
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00003942/00001
Material Information
 Title:
 On the automated optimal design of constrained structures
 Creator:
 Hornbuckle, Jerry Clyde, 1942
 Publication Date:
 1974
 Language:
 English
 Physical Description:
 ix, 254 leaves. : illus. ; 28 cm.
Subjects
 Subjects / Keywords:
 Approximation ( jstor )
Boundary conditions ( jstor ) Buckling ( jstor ) Cost functions ( jstor ) Eigenvalues ( jstor ) Initial guess ( jstor ) Mathematical variables ( jstor ) Mathematics ( jstor ) Maximum principle ( jstor ) Structural deflection ( jstor ) Buckling (Mechanics) ( lcsh ) Dissertations, Academic  Engineering Sciences  UF Engineering Sciences thesis Ph. D Girders ( lcsh ) Structural design ( lcsh ) Structural dynamics ( lcsh ) Structural engineering ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis  University of Florida.
 Bibliography:
 Bibliography: leaves 244253.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 Jerry C. Hornbuckle.
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 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
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14184224 ( OCLC ) ADB3751 ( NOTIS )

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Full Text 
ON THE AUTOMATED OPTIMAL DESIGN
OF
CONSTRAINED STRUCTURES
BY
JERRY C. HORNBUCKLE
A Mi'.S;FTATION PRESENTED TO THE GR\DU.LITF CLOU'NCE OF
THE UNIVERSITY OF FLOr.IDA IN PARTIAL
FULFILLMENT OF THE REQLIRC1;S r THE D'LE OF
DOCTOR OF PHILOSOPHY
[lilVEPSITY OF FLORIOA
1974
Copyright by
Jerry C. Hornbuckle
1974
DEDICATION
To my grandmother, Mrs. Florence Hornbuckle,
and my wife, Carolyn. Without the love, confidence, and
patient understanding of Granny Hornbuckle and Carolyn
my graduate studies would never have been attempted.
ACKNOWLEDGMENTS
To Dr. Robert L. Sierakowski and Dr. William H. Boykin, Jr.,
for guiding my research and for being more than just advisors.
To Dr. Gene W. Hemp, Dr. Ibrahim K. Ebcioglu, and Dr. John
M. Vance, for their assistance, support, and for serving on my advisory
committee.
To Dr. Lawrence E. Malvern and Dr. Martin A. Eisenberg, for
always finding the time to offer advice and explanations on questions
related to solid mechanics and academics.
To the departmental office staff for their kind assistance
with administrative problems and clerical support.
To Randell A. Crowe, Charles D. Myers, and J. Eric Schonblom
for attentive discussions of many little problems and for assistance
in preparing for the qualifying examination.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . iv
ABSTRACT . viii
CHAPTER
I INTRODUCTION ,. .. 1
1.0 Survey Papers .. .. 1
1.1 Historical Development: Optimal Columns 6
1.2 Historical Development: Optimal Static Beams 9
1.3 Historical Development: Optimal Dynamical Beams 12
1.4 Scope of the Dissertation .. 14
II GENERAL PROBLEMS AND METHODS IN STRUCTURAL OPTIMIZATION 16
2.0 Introduction . .. 16
2.1 Problem Classification Criteria 16
2.1.0 Problem Classification Guidelines .. 18
2.1.1 Governing Equations of the System .. 18
2.1.2 Constraints ... 19
2.1.3 Cost Functionals ... 21
2.2 Methods: Continuous Systems ... 26
2.2.0 Special Variational Methods .. 27
2.2.1 Energy Methods .. 27
2.2.2 Pontryagin's Maximum Principle .. 29
2.2.3 Method of Steepest Ascent/Descent .. 31
2.2.4 Transition Matrix: Aeroelasticity Problems 31
2.2.5 Other Miscellaneous Methods .. 32
2.3 Methods: Discrete Systems ... 33
2.3.0 Mathematical Programming .. 34
2.3.1 Discrete Solution Approximations .. 35
2.3.2 SegmentwiseConstant Approximations .. 37
2.3.3 Complex Structures with Frequency
Constraints 38
2.3.4 Finite Element Approximations .. 40
2.4 Closure . .. 41
TABLE OF CONTENTS (Continued)
CHAPTER Page
III THEORETICAL DEVELOPMENT .. 43
3.0 Introduction . .. 43
3.1 Problem Statement and Necessary Conditions ... 44
3.2 Mathematical Programming: Gradient
Projection Method ... 48
3.3 Gradient Projection Methods Applied to
the Maximum Principle ... 53
3.4 Maximum Principle Algorithm .. 60
3.5 Solution Methods .. 63
IV CONSTRAINED DESIGN OF A CANTILEVER BEAM
BELiDIING DUE TO ITS OWN WEIGHT .. 66
4.0 Introduction . 66
4.1 Problem Statement .. 66
4.2 Structural System. .... 67
4.3 Unmodified Application of the Maximum Principle 70
4.4 Results: Geometric Control Constraints .. 79
4.5 Inequality Stress Constraints .. 89
4.6 Results: Stress Constraints Included. .. 93
V CONSTLr.I[:i) DESIGN FOR AN OPTIMAL EIGENVALUE P.ODLLF1I 101
5.0 Introduction . .. .. 101
5.1 Problem Statement .. 101
5.2 Structural System .. 102
5.3 Analysis of the Problem ... 109
5.4 Application of the Maximum Principle .. 121
5.5 Results: Geometric Control Constraints 132
5.6 Inequality Stress Constraints ... 148
VI FINITE ELEMENT METHODS IN ST U'ii i.i. OPf it...'T L.__:
AN EXAMPLE . 155
6.0 Introduction . .. 155
6.1 Finite Element Problem Statement. ... 155
6.2 Mathematical Programming: Gradient
Projection Method .. 157
6.3 Results ... 162
VII COMMENTS ON NUMERICAL INSTABILITY IN THE
(l'ir'SILINEARIZATION ALGORITHM. ... .. 173
7.0 Introduction .. 173
7.1 Computer Program Convergence Features .. 1.73
7.2 :!,.:rical Instatilibies for Cantilever
Beam Example . 175
7.3 Numerical Instabilities for Column
Buckling Example .... 183
TABLE OF CONTENTS (Continued)
CHAPTER Page
VIII CONCLUSIONS AND RECOMMENDATIONS ... 186
8.0 Summary and Conclusions ... 186
8.1 Recommendations ... 186
APPENDIXES
A HISTORICAL DEVELOPMENTS . 191
B A SIMPLE PROOF OF THE KUHNTUCKER THEOREM .. 206
C COMPUTER SUBROUTINE LISTINGS ... 214
BIBLIOGRAPHY ..... .... 244
BIOGRAPHICAL SKETCH . ... 254
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
ON THE AUTOMATED OPTIMAL DESIGN
OF CONSTRAINED STRUCTURES
By
Jerry C. Hornbuckle
August, 1974
Chairman: Dr. William H. Boykin, Jr.
CoChairman: Dr. Robert L. Sierakowski
Major Department: Engineering Sciences
Pontryagin's Maximum Principle is applied to the optimal design
of elastic structures, subject to both hard inequality constraints and
subsidiary conditions. By analyzing the maximum principle as a non
linear programming problem, an explicit formulation is derived for the
Lagrangian multiplier functions that adjoin the constraints to the cost
functional. With this result the usual necessary conditions for opti
mality can themselves be used directly in an algorithm for obtaining
a solution.
A survey of general methods and problems in the optimal design
of elastic structures shows that there are two general problem types
depending upon whether or not the cost functional is an eigenvalue.
An example problem of each type is included with the solutions obtained
by the method of quasilinearization. In the first example, a minimum
deflection beam problem, classical Maximum Principle techniques are
used. The eigenvalue problem is exemplified by the maximization of the
buckling load of a column and uses the explicit multiplier function
viii
formulation mentioned above. Since the problem considered is conserva
tive, it is therefore described mathematically by a selfadjoint system;
under this condition it is shown that the minimum weight problem is
identical to the maximum buckling load problem.
In order to demonstrate the theory for the programming techniques
used, the beam problem is also solved by using a finite element repre
sentation of the structure. From a comparison to the maximum principle
solution it is shown that the form of the optimal solution obtained is
dependent upon the magnitude of the tolerance used with the numerical
solution scheme. Furthermore, it is shown that convergence by the
quasilinearization algorithm is related to the respective curvatures of
the initial guess and the solution.
Recommendations for additional investigations pertinent to this
study are also included.
CHAPTER I
INTRODUCTION
1.0 Survey Papers
It is exceedingly difficult to write a general introduction to
the field of structural optimization for two basic reasons: (i) there
is no conventionally accepted nomenclature, and (ii) there is also no
conventionally accepted classification of problem types or character
istics. In marked contrast, when one considers the calculus of varia
tions, "cost functional, system equations, kinematic and natural bound
ary conditions, adjoint variables, Hamiltonian, etc.," all have well
defined, universally accepted meanings. Additionally, there is no con
fusion when speaking of the problem types of Mayer, Lagrange, and Bolza.
This common language and categorization of problems does not exist in
structural optimization.
Instead, the field tends to branch and fragment into very special
ized subdisciplines that are oriented towards applications. ':hile these
branches are related to the general field, the techniques and methods of
one branch can seldom be applied to another. Moreover, as a result of
the tendency to an applications orientation, solutions are generally ad
hoc and not useful for other problems even within the same branch. The
lack of any definitive unification of the subject cannot be blamed either
on being recently developed or in receiving too little attention.
This is readily seen by considering the survey papers described in the
following paragraphs.
The earliest comprehensive survey paper is Wasiutynski and
Brandt (1963). Although their excellent historical development is
dominated by Russian and Eastern European references, the authors do
include a higher percentage of papers by Western authors than is
encountered in the typical paper from Eastern Europe. A more funda
mental criticism is that too little is said regarding problem types or
solution methods.
Chronologically, the next survey paper is Gerard (1966).
The theme of this review is aerospace applications, with a particular
orientation to the designmanagement, decisionmaking process. Most of
the papers cited treat specialized aerospace structures and applica
tions; however, the author does try to generalize by introducing a
design index D, a material efficiency parameter M, and a structural
efficiency parameter S. After defining the expressions for M and S
corresponding to several structural elements, design charts are pre
sented which show regions of possible application for various materials.
Unfortunately the design charts do not satisfy expectations aroused by
the introduction of the three general parameters.
Rozvany (1966) presents a similar paper pertaining to structures
in civil engineering. This paper is less comprehensive and more ori
ented towards specific structural applications than either of the pre
ceding survey papers. The author postulates several "interrelated
qu3nti.ies (or parameters)" which could perhaps be used to generalize
structural optimization into a more rational methodology. However,
these quantitiesloading (L), material (M), geometry (G), initial
behavior (IB), and design behavior (SB)are only applied to an abstract
discussion of concepts.
Barnett (1966) has a readable short survey of the field that
dwells more upon theoretical aspects. He postulates a general problem
in which the cost is minimized subject to a system in equilibrium with
its loads, while "behavior constraints on strength, stiffness, and
stability" are satisfied. Uniform strength design introduces the dis
cussion of optimal trusses; virtual work theorems that were derived
originally for trusses are then applied to simultaneous plastic collapse
problems. Following a brief discussion of the plastic collapse load
bounding theorems, there is a short treatment of elastic stability
problems and material merit indices. Barnett's stiffness design example
exhibits several important features worth noting. Specifically, the
example is to minimize the weight of a beam subject to some given load
where the deflection at a certain point is specified. Virtual work is
used to handle the subsidiary deflection condition. Necessary condi
tions are obtained from the calculus of variations, but more signif
icantly, the Schwarz inequality provides a sufficient condition for
global optimality. Barnett concludes his survey with a section stating
that multiload designs satisfying "all three behavior criteria" are more
easily solved in "design space" by mathematical programming techniques
described by Schmidt (1966).
As a sequel to the comprehensive survey (1638 to 1962) by
Wasiutynski and Brandt, Sheu and Prager (1968b) present a complete review
of developments from 1962 to 1968. This paper contains three major
sections: general background, methodology, and specific problems.
In the first section they state that the "wellposed problem of optimal
structural design" requires specification of the
(1) purpose of the structure (load and environment),
(2) geometric design constraints (limits to design
parameters),
(3) behavioral design constraints (limits to the "state" of
the structure),
(4) design objective (cost functional).
The methodology section is not noteworthy, but the third section lists
what is in their opinion the specific problem types: static compliance,
dynamic compliance, buckling load, plastic collapse load, multipurpose/
multiconstraint structures, optimal layout (e.g., trusses), reinforced/
prestressed structures, and from the background section, probabilistic
problems. Their concluding remarks succinctly summarize the paramount
difficulty of the subject: realistic problems are too complicated for
precise analytical treatment. While progress is being made in analyt
ical treatment of simple structures, the authors opine that realistic
problems require mathematical programming techniques. However, they do
Feel that analytical treatment is desirable to provide "a deeper insight
into the analytical nature of optimality."
A related survey by Wang (1968) on distributed parameter systems,
". .. whose dynamical behaviors are describable by partial differential
equations, integral eqlujtiLns, or functional differential equations,"
consists entirely of a bibliography. While not pertinent to the disser
tation, it is mentioned here for completeness.
Prager (1970) provides another survey which is not comprehensive,
nor does he present any new results as claimed. For example, Prager
and Taylor (1968) used the principle of minimum potential energy
and the assumption that stiffness is proportional to specific mass to
prove global optimality. It could hardly be called a new development
in 1970. At the same time, the author does present an excellent example
of a multipurpose optimal design problem. Prager also treats "segment
wise constant" approximations and the optimal layout of trusses.
The final survey paper, Troitskii (1971), is an unusual review
of methods in the calculus of variations. Whereas some of the earlier
surveys present lengthy lists of references but contain little method
ology or theory, this survey is just the opposite. Part of what makes
it unique is that the author believed only eight articles merited cita
tionall of them by Troitskii. This shortcoming is more than overcome
by a thorough classification of optimal control problems in the calculus
of variations. Troitskii bases the classification on "certain character
istics of control problems": types of constraints, properties of the
governing dynamical equations of the system, type of cost functional, and
possible state discontinuities. From these four criteria he postulates
five principal classes of problems; however, it is the criteria that
are important and not the specific problem type.
By comparing what the authors of the aforementioned surveys
believe to be the important types of problems, it is readily apparent
that there is little agreement on which characteristics of structural
optimization problems are significant.
1.1 Historical Development:
Optimal Columns
The beginning of structural optimization is generally attributed
to Galileo's studies in 1638 of the bending strength of beams. Accord
ing to Barnett (1968), Galileo considered a constantwidth cantilever
beam under a tip load as part of a study of "solids of equal resistance."
In requiring the maximum stress in each cross section to be constant
throughout the beam, the height must be a parabolic function of position
along the beam. While this appears to be the origin of the field, a prob
lem that received more attention is the buckling of a column.
Using the newly developed calculus of variations, in 1773 Lagrange
attempted to apply variational techniques to the problem of finding that
distribution of a homogeneous material along the length of a column which
maximizes the buckling load. Truesdell (1968) relates that through an
insufficient mathematical formulation Lagrange showed the optimal form
to be a circular cylinder. Clausen (1851) provides the earliest known
solution to this problem for the simply supported case. As described in
Todhunter and Pearson (1893, pp. 325329), Clausen minimized the volume
of the column with the differential equation for buckled deflection
treated as a subsidiary condition. Assuming all cross sections to be
similar, after several variable transformations and complicated manip
ulation, he obtained an implicit, analytical solution.
The next development was Greenhill (1881), according to Keller
and Niordson (1966). Greenhill determined the height of a uniform
prismatic column, beyond which the column buckled due solely to its own
weight. Timoshenko and Gere (1961) reproduce the solution in which the
deflection is expressed as the integral of a Bessel function of the
first kind (of the negative onethird order).
Blasius (1913) introduces his paper with a uniform strength and
a minimum deflection beam problem. For a given load and amount of mate
rial, the crosssectional area distribution is determined which maxi
mizes the buckling load of a circular column. The solution is identical
to that obtained by Clausen. In addition, Blasius also obtained solu
tions for columns having rectangular cross sections and discussed the
effect of different boundary conditions on the results.
For the next few decades, structural optimization appears to
have been directed towards applications in the aircraft industry, where
aircraft structural problems and results are presented in the format of
a design handbook. Feigen (1952) is a good example of this, consider
i:; the buckling of a thinwall column. Given a constant load and wall
thickness, he required the variable inside diameter to be chosen such
that the buckling load is maximized. Wall thickness is selected to make
local buckling and Euler buckling occur at the same load. Solid tapered
columns having blunt ends are also treated for assumed stiffness dis
tributions.
Renewed interest was aroused by Keller (1960), who examined the
problem from the point of view of the theory of elasticity, and in
choosing the crosssectional shape to give the maximum stiffness.
Neglecting the r.bht of the column, he obtained via the former that
twisting the column does not affect the buckling. Of all convex cross
sections, the equilateral triangle is shown to have the largest second
moment of area relative to a centroidal axis. Hence, from the defin
ition of buckling load, the "best" crosssectional shape is the equi
lateral triangle. Keller also obtained Clausen's implicit, analytical
solution. Subsequently, Tadjbakhshand Keller (1962) generalized the
problem to a general eigenvalue problem and boundary conditions subject
to a subsidiary equality constraint. The latter corresponds to spec
ifying the volume (or weight) of material to be distributed in an opti
mal manner. Using the H6lder inequality they demonstrate global opti
mality of the eigenvalue for the hingedhinged column.
Keller and Niordson (1966) examine the case of a vertical column
fixed at the base, subject to a vertical load at the tip and the column's
own weight. It is also assumed that all cross sections are similar.
Their approach is to state the problem as a simultaneous, dual optimiza
tion of the Rayleigh quotient. The eigenvalue is minimized with respect
to the eigenfunction and maximized with respect to crosssectional area
distribution. Specifying the volume of material available is treated
as a subsidiary equality constraint. From the maximum lowest eigenvalue
the maximum height at buckling is determined. Solutions are obtained by
an iterative technique employed with integral equations.
In a brief note, Taylor (1967), suggests that ener, methods may
link optimum column problems to the traditional eigenvalue problems of
mechanics. Prager and Taylor (1968) classify problems in optimal struc
tural design and demonstrate global optimality using energy principles.
Unfortunately their assumption of thinwall construction limits the
results to structures where the stiffness is proportional to the specific
mass density. The consequence of this assumption is that in the energy
formulation the resulting control law and governing equations are
decoupled, and hence easily solved. Huang and Sheu (1968) apply this
same thinwall assumption to the problem treated by Keller and Niordson.
However, the former seek the maximum end load instead of the maximum
height. The authors also attempt to limit the maximum allowable stress
and obtain solutions by a finitedifference method. Further discussion
of sandwich (thinwall construction) columns is given by Taylor and Liu
(1968). Basically, this paper is an elaboration of the techniques
described by Prager and Taylor when applied to columns. Extensive
results are provided for various cases.
Postbuckling behavior for columns subject to conservative
loads is considered by Gajewski and Zyczkowski (1970). A nonconcerva
tive problem is treated by Plaut (1971b). The first of these two papers
is lengthy but is much too narrow in scope to be particularly useful.
In the second paper, the Ritz method is applied to an energy functional,
obtaining the "best" form of the assumed approximation to the optimal
solution.
1.2 Historical Development:
ODtimal Static Beams
That beam problems played a role in the early developments of
structural optimization has already been indicated in the preceding sec
tion. No attempt is made in what follows to present a complete history,
but merely to outline the type of problems that have been considered.
Opatowski (1944) has an outstanding paper that deals with
cantilever beams of uniform strength. Besides providing numerous refer
ences to earlier studies, the author treats the problem with impres
sive mathematical rigor. The beam is considered to deflect under its
own weight and a transverse tip load; bending deflection is described
by a Volterra integral equation which is solved exactly for various
assumed classes of variable crosssectional geometry. This paper is
representative of earlier papers in that it contains extensive analysis
and analytical results, but little numerical data. Barnett's work (1961)
and its sequal (1963a) apply the calculus of variations to more real
istic Ibeams. One problem considered is maximizing the weight sub
ject to general, unspecified loads for a specified deflection at a given
point. The Schwarz inequality is used to derive a sufficient condition
for global optimality. Also included is a comparison of uniform strength
beams to the minimum deflection beam for several different cases of
applied load and geometry. The paper is concluded with various minimum
weight design examples in which both bending and shear stiffness are con
sidered.
Haug and Kirmser's (1967) work is one of the most comprehensive
studies of minimumweightbeam problem published. Wlnilc it may succeed in
handling any conceivable problem and in employing the most realistic
stress constraints, this very generality requires so many variables and
conditions that the mathematics is complicated almost beyond reason.
Another study of minimum weight beams (Huang and Tang, 1969) is
important for several reasons. By dividing the beam into many segments
having constant properties, and determining the necessary conditions
that must be satisfied by every segment, it appears that the authors
are using the same methods that were used to solve the Brachistochrone
problem in the seventeenth century. In their treatment, multiple
loads must produce a specified deflection at a given point; these
"segmentwise constant" approximations and multiple load problems have
recently received more attention. Of further interest in this paper is
the derivation of the multiple load optimality condition using
Pontryagin's Maximum Principle.
While no new results or techniques are contained in Citron's
(1969, pp. 154166), the author gives a very readable minimum weight
beam example. The problem is simple, described in detail completely,
and provides an excellent example of how control theory is applied to
an optimal problem in structural design. An analysis of intermediate
beams which may form plastic hinges is provided by Gjelsvik (1971).
It shows that if hinges are placed at all points of the beam where the
bending moment is zero, this makes the plastic or elastic minimum
weight beam statically determinate. Both the elastic and plastic beam
designs are shown to be fully stressed, i.e., are uniform strength beams.
Application of generalized vector space techniques is character
istic of more recent papers. Bhargava and Duffin's (1973) is such a study.
It treats the maximum strength of a cantilever beam on an elastic
Since these early papers are not readily available, this statement
is based upon descriptions of them given by historical references
cited in Appendix A.
foundation subject to an upper bound on weight. Although involving
more advanced mathematics than normally required by variational tech
niques, vector space methods may also provide more powerful analyt
ical tools.
1.3 Historical Development:
Optimal Dynamical Beams
The title of this section is a misnomer. On the basis of the
related literature, a more realistic terminology would be "Optimal
Quasistatic Beams." Papers dealing with the optimization of dynamical
beams ultimately contain some assumption or given condition that effec
tively transforms the problem to an equivalent static case. Simple
harmonic motion is frequently assumed to remove time dependence from the
governing equation. For example, Barnett (1963b)minimizes the tip of
deflection of a cantilever beam accelerating uniformly upwards, with the
total '.Aicht of the beam specified. The optimal solution for cross
sectional area distribution is specified by a nonlinear integral equa
tion solved by successive approximations. Dynamics enters the problem
only as a time invariant inertia load which converts the problem to a
static beam subject to a body force.
Niordson (1964) finds the tapering of a simply supported beam
of given volume and length which maximizes the fundamental frequency of
free vibration. Assuming that all cross sections are similar, Niordson
expresses the desired frequency as the Rayleigh quotient. This is
obtained by the equation for spatial dependence associated with the
usual separation of variables technique. In this case it is assumed
that deflection, shear, and rotational inertia effects are small.
Solution of the conditions for optimality were obtained by successive
approximations. This approach results in a problem identical in form
to the eigenvalue problem associated with maximizing the Euler buckling
load of a column.
A very specialized problem is treated in Brach (1968). Cross
sectional area and stiffness are proportional, and have both upper and
lower bounds; material properties and length of the beam are specified
constants. The object is to make the fundamental frequency of free
vibration stationary, without any constraints on the weight. Instead
of using the Rayleigh quotient, Brach uses the total potential energy.
His solution method is ad hoc and not generally useful. A more useful
approach is described by Icerman (1969) for structures subject to per
iodic loads. Necessary and sufficient conditions for minimum weight
are obtained from the principle of minimum potential energy. Amplitude
and frequency of the applied load is specified as well as "dynamic
response," defined as the potential energy associated with the load
amplitude displaced a distance equal to the displacement amplitude at
the point of application. It is also required that the load's frequency
be less than the fundamental natural frequency. Subject to these con
straints, the structure's weight is to be minimized. Trusses and
segmentwise constant approximations are also treated. Once again a
dynamical problem is effectively transformed into a quasistatic problem.
Realistic treatment of the dynamic response of beams subject to
time dependent loads is given in Plaut (1970). An upper bound on the
response of the beam is obtained from the "largest possible displace
ment of the beam under a static concentrated unit load." This inequal
ity is determined from the time derivative of the total energy of the
beam and the Schwarz inequality. Plaut minimizes the upper bound on
the response for specified total weight and relationship between specific
mass and stiffness. Despite the consideration of a truly dynamical
problem, this approach has two weaknesses indicated by the author.
First, the upper bound need not be close to the exact answer; secondly,
there is no demonstration that minimizing the upper bound also minimizes
the actual response of the beam subject to dynamic loading.
Another paper worthy of mention is Brach and Walters (1970).
They maximize the fundamental natural frequency which is expressed as
a Rayleigh quotient that includes the effect of shear. Standard varia
tional methods are employed to derive necessary conditions, but no
solutions nor examples are given. The authors do, however, suicest
using the method of quasilinearization. This paper is another example
of a quasistatic, dynamical beam problem.
