
J.S. (ed'): Optimization in Computer-Aided Design. North
Holland Amsterdam, New York, Oxford: Elsevier Science Publi
shing Company (1984) 47-80
Sons (1984) 459-481
Atrek, Gallagher, Zienkiewicz (ed.): New Directions in
Optimum
Structural Design. Chichester, New York, Brisbane, Toronto,
Singapore: J. Wiley & Sons (1984) 483-503
References 31
Structul'al Optimization of Engineer-ing Designs. DFG-Report
of
the Research Laboratory for Applied Structural Optimization,
Univel'sity of Siegen, May 1985
[17] Eschenauer, H.: Parabolantennen fUr Satellitenfunk und
Radio
astronomie im Millimeterwellenbereich-Forderungen und Auf
gaben an den Ingenieur. In: Kreuzel', H., Bonfig, K. W.:
Entwick
lungen del' siebziger Jahre, Gerabronn: Hohenlohel' Druck-
und
VerIagshaus, (1978) 531-54-9
Accurate Focusing Systems. In: W. Stadler: Application of
Mul
ticriteria Optimization in Engineel'ing and the Sciences.
Plenum
Publishing Corporation. (1988) 309-354-
[19] Eschenauer, H.; Vermeulen. P.: Contribution to the
Optimization
of a Novel Solar Energy Collector. ZFW, Bd. to, H.3, (1986)
190-198
of Structural Optimization 1, (1989) 171-180
[21] Kneppe, G.: Dil'ekte Losungsstrategien zur
Gestaltsoptimierung
von FHi.chentragwerken. Dissertation, Universiti:it-GH-Siegen,
1985
[22] Eschenauer, H.; Kneppe. G.: Min-Max-Formulierungen als
Strate
gie in del' Gestaltsoptimierung. ZAMM 6 (1985) T344--T34-5
[23] Pierre, D.A.: Lowe, M.J.: Mathematical Programming via
Augmen
ted Lagrangian. London: Addison-Wesley, 1975
[24-] Kuhn, H.W.; Tucker, A.W.: Nonlinear Progl'amming.
Proceedings
of the 2nd Berkeley Symposium on Mathematical Statistics and
Probability, University of California, Berkeley, California,
1951
[25] Hettich, R.: Charakterisierung lokalel' Pareto-Optima.
Optimiza
tion and Operations Research. In: Oettli, W.; Ritter, K.
(eds,):
Lecture Notes in Economics and Mathematical Systems. No. 117.
Berlin: Springer-Verlag, (1976) 127-141
3-26
Vieweg- Verlag, 1979
In: Lecture Notes in Economics and Mathematical Systems. No.
76.
Bedin: Spl'inger-Verlag, 1972
[29] Sattler, H.J.: Eine Herleitung der Zielgewichtung in der
Vektor
optimierung aus einer Abstandsfunktionsformulierung. In:
Zeit
schrift fUr Angewandte Mathematik und Mechanik. ZAMM 62
(1982) T382-T384 [30] Charnes, A.; Cooper, W. W.: Management Models
and Industrial
Application of Linear Programming. Vol. 1. New York: Wiley
1961
[31] Fox, RL.: Optimization Methods for Engineering Design.
London:
Addison-Wesley, 1971
blems for Engineering Design, Computational Methods in
Applied
Mech. and Eng., Vol. 15, (1978) 309-333
[33] Osyczka, A.: An Approach to Multicritel'ion Optimization
for
Structural Design, Proceedings of International Symposium on
Optimum Structural Design, University of Arizona, 1981
[34] Osyczka, A.: Multicriterion Optimization for Engineering
Design,
In: Gero, J.S. (ed): Design Optimization. New York: Academic
Press
Inc., (1985) 193-227
[35] Stadler, W.: Multicriteria Optimization in Engineering and in
the
Sciences. New York and London: Plenum Press. 1988
[36] Stadler, W.: Multicriteria Optimization in Mechanics (A
Survey).
Applied Mechanics Rewievs, Vol. 20, (1984) 1442-1471
[37] VOl-Guideline 2212: Systematisches Suchen und Optimieren
Konstruktiver Losungen. VDI-Handbuch Konstruktion, DK631:
658,512,2 (083,132)
[38] Koski, J.; Silvennoinen, R.: Norm Methods and Partial
Weighting
in Multicriterion Optimization of Structures. International
Journal
for Numerical Methods in Engineering. Vol. 24 (1987)
1101-1121
[39] Koski, J.: Defectiveness of Weighting Method in
Multicriterion
Optimization of Structures. Communications in Applied
Numerical
Methods, Vol. 1 (t985) 333-337
[40] Eschenauer, H.; Post, P.U.; Bremicker, M.: Einsatz del'
Optimie
rungsprozedur SAPOP zur Auslegung von Bauteilkomponenten.