1.4 Scope of the Dissertation
This dissertation is primarily concerned with the application
of Pontryagip,'s Maximum Principle to problems in structural optimization.
Only elastic materials are considered; however, various types of con
straints are treated.
The theoretical development of the dissertation pertains to
problems described by an ordinary differential equation but is based
upon a numerical technique normally used with systems described by
a finite number of discrete quantities. For this reason, examples are
included for both types of systems.
No new solution techniques have been developed for the nonlinear
two point boundary value problems which characteristically arise in
optimization problems for continuous systems. Solutions are obtained
by a standard quasilinearization method. However, a modified "feasible
direction" numerical algorithm for use with discrete systems is
described and an example included to demonstrate its operation. This
serves to illustrate the application of the theory on which the algo
rithm is based to the theoretical development associated with contin
uous systems. Furthermore, it provides a comparison between the
solution to a problem described as a continuous system, and alternately
by a discrete approximation.
Additionally, it is the intent of this dissertation to replace
some of the confusion in classification of problem types contained in
the various survey papers with an organization based upon mathematical
attributes. The result is a logical approach to the formulation of
optimization problems for elastic structures.
CHAPTER II
GENERAL PROBLEMS AND METHODS
IN STRUCTURAL OPTIMIZATION
2.0 Introduction
The first chapter presents a broad view of structural optimiza
tion and the historical development of two general types of problems
that are used as examples in later chapters. Before the mathematical
theory is developed in the next chapters, general problems and methods
in structural optimization itself are briefly outlined in this chapter.
The classification of problem types visavis mathematical attributes
is discussed first. This is followed by short descriptions of the
major methods of structural optimization for both continuous and dis
crete systems. Appropriate references are cited for each section.
2.1 Problem Classification Criteria
Perhaps the major source of difficulty in classifying struc
tural optimization problems lies in the translation from the physics
involved to a mathematical representation. A single physical concept
when transformed to mathematics may become more than one mathematical
attribute. For example, consider the class of conservative problems.
Since energy is conserved this immediately prohibits dissipative mate
rials, time varient constraints, and nonholonomic constants. More
important, consider the following statements from Lanczos (1962,
p. 226):
all the equations of mathematical physics which
do not involve any energy losses are deducible from
a "principle of least action," that is the principle
of making a certain scalar quantity a minimum or a
maximum .all the differential equations which are
selfadjoint, are deducible from a minimummaximum
principle and vice versa.
However, it is shown in Chapter V that to be selfadjoint systems places
requirements on both the differential equation and the boundary condi
tion. Thus, the single physics attribute of being a conservative sys
tem is described mathematically by expressions involving the cost func
tional, the governing system of differential equations, boundary condi
tions, and constraints.
As a result of this lack of similarity in descriptions, a choice
must be made as to which realm will be used for classification of prob
lems. Since all problems are ultimately transformed to mathematics,
mathematical characteristics are selected as the criteria. On the basis
of survey paper contents and the many related papers, it is felt that
the proper (not necessarily the best, nor all inclusive) characteristics
for classification of problem types are:
(i) cost functional
(ii) system equation and boundary conditions
(iii) control constraints
(iv) behavioral constraints (state and/or control constraints)
Subsequent discussion is in terms of these four characteristics.
Exceptions to these descriptors are readily acknowledged, e.g., whether
the problem is deterministic or probabilistic. An example of the latter
may be seen in Moses and Kinser (1967). These exceptions do not serve
as negating counterexamples but instead indicate the requirement for
additional descriptors and verify the difficulty of the task, suggest
ing the need for further comprehensive study.
2.1.0 Problem Classification Guidelines
In the following sections each criterion is briefly discussed.
Some of the various characteristics of each are mentioned, and where
appropriate references exist the citation is given. It must again be
emphasized that the following is not all inclusive; it is an attempt to
categorize the types of problems existing in the literature according
to the four mathematical descriptors postulated. Moreover, the descrip
tors are not discussed in the order given but instead are treated in
the order normally encountered during a problem solution.
2.1.1 Governing Equations of the System
The immediate question to be answered is whether the structural
system is described by a set of continuous functions or a set of dis
crete constants. Bending deflection of simple structural elements is
an example of the former; design of trusses is a good example of the
latter. In general, variational techniques are employed with contin
uous systems while mathematical progrirming techniques are most fre
quently applied to the discrete systems. However, variational methods
can be applied to the approximation of continuous systems by discrete
elements. This usually takes the form of either a finite element or
"segmentwise constant" approximation of the continuous structure.
References treating the different systems described above are included
in the section on methods.
2.1.2 Constraints
Two of the descriptors are postulated to be control constraints
and behavioral constraints. A further consideration is whether the
constraint is defined by an equality relationship or by an inequality
expression. Equality constraints are handled by a longknown technique
entitled "isoperimetric constraints." Valentine's (1937) work is known
to contain the initial development of a technique for converting inequal
ity constraints to equality constraints. The introduction of slack
variables increases the number of variables in the problem to be solved
but at the same time permits all of the isoperimetric techniques to be
used. A detailed application of this approach is presented in Appendix B.
It should also be noted that isoperimetric constraints are sometimes
referred to as "accessory or subsidiary conditions."
Most real structural optimization problems possess an isoperi
metric constraint as well as inequality constraints dependent upon the
control u and/or the state x of the structural system. Typically the
control constraints are the result of geometrical limitations or
restrictions to the types of available materials. Behavioral constraints
are related to the state of the system and may depend solely upon the
state (of deformation), or in the case of most stress constraints,
jointly upon the state and control variables. With this distinction the
constraints may be classified visavis the two criteria and optimal
control characteristics as follows:
By convention all vectors are column vectors unless indicated otherwise.
(i) unconstrained
(ii) 4(u) < 0 control constraint
(iii) O(x,u) < 0
Behavioral constraints
(iv) 0(x) 0)
All of this discussion pertains to both continuous and discrete systems.
No references for unconstrained optimization problems are given.
It may be that in some problems the unconstrained structural optimiza
tion solutions have either infinite stiffness and finite weight, or
finite stiffness and zero weight. A discussion of this can be found in
Salinas (1968, pp. 2326).
Investigation of control constraints led to the development of
the maximum principle. Although reserving discussion of the method for
a later section, the classical reference detailing the derivation of
the principle is given here for completeness. Rozonoer (1959) treats con
trol constraints but only as related to the development of the maximum
principle.
Most of the literature concerns either bounded control problems
or a more general form of constraint which can be classified as a behav
ioral constraint. The latter is a mixed constraint which depends upon
both the state and control variables. Breakwell (1959) is a lucid paper
dealing with this type of constraint. References which treat state
and/or mixed constraints are Bryson et al. (1963), and Speyer and
Bryson (1968). Constraints which depend upon only the state are not
treated in this dissertation; an example of such a constraint is to
determine the optimal solution for some problem subject to an upper
bound on deflection of the structure at any point.
2.1.3 Cost Functionals
There are two basic types of cost functionals that occur in the
field of structural optimization. They were first identified by Prager
(1969) but not for the proper reasons. Using Prager's notation, they are
J = Min f F(p) dV
V
f G(W) dV
JQ = Min
f H(J ) dV
V
where F, G, and H are scalar functionals of . The latter functional
represents a Rayleigh quotient associated with an eigenvalue problem.
It can be reduced to the first type of functional shown above by choos
ing a normalization of the eigenfunction such that the numerator
equals unity for all admissible variations (see section 5.3). This
normalization is thereafter treated as a subsidiary constraint.
What actually distinguishes the second functional from the
first is not that the functional is a quotient, but that the extrem
ization of an eigenvalue requires a dual extremization (see section 5.3).
In terms of a state x and control u, the fundamental eigenvalue is
given by minimization of the Rayleigh quotient with respect to the
eigenfunction x, or
f G(x,u) dV
J = Min
x f H(x,u) dV
V
where u represents some specified design parameter. If the desired
result is to maximize the cost JQ with respect to all admissible u, it
is observed that a second extremization is required; for example, see
Keller and Niordson (1966). Thus, a more appropriate manner of classi
fying cost functionals is on the basis of whether the problem statement
implies a single or a double extremization. Hence the two basic types
of cost functionals encountered in structural optimization are
J = Min f F(x,u) dV
u V
f G(x,u) dV
V
J = Max Minr
u x f H(x,u) dV
V
where x must satisfy an equilibrium condition of the state, and u is
subject to some admissibility requirements.
There is a special case related to these two in which the weight
is to be minimized for a specified eigenvalue. This problem is
treated in Icerman (1969) with a mathematical discussion of such a
variational problem presented in Irving and Mullineux (1959, p. 394).
In terms of the two cost functionals, the special case is
J = Min f {G(x,u) J H(x,u)} dV
uV V
where J is a specified constant. This approach is frequently employed
in eigenvalue problems to avoid the inherent difficulties associated
with the dual extremization problem.
There have also been many papers published that consider
"multipurpose structures," e.g., Prager (1969), Martin (1970), and
Prager and Shield (1968). The cost function for such problems is
defined as
k
J = Z a.J.(x,u)
i=l
where a. are positive constants, serving as weighting parameters.
While perhaps demonstrating much potential, no significant results
obtained with this approach have so far been published. What problems
have been solved are too simple; indeed the authors indicate the need
for using a discrete approximation and mathematical programming tech
niques in realistic applications.
A subject closely related to "multipurpose structures" is
that of multiple constraints. It is mentioned here only because most
papers on the latter also include the formersee Martin (1971). The
idea of multiple constraints is not new; in both variational and math
ematical programming fields there exist standard techniques for han
dling multiple constraints.
A recent Russian paper (Salukvadze, 1971) suggests an alter
nate to the "multipurpose" cost functional. Instead of treating a
vector functional that requires the choosing of weighting coefficients,
it is suggested that the several functionals be combined into one.
Given a system and vector cost functionals
x = f(x,u,t)
J.[u] = J.(x,u,t) i = l,...,k
(i)
Let uOPT denote the optimal solution for which J. assumes the optimal
OPT 1
value on the trajectory of the system. For each of the J. there is a
(i)
different u PT. These k values J. can be thought of as components of
OPT 1
t
a vector r where
= (1)1 [ (kl )T
"I PTJ k"'"
For any arbitrary u the result
T
r[u] = {J [u] ... J [u]}
is just some vector functional.
*
Vector r represents a constant point in the space of
(J ,...'Jk). Since no choice of u can optimize all of the J. simul
taneously, that is, to attain the point r in J.space, the best alter
*
native is to minimize the distance between r[u] and r That distance
is defined by the Euclidean norm. To avoid the question of incon
sistent dimensions the functionals are reduced to dimensionless form.
Thus,
_\
k k J[u] J.M
J[u] = i  ___
(i)
i=1 J.
i PT
and PyT is that function u which minimizes the functional J[uj.
Mathematically speaking,
uOT = ARGMIN {J[u]}
This type of vector cost functional is much more appealing
than the type treated in the papers on multipurpose structures.
It also suggests an entirely new field of study: the more realistic
t T
Superscript "T", e.g., u denotes the transpose of the vector u.
choice of cost functionals. The mathematics of a problem seldom
accommodates financial considerations. For example, the design which
requires the least material may reduce the cost of materials at an
overwhelming expense in manufacturing or fabrication. When aesthetic
appeal and environmental impact are includedas must be done in any
real, commercial applicationthe selection of an appropriate cost
functional is an almost insurmountable task. However, a simple exten
sion of Salukvadze's composite cost functional may reduce the difficul
ties to operations research considerations.
In problems where it is desired to optimize simultaneously
several different functionals, not all having the same dimensions, the
concept of a generalized inner product may prove useful. It is defined
in terms of a metric operator A; for some general vector z
z 2= (z,z)A =TAz
and symbol means "is defined by." With reference to the vector
cost functional, A represents a set of scale factors which converts
all of the separate cost functionals to a common dimension. This is
where the operations research entersrelating material expense to
fabrication to sociological considerations and so forthto determine
the metric A. For a vector c whose elements are functionals,
E = r[u] r
where r[u] and r are defined above, the composite cost functional is
J[u] = (E,Ac)
The weakness in this method is that an optimal solution must be
obtained for each individual cost functional prior to attempting a
solution to the composite problem. Additionally, some of the more
abstract cost objectives may be difficult to quantify in a meaningful
manner. Despite these shortcomings this approach does suggest inter
esting applications.
2.2 Methods: Continuous Systems
The problems characterized by mathematical functions, in con
trast to those represented by a set of discrete constants, are normally
treated by variational techniques. Many books on this subject have
been published; the better authors include Elsgolc (1961), Gelfand and
Fomin (1963), Dreyfus (1965), Hestenes (1966), Denn (1969), Luenberger
(1969), and Bryson and Ho (1969). An excellent summary paper is avail
able in Berg (1962).
To see how these techniques are applied, three papers are
recommended. The first is Blasius (1913), which provides sufficient
detail and explanation to make it quite worthwhile. Although it does
include several examples that involve subsidiary conditions, no inequal
ity constraints are treated. An example that includes inequality con
straints to the control variable is contained in Brach (1968). A much
more general application of variational principles is presented in
Oden and Reddy (1974). In this paper a dualcomplementary variational
principle is developed for a particular class of problems. It is shown
that the canonical equations obtained are the Euler1.arange equations
for a certain functional.
2.2.0 Special Variational Methods
Besides the ordinary variational method, more specialized
techniques have been developed to the point where they are recognized
as independent methods in their own right. In the following sections
these methods are identified and a number of representative references
given.
2.2.1 Energy Methods
The oldest of these methods is the energy method. It originated
with the principle of minimum potential energy, and was later extended
to include the representation of eigenvalues through the energy func
tional. A good discussion of the former is available in Fung (1965) or
Przemieniecki (1968); the best general treatment of the latter is avail
able in either Gould (1957, Chapter 4) or in Mikhlin and Smolitskiy
(1967, Chapter 3).
The principle of minimum potential energy is frequently used
with simple problems to prove that a necessary condition for optimality
is also sufficient. Prager and Taylor (1968) contains such a proof
for the global, maximum stiffness design of an elastic structure of
sandwich construction; two papers that also consider this problem are
Huang (1968) and Taylor (1969). Specific application of the energy
method to an eigenvalue problem is demonstrated in Taylor and Liu (1968).
A much more general discussion of the energy method is provided in
Salinas (1968). Further extensions of the method are presented in
Masur (1970), in which the principle of minimum complementary energy
is applied to problems of the optimum stiffness and strength of elastic
structures. In these problems a necessary condition for optimality
is that the strain energy density be constant throughout the structure.
This condition is also sufficient for optimality in certain classes of
structures that satisfy a specific relationship between the strain
energy density and design variables.
Many of the energy problems belong to the class of problems
having quadratic cost functionals. The significance of this character
istic is that the EulerLagrange equations derived from such functionals
are linear.
A more recent energy method development is the concept of
"mutual potential energy." Mutual potential energy techniques resemble
those of the principle of minimum potential energy. In both methods
a cost functional is defined over the entire domain occupied by the
structure and optimized with respect to the control variable. If it
is desired that the optimal solution be required to have a specified
deflection at a certain point, this condition corresponds to a sub
sidiary state constraint when using the principle of minimum potential
energy. The mutual potential energy method incorporates this type of
localized constraint into the cost functional which is defined over
the entire domain of the structure. By itself this alone is advanta
geous; however, for certain types of problems the mutual potential
energy method also provides both a necessary and sufficient condition
for global optimality. In the way of a critical comment, either the
method has received too little attention, or else it does not effi
ciently handle problems more difficult than the simple examples presented.
Four papers that are representative of the literature associated with
this method are Shield and Prager (1970), Chern (1971a, 1971b), and
Plaut (1971c).
Another recent development is that the class of problems for
which energy methods is applicable has been expanded to include cer
tain types of nonconservative systems. Together with the mutual poten
tial energy concepts, this suggests that perhaps the classical energy
method is a special case of a more generalized method. If a technique
can be developed which uses the adjoint variables to transform a gen
eral nonconservative system into an equivalent selfadjoint form, the
method might be deduced. Some papers pertaining: to the subject are
Prasad and Herrmann (1969), Wu (1973), and Barston (1974).
2.2.2 Pontryagin's Maximum Principle
There are many textbooks which derive, explain, and give examples
for the maximum principle. The original (Pontryagin et al., 1962)
requires a knowledge of functional analysis. A condensed form of this
same material is available in Rozonoer (1959). Denn (1969) provides
another point of view in which the principle is derived from Green's
functions. In this manner, the sensitivity to variations is readily
observed. To understand Denn's treatment requires only a knowledge of
the solution of differential equations.
Shortly after Pontryagin's book was published, many papers
devoted to the theoretical aspects of the maximum principle were pub
lished. Some of the more readable ones are Kopp (1962, 1963), Roxin
(1963), and Halkin (1963). Another early paper (Breakwell, 1959)
appears to be a completely independent derivation of the maximum prin
ciple. Although quite general in the mathematical sense, the examples
presented are trajectory optimization problems and not a general type
of mathematical problem. This may be an explanation for what seems to
be a lack of recognition for a significant achievement.
The application of PMP to problems in structural optimization
is relatively recent. When the method is used, one of two difficulties
is often encountered. The first is errors in the formulation of the
optimal control problem; the second is that once a wellposed, non
linear twopoint boundary value problem (TPBVP) is obtained, it is dif
ficult to solve. An example of the first is provided by Dixon (1968)
the correction was given in Boykin and Sierakowski (1972). De Silva
(1972) provides a clear presentation on the application of PTIP to a
specific problem, but includes no data because a solution could not be
obtained. Despite the failure to determine the solution, this paper
is worthwhile for its lucid discussion of the PMP application. Another
paper that gives a good specific application of PMP is Maday (1973).
Although much analysis is presented very little is said regarding the
solution techniques.
All of the above references are applicable only to systems
that are described by ordinary differential equations, in contrast with
the calculus of variations which also handles problems described by
partial differential equations. Since many of the problems of mathe
matical physics involve partial differential equations, an extension of
PMP to include this class of problems is the next logical development.
Some work has already been done, for example, Barnes (1971) and
Komkov (1972). A survey of these "distributed parameter systems"
see Section 1.0is presented in Wang (1968).
2.2.3 Method of Steepest Ascent/Descent
This method is frequently cited in the literature for trajec
tory optimization, and occasionally in references related to optimal
structures. When the method of quasilinearization converged for the
dissertation example problems, there was no need to investigate other
methods such as the method of steepest ascent. Consequently, little is
said about it. According to the references, it is applied in a straight
forward manner. Furthermore, the example problem solutions presented
seem to be real problems and not academically simple. The following
four papers treat the method in general with trajectory optimization
applications: Bryson and Denham (1962, 1964), Bryson et al. (1963), and
Hanson (1968). In Haug et al. (1969), the method of steepest ascent is
derived in detail, completely discussed, and compared to the maximum
principle. Several structural optimization problems are then solved
by the method of steepest ascent. Although no exciting results are
obtained, use of the method is clearly illustrated by the applications
to realistic structural problems.
2.2.4 Transition Matrix:
Aeroelasticity Problems
For the past few years a group at Stanford University has studied
the optimization of structures subject to dynamic or aerodynamic con
straints. The general problem of their interest is that of minimizing
the weight of a given structure for specified eigenvalue, subject to
inequality constraints on control.
Three types of solution techniques are used once the necessary
conditions for minimum weight are determined. Exact solutions are
obtained for most of the problems because they are so simple that
analytical methods are applicable. More complicated problems are
solved by a "transition matrix" method described in Bryson and Ho (1969).
On the basis of convergence difficulties reported in the references,
this method should be used with caution. Results have been obtained
only for very simple problems. However, these results are corroborated
by data obtained from a discrete approximation method. Five papers
that are representative of this work are McIntosh and Eastep (1968),
Ashley and McIntosh (1968), McIntosh et al. (1969), Ashley et al. (1970),
and Weisshaar (1970).
2.2.5 Miscellaneous Methods
The preceding sections have briefly outlined the methods of
structural optimization most frequently encountered in the literature.
Appropriate, representative references have also been given. Not all
methods are listed; while some are omitted for not being generally use
ful, others are omitted for not being generally used. Two examples of
the latter are the "modified quasilinearization" and "sequential gradient
restoration" algorithms described in Miele et al. (1972) and in Hennig
and Miele (1972). At some later time these methods may be acknowledged
as major methods that are applicable to many different or important prob
lems, but for now they are mentioned only in passing.
2.3 Methods: Discrete Systems
Discrete systems are described by a set of discrete constants
instead of the set of functions associated with continuous systems.
The classic example of a discrete system is a pinconnected truss, where
the state x. and control u. are the stress and crosssectional area,
1 1
respectively, for each member i. A discrete system also arises in the
approximation of a continuous system.
Several references that present a good discussion of general
methods applied to discrete systems are available. Most of these exist
in the form of an edited collection of papers by various authors on the
topics of their acknowledged expertise. Four such publications are
Gellatly (1970), Gellatly and Berke (1971), Pope and Schmidt (1971),
and Gallagher and Zienkiewicz (1973). Another report, Melosh and Luik
(1967), provides a good exposition of the difficulties associated with
the analysis portion of least weight structural design. It also con
tains a brief comparison of various mathematical programming methods.
McNeill (1971) is the last reference to be cited in the section
on general methods for the optimization of discrete systems. Minimum
weight design of general structures is treated in a mathematically
precise formulation. Legendre's necessary condition is combined with
the concepts of convex functions and sets to derive the necessary and
sufficient conditions for global optimality. Fully stressed designs
and constraints to eigenvalues are also discussed. In summary, this
paper provides a good example of the general mathematical problem that
must be solved in the optimization of discrete systems.
While certain variational methods may be applied to discrete
systems, the most frequently used technique is mathematical programming.
In the following sections, this method and other major methods are dis
cussed and representative references cited.
2.3.0 Mathematical Programming
The general method of mathematical programming is discussed
in section 3.2 of the dissertation, and the solution of an example
problem using this method is detailed in Chapter VL. In the literature
related to this subject, a very readable textbook is availableFox
(1971). This book complements the theory with numerous discussions
pertaining to numerical techniques and methods that can be employed to
overcome certain difficulties that may arise. Although it does contain
flowcharts of several algorithms, there are few specific examples given.
For a discussion of the general theory, two alternatives to this book
exist in the form of papers: Schmidt (1966, 1968). The first is
written in a conversational style, contains no mathematics, and is
intended to provide only a general description of the subject. The
latter paper is theoretical in content.
An excellent application to a realistic problem is to be found
in Stroud et al. (1971). Ths paper contains little discussion of the
method itself, but does demonstrate an application that allows a concep
tual visualization of the solution. The approach is to assume the
solution to be a linear combination of specified functions, and to
choose the weighting coefficients to minimize the cost. Mathematical
programming is employed to determine the optimal set of coefficients.
This approach resembles Galerkin's method, and though not mathemat
ically rigorous, it may provide a useful approximation to large,
unwieldy problems.
2.3.1 Discrete Solution Approximations
In the previous section a paper is cited that contains an
approximate solution obtained by Galerkin's method. The use of the
Galerkin or RayleighRitz approximate solution techniques is suffi
ciently widespread to be considered a general method. For both methods,
the solution is assumed to be a linear combination of the solution to
the linear part of the governing equation and a set of prescribed func
tions. This approximate solution does not satisfy the given equation
exactly but produces some residual function. A cost function that
depends upon the residual is then minimized with respect to the unknown
coefficients. The two weighted residual methods mentioned above have
different cost functions, but the methods are identical for linear
equationssee Cunningham (1958, p. 158).
The advantage to using these methods is that after assuming
the particular form of solution, the problem of solving for the weight
ing coefficients may be much simpler than the original problem. In the
case of Stroud et al. (1971), the coefficients were obtained by mathe
matical programming techniques. However, the weakness of the method
is the restricted function space of possible solutions. With the coef
ficents obtained by these methods the resulting solution is the best
approximation that is possible from the set of solution functions
prescribed. There is no guarantee that the approximation even resembles
the true solution.
The flutter of a panel is solved using Galerkin's method in
Plaut (1971a). No general developments are presented and the assumed
solution functions are trivially simple. However, this paper does
provide an application of the method to obtain an approximate solution
to a very difficult optimization problem involving the stability of
a nonconservative system. A similar problem is treated in a more theo
retical manner in Plaut (1971b) using a modified RayleighRitz method.
"Segmentwiseconstant" control functions are assumed also; this partic
ular approximation is discussed with more detail in the following sec
tion. Additional nonconservative problems are treated in Leipholz
(1972), applying Galerkin's approximate solution to the energy method.
Several simple examples are included.
Nonconservative elastic stability problems of elastic continue
are treated in Prasad and Herrmann (1969) using adjoint systems. This
approach is more realistic than the segmentwiseconstant control assump
tion described in the following section. Solutions for the state and
adjoint system are assumed, such that approximation process resembles
the RayleighRitz method. However, only a single type of nonconserva
tive system is considered. Extension to several other types of noncon
servative elastic continue problems is given in Dubey (1970). Varia
tional equations corresponding to both the Galerkin and RayleighRitz
methods are derived. Furthermore, the condition for equivalence of the
two methods is shown to be that the admissible velocity field must
satisfy a natural boundary condition over that portion of the body's
surface where tractions are prescribed.
2.3.2 SegmentwiseConstant Approximations
The definitive characteristic of this method is approximating
the structural system by a number of discrete segments, where within
each segment the control function has a constant value. In general,
the constant value of the control differs from segment to segment. For
the many papers on this method that have been published, the procedure
is the same. An optimality condition (necessary in all cases but also
sufficient in some) or cost functional is derived for the continuous
system. After defining the segmentwiseconstant approximation, the
condition or functional is reformulated in terms of the discrete param
eters. Most of the papers use so few elements that solving for the
discrete values of the control parameter poses no difficulties. Although
this method does simplify the mathematical problem to be solved, the
crudeness of the approximation is not appealing. Five papers which
treat a variety of problems using this approximation are cited below.
Minimum weight of sandwich structures subject to static loads
is discussed in Sheu and Prager (1968a). In Sheu (1968) the same type
of structure is considered. It differs from the first problem by
requiring point masses to be supported such that the total structure has
a prescribed fundamental frequency of free vibration. Icerman (1969)
treats the problem of elastic structures subject to a concentrated load
of harmonically varying amplitude. The minimum weight design is obtained
subject to a compliance constraint related to the applied load, and
which is effectively a boundary condition on displacement at the point
of application. A truss problem is also included.
The concept of a compliance constraint is pursued further in
Chern and Prager (1970). The minimum weight design for sandwich con
struction beams under alternative loads is found, subject to this type
of constraint. The paper uses up to eight segments, thereby obtaining
a more realistic approximation to the continuous problem. Minimum weight
design of elastic structures subject to body forces and a prescribed
deflection is discussed in Chern (1971a). This investigation is no
table in that it considers applied loads that are functions of the design
functions.
2.3.3 Complex Structures with
Frequency Constraints
On the basis of useful application, perhaps the most important
class of discrete structural optimization problems is the minimum weight
design of complex structures subject to natural frequency constraints.
Since most real structures are built with many structural elements of
various types, and are not realistically described by any single type,
this approach is more appropriate from the aspect of modeling the struc
ture. Furthermore, many structures must be designed to avoid certain
natural frequencies because of resonance or selfinduced oscillations;
this situation indicates that the natural frequency constraint is also
appropriate.
Many different solution schemes have been developed which are
usually based upon general mathematical programming techniques.
Typically, a design is iteratively altered to minimize the weight with
a subsequent increase in frequency until a constraint is violated.
At that point the design process uses an iteration which simultaneously
reduces both weight and frequency. These two processes are repeated
sequentially until no further weight reduction is possible.
Although circumstances may require the use of many elements,
the number of them may itself be a critical factor. Some of the schemes
require a matrix inversion as part of the eigenvalue problem solution
associated with the frequency constraint. If the number of elements
becomes too large, the size of the matrix to be inverted likewise be
comes excessively large. When that occurs the matrix inversion can
require excessive amounts of computer time. Another possible difficulty
is that the inverse matrix itself is not sufficiently accurate, such
that the subsequent calculations are not acceptable. However, for
structures such as reinforced shells composed of different types of
structural elements, this method may be the most applicable.
Many papers have been published pertaining to this class of
structural optimization problem. Because the method is inherently
oriented towards applications, the references are cited in chronolog
ical order without additional comments. Interested readers are referred
to: Turner (1967), Zarghamee (1968), Turner (1969), De Silva (1969),
Rubin (1970), Fox and Kapoor (1970), McCart et al. (1970), and Rudisill
and Bhatia (1971).
2.3.4 Finite Element Approximations
There is an unfortunate ambiguity to the label "finite elements"
that occurs because these words are used to describe two completely
different entities. In papers cited in the preceding section they are
used to indicate the discrete structural elements of finite dimensions
which comprise the complex structure. The analysis of such systems of
structural elements has been accomplished by ordinary matrix methods
during the last three decades. However, during the past decade another
method has been developed and named "the finite element method."
In this method a continuum is divided into small, finite ele
ments over which a particular form of approximation of either the dis
placement and/or force is assumed. A number of nodes common to one or
more element is prescribed; continuity is required to exist at these
nodes but not necessarily elsewhere. An equilibrium equation is derived
for each element, and then all of the individual equations are combined
into a single equilibrium equation for the entire system. lihc result
ing equation is a linear algebraic equation whose unknowns are displace
ments and/or forces at the nodes. Once the matrix equation is inverted,
the nodal displacements and/or forces are used with the assumed approx
imation form to describe the state of the structure throughout each and
every element, and hence the system. Hereafter this method is referred
to as the "finite element method."
The most frequent application of the finite element method is
to problems having complicated loads, geometry, and response. Generally
speaking, the method is employed wherever the physical system is too
complex to be described adequately by a single differential equation
and boundary conditions. For a complete theoretical development of the
finite element method and numerous examples, see Zienkiewicz (1971).
With respect to structural optimization the method is employed
to simplify the problem to be solved. Very little has been published
on this subject, but the papers available cover a wide spectrum of tech
niques. For example, Dupuis (1971) combines the finite element and
segmentwiseconstant methods as applied to minimum weight beam design.
A similar application to column buckling is contained in Simitses et al.
(1973). Another paper, Wu (1973), is a study of two classical noncon
servative stability problems. Although adapted to stability consider
ations, this presentation is the best exposition available in the open
literature.
In Chapter VI a minimum deflection beam problem is solved with
the combined methods of finite elements and mathematical programming.
2.4 Closure
In the preceding sections of this chapter, general problem
types and methods are discussed. Only those methods that appear to
have attained some standard of acceptance are presented. It must be
acknowledged that other areas of important study exist but are perhaps
overlooked as not being pertinent to the general subject area of the
dissertation. As an example, Dorn et al. (1964) treats the optimal
layout of trussesan important subject but not related to the general
problem to be considered in this dissertation. In addition only
elastic structures have been considered although there are numerous
publications on optimal design of inelastic structures. References
that are representative of this subject are: Drucker and Shield
(1957a, 1957b), Hu and Shield (1961), Shield (1963), Prager and Shield
(1967), and Mayeda and Prager (1967).
On considering the various references mentioned above it would
appear that there are two possible pitfalls in structural optimization
that should be avoided. The first is the confusing of method of opti
mization with the solution techniques employed to obtain a solution to
the resulting TPBVP. In order to avoid possible errors the two should
be dealt with independently, unless it is clearly advantageous to
relate one to the other. Besides this it must be recognized that any
solution obtained is "optimal" only with respect to the given condi
tions of the particular problem. Any change in the problem statement
invalidates the applicability of that solution. The change may lead to
a more desirable solution, but the original solution is no less valid.
Simitses (1973) is an example where this situation is not acknowledged.
In this paper the thickness of a thin reinforced circular plate of spec
ified weight and diameter is determined such that the average deflec
tion due to a uniform load is minimized. An earlier paper which did
not include stiffening is cited with the implication that the optimum
solution for the unstiffened plate is not correct. The point made above
is that both of these solutions are optimum under the respective condi
tions of the two problems. Neither solution is more, or less, valid
than the other.
CHAPTER III
THEORETICAL DEVELOPMENT
3.0 Introduction
This chapter contains the development of two distinct methods
used in the theory of optimal processes, into a more general method.
The first section defines precisely the problem to be considered.
This includes the necessary conditions for an optimal solution given by
the calculus of variations. Several mathematical programming techniques
are described in the second section along with a numerical algorithm
called the gradient projection method. The application of this numer
ical method to the solution of the necessary conditions from Pontryagin's
Maximum Principle (PMP) is detailed in Section 3.3. Results of this
approach are shown to be consistent with the necessary conditions, given
in Section 3.1; these results provide a clarifying insight to the math
ematical processes entailed in the maximum principle, and an explicit
formulation for the Lagrangian multiplier functions. This explicit for
mulation is used in the next section to show the necessary conditions
may then be regarded as an algorithm. The final section contains a brief
summary of solution methods.
The main theoretical development of the dissertation is contained
in the first three sections. It is well known that the problems encoun
tered in the calculus of variations are equivalent to the optimization of
a functional (in the sense of mathematical programming problems) under
certain restrictions upon the variations. A good exposition of this is
available in Luenberger (1969). With this equivalence in mind, it is
noted that the PMP is itself worded as a constrained optimization prob
lem. When treated with what is normally regarded as a numerical method,
the gradient projection method, an explicit formulation of the atten
dant Lagrangian multipliers is obtained. This form satisfies all of the
calculus of variation necessary conditions and allows one to use them
in a most straightforward fashion. As a result, these necessary condi
tions may be directly used in the form of an algorithm to obtain a solu
tion. Furthermore, it is believed that treating the PMP as a mathemat
ical programming problem in conjunction with the gradient projection
method helps to explain the effect of combined controlstate constraints
upon the maximum principle.
3.1 Problem Statement and
Necessary Conditions
A general problem which represents a large class of structural
optimization problems is treated in the sequel. The functional
tF
J = / L0(x,u) dt (3.1.1)
0
is to be minimized with respect to the control u(t) where the state
x(t) must satisfy certain boundary conditions and a differential con
straint; in addition, an inequality constraint involving both the state
and control must be satisfied. For
uT(t) = [u (t) u2(t) ... u (t)] (3.1.2)
 1 2 m
x (t) = [xl(t) x2(t) ... x (t)] (3.1.3)
the subsidiary conditions to minimizing the cost function J are:
x = f(x,u) (3.1.4)
Specified Boundary Conditions on x(t) (3.1.5)
(x,u) < 0 = l,...,q (3.1.6)
Terminal time tF is considered to be constant; allowing it to be
unspecified requires only a slight modification to the following
derivation.
This problem is a particular form of a very general one treated
by Hestenes (1966). His results are a set of necessary conditions which
must be satisfied by the optimal solution and include the maximum prin
ciple. To obtain the necessary conditions, the inequality constraints
are converted to equality constraints in the manner of Valentine (1937).
These constraints and the differential constraints are then adjoined to
the cost function via Lagrangian multiplier functions ,j(t) and Pi(t)
respectively.
(x,u) + s2(t) = 0
where the slack variables s (t) are defined such that
The symbol denotes "is defined by."
The symbol "= denotes "is defined by."
A 1 ]
s(t) = [(P(x,u)] 0 0
tF t
J = J Lo(x,.u)dt + p (t) [xf(x,u)] dt +
0 0
tF
+ f P,(t) [PQ(x,u) + S2(t)] dt
0
Implied summation convention is used whenever a vector formulation leads
to possible ambiguities in later developments. Integrating the second
integral by parts gives a result that leads to the variational Hamiltonian.
t t
=T. TF dF
J = P +/ [Lop + x 2] dt
0 0
Define:
A T
H(x,u,2) = L (x,u) p (t) f(x,u) (3.1.7)
the variational Hamiltonian, and H which will include terms arising
from the inequality constraint.
H = H H
or
H = (t) f(x,u) Lo(x,u) Zi(t) pk(x,u)
Hence
J = T x F [H* + x pIs ] dt
0 0
With the e'.c:eption of the maximum principle, all of Hestenes' necessary
conditions are obtained from the requirement that the first variation
of the cost function vanish. In the following, "6x" designates
47
"the variation of x"; a subscript vector designates the partial deriva
tive with respect to that vector, with the result itself a column vector.
Thus,
tF t
T
6J = p 6x f
0 0
[x (H* ) +
x
T *
+ 6u H 2s6s] dt = 0
u k z k
To derive the PMP requires an extensive mathematical development and is
not included since it contributes nothing to the present discussion.
However, the necessary conditions are listed in order to be available
for later reference.
x = H = f(x,u)
P=Hx
0 = Px t
1 1
0 = p.6x. t
Specified Boundary
= 0
 i
= t o
tF
Conditions on x(
0 =H
U
0 = uz(t) p((x,u) P(t) ( 0
H(x PT' ~PT' ) H(OpT' u, e)
The optimal solution must satisfy these six conditions together with the
inequality constraint (3.1.6).
The PMP states that along the optimal trajectory, each instant
of time t, state XOPT(t) and adjoint state p(t), treated as fixed, the
optimal control UPT(t) is that admissible control which minimizes the
variational Hamiltonian. In the present context, admissibility requires
that u(t) be piecewise continuous, the set of admissible controls being
denoted by Q. Hence the PMP indicates that
u (t) = ARGMIN [H(x O, u p)] (3.1.8)
Notice that the necessary conditions suggest nothing about how
a solution is obtained, but merely indicate certain functional relation
ships that must be satisfied. However, equation (3.1.8) seems to inti
mate that solution of the necessary condition of P!1P involves a mathe
matical programming problem.
3.2 Mathematical Programming:
Gradient Projection Method
Having shown that the PMP from the calculus of variations
approach to an optimization problem may perhaps be related to a mathe
matical programming problem, the latter will be discussed in general
terms. Consider a nonlinearly constrained optimization problem
xo = ARGECMIN [F(x)]
xE~
subject to
gj(x) 0 j = l,...,m
where Q denotes the set of admissible state components x., i = l,...,n,
and to be admissible requires only the satisfaction of the m inequalities.
Necessary conditions which xOPT must satisfy are given in the KuhnTucker
theorem:
(i) constraints are satisfied gj (xPT) < 0
(ii) multipliers exist such that X. 0
and for all j = l,...,m AXg (xPT) = 0
m
(iii) and VF(x T)+ E V g(x ) =0
Sj=l 
Observe that if I denotes the set of indices associated with active
constraints, the first two conditions may be written as
j e IA g.(x) = 0 and A. > 0
j IA gj.(x) < 0 and X. = 0
A J J
Fox (1971, pp. 168176) presents a very readable proof of this theorem;
a more mathematical proof using vector space concepts is available in
Luenberger (1969).
Many methods for obtaining a numerical solution to the nonlinear
programming problem described by the first two equations of this section
have been developed. The gradient projection method by Rosen (1960) is
used frequently in structural optimization. Basic to the method is the
orthogonal projection of the cost function gradient onto a subspace
defined by the normal vectors of the active constraints. An inherent
part of the algorithm is the concept of a "feasible," "usable" direction.
Any direction d is feasible if an increment x in that direction improves
the cost function, i.e., decreases F(x). Direction d is said to be
usable if it also satisfies the constraints. As long as a feasible,
usable direction exists, the cost function may be improved. A constrained
optimal solution xPT occurs at that point where no feasible direction
is also usable, i.e., any attempt to improve the cost violates a con
straint. In Appendix B these concepts are used in a concise proof of the
KuhnTucker conditions.
Fox (1971) derives the matrix P which projects the cost function
gradient into the subspace defined by vectors normal to the active con
straints. This is equivalent to subtracting all components parallel to
vectors that are normal to surfaces of active constraints from the nega
tive gradient of the cost function. Recalling the definition of set IA,
consider r constraints to be active such that
IA = {al ,a2... ,}
Define a vector whose elements are the corresponding nonzero
Lagrangian multipliers, and another vector whose elements are the active
constraint functions
A1 = [\ \ ... \ ]
1 C2 r
T
N [g g, ... g]
1 2 r
From the N vector, a matrix N is introduced, each column of xhiCh is the
gradient of an active constraint. Hence, N is an (nx r) matrix where
T
N N = [N..] i = 1,...,n
and (3.2.1)
N = = j = 1,. ,r
ij 1x. x.
1 1
With these definitions of A and N, the third KuhnTucker condition can
be written as
V F(x) + N X = 0
At any feasible point x where g (x) : 0, the direction which best
improves the cost function is the negative gradient of the cost. If
those directions which lead to constraint violations are subtracted from
V F(x), the projection matrix P is obtained. Directions causing a
constraint violation are specified by the gradients of active constraints,
T
i.e., the columns of N What is required of S, the projection of the
x
gradient, is that
S = (V F(x)) Nx A (3.2.2)
x x
where A are scalar coefficients to be determined such that S is ortho
T
gonal to each column of N or
x
(N)T S=0
(N )r S = 0
x
T
When the matrix equivalent to N is used together with the S expres
x
sion (3.2.2), the result is
N (V F(x) N X) = 0
such that the X which satisfies this orthogonality condition is:
A = (NTN)1 NT(V F(x)) (3.2.3)
Unless the active boundary surface normals V xg(x) are linearly depen
dent, the matrix (N N) is nonsingular. Conversely, if this matrix is
singular the active constraints are not linearly independent; however,
this is not a condition encountered in most real cases.
Substitution of the X expression into the S equation leads
directly to the projection matrix P:
S= P V F(x)
X 
where
P = I N(NTN)1 NT (3.2.4)
where I is the identity matrix. The direction S which best improves
the cost is given in terms of P, where P and N are given by (3.2.4)
and (3.2.1). If no constraints are active at a point x, then N is a
null matrix, P reduces to an identity matrix, and the direction of best
improvement is coincident with the direction of steepest descent.
In the algorithm associated with this method the starting point
must be a feasible point where g.(x) : 0 for all j = l,...,m. The
design then proceeds in the S direction until the solution is satisfied
to within a specified position tolerance E. Necessary conditions
generally programmed in a computer program are:
Is. i i = 1,...,n
A. > 0 j A
A. = 0 j I
It is readily seen that for S, P, and A defined as above, these are
completely equivalent to the KuhnTucker conditions.
3.3 Gradient Projection Method Applied
to the Maximum Principle
Based upon the preceding discussion, the similarity between PMP
and themathematical programming problem can be discussed. The maximum
principle states that the optimal control uOPT minimizes the variational
Hamiltonian with respect to all admissible u. Or, at each time
0 t t tF, uPT minimizes H(x,u,p) with respect to u for given x and p
and where OPT is subject to constraints 4 (x,u) < 0, k = l,...,q.
Treating this as a mathematical programming problem, the following
correspondences are recognized
x U
F(x) % H(x,u,p)
gj(x) % (x,u)
j v<(t)
V F(x) 'b H
x u
S H plus constraints
u
Continuing to identify corresponding quantities, at each time t, let
IA denote the set of active constraints, taken to be r in number.
IA = {al, 02' ...' r }
Then NT T = [ U ..1. ** ]
T T
A' = [) (t) PQ (t) ... (t)0
1 2 r
N T H j] = Jj = l,...,n
3 j = 1,... ,m
54
Furthermore, define (H ) to be the gradient of H with respect to u
where all components that cause a constraint violation have been removed.
Since projection matrix P removes cost function gradient components that
lead to constraint violations, consider its use in the maximum principle.
T
With the correspondent to N identified as then
U'
P = I Tu (3.3.1)
u  u
and
(Hu) = PHu (3.3.2)
From the KuhnTucker conditions, this implies that along the
optimal trajectory (t,XOPT, u OPT)
= (Hu)p = 0 (3.3.3)
Similarly, at each time
g(t) ( T ) T H (3.3.4)
T T u
u u 
from which it follows
(. T )) + H T Hu = 0
u u 
T
u + Hu = 0
u
(H + _To)u = 0 (3.3.5)
( H ) 0
Or, H = 0 (3.3.6)
u
Hence the control law from Hestenes' necessary conditions can be derived
from the PMP condition by treating it as a nonlinearly constrained math
ematical programming problem. While using the gradient projection method
in the derivation, it is seen that equation (3.3.5) is equivalent to the
third KuhnTucker condition. The second KuhnTucker condition is iden
tical to Hestenes' necessary condition on the Lagrangian multipliers used
to adjoin the inequality constraints to the cost function. Satisfaction
of the inequality is implied by requiring the first KuhnTucker condition
to be fulfilled, where
X. > 0 i(t) > 0 (3.3.7)
Sj g(x) =0 + (t)>(x,u) = 0 (3.3.8)
g.(x) W 0 z 4 (x,u) : 0 (3.3.9)
Thus by treating the solution of the necessary conditions of the max
imum principle as a programming problem with inequality constraints,
using the gradient projection matrix, and by requiring satisfaction of
the KuhnTucker conditions, an explicit formula for Hestenes'
Lagrangian multiplier functions has been derived. It is further demon
strated that with the u (t) so defined satisfaction of the extremum con
trol law condition is implied. However, before this treatment can be
accepted as valid, it must also be shown that the system of canonical
differential equations is unchanged.
Consider the state system equations
x = H = f(x,u)

where
H = T (t)f(x,u) L (xu) (t) (x,u)
It is obvious that the explicit form of p(t) has absolutely no effect
upon the state system equation expressed in canonical form.
Demonstrating that the adjoint system equation is unchanged
requires the method described by Bryson et al. (1964). Consider the
general problem of Section 3.1 again, but with only the differential
constraints adjoined to the cost function, i.e.,
tF t
Min {J = px + f (H xT)dt} (3.3.10)
u 0 0
subject to: pk(x,u) < 0 9 = l,...,q (3.1.6)
where H(x,u,p) = L (x,u) T (t)f(x,u) (3.1.7)
and x = f(x,u) (3.1.4)
Again let IA denote the set of indices associated with r active
constraints at any time t
IA = {al a, ...,ar
4 (
k(x,u) = 0 E IA A> v(t) > 0
and p is defined as before
T
1 a 2 r
The problem can then be thought of as minimizing (3.3.10) subject to
p(x,u) = 0
While on the constraint surfaces defined by this equation the variations
in control 6u(t) and state 6x(t) are not independent but instead are
related through the subsidiary requirement that
6O(x,u) = 0
or
Sx 4 (x,u) + 6uT T(x,u) = 0 (3.3.11)
  __U 
This imposes a restriction to the admissible variations. For cost func
tion (3.3.10) to be a minimum, it is necessary that its first variation
vanish, i.e.,
tF tI
6J = 6x + f 6xT H + 6u H 6xT] dt = 0 (3.3.12)
0 0 u
It has already been shown that
T
H = (H + ) = 0
u u
which will be used to advantage shortly, after having added and sub
T T
tracted the term 6u (1u 4T) from the integrand of (3.3.12).
+ 6UT RT1 6uT(u
t t
T F ^F
0=E6x + f 6xT(H uTH +
+ uT ~( )u ~T (T4) )udt
Rearranging terms gives
tF tF
T F F
0 =px +f pxT(H i) +
0 0
T T T T).dt
+ 5u (H + (j 4i)u) cu (ii) dt
58
But,
T T *
(H + (iT) = (H + I )u =(H) =0
0
and
T T
(Ti )u =
Hence,
0 = T tF + F 6x T(H ) u dt
0 0x 
It is here that the restrictions imposed by the active constraints are
applied; from (3.3.11)
T T T T
Su t = 6x x
such that
T~6x + F _xT(Hx ) + x _A dt
0 0
t t
0 = x + f/ 6x (H p + ) dt
+f x x
0 0 
Applying Euler's lemma, for arbitrary variations in the state which
satisfies the constraints,
(H + P) =
which by the following manipulations is shown to be the adjoint system
equation of Hestenes.
i = + (T)x
x
(H _Tx
p=H
x
Thus, the explicit formulation for pL(t) obtained by applying the gradient
projection method to the PMP satisfies all the necessary conditions of
Hestenes.
It may happen that in some cases the constraint upon control
does not depend upon the state. It can be shown that the y(t) explicit
formulation is equally valid in this instance. By examination of
equations (3.3.1) through (3.3.9) it can be verified that all the neces
sary conditions except the adjoint system equation are satisfied. To
demonstrate the latter, recall that when on a constraint boundary the
first variation of both the cost functional and the constraint function
must vanish. That is, for
K(u) = 0
both
J = 0
and
65 = uT 6 T = 0 (3.3.13)
u
To derive the desired equivalence, the same term must be added and
subtracted from the integrand of 6J as before, again arriving at the
result
t t
T I / T T T
0 = p 6x, + f Fx (H P) 6Tu I dt
0 0 0
When the constraint variation (3.3.13) is introduced into this last
equation, then by Euler's lemma
(Hx p) = 0
A
Since it was stipulated that (u) is not a function of x, the equation
may be written
T
(H +1 4)_ = 0
SH )
x
Thus, the expression for I(t) is valid when the constraint inequality
depends only upon the control u(t).
3.4 Maximum Principle Algorithm
In the introduction to this chapter it was stated that the
Lagrange type problem from the calculus of variations is equivalent to
an ordinary mathematical programming problem based on the KuhnTucker
conditions. Furthermore, when inequality constraints are present the
necessary conditions are equivalent to the KuhnTucker conditions.
It was demonstrated in the preceding section that if the PMP is itself
treated as a mathematical programming problem, application of the
gradient projection method provides an explicit solution for the
Lagrangian multipliers associated with active constraints. This explicit
solution for P(t) also satisfies all of the other necessary conditions
for an optimal solution. The ability to determine i (t) explicitly
in terms of parameters and functions that describe the problem suggests
the possibility of converting the necessary conditions of an optimal
solution into an algorithm for obtaining it.
Ensuing discussion of the algorithmic form of the necessary
conditions contains the implicit assumption that all equations are valid
along the optimal trajectory. It is further assumed that the problem
under consideration is that one described in equations (3.1.1 3.1.6).
The algorithm requires that x(t) and p(t) be known at each time
0 ( t < tF for which the solution procedure is as follows.
(i) Use PMP on the variational Hamiltonian to determine an optimal
control u (t) independent of constraints.
u (t) = ARGMIN [H(x,u,p)]
Evaluating the inequality constraints with u = u reveals which of
the k = l,...,q constraints are active. Let r denote the number
of active constraints and IA the set of indices associated with
them.