Bauingenieur 63, II (1988) 515-526
[41] Rozvany, G.I.N.: Structural Design via Optimality Criteria.
001'
drecht/Bostonl London: Kluwer Academic Publishers, 1989
PART I
2 OPTIMIZATION PROCEDURE SAPOP - A GENERAL TOOL FOR MULTICRITERIA
STRUCTURAL DESIGNS
M. Bremicker, H.A. Eschenauer, P. U. Post
2.1 Demands on an Optimization Procedure
As presented in Chapter 1, it is an important goal of
engineering
activities to improve and optimize technical designs, structural
assem
blies and stl'uctural components. The task of stl'uctul'al
optimization is
to SUppOI't the engineer in searching fOl' the best possible design
alter
natives of specific structul'es. The "best possible" or "optimal"'
structul'e
here applies to that structure which mostly corresponds to the
designer's
desired concept and his objectives meeting at the same time
operational,
manufactul'ing and application demands. Compared with the "Tl'ial
and
Error"-method generally used in engineering practice and based on
an
intuitive empirical approach, the determination of optimal
solutions by
applying mathematical optimization procedures is more reliable
and
efficient. These procedures can be expected to be more
frequently
applied in industl'ial practice. In order to apply structural
optimization
methods to an optimization task, both the design objectives and
the
J'elevant constraints must be expl'essed by means of mathematical
func
tions. One example of a design objective is the demand fOl' the
maximum
degree of stiffness of a stl'ucture which can be described by the
objective
"minimization of the maximum structural deformation". The
design
val'iables al'e the parameters of the structure, for example the
CI'OSS
sectional and geometl'ical quantities, which should be selected in
a way
that the objective function can be minimized by considel'ing
additional
conditions. These conditions or constraints al'e equality and
inequality
equations which include the mathematical formulation of demands
such
as permissible stresses, stability critel'ia etc. The formulation
of the
scalar design pl'Oblem is generally given by (1-0:
Min {f(x) I h(x) = 0 ; g (x) :s; 0 } . xEIRn
(2-1)
The solution of optimization problems requires software
systems
which are easy to use, provide sufficient efficiency, and are
available for
practical application. Several optimization algorithms should be
linked
to structural analysis procedures in a suitable manner by means
of
optimization model processors [1,2,45].
In general, a software system should meet the following
requirements:
- possibility of selecting the suitable optimization algorithm for
an
optimization problem from a number of efficient methods,
- use of different methods for structural analysis such as finite
and
analytical methods,
- application of automatic design and evaluation models (pre- and
post
processors) for a wide range of standard problems in
optimization
modelling; simple integration of special optimization models if
re
quired,
rent program modules,
without comprehensive implementation work,
optimization tasks by applying efficient algorithms (e.g.
sensitivity
analysis of FE-structures, solution methods for linear and
nonlinear
equation systems etc.),
systems), utilization of modern programming techniques
(parallel
computing>,
mentation.
2.2.1 Definitions
Before describing an optimization procedure and its practical
realiz
ation some of the terms frequently used in this chapter shall be
defined
(see Figs. 211-213):
Optimization algorithm : mathematical procedure for
constrained/
unconstrained optimization (optimality critel"ia
methods. mathematical programming methods),
optimization problem,
problems to simplified substitute problems or
smaller subproblems, respE'ctively,
or, in accordance with the given requirements,
of several design objectives.
into one scalar substitute objective function,
Constraints mathematically formulated design requirements
which are not covered by the objective
function(s),
behaviour (mathematical-physical modell,
Analysis variables structural parameters which can be varied
during optimization computations,
Initial design initial values of the design variables at the
beginning of the optimization process,
Design model
analysis variables.
tion concepts,
objective function and constrain t values under
consideration of optimization strategies,
mode\(s>.
When dealing with a structural optimization problem, it is
recommen
dable to proceed following the "Three-Columns Concept" [1] (Fig.
2/2.>'
38 2 Optimization P."ocec\ure SAPOP
r----, Data I Designer t--------, L.. • Input -,-- I
Optimal
Design
x·
I
('-'---'-- -'-'--'-'-i . Transformed i I AnalysIs
Variables .i J Transformation L LI Design Model I i Variables
Z I I Z - x IDe~ I x - y I I y
I Variables i i x i i i i i i OPTIMIZATION i Structural
Optimization
Algorithm i MODEL i i i I i i i jl i
f,p.g I Evaluation I i U
Preference Function! L Model I I State
Model
Fig 2/1. Structure of an optimization loop
The first step is the theoretical formulation of the optimization
problem
taking into account all relevant demands on the structure. The next
step
involves the solution of the subproblems "structural modelling"
and
"optimization modelling". From the third column an optimization
algo
rithm is selected and linked with the structural and the
optimization
model to form an optimization procedure. In the following a
detailed
description of the columns is given.