IA = {a a2, ., a }
9 (x,u) = 0 C IA
P (x,u) < 0 I1 IA
From this the vectors whose elements are the nonzero Lagrangian
multipliers and corresponding constraint functions are defined,
respectively, at the instant of time t.
T
PT(t) = [u J ... ]
a 1 42 r
T (x'u) = [ a1 a2 a' a I
(ii) Having identified which of the q constraints are active, r
components of OPT are specified by _(x,u) = 0. They may be
solved by using the implicit function theorem, which requires
T to be of rank r. This in turn requires the r constraints
Iu
which are active at point x(t) to be linearly independent.
To determine the remaining (m r) components of uOPT requires
that be known at time t, but
(t) = ( T )1 T Hu
T  T u
u U 
This value of is used to determine the "constrained" Hamil
tonian,
T
H = (H + P )
*
(iii) With the nonzero Lagrangian multipliers V known and H conse
quently defined, the remaining (mr) unknown components of
OPT are determined from the control law for the constrained
system, i.e,, from
H =0
u
Once uOPT is completely known, the adjoint system equations
are determined by
x
The process outlined above then allows uOPT to be written as
u = ARGMIN [H(x Tu,)
OPT OPT
since the u obtained in this fashion satisfies (x,u) < 0 which is the
only requirement for being admissible. However it must be recalled
that these equations are valid along the optimal trajectory; it remains
to be shown that this algorithm may be employed in some manner to obtain
that optimal trajectory and to demonstrate their satisfaction along it.
3.5 Solution Methods
Necessary conditions from the calculus of variations provides
a Two Point Boundary Value Problem (TPBVP) to be solved, which is in
general, nonlinear. For all but the most simple problems no analytical
solution is possible and if any solution is to be obtained a computer
must be used with some numerical method. A discussion of the available
methods and their relative advantages/disadvantages is not included here
due to the availability of such discussions in the literature, e.g.,
Bullock (1966). All of the methods involve some iterative scheme, and
for optimal control can be separated into two general categories.
(i) Indirect methods. Schemes which require an initial guess of
the state's solution: In these methods the starting point is
an initial guess of the time history of the solution. The con
trol associated with the solution is a subsequent calculation.
Iteration continues until the state satisfies some criterion
connoting convergence; the final control history at conver
gence is the optimal control.
(ii) Direct methods. Schemes which require an initial guess of the
control function: The starting point for these methods is an
initial guess of the control time history. For this class of
methods the state associated with the control is a subsequent
calculation. Iteration continues until the control satisfies
some convergence criterion.
The method of quasilinearization was selected, based upon the
success of Boykin and Sierakowski (1972) in applying it to a constrained
structural optimization problem. Excellent convergence for their
problem, the capability to handle nonlinear systems, and the avail
ability as an IBM SHARE program, ABS QUAS1, dictated its selection.
In the application to the examples in Chapters IV and V the program
required no modification. As a result, a detailed discussion of the
method of quasilinearization is not included.
The problem discussed in preceding sections of this chapter
falls into the general class of problems that QUAS1 handles, that is,
Y = (Y,t)
with the boundary condition of the form
B Y(O) + B Y(t ) + C = 0
k r F Q
where tF, square matrices B and Br, and vector C are specified,
constant quantities. The specific form of B B and C depend upon the
2 r oQ
given boundary conditions. As described in algorithm form
x = f(x, oPT(xP)) = Gl(x,E)
P = H (x P (x,)) = G (x,p)
x OPT 2
In terms of the general QUASI nomenclature,
= g(Y, t) = _
1G2 (')
Boundary conditions are determined by those specified for the original
system and by the necessary conditions outlined in the first section
of this chapter.
In Chapter VI the problem treated by Boykin and Sierakowski (1972)
is solved by the gradient projection method applied to a finite element
65
formulation for the description of the structural system. This is
a method of the second kind mentioned above. Results of the two methods
are compared.
CHAPTER IV
CONSTRAINED DESIGN OF A CANTILEVER BEAM
BENDING DUE TO ITS OWN WEIGHT
4.0 Introduction
A structural optimization problem has been selected for its
simplicity and stated as an optimal control problem. The maximum prin
ciple is applied, giving a nonlinear TPBVP of the Mayer type. Among the
earliest expository papers on the maximum principle, Rozonodr (1959) gives
an excellent treatment to a similar type of problem; his technique is
used to obtain both the variational Hamiltonian and adjoint variable
boundary conditions. It is shown that no finite solution exists for the
situation of unconstrained control. Numerical solutions for constrained
control are obtained by the method of quasilinearization. Constraints
include both geometric limitations to control as well as maximum stress
limits that become mixed constraints depending upon both state and con
trol variables.
4.1 Problem Statement
A cantilever beam of variable rectangular cross section is to be
designed for minimum tip deflection due solely to its own weight. The
material is specified to the extent that the modulus E and density p are
constants. Length L is specified but the design variables, height h(x)
and width w(x), may be chosen independently of each other, subject to
hard constraints upon the allowable dimensions. That is
a < w(x) < c
(4.1.1)
b < h(x) d d
If Y(x) denotes the deflection of the centerline, the problem is:
given E, p, L, and the constraints, find h(x) and w(x) to minimize
Y(L). The particular form of differential constraints to be satisfied
will be derived in the next section.
4.2 Structural System
Small deflections are assumed in order to use linear Bernoulli
Euler bending theory. Basic conventions assumed for this example are
depicted in Figure 4.1; with these conventions the governing equation
is derived using standard strength of materials considerations. The
result is
EIB(x)Y"(x) = M(x) (4.2.1)
where
L
MB(x) = yf (Tx)w(T) h(T)d
x
1
IB(x) = 1 w(x) h3(x)
and y = pg. Kinematic boundary conditions to be satisfied by the solu
tion of (4.2.1) are:
Y(O) = 0
Y'(0) = 0
/ x h (x)
T
/ w(x)
dW = yw(x)h(x)dx
V (x)
MB (x)
Note: Y(x) is centerline deflection, positive downward.
Structural Conventions
Figure 4.1
Design variables and the related constraints (4.1.1) are put into
dimensionless form such that
h(x)
v= = h + b/d < v < 1
1 d 1
w(x)
v = W a/c s v2 1
2 c 2
IB(x) = 1 cd3 v 2
B 12 1 2
Replacing the independent variable with a dimensionless equivalent, and
using the control components allows the governing equation to be put
into a dimensionless form. For
x
t = 
L
1 E d2
(2)2 u = f (Tt) U1(T)u2(T)dT
yL2 t
where
(4.2.2)
u.(t) = vi(x(t))
':ien constant CB is defined, the usual kinematical relationships for
a beam may be written in a simple dimensionless form; that is, let
1 E d 2 1
C (d) (Units = Length )
B 12 yL2
x2 = CB
31 B 12
d
x3 = CBUU2Y
"l2 = CB 2t (u1u2Y)
u1d2 B (u1u2Y)
dt2
Deflection
Slope
Moment
Shear
Load
These state component definitions are used with the natural boundary
conditions to obtain
x3(1) = 0
x4(l) = 0
From (4.2.2), the state component definitions, and the above boundary
conditions it follows that
Xl = x2
2 = x3/uuu2
x3 = 4
x4 = UlU2
x1(0) = 0
x2(0) = 0
x3(1) = 0
x4(l) = 0
These equations and boundary conditions are used in the following section
to precisely state the problem. The solution and results are given later.
4.3 Unmodified Application of
the Maximum Principle
In terms of the state variables defined in the preceding section,
the problem can be stated with more mathematical precision. Find
Up (t) = ARGMIN [x (1)]
subject to:
subject to:
(i) differential constraints
(ii) kinematic boundary conditions
natural boundary conditions
(iii) hard geometric constraints
x = f(x,u)
xl(0) = x2(0) = 0
x3(1) = x4(l) = 0
b/d : ul(t) 5 1
a/c < u2(t) < 1
According to terminology in the calculus of variations this is a Mayer
type problem. Among the early papers concerning the PMP, Rozonoer (1959)
applies the PMP to a similar problem giving a geometric interpretation
to the function of the adjoint variables.
In Rozonoir's problem the cost is a generalization of the
ordinary Mayer problem, in the sense that the cost function is a linear
combination of the terminal state components. It can be shown via the
calculus of variations that to minimize
where c is
r
J = c x(tF)
r F
a vector of prescribed constants, the necessary conditions are
A T
H = T(t)f(x,u,t)
x= H = f
T
p = H =
0 =H
o = p (0) x(0)
0 = (tF) [r + P(tF)
H(OPT', uOPT' p,t) H(xOPT' u, t)
That is,
Min[J] Max [H]
u u
If we consider the cantilever beam problem, the forms in the
necessary conditions are
c = [1 0 0 ]T
r
H = pl2 + P2X3/U 2 + 3X4 + p UU2
t =1
rx2
x3/uu2
f(x,u)
x
from which
from which
l = 0
P2 = P1
= P2/ulU2
P4 = P3
pl(l) = 1
P2(1) = 0
P3(0) = 0
4(0) = 0
Adjoint variables Pl(t) and p2(t) can be integrated by inspection
Pl(t) = 1
p3(t) = (lt)
0 < t 1
such that
H = x (lt)x3/uu2 + P3X4 + PUU2
(4.3.1)
With this result, the necessary condition for control to minimize tip
deflection is
Hu = 3(lt)x3/uu2 + 2 = 0
H = (lt)x3/u3u2 + p41 = 0
u2 3 12
At first this appears to be a contradiction since the two equations can
be satisfied only by the trivial solution because they have equivalent
forms,
p4u = 3(1t)x3
P4 1 2 3
(4.3.2)
p u = (1t)x
P4UU2 3
Further examination, however, leads to the conclusion that when the
control is completely unconstrained there is no horizontal tangent
plane to the surface H = H(ul,u2).
When the geometric constraints to the control are included,
a constrained minimum may exist. If such is the case, the maximum
value of H occurs on the boundary of admissible control space. To that
end, PMP is employed along the control space boundary to determine
uPT (t) at each time t. Before detailing this procedure, it is neces
sary to first consider some structural aspects of the problem.
By definition the control components are positive, which in
turn implies
Load: 4 > 0 k i4 = u l2
Shear: x4 < 0 k x4 >0 x (1) = 0
Moment: x3 0 x < 0 x3(1) = 0
Furthermore, since p2(t) 5 0 from (4.3.1)
P3 > 0 3 > 0 P3(0) = 0
p4 < 0 4 P4 < 0 P4(0) = 0
This exercise makes it possible to use the information of the sense for
x3 and p4 to simplify the search for OPT on the control boundary. By
arranging the Hamiltonian in the following fashion
H = 2 + 34 + P4[u1U2 (lt)x3/p4ul2]
it is observed that both terms in the bracketed expression are positive.
This and the p4 outside the leading bracket allows the following
equivalence:
Max [H] I Min [ul2 (1t)x3/P4UU2]
uE 9U u eaU
or,
u =PT ARGMIN [$(u)]
where
[(u) = [u2 + F2(t)/uu2 ] (4.3.3)
and
F2(t) = (lt)x3/P4 > 0
At each position t along the beam, the state and adjoint variables must
satisfy the appropriate differential equations, and u is specified by
the preceding three equations.
Control space boundary rU is illustrated in Figure 4.2, where
the ul axis is treated as the ordinate since ul(t) and u2(t) correspond
b/d < u 1 1
a/c < u < 1
/
1
ul
U1
b/d 
0
0 a/c 1
U2
"i2
Figure 4.2 Admissible Control Space
to the height and width, respectively, of the cross section of the
beam at position t.
Along the constant u edges of 3U, let uI = u where u has the
1 1 c c
value of either b/d or unity. If $ (Uc,u2) has a minimum point
dl d2$
= 0 and  > 0
du2 du2
where
where
The value of
D(u ,u ) = u u + F2(t)/u3u
2 c 22 2
du u F (t)/u3u2
d c c 2
du2 c2
2
u2 which satisfies the first
u = F(t)/u2
2 c
Furthermore, it is observed that only one extremum of Q(u) exists along
ul = Uc and that it is a minimum. Hence, either 4(uc,U ) has a minimum
on the constant u1 edge or is monotonically decreasing/increasing. If
either
u 2 a/c
1 i u2
then along the constant ul edge, H has its maximum value at a corner of
the rectangular 3U. On the other hand, if
a/c < u2 < 1
then H has its maximum value on the line u = u interior to the
1 c
endpoints.
condition is
Similarly, along the u2 edges of U, denote u2 = u where u
has the value of either a/c or unity. If $(ul,uc) has a minimum point
de d2,
S0 and 7 > 0
du1 du
where
$(Ulu ) = 1u + F2(t)/u3u
d u 3F2(t) /uu
du1 c 1c
d21
d2 12F2(t)/ulu
du? c
1
It is observed that $(u1,uc) has only one extremum along u2 = uc,
it is a minimum, and occurs at the point ul = ul where
u = + {3F2(t)/u2}
1 c
Thus, by the same argument posed in the preceding paragraph, if either
*
U1 < b/d or 1 ul
then along the constant u2 edge, H has its maximum value at a corner of
the rectangular DU. Wherever
b/d < u1 < 1
H has its maximum value on the line u2 = u interior to the endpoints.
2 c
On the basis of these arguments, the following system was
solved by the method of quasilinearization:
X1 = x2 x1(0) = 0
2 = x3/uu2 x2(0) = 0
3 = x4 x3(1) = 0
4 = ul2 x4(1) = 0
3 = P2/3U2 ''P3(0)= 0
4 = P3 P4(0) = 0
u = ARGMIN [UU2 (lt)x3/p4ulu2]
U 1U
The beam is represented by 100 intervals composing the range 0 < t 5 1,
which is separated by 101 "mesh points." An initial guess of the
solution x(t) and p(t) is chosen; it is selected to satisfy the bound
ary conditions. This guess is not a solution and does not satisfy the
differential equations. The and j equations are linearized about the
initial guess, then the resulting linear TPBVP is solved to obtain new
x(t) and p(t) functions which more closely satisfy the differential
equations. At each time t corresponding to a mesh point, H is numer
ically evaluated along each of the four straight line segments com
posing 3U to determine 4OPT. The point (ul,u2) on 3U which gives
H(u;x,p,t) its maximum value is u T. This process is repeated until
the x(t) and p(t) iterate satisfies the differential equations to
within a specified tolerance. The equations necessary to use the IBM
program available are given in Appendix C, in the form of a subroutine
listing.
79
4.4 Results: Geometric Control Constraints
For the most part, no major difficulties were encountered in
using quasilinearization to obtain a solution to the sixth order sys
tem derived in the previous section. Certain parameter values did
engender numerical instability. These cases, the source of the diffi
culty, and its circumvention are discussed in Chapter VII. Moreover,
all calculations were done in double precision as necessitated by matrix
inversion accuracy requirements.
Parameter values selected to illustrate the solution method are:
u : b/d = 0.25
(4.4.1)
u2 : a/c = 0.20
The measure of error of satisfaction of the differential equations in
the TPBVP is in terms of the general system
dY.
= f.(Y,x) Y Y.(x), i = ,...,n
dx 1 1
E1. i = Max IdY. f.(Y,x)dx (4.4.2)
i
Deflection of a uniform beam due to its own weight was used to infer
an initial guess which satisfies all boundary conditions:
xl(t) = t4
x2(t) = t3
x3(t) = 1 t2
0 < t < 1 (4.4.3)
x4(t) = 1 + t
P3(t) = t2
P4(t)= t3
With these specified parameter values and initial guess of the solu
tion, the program converged to a solution in five iterations. From
this run a tolerance was selected for all subsequent cases; the follow
ing tabulation provides the data used in its selection:
Iteration ERROR Tip Deflection (Cost)
1 .2028 .7387749327 x 10
2 .1704 .3192152426 x 10
3 .6533 x 101 .2847993812 x 10
4 .3031 x 105 .2853731846 x 10
5 .1129 x 1010 .2853719983 x 101
It is seen from these tabular data that there is little improvement in
cost (tip deflection) as a result of the fifth iteration. For this
o
reason a value for the tolerance was selected as 0.5 x 10 which corre
sponds to about six significant digits in the cost functional.
Recall from the previous section that no unconstrained minimum
exists. With the control bounds included, the intuitive solution is
one in which the crosssectional area is maximum near the root, and
reduces to a minimum at the tip. Recalling that for the optimal control,
u =NMax [H] Min [i]
A sequence of illustrations in Figure 4.3 demonstrates the location of
uPT on 5U for several stations along the beam. Constant contours of
D(u) are plotted on the admissible control space at five distinct posi
tions. If an extremal point exists interior to DU some lines of constant
0(u) contours must be closed curves in uspace. This is impossible for
this example.
x/L .1
tI
0 1 1
1
U1
0
x/L = .7
O Minimum $(u) upT
O Maximum O(u)
> Direction of Increasing 0(u)
01
Figure 4.3 Contour Plots of 0(u) at Various Stations
Along the Beam
x/L = .5
01
u2
x/L = 1.
x/L = .3
The first illustration is for the t = 0.1 cross section, near
the root of the beam. Since uPT occurs at the point of minimum 0(u),
the optimum value for both ul and u2 is unity, the maximum allowable
dimensions for both height and width. Constant contour lines indicate
that ((u) is mathematically decreasing in either direction of uspace.
Lines of constant 0(u) are also plotted for the cross section of
t = 0.3. The optimum control has the maximum admissible value for
height ul but u2 has a value somewhat less than unity. However, there
are still no contour lines which are closed curves.
At the midpoint cross section the minimum 0(u) point occurs
at ul = 1 and u2 = a/c. Although the surface Q(u) forms a scooplike
shape, there are still no closed curve contours, and hence no extremal
interior to admissible uspace. The next cross section at which 0(u)
is displayed occurs at t = 0.7. On this section, u2 is still at its
lower bound but ul is no longer at the maximum allowable value of unity
as shown in Figure 4.3. In the last of the sequence, d(u) contours for
the cross section at the tip of the beam are displayed. The point of
minimum q(u) occurs where both components of control have their minimum
allowable values. Again, no contour lines of constant 0(u) form a
closed curve indicating the existence of an interior extremal point.
This sequence of illustrations indicates two things. First,
the lack of closed curve contour lines of <(u) verifies that uPT exists
on U. With further study it may be possible to obtain some condition
on H = 0 which implies the equations corresponding to (4.3.2) can
never yield a finite, unconstrained optimum. Such a condition would
define the class of structures whose unconstrained solution is the
"zero volume solution" frequently described in the literature on struc
tural optimization. Secondly, at t = 0 the point u PT occurs at
T
u = [1,1] the point of maximum crosssectional area; as t increases
from zero to one, the point uPT moves along the ul = 1 boundary of aU
to the u2 lower bound, and then down the u2 = a/c boundary of DU to ul
lower bound. By the time t = 1 the optimal crosssectional area is the
minimum allowable area. As a result of the prescribed form of 3U,
if u (t) follows this particular path as t increases from zero to one,
OPT
each component of u (t) has its own distinct region of transition.
OPT
That is, at any value of t, if b/d < ul < 1, then u2 must be on either
its upper or lower bound. Conversely, if u2 is in transition where
a/c < u2 < 1, then ul must be on one of its bounds.
This effect is seen most clearly in Figure 4.4, where uOPT is
displayed for the example case parameter values specified by (4.4.1).
The profiles are displayed on a twoview drawing as a planform of the
beam might appear. State components corresponding to this beam are
shown in Figure 4.5, representing dimensionless deflection, slope,
moment, and shear, respectively. As observed in Figure 4.4, there are
five distinct regions of the beam:
(i) 0 < t .25
uI =1
controls onupper bounds
u2 = 1
(ii) .25 < t < .52
u = 1
u2 transition
a/c
x/L
a/c = .20
TOP VIEW
SIDE VIEW
Figure 4.4 Planform Views of Optimal Solution for
b/d = .20 and a/c = .25
"u2
2 u
Shear
0
.2
x4
.4
.10
.05
x3
0
.05
x1
0
.02
0
0 .2 .4 .6 .8 1.0
Figure 4.5
State Components of Optimal Solution for
b/d = .20 and a/c = .25

Moment
Slope
Deflection
Deflection
86
(iii) .52 < t < .59
uI = 1 on upper bound
u2 = a/c on lower bound
(iv) .59 < t < .90
b/d
1 u transition
u = a/c
(v) .90 < t < 1.00
ul = b/d
controls on lower bounds
u = a/c
The curves that show the intercept locations as a function of
parameter values b/d have been called "correlation curves" in earlier
studies. When the width is allowed to vary also, the second parameter
a/c is introduced. For the sake of comparison to previous studies, the
intercept/correlation curves are plotted as dependent upon b/d and
parametric in a/c. However, it would be just as correct to do the
opposite.
Intercept location curves described above are shown in
Figure 4.6. The heavy black curve is the case where a/c = 1, a beam
of constant uniform widththe case cited from earlier literature.
Another special case is represented by dashed lines, correspondin; to
a/c = 0 which is the case corresponding to a minimum allowable thick
ness equal to zero. Dashed lines are used because these data are an
extrapolation: convergence problems encountered for parameter values
less than 0.1 prevented obtaining numerical results.
O
0
rJ
o c
r1
a 0
co
(d 4
Lr)
0
A discussion on the convergence difficulties experienced by
the quasilinearization algorithm for parameter values approaching zero
is presented in the chapter on numerical instabilities. In that dis
cussion, isolation of the source of difficulty is reported; it is pos
sible that this difficulty may be a general result applicable to all
problems to be solved by the method of quasilinearization. A solu
tion for this case is later obtained by finite element techniques.
Note that since 0 < a/c < 1 these two cases represent limits to the
solutions of the problem. In addition, if the four intercept locations
are plotted versus a/c and parametric in b/d, curves r and r appear
a c
as "horizontal vees" with rb and rd lines that are nearly parallel.
It is interesting to note from the figure depicting the solu
tion of this case as a planform, that in the central region of the
beam, the height is greater than the width. This result can be antic
ipated since such a configuration gives a greater bending resistance
per unit weight.
With further reference to Figure 4.4, the transition of ui(t)
is seen to be almost a linear taper, whereas the u2(t) transition
exhibits a much more pronounced curvature. To generalize from this
specific case of given values of b/d and a/c to arbitrary values
requires the introduction of four quantities characterizing the solu
tion. These quantities are the values of t at the points where the
transitions intercept the bounds on ul and u2; since t represents a
normalized position x/L, these quantities can be thought of as an
intercept location expressed as percent of the beam's length. They
are defined with reference to the five distinct regions of the beam
previously given, where
rb design
rd design
ra design
r design
such that the five
(i) 0 < t
(ii) rc < t
(iii) r < t
a 
(iv) rd < t
(v) rb t
b 
tes u (t)
tes ul(t)
tes u2(t)
tes u2(t)
regions
< r
 c
< r
a
< rd
d
< rb
< 1.0
intercept with lower bound
intercept with upper bound
intercept with lower bound
intercept with upper bound
are:
control on upper bounds
u2 transition
control on upper/lower bounds
u1 transition
control on lower bounds
4.5 Inequality Stress Constraints
This section treats an inequality limit to allowable normal
and shear stresses associated with bending. Using the ordinary
strength of materials formulations it can be shown that for the rec
tangular cross section these constraints take the form
1 MB(x)
Sh(x) MAX
2 IB(X) MAX
1 VB(x)
8 IB(X) h2(x) MAX
B
When dimensionless quantities are introduced
6 Lx3 u u2 < MAX
3 < 2
2 YL x4/ulu2 MAX
the inequalities may be written in the required form for mixed con
straints, i.e., as a function of both control and state components:
i1(xu) = x3/u u2 0 (4.5.1)
h2(xu) = x4/uu2 To 0 (4.5.2)
where
1 MAX d
((
0 6 yL
2 'MAX
T0 3 yL
The two stress constraints place restrictions upon the minimum
crosssectional dimensions to keep the normal and shear stresses less
than prescribed values. Specifically, from the constraints (4.5.1)
and (4.5.2), two control inequalities must be satisfied at each station
t, and these inequalities depend upon the state of the structural sys
tem. The inequalities are:
ulu2> x3/o0
uu2 > x4/T x4(t) < 0
from which can be derived boundary arcs in uspace:
ul(u2) = (x3/O0u2) 0
(4.5.3)
uT(U2) = 4/TO > 0
Both of these boundary arcs are hyperbolas restricted to the first
quadrant of uspace. Depending upon the location of the arcs, visavis
the rectangular 3U, inclusion of stress constraints has one of three
effects in determining what u is admissible. At any station t in some
structural state x(t),
(i) if 0 ,T0 is too small the stress boundary arc lies entirely
above rectangular 3U; all geometrically admissible u violate
the stress constraints.
(ii) if 0,T0 is too large the stress boundary arc lies entirely
below rectangular DU; all geometrically admissible u satisfy
the stress constraints.
(iii) for some range of 0,T 0 the stress boundary arc divides the
rectangular DU into two regions: the upper region consists of
geometrically admissible u that satisfy the stress constraint,
u in the lower region are geometrically admissible but violate
the stress constraint.
Inclusion of stress constraints alters the admissible control
space from the rectangular shape previously considered to a shape that
may contain a stress boundary arc as part of its boundary. Consider the
normal stress boundary arc specified by (4.5.3) to be a part of 4U.