Column 1: Structural model
Any structural optimization requires the mathematical determination
of
the physical behaviour of the structure. In the case of
mechanical
systems, this refers to the typical structural response subject to
static
and dynamic loading such as deformations, stresses, eigenvalues,
etc.
Furthermore, information on the stability behaviour (buckling
loads) has
to be determined. All state variables required for the objective
function
and constraints have to be provided. The structural calculation is
carried
out using efficient analysis procedures such as the finite element
method
or transfer matrices methods. In order to ensure a wide field of
applic
ation. it should be possible to adapt several structural analysis
methods.
2.1 Demands on an Optirnization Procedure 39
Column 2: Optimization modelling
From an engineer's point of view, this column is the most
important
one of the optimization procedure. First of all, the analysis
variables
which are to be changed during the optimization process are
selected
from the structural parameters. The design model including
variable
linking. variable fi:\.ing. shape functions etc. provides a
mathematical
link between the analysis variables and the design variables. In
order to
increase efficiency and improve the convet'gence of the
optimization,
the optimization problem is adapted to meet the special
requirements
of the optimization algorithm by transforming the design variables
into
transformation variables. By using this approach, it is e.g.
possible to
almost linearize the stress constraints of a sizing optimization
problem.
Additionally, objective functions and constraints have to be
determined
by procedures that evaluate the structural response or state
variables.
\Vhen formulating the optimization model, the engineer has to
consider
the demands from the fields of design, material, manufacturing,
assem
bly and operation.
OPTIMIZATION PROCEDURE
I Oeslgn I I Matf'rlal I I Manufacturing I I Auembl) I '-..
......... '" ¥
r Structur-al MuJf'1 Optlmlzlltlon Model I Optimlz.atlon
Algorithms
I I I j j Anal}tical Discrete
"an.' J 1 r l 1 ~bthematical Special Methods Methods forme'!.-
Transfor- Design D('sign Anal}sis Programming Methods 'l mal;on J
-l Mod.1 J 'I - Rayligkl/Ritz - Finite Element Variables z _:0(
Variables " _) VariableS! - S('C]uential - OC- Mtothods
Method Unearization - Galerkln - Discrete
- Exact Element I E"aluation I Quadratic Solution Method State
Objectivt'S Programing - Dvnamic
Variables "
Method Gradients - Stochastic Optimization
- Transfer OptimiZation Strat~gles
V I r v.eto< I r u'g' Scal. I Sh.pe
V Optimization System Function
Optimization Decomposition Optimization
I
Column 3: Optimization algorithms
problems. These algorithms are iterative procedures which,
proceeding
from an initial design xo ' generally provide an improved design
variable
vector xk as a result of each iteration k. The optimization is
terminated
if a breaking-off criterion responds during an iteration.
Numerous
studies have demonstrated that the selection of the optimization
algo rithm is problem-dependent. This is particularly important
for a reliable
optimization and a high level of efficiency (computing time, rate
of
convergence>. If, for example, all iteration results have to lie
within the
feasible domain, an algorithm that iterates within the feasible
domain
(e.g. generalized reduced gradients (GRG» should be applied.
2.3. Basic Ideas of the Procedure SAPOP
On the basis of the "Three-Columns Concept" and on the
)'equirements
mentioned above, the software system SAPOP (Structural Analysis
Program and Optimization Procedure, Fig. 2/3,) was developed.
It
INPUT
Fawco3 I' l8---IDIWSAPI
IrORCE I [DiS"="I ~
lSAP
POST
WEIGHT DISPlACE 1JISPl..:E STRESS NlPlAT STRESS ElGEN EIGEN ElJ(KlE
FALCRIT ~
'viiPST OBJWEI COT ......... DISTANCE
GRAPH
consists of three independent parts communicating with each other
via
a Data Management System (data base>. Each of these parts is
divided
into individual blocks connected by standardized intel"faces to
ensure
the largest possible modularity. Each block contains a number of
inter
changeable modules. When carrying out an optimization
computation
only those modules which are actually needed are linked
together.
2.3.1 Problem Formulation and Input Data
The input system is used to prepare the input data provided by the
user
to be stored on a database. All quantities required to describe the
struc
tLII"al and the optimization model as well as the parameters for
control
ling the optimization pmcess are edited hel"e. The user has to
provide at
least two different data items. The data fi Ie OPTDA T incl udes
all data
necessary to control the optimization process as well as the
initial
values of the design variables and the input quantities for the
formu
lation of the optimization model. The data file STRDAT includes the
input
data fot" the software system which is applied to the structural
analysis.
The module MINBA includes a band-width-minimizer for
FE-stnlctlll"es.