Then to find uPT in the manner outlined in Section 4.3, (u) must be
evaluated along ul = u l(u2). If a minimum exists along the orthogonal
projection of u l(u2) on the O(u) surface, then
^a$(u)
S=0
Min (u) U 
ul ul a20(u)
> 0
lu2
1

Full Text 
ON THE AUTOMATED OPTIMAL DESIGN
OF
CONSTRAINED STRUCTURES
BY
JERRY C. HORNBUCKLE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1974
Copyright by
Jerry C. Hornbuckle
1974
DEDICATION
To my grandmother, Mrs. Florence Hornbuckle,
and my wife, Carolyn. Without the love, confidence, and
patient understanding of Granny Hornbuckle and Carolyn
my graduate studies would never have been attempted.
ACKNOWLEDGMENTS
To Dr. Robert L. Sierakowski and Dr. William H. Boykin, Jr.,
for guiding my research and for being more than just advisors.
To Dr. Gene W. Hemp, Dr. Ibrahim K. Ebcioglu, and Dr. John
M. Vance, for their assistance, support, and for serving on my advisory
committee.
To Dr. Lawrence E. Malvern and Dr. Martin A. Eisenberg, for
always finding the time to offer advice and explanations on questions
related to solid mechanics and academics.
To the departmental office staff for their kind assistance
with administrative problems and clerical support.
To Randell A. Crowe, Charles D. Myers, and J. Eric Schonblom
for attentive discussions of many little problems and for assistance
in preparing for the qualifying examination.
iv
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iv
ABSTRACT viii
CHAPTER
I INTRODUCTION , 1
1.0 Survey Papers 1
1.1 Historical Development: Optimal Columns 6
1.2 Historical Development: Optimal Static Beams ... 9
1.3 Historical Development: Optimal Dynamical Beams . . 12
1.4 Scope of the Dissertation 14
II GENERAL PROBLEMS AND METHODS IN STRUCTURAL OPTIMIZATION . 16
2.0 Introduction 16
2.1 Problem Classification Criteria 16
2.1.0 Problem Classification Guidelines 18
2.1.1 Governing Equations of the System 18
2.1.2 Constraints 19
2.1.3 Cost Functionals 21
2.2 Methods: Continuous Systems 26
2.2.0 Special Variational Methods 27
2.2.1 Energy Methods 27
2.2.2 Pontryagin's Maximum Principle 29
2.2.3 Method of Steepest Ascent/Descent 31
2.2.4 Transition Matrix: Aeroelasticit.y Problems . 31
2.2.5 Other Miscellaneous Methods 32
2.3 Methods: Discrete Systems 33
2.3.0 Mathematical Programming 34
2.3.1 Discrete Solution Approximations 35
2.3.2 SegmentwiseConstant Approximations 37
2.3.3 Complex Structures with Frequency
Constraints 38
2.3.4 Finite Element Approximations 40
2.4 Closure 41
v
TABLE OF CONTENTS (Continued)
CHAPTER Page
IIITHEORETICAL DEVELOPMENT 43
3.0 Introduction 43
3.1 Problem Statement and Necessary Conditions 44
3.2 Mathematical Programming: Gradient
Projection Method . . . . 48
3.3 Gradient Projection Methods Applied to
the Maximum Principle 53
3.4 Maximum Principle Algorithm 60
3.5 Solution Methods 63
IVCONSTRAINED DESIGN OF A CANTILEVER BEAM
BENDING DUE TO ITS OWN WEIGHT 66
4.0 Introduction . 66
4.1 Problem Statement 66
4.2 Structural System 67
4.3 Unmodified Application of the Maximum Principle . . 70
4.4 Results: Geometric Control Constraints 79
4.5 Inequality Stress Constraints 89
4.6 Results: Stress Constraints Included 93
VCONSTRAINED DESIGN FOR AN OPTIMAL EIGENVALUE PROBLEM . . 101
5.0 Introduction 101
5.1 Problem Statement 101
5.2 Structural System 102
5.3 Analysis of the Problem 109
5.4 Application of the Maximum Principle 121
5.5 Results: Geometric Control Constraints 132
5.6 Inequality Stress Constraints 148
VIFINITE ELEMENT METHODS IN STRUCTURAL OPTIMIZATION:
AN EXAMPLE 155
6.0 Introduction 155
6.1 Finite Element Problem Statement . 155
6.2 Mathematical Programming: Gradient
Projection Method 157
6.3 Results 162
VIICOMMENTS ON NUMERICAL INSTABILITY IN THE
QUASILINEARIZATION ALGORITHM 173
7.0 Introduction 173
7.1 Computer Program Convergence Features 173
7.2 Numerical Instatilibies for Cantilever
Beam Example 175
7.3 Numerical Instabilities for Column
Buckling Example 183
vi
TABLE OF CONTENTS (Continued)
CHAPTER Page
VIII CONCLUSIONS AND RECOMMENDATIONS 186
8.0 Summary and Conclusions 186
8.1 Recommendations 186
APPENDIXES
A HISTORICAL DEVELOPMENTS 191
B A SIMPLE PROOF OF THE KUHNTUCKER THEOREM 206
C COMPUTER SUBROUTINE LISTINGS 214
BIBLIOGRAPHY 244
BIOGRAPHICAL SKETCH 254
vii
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
ON THE AUTOMATED OPTIMAL DESIGN
OF CONSTRAINED STRUCTURES
By
Jerry C. Hornbuckle
August, 1974
Chairman: Dr. William H. Boykin, Jr.
CoChairman: Dr. Robert L. Sierakowski
Major Department: Engineering Sciences
Pontryagin's Maximum Principle is applied to the optimal design
of elastic structures, subject to both hard inequality constraints and
subsidiary conditions, By analyzing the maximum principle as a nonÂ¬
linear programming problem, an explicit formulation is derived for the
Lagrangian multiplier functions that adjoin the constraints to the cost
functional. With this result the usual necessary conditions for optiÂ¬
mality can themselves be used directly in an algorithm for obtaining
a solution.
A survey of general methods and problems in the optimal design
of elastic structures shows that there are two general problem types
depending upon whether or not the cost functional is an eigenvalue.
An example problem of each type is included with the solutions obtained
by the method of quasilinearization. In the first example, a minimum
deflection beam problem, classical Maximum Principle techniques are
used. The eigenvalue problem is exemplified by the maximization of the
buckling load of a column and uses the explicit multiplier function
viii
formulation mentioned above. Since the problem considered is conservaÂ¬
tive, it is therefore described mathematically by a selfadjoint system;
under this condition it is shown that the minimum weight problem is
identical to the maximum buckling load problem.
In order to demonstrate the theory for the programming techniques
used, the beam problem is also solved by using a finite element repreÂ¬
sentation of the structure. From a comparison to the maximum principle
solution it is shown that the form of the optimal solution obtained is
dependent upon the magnitude of the tolerance used with the numerical
solution scheme. Furthermore, it is shown that convergence by the
quasilinearization algorithm is related to the respective curvatures of
the initial guess and the solution.
Recommendations for additional investigations pertinent to this
study are also included.
ix
CHAPTER I
INTRODUCTION
1.0 Survey Papers
It is exceedingly difficult to write a general introduction to
the field of structural optimization for two basic reasons: (i) there
is no conventionally accepted nomenclature, and (ii) there is also no
conventionally accepted classification of problem types or characterÂ¬
istics. In marked contrast, when one considers the calculus of variaÂ¬
tions, "cost functional, system equations, kinematic and natural boundÂ¬
ary conditions, adjoint variables, Hamiltonian, etc.," all have well
defined, universally accepted meanings. Additionally, there is no conÂ¬
fusion when speaking of the problem types of Mayer, Lagrange, and Bolza.
This common language and categorization of problems does not exist in
structural optimization.
Instead, the field tends to branch and fragment into very specialÂ¬
ized subdisciplines that are oriented towards applications. While these
branches are related to the general field, the techniques and methods of
one branch can seldom be applied to another. Moreover, as a result of
the tendency to an applications orientation, solutions are generally ad
hoc and not useful for other problems even within the same branch. The
lack of any definitive unification of the subject cannot be blamed either
on being recently developed or in receiving too little attention.
1
2
This is readily seen by considering the survey papers described in the
following paragraphs.
The earliest comprehensive survey paper is Wasiutynski and
Brandt (1963). Although their excellent historical development is
dominated by Russian and Eastern European references, the authors do
include a higher percentage of papers by Western authors than is
encountered in the typical paper from Eastern Europe. A more fundaÂ¬
mental criticism is that too little is said regarding problem types or
solution methods.
Chronologically, the next survey paper is Gerard (1966) .
The theme of this review is aerospace applications, with a particular
orientation to the designmanagement, decisionmaking process. Most of
the papers cited treat specialized aerospace structures and applicaÂ¬
tions; however, the author does try to generalize by introducing a
design index D, a material efficiency parameter M, and a structural
efficiency parameter S. After defining the expressions for M and S
corresponding to several structural elements, design charts are preÂ¬
sented which show regions of possible application for various materials.
Unfortunately the design charts do not satisfy expectations aroused by
the introduction of the three general parameters.
Rozvany (1966) presents a similar paper pertaining to structures
in civil engineering. This paper is less comprehensive and more oriÂ¬
ented towards specific structural applications than either of the preÂ¬
ceding survey papers. The author postulates several "interrelated
quantities (or parameters)" which could perhaps be used to generalize
3
structural optimization into a more rational methodology. However,
these quantitiesâ€”loading (L), material (M), geometry (G), initial
behavior (IB), and design behavior (SB)â€”are only applied to an abstract
discussion of concepts.
Barnett (1966) has a readable short survey of the field that
dwells more upon theoretical aspects. He postulates a general problem
in which the cost is minimized subject to a system in equilibrium with
its loads, while "behavior constraints on strength, stiffness, and
stability" are satisfied. Uniform strength design introduces the disÂ¬
cussion of optimal trusses; virtual work theorems that were derived
originally for trusses are then applied to simultaneous plastic collapse
problems. Following a brief discussion of the plastic collapse load
bounding theorems, there is a short treatment of elastic stability
problems and material merit indices. Barnett's stiffness design example
exhibits several important features worth noting. Specifically, the
example is to minimize the weight of a beam subject to some given load
where the deflection at a certain point is specified. Virtual work is
used to handle the subsidiary deflection condition. Necessary condiÂ¬
tions are obtained from the calculus of variations, but more signifÂ¬
icantly, the Schwarz inequality provides a sufficient condition for
global optimality. Barnett concludes his survey with a section stating
that multiload designs satisfying "all three behavior criteria" are more
easily solved in "design space" by mathematical programming techniques
described by Schmidt (1966).
As a sequel to the comprehensive survey (1638 to 1962) by
Uasiutynski and Brandt, Sheu and Prager (1968b) present a complete review
4
of developments from 1962 to 1968. This paper contains three major
sections: general background, methodology, and specific problems.
In the first section they state that the "wellposed problem of optimal
structural design" requires specification of the
(1) purpose of the structure (load and environment),
(2) geometric design constraints (limits to design
parameters),
(3) behavioral design constraints (limits to the "state" of
the structure),
(4) design objective (cost functional).
The methodology section is not noteworthy, but the third section lists
what is in their opinion the specific problem types: static compliance,
dynamic compliance, buckling load, plastic collapse load, multipurpose/
multiconstraint structures, optimal layout (e.g., trusses), reinforced/
prestressed structures, and from the background section, probabilistic
problems. Their concluding remarks succinctly summarize the paramount
difficulty of the subject: realistic problems are too complicated for
precise analytical treatment. While progress is being made in analytÂ¬
ical treatment of simple structures, the authors opine that realistic
problems require mathematical programming techniques. However, they do
feel that analytical treatment is desirable to provide "a deeper insight
into the analytical nature of optimality."
A related survey by Wang (1968) on distributed parameter systems,
". . . whose dynamical behaviors are describable by partial differential
equations, integral equations, or functional differential equations,"
consists entirely of a bibliography. While not pertinent to the disserÂ¬
tation, it is mentioned here for completeness.
5
Prager (1970) provides another survey which is not comprehensive,
nor does he present any new results as claimed. For example, Prager
and Taylor (1968) used the principle of minimum potential energy
and the assumption that stiffness is proportional to specific mass to
prove global optimality. It could hardly be called a new development
in 1970. At the same time, the author does present an excellent example
of a multipurpose optimal design problem. Prager also treats "segment
wise constant" approximations and the optimal layout of trusses.
The final survey paper, Troitskii (1971), is an unusual review
of methods in the calculus of variations. Whereas some of the earlier
surveys present lengthy lists of references but contain little methodÂ¬
ology or theory, this survey is just the opposite. Part of what makes
it unique is that the author believed only eight articles merited citaÂ¬
tionâ€”all of them by Troitskii. This shortcoming is more than overcome
by a thorough classification of optimal control problems in the calculus
of variations. Troitskii bases the classification on "certain characterÂ¬
istics of control problems": types of constraints, properties of the
governing dynamical equations of the system, type of cost functional, and
possible state discontinuities. From these four criteria he postulates
five principal classes of problems; however, it is the criteria that
are important and not the specific problem type.
By comparing what the authors of the aforementioned surveys
believe to be the important types of problems, it is readily apparent
that there is little agreement on which characteristics of structural
optimization problems are significant.
6
1.1 Historical Development:
Optimal Columns
The beginning of structural optimization is generally attributed
to Galileo's studies in 1638 of the bending strength of beams. AccordÂ¬
ing to Barnett (1968), Galileo considered a constantwidth cantilever
beam under a tip load as part of a study of "solids of equal resistance."
In requiring the maximum stress in each cross section to be constant
throughout the beam, the height must be a parabolic function of position
along the beam. While this appears to be the origin of the field, a probÂ¬
lem that received more attention is the buckling of a column.
Using the newly developed calculus of variations, in 1773 Lagrange
attempted to apply variational techniques to the problem of finding that
distribution of a homogeneous material along the length of a column which
maximizes the buckling load. Truesdell (1968) relates that through an
insufficient mathematical formulation Lagrange showed the optimal form
to be a circular cylinder. Clausen (1851) provides the earliest known
solution to this problem for the simply supported case. As described in
Todhunter and Pearson (1893, pp. 325329), Clausen minimized the volume
of the column with the differential equation for buckled deflection
treated as a subsidiary condition. Assuming all cross sections to be
similar, after several variable transformations and complicated manipÂ¬
ulation, he obtained an implicit, analytical solution.
The next development was Greenhill (1881), according to Keller
and Niordson (1966). Greenhill determined the height of a uniform
prismatic column, beyond which the column buckled due solely to its own
weight. Timoshenko and Gere (1961) reproduce the solution in which the
7
deflection is expressed as the integral of a Bessel function of the
first kind (of the negative onethird order).
Blasius (1913) introduces his paper with a uniform strength and
a minimum deflection beam problem. For a given load and amount of mateÂ¬
rial, the crosssectional area distribution is determined which maxiÂ¬
mizes the buckling load of a circular column. The solution is identical
to that obtained by Clausen. In addition, Blasius also obtained soluÂ¬
tions for columns having rectangular cross sections and discussed the
effect of different boundary conditions on the results.
For the next few decades, structural optimization appears to
have been directed towards applications in the aircraft industry, where
aircraft structural problems and results are presented in the format of
a design handbook. Feigen (1952) is a good example of this, considerÂ¬
ing the buckling of a thinwall column. Given a constant load and wall
thickness, he required the variable inside diameter to be chosen such
that the buckling load is maximized. Wall thickness is selected to make
local buckling and Euler buckling occur at the same load. Solid tapered
columns having blunt ends are also treated for assumed stiffness disÂ¬
tributions .
Renewed interest was aroused by Keller (1960), who examined the
problem from the point of view of the theory of elasticity, and in
choosing the crosssectional shape to give the maximum stiffness.
Neglecting the weight of the column, he obtained via the former that
twisting the column does not affect the buckling. Of all convex cross
sections, the equilateral triangle is shown to have the largest second
8
moment of area relative to a centroidal axis. Hence, from the definÂ¬
ition of buckling load, the "best" crosssectional shape is the equiÂ¬
lateral triangle. Keller also obtained Clausen's implicit, analytical
solution. Subsequently, Tadjbakhsh and Keller (1962) generalized the
problem to a general eigenvalue problem and boundary conditions subject
to a subsidiary equality constraint. The latter corresponds to specÂ¬
ifying the volume (or weight) of material to be distributed in an optiÂ¬
mal manner. Using the Holder inequality they demonstrate global optiÂ¬
mality of the eigenvalue for the hingedhinged column.
Keller and Niordson (1966) examine the case of a vertical column
fixed at the base, subject to a vertical load at the tip and the column's
own weight. It is also assumed that all cross sections are similar.
Their approach is to state the problem as a simultaneous, dual optimizaÂ¬
tion of the Rayleigh quotient. The eigenvalue is minimized with respect
to the eigenfunction and maximized with respect to crosssectional area
distribution. Specifying the volume of material available is treated
as a subsidiary equality constraint. From the maximum lowest eigenvalue
the maximum height ac buckling is determined. Solutions are obtained by
an iterative technique employed with integral equations.
In a brief note, Taylor (1967), suggests that energy methods may
link optimum column problems to the traditional eigenvalue problems of
mechanics. Prager and Taylor (1968) classify problems in optimal strucÂ¬
tural design and demonstrate global optimality using energy principles.
Unfortunately their assumption of thinwall construction limits the
results to structures where the stiffness is proportional to the specific
9
mass density. The consequence of this assumption is that in the energy
formulation the resulting control law and governing equations are
decoupled, and hence easily solved. Huang and Sheu (1958) apply this
same thinwall assumption to the problem treated by Keller and Niordson.
However, the former seek the maximum end load instead of the maximum
height. The authors also attempt to limit the maximum allowable stress
and obtain solutions by a finitedifference method. Further discussion
of sandwich (thinwall construction) columns is given by Taylor and Liu
(1968). Basically, this paper is an elaboration of the techniques
described by Prager and Taylor when applied to columns. Extensive
results are provided for various cases.
Postbuckling behavior for columns subject to conservative
loads is considered by Gajewski and Zyczkowski (1970). A nonconcerva
tive problem is treated by Plaut (1971b). The first of these two papers
is lengthy but is much too narrow in scope to be particularly useful.
In the second paper, the Ritz method is applied to an energy functional,
obtaining the "best" form of the assumed approximation to the optimal
solution.
1.2 Historical Development:
Optimal Static Beams
That beam problems played a role in the early developments of
structural optimization has already been indicated in the preceding secÂ¬
tion. No attempt is made in what follows to present a complete history,
but merely to outline the type of problems that have been considered.
10
Opatowski (1944) has an outstanding paper that deals with
cantilever beams of uniform strength. Besides providing numerous referÂ¬
ences to earlier studies, the author treats the problem with impresÂ¬
sive mathematical rigor. The beam is considered to deflect under its
own weight and a transverse tip load; bending deflection is described
by a Volterra integral equation which is solved exactly for various
assumed classes of variable crosssectional geometry. This paper is
representative of earlier papers in that it contains extensive analysis
and analytical results, but little numerical data. Barnett's work (1961)
and its sequal (1963a) apply the calculus of variations to more realÂ¬
istic Ibeams. One problem considered is maximizing the weight subÂ¬
ject to general, unspecified loads for a specified deflection at a given
point. The Schwarz inequality is used to derive a sufficient condition
for global optimality. Also included is a comparison of uniform strength
beams to the minimum deflection beam for several different cases of
applied load and geometry. The paper is concluded with various minimum
weight design examples in which both bending and shear stiffness are conÂ¬
sidered .
Haug and Kirmser's (1967) work is one of the most comprehensive
studies of minimum weight beam problem published. While it may succeed in
handling any conceivable problem and in employing the most realistic
stress constraints, this very generality' requires so many variables and
conditions that the mathematics is complicated almost beyond reason.
Another study of minimum weight beams (Huang and Tang, 1969) is
important for several reasons. By dividing the beam into many segments
11
having constant properties, and determining the necessary conditions
that must be satisfied by every segment, it appears that the authors
are using the same methods that were used to solve the Brachistochrone
JL
A
problem in the seventeenth century. In their treatment, multiple
loads must produce a specified deflection at a given point; these
"segmentwise constant" approximations and multiple load problems have
recently received more attention. Of further interest in this paper is
the derivation of the multiple load optimality condition using
Pontryaginâ€™s Maximum Principle.
While no new results or techniques are contained in Citron's
(1969, pp. 154166), the author gives a very readable minimum weight
beam example. The problem is simple, described in detail completely,
and provides an excellent example of how control theory is applied to
an optimal problem in structural design. An analysis of intermediate
beams which may form plastic hinges is provided by Gjelsvik (1971).
It shows that if hinges are placed at all points of the beam where the
bending moment is zero, this makes the plastic or elastic minimum
weight beam statically determinate. Both the elastic and plastic beam
designs are shown to be fully stressed, i.e., are uniform strength beams
Application of generalized vector space techniques is characterÂ¬
istic of more recent papers. Bhargava and Duffin's (1973) is such a study
It treats the maximum strength of a cantilever beam on an elastic
Since these early papers are not readily available, this statement
is based upon descriptions of them given by historical references
cited in Appendix A.
12
foundation subject to an upper bound on weight. Although involving
more advanced mathematics than normally required by variational techÂ¬
niques, vector space methods may also provide more powerful analytÂ¬
ical tools.
1.3 Historical Development:
Optimal Dynamical Beams
The title of this section is a misnomer. On the basis of the
related literature, a more realistic terminology would be "Optimal
Ouasistatic Beams." Papers dealing with the optimization of dynamical
beams ultimately contain some assumption or given condition that effecÂ¬
tively transforms the problem to an equivalent static case. Simple
harmonic motion is frequently assumed to remove time dependence from the
governing equation. For example, Barnett (1963b) minimizes the tip of
deflection of a cantilever beam accelerating uniformly upwards, with the
total weight of the beam specified. The optimal solution for cross
sectional area distribution is specified by a nonlinear integral equaÂ¬
tion solved by successive approximations. Dynamics enters the problem
only as a time invariant inertia load which converts the problem to a
static beam subject to a body force.
Niordson (1964) finds the tapering of a simply supported beam
of given volume and length which maximizes the fundamental frequency of
free vibration. Assuming that all cross sections are similar, Niordson
expresses the desired frequency as the Rayleigh quotient. This is
obtained by the equation for spatial dependence associated with the
usual separation of variables technique. In this case it is assumed
13
that deflection, shear, and rotational inertia effects are small.
Solution of the conditions for optimality were obtained by successive
approximations. This approach results in a problem identical in form
to the eigenvalue problem associated with maximizing the Euler buckling
load of a column.
A very specialized problem is treated in Brach (1968). Cross
sectional area and stiffness are proportional, and have both upper and
lowed bounds; material properties and length of the beam are specified
constants. The object is to make the fundamental frequency of free
vibration stationary, without any constraints on the weight. Instead
of using the Rayleigh quotient, Brach uses the total potential energy.
His solution method is ad hoc and not generally useful. A more useful
approach is described by Icerman (1969) for structures subject to perÂ¬
iodic loads. Necessary and sufficient conditions for minimum weight
are obtained from the principle of minimum potential energy. Amplitude
and frequency of the applied load is specified as well as "dynamic
response," defined as the potential energy associated with the load
amplitude displaced a distancesqual to the displacement amplitude at
the point of application. It is also required that the load's frequency
be less than the fundamental natural frequency. Subject to these conÂ¬
straints, the structure's weight is to be minimized. Trusses and
segmentwise constant approximations are also treated. Once again a
dynamical problem is effectively transformed into a quasistatic problem.
Realistic treatment of the dynamic response of beams subject to
time dependent loads is given in Plaut (1970). An upper bound on the
14
response of the beam is obtained from the "largest possible displaceÂ¬
ment of the beam under a static concentrated unit load." This inequalÂ¬
ity is determined from the time derivative of the total energy of the
beam and the Schwarz inequality. Plaut minimizes the upper bound on
the response for specified total weight and relationship between specific
mass and stiffness. Despite the consideration of a truly dynamical
problem, this approach has two weaknesses indicated by the author.
First, the upper bound need not be close to the exact answer; secondly,
there is no demonstration that minimizing the upper bound also minimizes
the actual response of the beam subject to dynamic loading.
Another paper worthy of mention is Brach and Walters (1970) .
They maximize the fundamental natural frequency which is expressed as
a Rayleigh quotient that includes the effect of shear. Standard variaÂ¬
tional methods are employed to derive necessary conditions, but no
solutions nor examples are given. The authors do, however, suggest
using the method of quasilinearization. This paper is another example
of a quasistatic, dynamical beam problem.
1.4 Scope of the Dissertation
This dissertation is primarily concerned with the application
of Pontryagin,'s Maximum Principle to problems in structural optimization.
Only elastic materials are considered; however, various types of conÂ¬
straints are treated.
The theoretical development of the dissertation pertains to
problems described by an ordinary differential equation but is based
upon a numerical technique normally used with systems described by
15
a finite number of discrete quantities. For this reason, examples are
included for both types of systems.
No new solution techniques have been developed for the nonlinear
two point boundary value problems which characteristically arise in
optimization problems for continuous systems. Solutions are obtained
by a standard quasilinearization method. However, a modified "feasible
direction" numerical algorithm for use with discrete systems is
described and an example included to demonstrate its operation. This
serves to illustrate the application of the theory on which the algoÂ¬
rithm is based to the theoretical development associated with continÂ¬
uous systems. Furthermore, it provides a comparison between the
solution to a problem described as a continuous system, and alternately
by a discrete approximation.
Additionally, it is the intent of this dissertation to replace
some of the confusion in classification of problem types contained in
the various survey papers with an organization based upon mathematical
attributes. The result is a logical approach to the formulation of
optimization problems for elastic structures.
CHAPTER II
GENERAL PROBLEMS AND METHODS
IN STRUCTURAL OPTIMIZATION
2.0 Introduction
The first chapter presents a broad view of structural optimizaÂ¬
tion and the historical development of two general types of problems
that are used as examples in later chapters. Before the mathematical
theory is developed in the next chapters, general problems and methods
in structural optimization itself are briefly outlined in this chapter.
The classification of problem types vishvis mathematical attributes
is discussed first. This is followed by short descriptions of the
major methods of structural optimization for both continuous and disÂ¬
crete systems. Appropriate references are cited for each section.
2.1 Problem Classification Criteria
Perhaps the major source of difficulty in classifying strucÂ¬
tural optimization problems lies in the translation from the physics
involved to a mathematical representation. A single physical concept
when transformed to mathematics may become more than one mathematical
attribute. For example, consider the class of conservative problems.
Since energy is conserved this immediately prohibits dissipative mateÂ¬
rials, time varient constraints, and nonholonomic constants. More
important, consider the following statements from Lanczos (1962,
P. 226):
16
17
. . . all the equations of mathematical physics which
do not involve any energy losses are deducible from
a "principle of least action," that is the principle
of making a certain scalar quantity a minimum or a
maximum . . . all the differential equations which are
selfadjoint, are deducible from a minimummaximum
principle . . . and vice versa.
However, it is shown in Chapter V that to be selfadjoint systems places
requirements on both the differential equation and the boundary condiÂ¬
tion. Thus, the single physics attribute of being a conservative sysÂ¬
tem is described mathematically by expressions involving the cost funcÂ¬
tional, the governing system of differential equations, boundary condiÂ¬
tions, and constraints.
As a result of this lack of similarity in descriptions, a choice
must be made as to which realm will be used for classification of probÂ¬
lems. Since all problems are ultimately transformed to mathematics,
mathematical characteristics are selected as the criteria. On the basis
of survey paper contents and the many related papers, it is felt that
the proper (not necessarily the best, nor all inclusive) characteristics
for classification of problem types are:
(i)cost functional
(ii)system equation and boundary conditions
(iii)control constraints
(iv)behavioral constraints (state and/or control constraints)
Subsequent discussion is in terms of these four characteristics.
Exceptions to these descriptors are readily acknowledged, e.g., whether
the problem is deterministic or probabilistic. An example of the latter
may be seen in Moses and Kinser (1967). These exceptions do not serve
18
as negating counterexamples but instead indicate the requirement for
additional descriptors and verify the difficulty of the task, suggestÂ¬
ing the need for further comprehensive study.
2.1.0 Problem Classification Guidelines
In the following sections each criterion is briefly discussed.
Some of the various characteristics of each are mentioned, and where
appropriate references exist the citation is given. It must again be
emphasized that the following is not all inclusive; it is an attempt to
categorize the types of problems existing in the literature according
to the four mathematical descriptors postulated. Moreover, the descripÂ¬
tors are not discussed in the order given but instead are treated in
the order normally encountered during a problem solution.
2.1.1 Governing Equations of the System
The immediate question to be answered is whether the structural
system is described by a set of continuous functions or a set of disÂ¬
crete constants. Bending deflection of simple structural elements is
an example of the former; design of trusses is a good example of the
latter. In general, variational techniques are employed with continÂ¬
uous systems while mathematical programming techniques are most freÂ¬
quently applied to the discrete systems. However, variational methods
can be applied to the approximation of continuous systems by discrete
elements. This usually takes the form of either a finite element or
"segmentwise constant" approximation of the continuous structure.
References treating the different systems described above are included
in the section on methods.
19
2.1.2 Constraints
Two of the descriptors are postulated to be control constraints
and behavioral constraints. A further consideration is whether the
constraint is defined by an equality relationship or by an inequality
expression. Equality constraints are handled by a longknown technique
entitled "isoperimetric constraints." Valentine's (1937) work is known
to contain the initial development of a technique for converting inequalÂ¬
ity constraints to equality constraints. The introduction of slack
variables increases the number of variables in the problem to be solved
but at the same time permits all of the isoperimetric techniques to be
used. A detailed application of this approach is presented in Appendix B.
It should also be noted that isoperimetric constraints are sometimes
referred to as "accessory or subsidiary conditions."
Most real structural optimization problems possess an isoperiÂ¬
metric constraint as well as inequality constraints dependent upon the
control _u and/or the state x of the structural system. Typically the
control constraints are the result of geometrical limitations or
restrictions to the types of available materials. Behavioral constraints
are related to the state of the system and may depend solely upon the
state (of deformation), or in the case of most stress constraints,
jointly upon the state and control variables. With this distinction the
constraints may be classified visÃ¡vis the two criteria and optimal
control characteristics as follows:
By convention all vectors are column vectors unless indicated otherwise.
20
(i)
uncons trained
(ii)
o
VI
/â€”s
0
(iii)
$ (x, u) < 0
(iv)
(x) < 0
control constraint
behavioral constraints
All of this discussion pertains to both continuous and discrete systems.
No references for unconstrained optimization problems are given.
It may be that in some problems the unconstrained structural optimizaÂ¬
tion solutions have either infinite stiffness and finite weight, or
finite stiffness and zero weight. A discussion of this can be found in
Salinas (1968, pp. 2326).
Investigation of control constraints led to the development of
the maximum principle. Although reserving discussion of the method for
a later section, the classical reference detailing the derivation of
the principle is given here for completeness. RozonoÃ©r (1959) treats conÂ¬
trol constraints but only as related to the development of the maximum
principle.
Most of the literature concerns either bounded control problems
or a more general form of constraint which can be classified as a behavÂ¬
ioral constraint. The latter is a mixed constraint which depends upon
both the state and control variables. Breakwell (1959) is a lucid paper
dealing with this type of constraint. References which treat state
and/or mixed constraints are Bryson et al. (1963), and Speyer and
Bryson (1968). Constraints which depend upon only the state are not
treated in this dissertation; an example of such a constraint is to
determine the optimal solution for some problem subject to an upper
bound on deflection of the structure at any point.
21
2.1.3 Cost Functionals
There are two basic types of cost functionals that occur in the
field of structural optimization. They were first identified by Prager
(1969) but not for the proper reasons. Using Prager's notation, they are
J = Min / F(ip) dV
V
/ G0I0 dV
V
JQ = Min
/ H(i>) dV
V
where F, G, and H are scalar functionals of ifi. The latter functional
represents a Rayleigh quotient associated with an eigenvalue problem.
It can be reduced to the first type of functional shown above by choosÂ¬
ing a normalization of the eigenfunction such that the numerator
equals unity for all admissible variations (see section 5.3). This
normalization is thereafter treated as a subsidiaryâ€™ constraint.
What actually distinguishes the second functional from the
first is not that the functional is a quotient, but that the extrem
ization of an eigenvalue requires a dual extremization (see section 5.3).
In terms of a state x and control u, the fundamental eigenvalue is
given by minimization of the Rayleigh quotient with respect to the
eigenfunction x, or
= Min
/ G(x,_u) dV
V
^ Ã¼ / H(x,u) dV
V
where ii represents some specified design parameter. If the desired
result is to maximize the cost with respect to all admissible _u, it
is observed that a second extremization is required; for example, see
22
Keller and Niordson (1966). Thus, a more appropriate manner of classiÂ¬
fying cost functionals is on the basis of whether the problem statement
implies a single or a double extremization. Hence the two basic types
of cost functionals encountered in structural optimization are
J = Min ( F(x>Jj) dV
u V
/ G(x,_u) dV
V
J = Max Minâ€”â€”â–  â– â– â– â– â€”â€”â€”
â€¢ u x / H(x,u) .dv
V
where _x must satisfy an equilibrium condition of the state, and u is
subject to some admissibility requirements.
There is a special case related to these two in which the weight
is to be minimized for a specified eigenvalue. This problem is
treated in Icerman (1969) with a mathematical discussion of such a
variational problem presented in Irving and Mullineux (1959, p. 394).
In terms of the two cost functionals, the special case is
J = Min J (G(x,u)  d^lKxju)} dV
u V ^
where is a specified constant. This approach is frequently employed
in eigenvalue problems to avoid the inherent difficulties associated
with the dual extremization problem.
There have also been many papers published that consider
"multipurpose structures," e.g., Prager (1969), Martin (1970), and
Prager and Shield (1968). The cost function for such problems is
defined as
23
k
J = Z a.J. (_x>iO
i=l 1 X
where a_^ are positive constants, serving as weighting parameters.
While perhaps demonstrating much potential, no significant results
obtained with this approach have so far been published. What problems
have been solved are too simple; indeed the authors indicate the need
for using a discrete approximation and mathematical programming techÂ¬
niques in realistic applications.
A subject closely related to "multipurpose structures" is
that of multiple constraints. It is mentioned here only because most
papers on the latter also include the formerâ€”see Martin (1971). The
idea of multiple constraints is not new; in both variational and mathÂ¬
ematical programming fields there exist standard techniques for hanÂ¬
dling multiple constraints.
A recent Russian paper (Salukvadze, 1971) suggests an alterÂ¬
nate to the "multipurpose" cost functional. Instead of treating a
vector functional that requires the choosing of weighting coefficients,
it is suggested that the several functionals be combined into one.
Given a system and vector cost functionals
x = f_(x,u, t)
J [u] = J^0x,u,t) i = l,...,k
Let uiP denote the optimal solution for which J. assumes the optimal
â€”OPT ^ i F
value on the trajectory of the system. For each of the J_^ there is a
different These k values J. can be thought of as components of
â€”OPT i
24
* t
a vector r where
For any arbitrary ui the result
rjuj = { J [u] ... Jk[u]}T
is just some vector functional.
k
Vector _r represents a constant point in the space of
(J , ...,J. ). Since no choice of u can optimize all of the J. simul
1 k â€” i
*
taneously, that is, to attain the point _r in J^space, the best alter
k
native is to minimize the distance between _r[u] and r_ . That distance
is defined by the Euclidean norm. To avoid the question of inconÂ¬
sistent dimensions the functionals are reduced to dimensionless form.
Thus,
J[u] =
k
Z
i=l
i[Hl 
J.
1
' (i)
HIPT
3
and u.__ is that function u which minimizes the functional J[uj.
â€”OPT â€” â€”
Mathematically speaking,
opt = ARGMIN
This type of vector cost functional is much more appealing
than the type treated in the papers on multipurpose structures.
It also suggests an entirely new field of study: the more realistic
f T
Superscript T , e.g., ju , denotes the transpose of the vector tu
25
choice of cost functionals. The mathematics of a problem seldom
accommodates financial considerations. For example, the design which
requires the least material may reduce the cost of materials at an
overwhelming expense in manufacturing or fabrication. When aesthetic
appeal and environmental impact are includedâ€”as must be done in any
real, commercial applicationâ€”the selection of an appropriate cost
functional is an almost insurmountable task. However, a simple extenÂ¬
sion of Salukvadze's composite cost functional may reduce the difficulÂ¬
ties to operations research considerations.
In problems where it is desired to optimize simultaneously
several different functionals, not all having the same dimensions, the
concept of a generalized inner product may prove useful. It is defined
in terms of a metric operator A; for some general vector _z
_Z  2 = (_Z, z)A = z^kz
and symbol "=" means "is defined by." With reference to the vector
cost functional, A represents a set of scale factors which converts
all of the separate cost functionals to a common dimension. This is
where the operations research entersâ€”relating material expense to
fabrication to sociological considerations and so forthâ€”to determine
the metric A. For a vector e_ whose elements are functionals,
e_ =  jr
where r[u] and r are defined above, the composite cost functional is
J t u] = (c,Ac)
26
The weakness in this method is that an optimal solution must be
obtained for each individual cost functional prior to attempting a
solution to the composite problem. Additionally, some of the more
abstract cost objectives may be difficult to quantify in a meaningful
manner. Despite these shortcomings this approach does suggest interÂ¬
esting applications.
2.2 Methods: Continuous Systems
The problems characterized by mathematical functions, in conÂ¬
trast to those represented by a set of discrete constants, are normally
treated by variational techniques. Many books on this subject have
been published; the better authors include Elsgolc (1961), Gelfand and
Fomin (1963), Dreyfus (1965), Hestenes (1966), Denn (1969), Luenberger
(1969), and Bryson and Ho (1969). An excellent summary paper is availÂ¬
able in Berg (1962).
To see how these techniques are applied, three papers are
recommended. The first is Blasius (1913), which provides sufficient
detail and explanation to make it quite worthwhile. Although it does
include several examples that involve subsidiary conditions, no inequalÂ¬
ity constraints are treated. An example that includes inequality conÂ¬
straints to the control variable is coontained in Brach (1968). A much
more general application of variational principles is presented in
Oden and Reddy (1974). In this paper a dualcomplementary variational
principle is developed for a particular class of problems. It is shown
that the canonical equations obtained are the EulerLagrange equations
for a certain functional.
27
2.2.0 Special Variational Methods
Besides the ordinary variational method, more specialized
techniques have been developed to the point where they are recognized
as independent methods in their own right. In the following sections
these methods are identified and a number of representative references
given.
2.2.1 Energy Methods
The oldest of these methods is the energy method. It originated
with the principle of minimum potential energy, and was later extended
to include the representation of eigenvalues through the energy funcÂ¬
tional. A good discussion of the former is available in Fung (1965) or
Przemieniecki (1968); the best general treatment of the 3_atter is availÂ¬
able in either Gould (1957, Chapter 4) or in Mikhlin and Smolitskiy
(1967, Chapter 3).
The principle of minimum potential energy is frequently used
with simple problems to prove that a necessary condition for optimality
is also sufficient. Prager and Taylor (1968) contains such a proof
for the global, maximum stiffness design of an elastic structure of
sandwich construction; two papers that also consider this problem are
Huang (1958) and Taylor (1969). Specific application of the energy
method to an eigenvalue problem is demonstrated in Taylor and Liu (1968).
A much more general discussion of the energy method is provided in
Salinas (1968). Further extensions of the method are presented in
Masur (1970), in which the principle of minimum complementary energy
is applied to problems of the optimum stiffness and strength of elastic
28
structures. In these problems a necessary condition for optimality
is that the strain energy density be constant throughout the structure.
This condition is also sufficient for optimality in certain classes of
structures that satisfy a specific relationship between the strain
energy density and design variables.
Many of the energy problems belong to the class of problems
having quadratic cost functionals. The significance of this characterÂ¬
istic is that the EulerLagrange equations derived from such functionals
are linear.
A more recent energy method development is the concept of
"mutual potential energy." Mutual potential energy techniques resemble
those of the principle of minimum potential energy. In both methods
a cost functional is defined over the entire domain occupied by the
structure and optimized with respect to the control variable. If it
is desired that the optimal solution be required to have a specified
deflection at a certain point, this condition corresponds to a subÂ¬
sidiary state constraint when using the principle of minimum potential
energy. The mutual potential energy method incorporates this type of
localized constraint into the cost functional which is defined over
the entire domain of the structure. By itself this alone is advantaÂ¬
geous; however, for certain types of problems the mutual potential
energy method also provides both a necessary and sufficient condition
for global optimality. In the way of a critical comment, either the
method has received too little attention, or else it does not effiÂ¬
ciently handle problems more difficult than the simple examples presented.
29
Four papers that are representative of the literature associated with
this method are Shield and Prager (1970), Chern (1971a, 1971b), and
Plaut (1971c).
Another recent development is that the class of problems for
which energy methods is applicable has been expanded to include cerÂ¬
tain types of nonconservative systems. Together with the mutual potenÂ¬
tial energy concepts, this suggests that perhaps the classical energy
method is a special case of a more generalized method. If a technique
can be developed which uses the adjoint variables to transform a genÂ¬
eral nonconservative system into an equivalent selfadjoint form, the
method might be deduced. Some papers pertaining: to the subject are
Prasad and Herrmann (1969), Wu (1973), and Barston (1974).
2.2.2 Pontryagin's Maximum Principle
There are many textbooks which derive, explain, and give examples
for the maximum principle. The original (Pontryagin et al., 1962)
requires a knowledge of functional analysis. A condensed form of this
same material is available in RozonoÃ©r (1959). Denn (1969) provides
another point of view in which the principle is derived from Green's
functions. In this manner, the sensitivity to variations is readily
observed. To understand Denn's treatment requires only a knowledge of
the solution of differential equations.
Shortly after Pontryagin's book was published, many papers
devoted to the theoretical aspects of the maximum principle were pubÂ¬
lished. Some of the more readable ones are Kopp (1962, 1963), Roxin
(1963), and Halkin (1963). Another early paper (Breakwell, 1959)
30
appears to be a completely independent derivation of the maximum prinÂ¬
ciple. Although quite general in the mathematical sense, the examples
presented are trajectory optimization problems and not a general type
of mathematical problem. This may be an explanation for what seems to
be a lack of recognition for a significant achievement.
The application of PMP to problems in structural optimization
is relatively recent. When the method is used, one of two difficulties
is often encountered. The first is errors in the formulation of the
optimal control problem; the second is that once a wellposed, nonÂ¬
linear twopoint boundary value problem (TPBVP) is obtained, it is difÂ¬
ficult to solve. An example of the first is provided by Dixon (1968)â€”
the correction was given in Boykin and Sierakowski (1972). De Silva
(1972) provides a clear presentation on the application of PMP to a
specific problem, but includes no data because a solution could not be
obtained. Despite the failure to determine the solution, this paper
is worthwhile for its lucid discussion of the PMP application. Another
paper that gives a good specific application of PMP is Maday (1973).
Although much analysis is presented very little is said regarding the
solution techniques.
All of the above references are applicable only to systems
that are described by ordinary differential equations, in contrast with
the calculus of variations which also handles problems described by
partial differential equations. Since many of the problems of matheÂ¬
matical physics involve partial differential equations, an extension of
PMP to include this class of problems is the next logical development.
31
Some work has already been done, for example, Barnes (1971) and
Komkov (1972). A survey of these "distributed parameter systems"â€”
see Section 1.0â€”is presented in Wang (1968).
2.2.3 Method of Steepest Ascent/Descent
This method is frequently cited in the literature for trajecÂ¬
tory optimization, and occasionally in references related to optimal
structures. When the method of quasilinearization converged for the
dissertation example problems, there was no need to investigate other
methods such as the method of steepest ascent. Consequently, little is
said about it. According to the references, it is applied in a straightÂ¬
forward manner. Furthermore, the example problem solutions presented
seem to be real problems and not academically simple. The following
four papers treat the method in general with trajectory optimization
applications: Bryson and Denham (1962, 1964), Bryson et al. (1963), and
Hanson (1968). In Haug et al. (1969), the method of steepest ascent is
derived in detail, completely discussed, and compared to the maximum
principle. Several structural optimization problems are then solved
by the method of steepest ascent. Although no exciting results are
obtained, use of the method is clearly illustrated by the applications
to realistic structural problems.
2.2.4 Transition Matrix:
Aeroelasticity Problems
For the past few years a group at Stanford University has studied
the optimization of structures subject to dynamic or aerodynamic conÂ¬
straints. The general problem of their interest is that of minimizing
32
the weight of a given structure for specified eigenvalue, subject to
inequality constraints on control.
Three types of solution techniques are used once the necessary
conditions for minimum weight are determined. Exact solutions are
obtained for most of the problems because they are so simple that
analytical methods are applicable. More complicated problems are
solved by a "transition matrix" method described in Bryson and Ho (1969) .
On the basis of convergence difficulties reported in the references,
this method should be used with caution. Results have been obtained
only for very simple problems. However, these results are corroborated
by data obtained from a discrete approximation method. Five papers
that are representative of this work are McIntosh and Eastep (1968),
Ashley and McIntosh (1968), McIntosh et al. (1969), Ashley et al. (1970),
and Weisshaar (1970).
2.2.5 Miscellaneous Methods
The preceding sections have briefly outlined the methods of
structural optimization most frequently encountered in the literature.
Appropriate, representative references have also been given. Not all
methods are listed; while some are omitted for not being generally useÂ¬
ful, others are omitted for not being generally used. Two examples of
the latter are the "modified quasilinearization" and "sequential gradient
restoration" algorithms described in Miele et al. (1972) and in Hennig
and Miele (1972). At some later time these methods may be acknowledged
as major methods that are applicable to many different or important probÂ¬
lems, but for now they are mentioned only in passing.
33
2.3 Methods: Discrete Systems
Discrete systems are described by a set of discrete constants
instead of the set of functions associated with continuous systems.
The classic example of a discrete system is a pinconnected truss, where
the state x. and control u. are the stress and crosssectional area,
i i
respectively, for each member i. A discrete system also arises in the
approximation of a continuous system.
Several references that present a good discussion of general
methods applied to discrete systems are available. Most of these exist
in the form of an edited collection of papers by various authors on the
topics of their acknowledged expertise. Four such publications are
Gellatly (1970), Gellatly and Berke (1971), Pope and Schmidt (1971),
and Gallagher and Zienkiewicz (1973). Another report, Melosh and Luik
(1967), provides a good exposition of the difficulties associated with
the analysis portion of least weight structural design. It also conÂ¬
tains a brief comparison of various mathematical programming methods.
McNeill (1971) is the last reference to be cited in the section
on general methods for the optimization of discrete systems. Minimum
weight design of general structures is treated in a mathematically
precise formulation. Legendre's necessary condition is combined with
the concepts of convex functions and sets to derive the necessary and
sufficient conditions for global optimality. Fully stressed designs
and constraints to eigenvalues are also discussed. In summary, this
paper orovides a good example of the general mathematical problem that
must be solved in the optimization of discrete systems.
34
While certain variational methods may be applied to discrete
systems, the most frequently used technique is mathematical programming
In the following sections, this method and other major methods are disÂ¬
cussed and representative references cited.
2.3.0 Mathematical Programming
The general method of mathematical programming is discussed
in section 3.2 of the dissertation, and the solution of an example
problem using this method is detailed in Chapter VL. In the literature
related to this subject, a very readable textbook is availableâ€”Fox
(1971). This book complements the theory with numerous discussions
pertaining to numerical techniques and methods that can be employed to
overcome certain difficulties that may arise. Although it does contain
flowcharts of several algorithms, there are few specific examples given
For a discussion of the general theory, two alternatives to this book
exist in the form of papers: Schmidt (1966, ]968). The first is
written in a conversational style, contains no mathematics, and is
intended to provide only a general description of the subject. The
latter paper is theoretical in content.
An excellent application to a realistic problem is to be found
in Stroud et al. (1971). Ths paper contains little discussion of the
method itself, but does demonstrate an application that allows a concep
tual visualization of the solution. The approach is to assume the
solution to be a linear combination of specified functions, and to
choose the weighting coefficients to minimize the cost. Mathematical
programming is employed to determine the optimal set of coefficients.
35
This approach resembles Galerkin's method, and though not mathematÂ¬
ically rigorous, it may provide a useful approximation to large,
unwieldy problems.
2.3.1 Discrete Solution Approximations
In the previous section a paper is cited that contains an
approximate solution obtained by Galerkin's method. The use of the
Galerkin or RayleighRitz approximate solution techniques is suffiÂ¬
ciently widespread to be considered a general method. For both methods,
the solution is assumed to be a linear combination of the solution to
the linear part of the governing equation and a set of prescribed funcÂ¬
tions. This approximate solution does not satisfy the given equation
exactly but produces some residual function. A cost function that
depends upon the residual is then minimized with respect to the unknown
coefficients. The two weighted residual methods mentioned above have
different cost functions, but the methods are identical for linear
equationsâ€”see Cunningham (1958, p. 158).
The advantage to using these methods is that after assuming
the particular form of solution, the problem of solving for the weightÂ¬
ing coefficients may be much simpler than the original problem. In the
case of Stroud et al. (1971), the coefficients were obtained by matheÂ¬
matical programming techniques. However, the weakness of the method
is the restricted function space of possible solutions. With the coef
ficents obtained by these methods the resulting solution is the best
approximation that is possible from the set of solution functions
36
prescribed. There is no guarantee that the approximation even resembles
the true solution.
The flutter of a panel is solved using Galerkin's method in
Plaut (1971a). No general developments are presented and the assumed
solution functions are trivially simple. However, this paper does
provide an application of the method to obtain an approximate solution
to a very difficult optimization problem involving the stability of
a nonconservative system. A similar problem is treated in a more theoÂ¬
retical manner in Plaut (1971b) using a modified RayleighRitz method.
"Segmentwiseconstant" control functions are assumed also; this particÂ¬
ular approximation is discussed with more detail in the following secÂ¬
tion. Additional nonconservative problems are treated in Leipholz
(1972), applying Galerkin's approximate solution to the energy method.
Several simple examples are included.