The input data for a multilevel optimization using a
decomposition
stl"ategy are provided by the file DECDAT; COMDAT supplies
the
relevant material specifications of fibre-reinforced composites. In
future,
the user will be supported by an expert system EXPERT when
generating
the input data.
The optimization computation is actually carried out by the
SAPOP
main module MAIN. First of all, an initialization phase is run, and
sub
sequently the optimization is stal"ted via the ONE-SYSTEM module
of
the DECOM block. Two decomposition strategies (cutting force
method
FORCE and deformation method DISPLACE [12]) allow to optimize
large
structures by optimizing substructures. A number of different
optimiz
ation algorithms can be called by the DECOM-modules. Apart from
the
seven mathematical programming methods. an optimality criteria
pl"oce
dure (stress- ratio method) is available. It is also possible to
couple
42 2 Optimization Procedure SAPOP
different algorithms by means of a series connection (serial
hybrid
approach), Among others, the following algorithms can be
applied:
COMBOX EXTREM SEQLI2
Direct Search Algorithm by Jacob [4-],
SEQuentialUnear Optimization Extended Version [2-6],
Variable Metric Method for Constrained Optimization
Including Watch Dog Technique [7],
Nonlinear Program with Quadratic Une Search [8],
Generalized REduced G.·adient Algorithm [9-11],
Quadratic Programming with Reduced Line-Search
Technique [12,13],
Lagrange Penalty Method for NonLinear Problems [14],
Optimality CRITeria Method (Stress Ratio Method) [15].
For each iteration the actual values of the objective function
and
constraints are required, and for mos,t of the algorithms the
gradients
have to be calculated with regard to the transformation variables.
The
control program FUNC for structural analyses and the control
programs
for sensitivity analyses (gradient calculations, Section 2.4.3) are
called
via the interface module COMBIN. Module FDYN is an algorithm
for
solving time-dependent optimization problems [41].
The transformation module TRANS shifts the transformation
variables
into design variables. The subsequently called PRE-processor
contains
different design models used to determine the analysis variables
from
the design variables. The design model SIZE includes variable
linking
and variable fixing for cross-section optimization (sizing). SHAPE,
GEOM
and MESH modules can be used for shape and geometry
optimization
tasks. As far as composite designs are concerned, the module
COMP
transforms design variables into layer thicknesses and ply angles
of a
fibre composite lamina, and the corresponding mateJ"ial
characteristics
(elasticity, stiffness, thermal and hygrothermal coefficients) are
calcu
lated. If a special design model is to be used to solve an
optimization
problem, a corresponding program module can be included. or the
entire
preprocessor can be exchanged.
The structural analysis is now carried out using the updated
analysis
variables. These are part of the structural parameters of the
mathema
tical-mechanical model which describes the physical behaviour of
the
2.3 Basic Ideas of the Procedure SAPOP 43
actual structure. Systems of algebraic or differential equations
are
solved by using efficient numerical methods. At present, the
following
structural analysis methods a.'e available in SAPOP:
SAPV-2
ORSAB
LSAP
NLPLAT
PAFEC
ANSYS
Method. The modules for linear displacement, stress
and eigenvalue analysis are integrated in SAPOP;
Orthotrope Rotationsschalen unte.' allgemeiner Belast
ung, Transfer-Matdx Method for arbitrary loaded iso
tl'Opic and orthotropic shells of revolution [1,46,47];
Laminated Shell Analysis Program [17], Finite Difference
Method for anisotropic shells of fj bel' composite
material;
Difference Method for anisotl'Opic composite plates with
time-dependent material behaviour and imperfections; Pt'ogram for
Automatic Finite Element Calculation [49];
Finite-Element-Pl'Og.'am of Swanson Analysis Systems.
Apart from these programs, the user can link his own
st.'uctu.'al
analysis modules to SAPOP. Thus, it is possible to deal with
structures
using analytical calculations or to deal with any examples from
inte.'
disciplinary fields. For the latter, however, other pre- and
postproces
sors a.'e usually .'equit'ed in orde.' to formulate the design and
evaluation
model.
The computed state vadables a.'e transferred to the postprocessor
in
orde.' to determine the objective function{s) and constraints.
Modules
are available for computing weight as well as stress,
defo.'mation,
and eigenvalue evaluations. In the case of composite structures
the
failure criteria of laminate composites are determined
[20-23].
Multicriteria optimization problems are solved by transforming
the
objective function into a scalar substitute function {preference
fUnctions,
see (1-10) to (1-24)).
If the range of performace of the postprocessor is not sufficient
for a
special application, user-defined programs for the fomlUlation of
objec
tive functions and constraints can be linked via standa.'dized
interfaces.
The actualized objective functions and constraints a.'e t.'ansfen'