Nonconservative elastic stability problems of elastic continua
are treated in Prasad and Herrmann (1969) using adjoint systems. This
approach is more realistic than the segmentwiseconstant control assumpÂ¬
tion described in the following section. Solutions for the state and
adjoint system are assumed, such that approximation process resembles
the RayleighRitz method. However, only a single type of nonconservaÂ¬
tive system is considered. Extension to several other types of nonconÂ¬
servative elastic continua problems is given in Dubey (1970). VariaÂ¬
tional equations corresponding to both the Galerkin and RayleighRitz
methods are derived. Furthermore, the condition for equivalence of the
two methods is shown to be that the admissible velocity field must
37
satisfy a natural boundary condition over that portion of the bodyâ€™s
surface where tractions are prescribed.
2.3.2 SegmentwiseConstant Approximations
The definitive characteristic of this method is approximating
the structural system by a number of discrete segments, where within
each segment the control function has a constant value. In general,
the constant value of the control differs from segment to segment. For
the many papers on this method that have been published, the procedure
is the same. An optimality condition (necessary in all cases but also
sufficient in some) or cost functional is derived for the continuous
system. After defining the segmentwiseconstant approximation, the
condition or functional is reformulated in terms of the discrete paramÂ¬
eters. Most of the papers use so few elements that solving for the
discrete values of the control parameter poses no difficulties. Although
this method does simplify the mathematical problem to be solved, the
crudeness of the approximation is not appealing. Five papers which
treat a variety of problems using this approximation are cited below.
Minimum weight of sandwich structures subject to static loads
is discussed in Sheu and Prager (1968a). In Sheu (1968) the same type
of structure is considered. It differs from the first problem by
requiring point masses to be supported such that the total structure has
a prescribed fundamental frequency of free vibration. Icerman (1969)
treats the problem of elastic structures subject to a concentrated load
of harmonically varying amplitude. The minimum weight design is obtained
38
subject to a compliance constraint related to the applied load, and
which is effectively a boundary condition on displacement at the point
of application. A truss problem is also included.
The concept of a compliance constraint is pursued further in
Chem and Prager (1970). The minimum weight design for sandwich conÂ¬
struction beams under alternative loads is found, subject to this type
of constraint. The paper uses up to eight segments, thereby obtaining
a more realistic approximation to the continuous problem. Minimum weight
design of elastic structures subject to body forces and a prescribed
deflection is discussed in Chern (1971a). This investigation is noÂ¬
table. in that it considers applied loads that are functions of the design
functions.
2.3.3 Complex Structures with
Frequency Constraints
On the basis of useful application, perhaps the most important
class of discrete structural optimization problems is the minimum weight
design of complex structures subject to natural frequency constraints.
Since most real structures are built with many structural elements of
various types, and are not realistically described by any single type,
this approach is more appropriate from the aspect of modeling the strucÂ¬
ture. Furthermore, many structures must be designed to avoid certain
natural frequencies because of resonance or selfinduced oscillations;
this situation indicates that the natural frequency constraint is also
appropriate.
Many different solution schemes have been developed which are
usually based upon general mathematical programming techniques.
39
Typically, a design is iteratively altered to minimize the weight with
a subsequent increase in frequency until a constraint is violated.
At that point the design process uses an iteration which simultaneously
reduces both weight and frequency. These two processes are repeated
sequentially until no further weight reduction is possible.
Although circumstances may require the use of many elements,
the number of them may itself be a critical factor. Some of the schemes
require a matrix inversion as part of the eigenvalue problem solution
associated with the frequency constraint. If the number of elements
becomes too large, the size of the matrix to be inverted likewise beÂ¬
comes excessively large. When that occurs the matrix inversion can
require excessive amounts of computer time. Another possible difficulty
is that the inverse matrix itself is not sufficiently accurate, such
that the subsequent calculations are not acceptable. However, for
structures such as reinforced shells composed of different types of
structural elements, this method may be the most applicable.
Many papers have been published pertaining to this class of
structural optimization problem. Because the method is inherently
oriented towards applications, the references are cited in chronologÂ¬
ical order without additional comments. Interested readers are referred
to: Turner (1967), Zarghamee (1968), Turner (1969), De Silva (1969),
Rubin (1970), Fox and Kapoor (1970), McCart et al. (1970), and Rudisill
and Bhatia (1971).
40
2.3.4 Finite Element Approximations
There is an unfortunate ambiguity to the label "finite elements"
that occurs because these words are used to describe two completely
different entities. In papers cited in the preceding section they are
used to indicate the discrete structural elements of finite dimensions
which comprise the complex structure. The analysis of such systems of
structural elements has been accomplished by ordinary matrix methods
during the last three decades. However, during the past decade another
method has been developed and named "the finite element method."
In this method a continuum is divided into small, finite eleÂ¬
ments over which a particular form of approximation of either the disÂ¬
placement and/or force is assumed. A number of nodes common to one or
more element is prescribed; continuity is required to exist at these
nodes but not necessarily elsewhere. An equilibrium equation is derived
for each element, and then all of the individual equations are combined
into a single equilibrium equation for the entire system. The resultÂ¬
ing equation is a linear algebraic equation whose unknowns are displaceÂ¬
ments and/or forces at the nodes. Once the matrix equation is inverted,
the nodal displacements and/or forces are used with the assumed approxÂ¬
imation form to describe the state of the structure throughout each and
every element, and hence the system. Hereafter this method is referred
to as the "finite element method."
The most frequent application of the finite element method is
to problems having complicated loads, geometry, and response. Generally
speaking, the method is employed wherever the physical system is too
41
complex to be described adequately by a single differential equation
and boundary conditions. For a complete theoretical development of the
finite element method and numerous examples, see Zienkiewicz (1971).
With respect to structural optimization the method is employed
to simplify the problem to be solved. Very little has been published
on this subject, but the papers available cover a wide spectrum of tech
Ã±iques. For example, Dupuis (1971) combines the finite element and
segmentwiseconstant methods as applied to minimum weight beam design.
A similar application to column buckling is contained in Simitses et al
(1973). Another paper, Wu (1973), is a study of two classical nonconÂ¬
servative stability problems. Although adapted to stability considerÂ¬
ations, this presentation is the best exposition available in the open
literature.
In Chapter VI a minimum deflection beam problem is solved with
the combined methods of finite elements and mathematical programming.
2.4 Closure
In the preceding sections of this chapter, general problem
types and methods are discussed. Only those methods that appear to
have attained some standard of acceptance are presented. It must be
acknowledged that other areas of important study exist but are perhaps
overlooked as not being pertinent to the general subject area of the
dissertation. As an example, Dorn et al. (1964) treats the optimal
layout of trussesâ€”an important subject but not related to the general
problem to be considered in this dissertation. In addition only
elastic structures have been considered although there are numerous
42
publications on optimal design of inelastic structures. References
that are representative of this subject are: Drucker and Shield
(1957a, 1957b), Hu and Shield (1961), Shield (1963), Prager and Shield
(1967), and Mayeda and Prager (1967).
On considering the various references mentioned above it would
appear that there are two possible pitfalls in structural optimization
that should be avoided. The first is the confusing of method of optiÂ¬
mization with the solution techniques employed to obtain a solution to
the resulting TPBVP. In order to avoid possible errors the two should
be dealt with independently, unless it is clearly advantageous to
relate one to the other. Besides this it must be recognized that any
solution obtained is "optimal" only with respect to the given condiÂ¬
tions of the particular problem. Any change in the problem statement
invalidates the applicability of that solution. The change may lead to
a more desirable solution, but the original solution is no less valid.
Simitses (1973) is an example where this situation is not acknowledged.
In this paper the thickness of a thin reinforced circular plate of specÂ¬
ified weight and diameter is determined such that the average deflecÂ¬
tion due to a uniform load is minimized. An earlier paper which did
not include stiffening is cited with the implication that the optimum
solution for the unstiffened plate is not correct. The point made above
is that both of these solutions are optimum under the respective condiÂ¬
tions of the two problems. Neither solution is more, or less, valid
than the other.
CHAPTER III
THEORETICAL DEVELOPMENT
3.0 Introduction
This chapter contains the development of two distinct methods
used in the theory of optimal processes, into a more general method.
The first section defines precisely the problem to be considered.
This includes the necessary conditions for an optimal solution given by
the calculus of variations. Several mathematical programming techniques
are described in the second section along with a numerical algorithm
called the gradient projection method. The application of this numerÂ¬
ical method to the solution of the necessary conditions from Pontryagin's
Maximum Principle (PMP) is detailed in Section 3.3. Results of this
approach are shown to be consistent with the necessary conditions, given
in Section 3.1; these results provide a clarifying insight to the mathÂ¬
ematical processes entailed in the maximum principle, and an explicit
formulation for the Lagrangian multiplier functions. This explicit forÂ¬
mulation is used in the next section to show the necessary conditions
may then be regarded as an algorithm. The final section contains a brief
summary of solution methods.
The main theoretical development of the dissertation is contained
in the first three sections. It is well known that the problems encounÂ¬
tered in the calculus of variations are equivalent to the optimization of
43
a functional (in the sense of mathematical programming problems) under
certain restrictions upon the variations. A good exposition of this is
available in Luenberger (1969). With this equivalence in mind, it is
noted that the PMP is itself worded as a constrained optimization probÂ¬
lem. When treated with what is normally regarded as a numerical method,
the gradient projection method, an explicit formulation of the attenÂ¬
dant Lagrangian multipliers is obtained. This form satisfies all of the
calculus of variation necessary conditions and allows one to use them
in a most straightforward fashion. As a result, these necessary condiÂ¬
tions may be directly used in the form of an algorithm to obtain a soluÂ¬
tion. Furthermore, it is believed that treating the PMP as a mathematÂ¬
ical programming problem in conjunction with the gradient projection
method helps to explain the effect of combined controlstate constraints
upon the maximum principle.
3.1 Problem Statement and
Necessary Conditions
A general problem which represents a large class of structural
optimization problems is treated in the sequel. The functional
tF
J = / L (x,u) dt (3.1.1)
0
is to be minimized with respect to the control u.(t) where the state
x(t) must satisfy certain boundary conditions and a differential conÂ¬
straint; in addition, an inequality constraint involving both the state
and control must be satisfied. For
45
uT(t) = [u (t) uâ€ž(t) ... u (t)] (3.1.2)
â€” l Â¿ m
xT(t) = [x^Ct) x2(t) ... *n(t)] (3.1.3)
the subsidiary conditions to minimizing the cost function J are:
x = f_(x,_u) (3.1.4)
Specified Boundary Conditions on x(tz) (3.1.5)
(J>Â£ (x,u) <0 l = 1,. ..,q (3.1.6)
Terminal time t is considered to be constant; allowing it to be
r
unspecified requires only a slight modification to the following
derivation.
This problem is a particular form of a very general one treated
by Hestenes (1966). His results are a set of necessary conditions which
must be satisfied by the optimal solution and include the maximum prinÂ¬
ciple. To obtain the necessary conditions, the inequality constraints
are converted to equality constraints in the manner of Valentine (1937).
These constraints and the differential constraints are then adjoined to
the cost function via Lagrangian multiplier functions p (t) and p.(t)
A/ 1
respectively.
<^Â£(x,u) + s^(t) = 0
J.
where the slack variables are defined such that
The symbol "="
denotes "is defined by."
46
sÂ£(t) = [Â£(x,u) ] 2 > 0
tp tp t
J = / Ln(x,u)dt + / p (t) [xf(x,u)]
0 U 0
tF
+ / PÂ£(t) [uA(x,u) + s~(t)] dt
dt +
Implied summation convention is used whenever a vector formulation leads
to possible ambiguities in later developments. Integrating the second
integral by parts gives a result that leads to the variational Hamiltonian.
J
0 0
[Iv
T
El +
dt
Define:
H(x,u,Â£) =
Lq(x>H.)
 P (t) f(x,u)
(3.1.7)
the variational Hamiltonian, and H which will include terms arising
from the inequality constraint.
H =  H 
or
H = Â£T(t) _f(x_,u)  Lq (x,u)  yÂ£(t)
Hence
J =
T
Â£. Ãœ
J F EH* + xTp
0
dt
With the exception of the maximum principle, all of Hestenesâ€™ necessary
conditions are obtained from the requirement that the first variation
of the cost function vanish. In the following, "Sjx" designates
47
"the variation of _x"5 a subscript vector designates the partial derivaÂ¬
tive with respect to that vector, with the result itself a column vector.
Thus,
Ã“J = J3 6_x
tp tp T *
 / [6x (H  Â¿) +
0 0 
T *
+ Ã³u H
â€” u
yA2SÂ£6SÂ£] dt = Â°
To derive the PMP requires an extensive mathematical development and is
not included since it contributes nothing to the present discussion.
However, the necessary conditions are listed in order to be available
for later reference.
x = H = f (x, u)
 _p_
*
0 = p ,<5x. , t = 0 ^
1 1 i
0=p.6x. , t = r l
i F f
Specified Boundary Conditions on x_^(t) ,
0 = H"
jj
0 = u^Ct) > PÂ£(t) > 0
d OPT â€™ ^OPT â€™  11 ^OPTâ€™ â€” â€™ ^
The optimal solution must satisfy these six conditions together with the
inequality constraint (3.1.6).
48
The PMP states that along the optimal trajectory, each instant
of time t, state x^ (t) and adjoint state p_(t), treated as fixed, the
optimal control u^p^Ct) is that admissible control which minimizes the
variational Hamiltonian. In the present context, admissibility requires
that _u(t) be piecewise continuous, the set of admissible controls being
denoted by Q. Hence the PMP indicates that
UQPpCt) = ARGMIN [HCxQpp, u , p)] (3.1.8)
uefi
Notice that the necessary conditions suggest nothing about how
a solution is obtained, but merely indicate certain functional relationÂ¬
ships that must be satisfied. However, equation (3.1.8) seems to intiÂ¬
mate that solution of the necessary condition of PMP involves a matheÂ¬
matical programming problem.
3.2 Mathematical Programming:
Gradient Projection Method
Having shown that the PMP from the calculus of variations
approach to an optimization problem may perhaps be related to a matheÂ¬
matical programming problem, the latter will be discussed in general
terms. Consider a nonlinearly constrained optimization problem
^OPT
= ARGMIN [F(x)]
x e Ã2
subject to
gj (x) < 0 j = 1 m
where Ã2 denotes the set of admissible state components x^, i = l,...,n,
and to be admissible requires only the satisfaction of the m inequalities.
49
Necessary conditions which x^p^ must satisfy are given in the KuhnTucker
theorem :
(i)
(Ãœ)
(iii)
constraints are satisfied
multipliers exist such that
and for all j = 1,...,m
m
and
V(W
E
j=l
?j ^OPT^
A.
J
A. V
3
j sj (^opt
8 j ^OPT
< 0
> 0
= 0
= 0
Observe that if I. denotes the set of indices associated with active
A
constraints, the first two conditions may be written as
3 Â£ IA gj (x) = 0
and
A .
3
j i I, g. (x) < 0
A J
and
A .
3
Fox (1971, pp. 168176) presents a very readable proof of this theorem;
a more mathematical proof using vector space concepts is available in
Luenberger (1969).
Many methods for obtaining a numerical solution to the nonlinear
programming problem described by the first two equations of this section
have been developed. The gradient projection method by Rosen (1960) is
used frequently in structural optimization. Basic to the method is the
orthogonal projection of the cost function gradient onto a subspace
defined by the normal vectors of the active constraints. An inherent
part of the algorithm is the concept of a "feasible," "usable" direction.
Any direction d is feasible if an increment x in that direction improves
the cost function, i.e., decreases F(x). Direction d is said to be
usable if it also satisfies the constraints. As long as a feasible,
50
usable direction exists, the cost function may be improved. A constrained
optimal solution x^pp occurs at that point where no feasible direction
is also usable, i.e., any attempt to improve the cost violates a conÂ¬
straint. In Appendix B these concepts are used in a concise proof of the
KuhnTucker conditions.
Fox (1971) derives the matrix P which projects the cost function
gradient into the subspace defined by vectors normal to the active conÂ¬
straints. This is equivalent to subtracting all components parallel to
vectors that are normal to surfaces of active constraints from the negaÂ¬
tive gradient of the cost function. Recalling the definition of set I ,
consider r constraints to be active such that
1^ â€” , â€¢  â€¢ ,n^_}
Define a vector whose elements are the corresponding nonzero
Lagrangian multipliers, and another vector whose elements are the active
constraint functions
A = [A A ... A
â€” a. aâ€ž a
12 r
n " tg g ... ga ]
12 r
From the N vector, a matrix N is introduced, each column of whiÂ£h is the
gradient of an active constraint. Hence, N is an (n x r) matrix where
T
N = N
â€”x
and
3N,
N. . =
ij 9x.
1 =
[N. .]
ij
5gai
9x. J
i
i = 1, . . . , n
j = 1,â€¢â€¢â€¢,r
(3.2.1)
51
With these definitions of A_ and N, the third KuhnTucker condition can
be written as
V F(x) + N A_ = 0
At any feasible point x where g(x) < 0, the direction which best
improves the cost function is the negative gradient of the cost. If
those directions which lead to constraint violations are subtracted from
V^F(x), the projection matrix P is obtained. Directions causing a
constraint violation are specified by the gradients of active constraints,
T 
i.e., the columns of N^. What is required of S, the projection of the
gradient, is that
S = (V F(x))  NT A (3.2.2)
_x â€” â€”x â€”
where A_ are scalar coefficients to be determined such that S is ortho
T
gonal to each column of N^, or
T T *
(N ) S = 0
â€”x
T .
When the matrix equivalent to is used together with the S expresÂ¬
sion (3.2.2), the result is
NT(vxF(x)  N X) = 0
such that the which satisfies this orthogonality condition is:
A =  (NTN) 1 N1(V F(x))
â€” x â€”
(3.2.3)
Unless
dent,
the active boundary surface normals (x)
T
the matrix (N N) is nonsingular. Conversely,
are linearly depen
if this matrix is
52
singular the active constraints are not linearly independent; however,
this is not a condition encountered in most real cases.
Substitution of the _X expression into the S equation leads
directly to the projection matrix P:
S = P V F(x)
x â€”
where
P = I  N(NTN)1 NT (3.2.4)
where I is the identity matrix. The direction S which best improves
the cost is given in terms of P, where P and N are given by (3.2.4)
and (3.2.1). If no constraints are active at a point then N is a
null matrix, P reduces to an identity matrix, and the direction of best
improvement is coincident with the direction of steepest descent.
In the algorithm associated with this method the starting point
must be a feasible point where g^. (jc) < 0 for all j = l,...,m. The
design then proceeds in the S direction until the solution is satisfied
to within a specified position tolerance e. Necessary conditions
generally programmed in a computer program are:
isp
* E,
i
1, . . . , n
A .
J
> o ,
j
Â£
XA
A.
3
= o ,
j
i
*A
It is readily seen that for S, P, and _A defined as above, these are
completely equivalent to the KuhnTucker conditions.
53
3.3 Gradient Pro jection Method Applied
to the Maximum Principle
Based upon the preceding discussion, the similarity between PMP
and the'mathematical programming problem can be discussed. The maximum
principle states that the optimal control minimizes the variational
Hamiltonian with respect to all admissible u_. Or, at each time
0 < t < tp, .Hqpp minimizes H(jc,ii,p) with respect to u for given _x and j)
and where _ti is subject to constraints p < 0, Â£ = l,...,q.
Treating this as a mathematical programming problem, the following
correspondences are recognized
y: 'v _u
F(x) ^ H(x,u,jp)
gj (x) ^ Â£ (x,u)
A. ^ p (t)
J 1
V F (x) H
â€” U
S ^ H plus constraints
_u
Continuing to identify corresponding quantities, at each time t, let
I. denote the set of active constraints, taken to be r in number.
A
1^ â€¢ â€¢ â€¢ j
Then
N1 'v Ã©T = [4 Ã³ ... 4i ]
a, aâ€ž a
12 r
at  PT = [p (t) P (t) ... P (t)]
â€” a, ctâ€ž a
12 r
N ^
^ aj
3x.
L J 
j = 1,. . ., n
j l,...,m
54
Furthermore, define (H )p to be the gradient of H with respect to u
where all components that cause a constraint violation have been removed
Since projection matrix P removes cost function gradient components that
lead to constraint violations, consider its use in the maximum principle
T
With the correspondent to N identified as
and
P  1  T^1 4. T
â€” u â€” u
(3.3.1)
(H ) = P H (3.3.2)
u P _u
From the KuhnTucker conditions, this implies that along the
optimal trajectory (t.^pp, jj0PT)
Â§ =  (Hu)p = 0 (3.3.3)
Similarly, at each time
1*00  (A T JÃÃV1 1 T Hu (3.3.4)
u u â€”
from which it follows
(i T & M + 1 T Hu = 0
u â€” u â€”
â– *u * + Hu  0
(H + /<{>) = 0
u
Or,
(  H )
0
h'â€˜
u
0
(3.3.5)
(3.3.6)
55
Hence the control law from Hestenes' necessary conditions can be derived
from the PMP condition by treating it as a nonlinearly constrained mathÂ¬
ematical programming problem. While using the gradient projection method
in the derivation, it is seen that equation (3.3.5) is equivalent to the
third KuhnTucker condition. The second KuhnTucker condition is idenÂ¬
tical to Hestenesâ€™ necessary condition on the Lagrangian multipliers used
to adjoin the inequality constraints to the cost function. Satisfaction
of the inequality is implied by requiring the first KuhnTucker condition
to be fulfilled, where
>*
(â€”1.
IV
o
> o
(3.3.7)
Ajg.Cx) = o
py (t)Â£(x,u)
= 0
(3.3.8)
g.(x) < 0
< o
(3.3.9)
Thus by treating the solution of the necessary conditions of the maxÂ¬
imum principle as a programming problem with inequality constraints,
using the gradient projection matrix, and by requiring satisfaction of
the KuhnTucker conditions, an explicit formula for Hestenes'
Lagrangian multiplier functions has been derived. It is further demonÂ¬
strated that with the pÂ£(t) so defined satisfaction of the extremum conÂ¬
trol law condition is implied. However, before this treatment can be
accepted as valid, it must also be shown that the system of canonical
differential equations is unchanged.
Consider the state system equations
*k
x = H = f(x,u)

56
where
Â«U rp rj1
H = p_ (t)_f (x,u)  Lq(x,u)  jj_ (t)Â¿(x,u)
It is obvious that the explicit form of y(t) has absolutely no effect
upon the state system equation expressed in canonical form.
Demonstrating that the adjoint system equation is unchanged
requires the method described by Bryson et al. (1964). Consider the
general problem of Section 3.1 again, but with only the differential
constraints adjoined to the cost function, i.e.,
Min {J =
u
(H  xTp.)dt}
(3.3.10)
subject to:
(3.1.6)
where
T
H(x,u,Â¿) = LQ(x,ji)  2. (t)_f(x,u)
(3.1.7)
and
X = f (x, u)
(3.1.4)
Again let I. denote the set of indices associated with r active
A
constraints at any time t
ta ~ {ai3
2â€™
,a }
r
< 0 > & ^ TA = Â°
4>^= 0 > Â£ E IA > yÂ£(t) > 0
and jj>_ is defined as before
â€¢ <Ã>0 ]
r
â€œl â€œ2
The problem can then be thought of as minimizing (3.3.10) subject to
(j>(x,u) = 0
57
While on the constraint surfaces defined by this equation the variations
in control 6_u(t) and state <5x(t) are not independent but instead are
related through the subsidiary requirement that
6^(2Â£,_u) = 0
or
<5x%T (x,u) + Ã³uT<}>T(x,u) = 0
X â€” â€”U â€” â€”
(3.3.11)
This imposes a restriction to the admissible variations. For cost funcÂ¬
tion (3.3.10) to be a minimum, it is necessary that its first variation
vanish, i.e.,
F
F
, T
, T T
â€ž T
+
Ã³x H
+ 5u H 
 6x p
0
0
â€” X
â€” u

<5J = p 6 x.
It has already been shown that
dt = 0
(3.3.12)
H* =  (H + yT) = 0
u u
which will be used to advantage shortly, after having added and sub
T T
tracted the term 6u (_U 40 from the integrand of (3.3.12).
0 = p 6 x
ÃœF tF
+ Ã
0 0
Ã³x^(H â€” p) + Su H +
â€” x â€” u
+ SuT(pTÂ¿)  6uT(pTD
â€” cu IU
dt
Rearranging terms gives
0 = p o >c
fcF fcF
0 0
+ / 6xT(H  p) +
+ 6uT(H + (yTj.) )  5u1(pTÂ£)
â€” u ^ u â€” L u
dt
58
But,
and
Hence,
(H + (yT<{>) = (H + yT!) = (hâ€œ) = 0
u ... â€” u u
(lLTi)u = ^ Â£
0 = R <5x
i F F
T. â€ž T T
_^o
+
o
6x (H â€” j>) â€” 6u y
dt
It is here that the restrictions imposed by the active constraints are
applied; from (3.3.11)
T T T T
 ou
â€” Hi â€” â€”x
such that
0 = jp. 6x
0 = ja Ã¡^x
+ / Ã“XT(H  V) + 5xT y
0 0 L   
dt
tF Ã¼f f
+ /
6xT(H  p + 4>T y)
â€” x x â€”
dt
0 0 L
Applying Euler's lemma, for arbitrary variations in the state which
satisfies the constraints.
(Hx â€œ P + Â£) =
0
which by the following manipulations is shown to be the adjoint system
equation of Hestenes.
Â¿ = Hx + (y Â±)x
 (H  V ^
R =  H
X
59
Thus, the explicit formulation for y_(t) obtained by applying the gradient
projection method to the PMP satisfies all the necessary conditions of
Hestenes.
It may happen that in some cases the constraint upon control
does not depend upon the state. It can be shown that the _p(t) explicit
formulation is equally valid in this instance. By examination of
equations (3.3.1) through (3.3.9) it can be verified that all the necesÂ¬
sary conditions except the adjoint system equation are satisfied. To
demonstrate the latter, recall that when on a constraint boundary the
first variation of both the cost functional and the constraint function
must vanish. That is, for
$(u) = 0
both
5 J = 0
and
6~ = 0 (3.3.13)
â€” â€” â€”u
To derive the desired equivalence, the same term must be added and
subtracted from the integrand of 6J as before, again arriving at the
result
Wien the constraint variation (3.3.13) is introduced into this last
equation, then by Euler's lemma
0 = p_ <5x
6xT(H
 p) 
* T T
6u * p
(H  p) = 0
x
60
Since it was stipulated that jÂ£(u) is not a function of x, the equation
may be written
(H + yV>x  Â£ = 0
(  H* )x  Â¿ = 0
Thus, the expression for _p(t) is valid when the constraint inequality
depends only upon the control ji(t) .
3.4 Maximum Principle Algorithm
In the introduction to this chapter it was stated that the
Lagrange type problem from the calculus of variations is equivalent to
an ordinary mathematical programming problem based on the KuhnTucker
conditions. Furthermore, itfhen inequality constraints are present the
necessary conditions are equivalent to the KuhnTucker conditions.
It was demonstrated in the preceding section that if the PMP is itself
treated as a mathematical programming problem, application of the
gradient projection method provides an explicit solution for the
Lagrangian multipliers associated with active constraints. This explicit
solution for 1^(0 also satisfies all of the other necessary conditions
for an optimal solution. The ability to determine p^(t) explicitly
in terms of parameters and functions that describe the problem suggests
the possibility of converting the necessary condtions of an optimal
solution into an algorithm for obtaining it.
61
Ensuing discussion of the algorithmic form of the necessary
conditions contains the implicit assumption that all equations are valid
along the optimal trajectory. It is further assumed that the problem
under consideration is that one described in equations (3.1.1  3.1.6).
The algorithm requires that jx(t) and ]3(t) be known at each time
0 < t < tâ€ž for which the solution procedure is as follows.
(i) Use PMP on the variational Hamiltonian to determine an optimal
JL
control _u (t) independent of constraints.
u"(t) = ARGMIN [H(x,u,p)]
Evaluating the inequality constraints with jj = jj reveals which of
the Â£ = 1,...,q constraints are active. Let r denote the number
of active constraints and I. the set of indices associated with
^ = 0 1 e IA
? (x,u) <0 a i lA
From this the vectors whose elements are the nonzero Lagrangian
multipliers and corresponding constraint functions are defined,
respectively, at the instant of time t.
(ii)
P (t) = [u p ... p ]
a a a
12 r
(x,u) = [
al a2
Having identified which of the q constraints are
components of u^ are specified by jj>_(x,u) = 0.
active, r
They may be
solved by using the implicit function theorem, which requires
62
T
â€”u
which are active at point _x(t) to be linearly independent.
To determine the remaining (mr) components of u^pT requires
that p be known at time t, but
y (t) =  (i T 1
â€” â€” I u â€” i u
u â€” u â€”
This value of _p is used to determine the "constrained" HamilÂ¬
tonian,
H* =  (H + uTÂ£)
k
(iii) With the nonzero Lagrangian multipliers jj known and H conseÂ¬
quently defined, the remaining (mr) unknown components of
are determined from the control law for the constrained
system, i.e,, from
*
H = 0
ij
Once ÃœQp.p is completely known, the adjoint system equations
are determined by
JL
â€¢ o
Â£. = ~ H
x
The process outlined above then allows UqPP to be written as
uQpT = ARGMIN [H(xopT,u,Â£)]
u eSJ
since the _u obtained in this fashion satisfies <>Â£ (x,_u) Â¿ 0 which is the
only requirement for being admissible. However it must be recalled
that these equations are valid along the optimal trajectory; it remains
to be shown that this algorithm may be employed in some manner to obtain
that optimal trajectory and to demonstrate their satisfaction along it.
63
3.5 Solution Methods
Necessary conditions from the calculus of variations provides
a Two Point Boundary Value Problem (TPBVP) to be solved, which is in
general, nonlinear. For all but the most simple problems no analytical
solution is possible and if any solution is to be obtained a computer
must be used with some numerical method. A discussion of the available
methods and their relative advantages/disadvantages is not included here
due to the availability of such discussions in the literature, e.g.,
Bullock (1966). All of the methods involve some iterative scheme, and
for optimal control can be separated into two general categories.
(i) Indirect methods. Schemes which require an intial guess of
the state's solution: In these methods the starting point is
an initial guess of the time history of the solution. The conÂ¬
trol associated with the solution is a subsequent calculation.
Iteration continues until the state satisfies some criterion
connoting convergence; the final control history at converÂ¬
gence is the optimal control.
(ii) Direct methods. Schemes which require an initial guess of the
control function: The starting point for these methods is an
initial guess of the control time history. For this class of
methods the state associated with the control is a subsequent
calculation. Iteration continues until the control satisfies
some convergence criterion.
The method of quasilinearization was selected, based upon the
success of Boykin and Sierakowski (1972) in applying it to a constrained
64
structural optimization problem. Excellent convergence for their
problem, the capability to handle nonlinear systems, and the availÂ¬
ability as an IBM SHARE program, ABS QUASI, dictated its selection.
In the application to the examples in Chapters IV and V the program
required no modification. As a result, a detailed discussion of the
method of quasilinearization is not included.
The problem discussed in preceding sections of this chapter
falls into the general class of problems that QUASI handles, that is,
1 = Â£(X>t)
with the boundary condition of the form
BÂ£Y(0) + Br Y(tF) + = 0
where t_, square matrices B and B , and vector C , are specified,
r 3c r Q
constant quantities. The specific form of B^, B^ and depend upon the
given boundary conditions. As described in algorithm form
â€¢ J
p = h^(2Â£>R0PT(x>R)) = G2(x,p)
In terms of the general QUASI nomenclature,
G1 (*>Â£.)
G2
Y =
Â£(Y,t) =
Boundary conditions are determined by those specified for the original
system and by the necessary conditions outlined in the first section
of this chapter.
In Chapter VI the problem treated by Boykin and Sierakowski (1972)
is solved by the gradient projection method applied to a finite element
65
formulation for the description of the structural system. This is
a method of the second kind mentioned above. Results of the two methods
are compared.
CHAPTER IV
CONSTRAINED DESIGN OF A CANTILEVER BEAM
BENDING DUE TO ITS OWN WEIGHT
4.0 Introduction
A structural optimization problem has been selected for its
simplicity and stated as an optimal control problem. The maximum prinÂ¬
ciple is applied, giving a nonlinear TPBVP of the Mayer type. Among the
earliest expository papers on the maximum principle, RozonoÃ©r (1959) gives
an excellent treatment to a similar type of problem; his technique is
used to obtain both the variational Hamiltonian and adjoint variable
boundary conditions. It is shown that no finite solution exists for the
situation of unconstrained control. Numerical solutions for constrained
control are obtained by the method of quasilinearization. Constraints
include both geometric limitations to control as well as maximum stress
limits that become mixed constraints depending upon both state and conÂ¬
trol variables.
4.1 Problem Statement
A cantilever beam of variable rectangular cross section is to be
designed for minimum tip deflection due solely to its own weight. The
material is specified to the extent that the modulus E and density p are
constants. Length L is specified but the design variables, height h(x)
and width w(x), may be chosen independently of each other, subject to
66
67
hard constraints upon the allowable dimensions. That is
a < w(x) < c
(4.1.1)
b < h(x) < d
If Y(x) denotes the deflection of the centerline, the problem is:
given E, p, L, and the constraints, find h(x) and w(x) to minimize
Y(L). The particular form of differential constraints to be satisfied
will be derived in the next section.
4.2 Structural System
Small deflections are assumed in order to use linear Bernoulli
Euler bending theory. Basic conventions assumed for this example are
depicted in Figure 4.1; with these conventions the governing equation
is derived using standard strength of materials considerations. The
result is
where
L
Mg(x) = y/ (tx)w(t) h(t)d r
x
and y = pg. Kinematic boundary conditions to be satisfied by the soluÂ¬
tion of (4.2.1) are:
Y(0) = 0
Y' (0) = 0
68
Nota: Y(x) is centerline deflection, positive downward.
Figure 4.1 Structural Conventions
69
Design variables and the related constraints (4.1.1) are put into
dimensionless form such that
v = b/d < v < i
id 1
and
w(x) .
= â€”â€” > a/c < vâ€ž < 1
2 c 2
VX) ' 12 Cd3 V1V2
Replacing the independent variable with a dimensionless equivalent, and
using the control components allows the governing equation to be put
into a dimensionless form. For
x
 L
~(^)2 uu?Y = / (xt) u1(T)u2(x)dT
(4.2.2)
where
U.(t) = v.(x(t))
When constant C is defined, the usual kinematical relationships for
B
a beam may be written in a simple dimensionless form; that is, let
c = JL _JL, (Ã¡)2
b 12 yl2 V
(Units = Length B
xi = cby
X2 = CBY
Deflection
Slope
X3 = C8U1U2Y
x4 â– CB at (uiu2Y)
Moment
Shear
U1U2 = CB 77
dtz
Load
70
These state component definitions are used with the natural boundary
conditions to obtain
x^Cl) = 0
V1} = 0
From (4.2.2), the state component definitions, and the above boundary
conditions it follows that
x, = xâ€ž
xâ€ž =
*3/u2U2
= X,
x, =
ulu2
x1(0) = 0
x2(0) = 0
x3(l) = 0
x^(l) = 0
These equations and boundary conditions are used in the following section
to precisely state the problem. The solution and results are given later.
4.3 Unmodified Application of
the Maximum Principle
In terms of the state variables defined in the preceding section,
the problem can be stated with more mathematical precision. Find
UQpjCt) = ARGMIN [x1(l)]
UÂ£Ã1
subject to:
(i)differential constraints
(ii)kinematic boundary conditions
natural boundary conditions
X = _f (x,_u)
x^CO) = x2(0) = 0
x3(l) = x^(l) = 0
(iii)hard geometric constraints
b/d < u^(t) < 1
a/c < u2(t) < 1
71
According to terminology in the calculus of variations this is a Mayer
type problem. Among the early papers concerning the PMP, RozonoÃ©r (1959)
applies the PMP to a similar problem giving a geometric interpretation
to the function of the adjoint variables.
In RozonoÃ©r's problem the cost is a generalization of the
ordinary Mayer problem, in the sense that the cost function is a linear
combination of the terminal state components. It can be shown via the
calculus of variations that to minimize
J
where c is a vector of prescribed constants, the necessary conditions are
â€”r
H = Â£T(t)f_(x,u, t)
x = H
f
Â£
x
0 = H
u
0 = /(0) x(0)
0 = Â£ (tpMllj. + Â£^tp) ]
T
n^OPTâ€™ ^OPT
p,t) > H(xnpT, u, Â£,t)
That is
Min[J] Max [H]
u
u
72
If we consider the cantilever beam problem, the forms in the
necessary conditions are
c = [1 0 0 0]T
â€”r
H = P;Lx2 + P2x3/uu2 + P3x4 + P^U1U9
from which
"N
f(x,u) /
x3/ulu2
VU1U2 J
pi = Â°
Pl(l) = 1
P2 = "Pi
P2(l) = 0
p3 = P2/u3u2
P3(0) = 0
P4 = p3
p4(0) = 0
Adjoint variables p^(t) and p2(t) can be integrated by inspection
pL(t) = â€œI
P3(t) =  (1t)
0 < t Â¿ 1
(4.3.1)
such that
H =  x2  (lt)x3/uju2 + p3x4 + P4U]Lu2
73
With this result, the necessary condition for control to minimize tip
deflection is
Hux = 3(lt)x3/u^u2 + p4u2 = 0
Hu2 = (lt)x3/uU2 + p4ux = 0
At first this appears to be a contradiction since the two equations can
be satisfied only by the trivial solution because they have equivalent
forms,
P4U1U2 = 3(1t)x3
4 2
P4U1U2 =
(lt)x.
(4.3.2)
Further examination, however, leads to the conclusion that when the
control is completely unconstrained there is no horizontal tangent
plane to the surface H = H(u3,u?).
When the geometric constraints to the control are included,
a constrained minimum may exist. If such is the case, the maximum
value of H occurs on the boundary of admissible control space. To that
end, PMP is employed along the control space boundary to determine
ucipT(t) at each time t. Before detailing this procedure, it is necesÂ¬
sary to first consider some structural aspects of the problem.
By definition the control components are positive, which in
turn implies
Load:
â€”4
>
0
f
U1U2
Shear:
X4
<
0
<
h *
0 ,
x4(l) = 0
Moment:
X3
>
0
X3
0 ,
x3(l) = 0
74
Furthermore, since p2(t)  0 from (4.3.1)
P3 > 0 e p3 > 0 , P3(0) = 0
p4 < 0 P4 < 0 > P4(Â°) = 0
This exercise makes it possible to use the information of the sense for
x3 and p^ to simplify the search for u^^ on the control boundary. By
arranging the Hamiltonian in the following fashion
H = x2 + p^ + p4[u3u2  (lt)x3/p4u3u2]
it is observed that both terms in the bracketed expression are positive.
This and the p^ outside the leading bracket allows the following
equivalence:
Max [H]
u e 9U
or,
^OPT
where
*(u)
Hir, [UjU2  (lt)x3/p4uu2]
US 3U
ARGMIN [$(u)]
u e 3U
[U;Lu2 + F2(t)/u3u? ]
and
F2(t) =  (lt)x3/p4 > 0
(4.3.3)
At each position t along the beam, the state and adjoint variables must
satisfy the appropriate differential equations, and u. is specified by
the preceding three equations.
Control space boundary SU is illustrated in Figure 4.2, where
the Up axis is treated as the ordinate since Up(t) and u.(t) correspond
75
b/d < < 1
a/c < u2 < 1
1 "
b/d
a/c
Figure 4.2 Admissible Control Space
76
to the height and width, respectively, of the cross section of the
beam at position t.
Along the constant u edges of 3U, let u, = u where u has the
1 1 c c
value of either b/d or unity. If $(uc,u2) has a minimum point
d$
duâ€ž
= 0
and
d2$
du2
> 0
where
$(u ,u ) = u u + F2(t)/u3u
c 2 c 2 c 2
= u  F (t)/u3u2
du0 c c 2
d~
du2
2F2(t)/u3u3
c z
The value of which satisfies the first condition is
u = F(t)/u2
2 c
Furthermore, it is observed that only one extremum of $(u) exists along
U, = u and that it is a minimum. Hence, either $(u ,uâ€ž) has a minimum
I c c Â¿
on the constant u^ edge or is monotonically decreasing/increasing. If
either
* *
u^ < a/c or 1 < u^
then along the constant u edge, H has its maximum value at a corner of
the rectangular 3U. On the other hand, if
a/c < u2 < 1
then H has its maximum value on the line u,= u interior to the
1 c
endpoints.
77
Similarly, along the edges of 3U, denote
has the value of either a/c or unity. If $(u.,u ) has
1 c
= u where u
c c
a minimum point
where
d2$
and â€”2" > 0
du
1
u,u + F2(t)/u2u
1 c 1 c
u  3F2(t)/uju
c 1 c
12F2(t)/u^u
1 c
It is observed that has only one extremum along u2 = uc5
it is a minimum, and occurs at the point where
U;L = + { 3F2 (t) /u2 }
Thus, by the same argument posed in the preceding paragraph, if either
k k
u^ < b/d or 1 < u^
then along the constant u? edge, H has its maximum value at a corner of
the rectangular 3U. Wherever
b/d < u^ < 1
H has its maximum value on the line uâ€ž = u interior to the endpoints.
2 c
78
On the basis of these arguments, the following system was
solved by the method of quasilinearization:
â– Â¿2
*3
*4
X2
x1(0) = 0
= x3/uu2
x2(0) = 0
il
X
x3(l) = 0
= U1U2
x4(l) = 0
= P2/u];u2
P3(0) = 0
= "P3
P4(0) = 0
OPT = ARGM1N ^uiu2 ~ (l_t)x3/P^uu2^
u e 9U
The beam is represented by 100 intervals composing the range 0 < t < 1,
which is separated by 101 "mesh points." An initial guess of the
solution x(t) and p(t) is chosen; it is selected to satisfy the boundÂ¬
ary conditions. This guess is not a solution and does not satisfy the
differential equations. The x and Â¿ equations are linearized about the
initial guess, then the resulting linear TPBVP is solved to obtain new
x(t) and p(t) functions which more closely satisfy the differential
equations. At each time t corresponding to a mesh point, II is numerÂ¬
ically evaluated along each of the four straight line segments comÂ¬
posing 3U to determine u^^. The point (u^,u^) on 9U which gives
H(u.;x,p,t) its maximum value is â€¢ This process is repeated until
the Ct) and _p(t) iterate satisfies the differential equations to
within a specified tolerance. The equations necessary to use the IBM
program available are given in Appendix C, in the form of a subroutine
listing.
79
4.4 Results: Geometric Control Constraints
For the most part, no major difficulties were encountered in
using quasilinearization to obtain a solution to the sixth order sysÂ¬
tem derived in the previous section. Certain parameter values did
engender numerical instability. These cases, the source of the diffiÂ¬
culty, and its circumvention are discussed in Chapter VII. Moreover,
all calculations were done in double
precision as necessitated by matrix
inversion accuracy requirements.
Parameter values selected to
illustrate the solution method are:
u : b/d = 0.25
(4.4.1)
u2 : a/c = 0.20
The measure of error of satisfaction of the differential equations in
the TPBVP is in terms of the general system
dY .
â€”^ = f. (Y,x) Y = Y. (x),
dx i â€” â€”i
i = 1,.. . ,n
ERROR = Max 1dY.  f.(Y,x)dx
1 l l â€”
X
(4.4.2)
Deflection of a uniform beam due to its own weight was used to infer
an initial guess xihich satisfies all boundary conditions:
xx(t) = t4
x2(t) = t3
x3(t) = 1  t2
0 < t < .1 (4.4.3)
x^(t) = 1 + t
P3(t) = t2
P4(t) = ~t3
80
With these specified parameter values and initial guess of the soluÂ¬
tion, the program converged to a solution in five iterations. From
this run a tolerance was selected for all subsequent cases; the followÂ¬
ing tabulation provides the data used in its selection:
Iteration ERROR Tip Deflection (Cost)
1
.2028
.7387749327
X
10â€œ1
2
.1704
.3192152426
X
10â€œ!
3
.6533 x
10â€œ1
.2847993812
X
10â€œ1
4
.3031 x
10â€œ 5
.2853731846
X
10_1
5
.1129 x
10â€œ10
.2853719983
X
101
It is seen from these tabular data that there is little improvement in
cost (tip deflection) as a result of the fifth iteration. For this
reason a value for the tolerance was selected as 0.5 x 10 Â° which correÂ¬
sponds to about six significant digits in the cost functional.
R.ecall from the previous section that no unconstrained minimum
exists. With the control bounds included, the intuitive solution is
one in which the crosssectional area is maximum near the root, and
reduces to a minimum at the tip. Recalling that for the optimal control,
IV Max [H] 5 Min [ ]
â€”OPT n
_u e 0 ti e Q
A sequence of illustrations in Figure 4.3 demonstrates the location of
Uqpt on for several stations along the beam. Constant contours of
<5(u) are plotted on the admissible control space at five distinct posiÂ¬
tions. If an extremal point exists interior to 3U some lines of constant
<&(u_) contours must be closed curves in jjspace. This is impossible for
this example.
81
O Minimum $(u.) uopT
â–¡ Maximum Ã¡>(u)
Direction of Increasing $(u)
Figure 4.3 Contour Plots of $(u) at Various Stations
Along the Beam
82
The first illustration is for the t = 0.1 cross section, near
the root of the beam. Since _u^p occurs at the point of minimum $(vi),
the optimum value for both u^ and u^ is unity, the maximum allowable
dimensions for both height and width. Constant contour lines indicate
that $(u) is mathematically decreasing in either direction of _uspace.
Lines of constant $ (u3 are also plotted for the cross section of
t = 0.3. The optimum control has the maximum admissible value for
height but has a value somewhat less than unity. However, there
are still no contour lines which are closed curves.
At the midpoint cross section the minimum $(u) point occurs
at Up = 1 and u^ = a/c. Although the surface $(_u) forms a scooplike
shape, there are still no closed curve contours, and hence no extremal
interior to admissible _uspace. The next cross section at which ?(_u)
is displayed occurs at t = 0.7. On this section, is still at its
lower bound but u^ is no longer at the maximum allowable value of unity
as shown in Figure 4.3. In the last of the sequence, $(u) contours for
the cross section at the tip of the beam are displayed. The point of
minimum i>(u) occurs where both components of control have their minimum
allowable values. Again, no contour lines of constant $ (_u) form a
closed curve indicating the existence of an interior extremal point.
This sequence of illustrations indicates two things. First,
the lack of closed curve contour lines of $(u) verifies that exists
on 3U. With further study it may be possible to obtain some condition
on = 0 which implies the equations corresponding to (4.3.2) can
never yield a finite, unconstrained optimum. Such a condition would
define the class of structures whose unconstrained solution is the
83
"zero volume solution" frequently described in the literature on strucÂ¬
tural optimization. Secondly, at t = 0 the point u^p occurs at
T
_u = [1,1] , the point of maximum crosssectional area; as t increases
from zero to one, the point u^p^ moves along the u^ = 1 boundary of 9U
to the Up lower bound, and then down the Up = a/c boundary of 3U to u^
lower bound. By the time t = 1 the optimal crosssectional area is the
minimum allowable area. As a result of the prescribed form of 8U,
if u^p,j,(t) follows this particular path as t increases from zero to one,
each component of u^p^(t) has its own distinct region of transition.
That is, at any value of t, if b/d < u^ < 1, then Up must be on either
its upper or lower bound. Conversely, if Up is in transition where
a/c < Up < 1, then u^ must be on one of its bounds.
This effect is seen most clearly in Figure 4.4, where u^p is
displayed for the example case parameter values specified by (4.4.1).
The profiles are displayed on a twoview drawing as a planform of the
beam might appear. State components corresponding to this beam are
shown in Figure 4.5, representing dimensionless deflection, slope,
moment, and shear, respectively. As observed in Figure 4.4, there are
five distinct regions of the beam:
(i) 0 < t < .25
(ii) .25 < t < .52
controls onupper bounds
a/c < Up < 1
Up transition
84
TOP VI Eli
SIDE VIEW
Figure 4.4 Planform Views of Optimal Solution for
b/d = .20 and a/c = .25
85
Figure 4.5 State Components of Optimal Solution for
b/d = .20 and a/c = .25
86
(iii) .52 < t < .59
U1 = 1
u2 = a/c
(iv) .59 < t < .90
b/d < < 1
u2 = a/c
(v) .90 < t < 1.00
u^ = b/d
u2 = a/c
on upper bound
on lower bound
u^ transition
controls on lower bounds
The curves that show the intercept locations as a function of
parameter values b/d have been called "correlation curves" in earlier
studies, liben the width is allowed to vary also, the second parameter
a/c is introduced. For the sake of comparison to previous studies, the
intercept/correlation curves are plotted as dependent upon b/d and
parametric in a/c. However, it would be just as correct to do the
opposite.
Intercept location curves described above are shown in
Figure 4.6. The heavy black curve is the case where a/c = 1, a beam
of constant uniform widthâ€”the case cited from earlier literature.
Another special case is represented by dashed lines, corresponding to
a/c = 0 which is the case corresponding to a minimum allowable thickÂ¬
ness equal to zero. Dashed lines are used because these data are an
extrapolation: convergence problems encountered for parameter values
less than 0.1 prevented obtaining numerical results.
88
A discussion on the convergence difficulties experienced by
the quasilinearization algorithm for parameter values approaching zero
is presented in the chapter on numerical instabilities. In that disÂ¬
cussion, isolation of the source of difficulty is reported; it is posÂ¬
sible that this difficulty may be a general result applicable to all
problems to be solved by the method of quasilinearization. A soluÂ¬
tion for this case is later obtained by finite element techniques.
Note that since 0 < a/c < 1 these two cases represent limits to the
solutions of the problem. In addition, if the four intercept locations
are plotted versus a/c and parametric in b/d, curves r^ and r^ appear
as "horizontal vees" with r. and r, lines that are nearly parallel.
b a
It is interesting to note from the figure depicting the soluÂ¬
tion of this case as a planform, that in the central region of the
beam, the height is greater than the width. This result can be anticÂ¬
ipated since such a configuration gives a greater bending resistance
per unit weight.
With further reference to Figure 4.4, the transition of u* (t)
is seen to be almost a linear taper, whereas the u^(t) transition
exhibits a much more pronounced curvature. To generalize from this
specific case of given values of b/d and a/c to arbitrary values
requires the introduction of four quantities characterizing the soluÂ¬
tion. These quantities are the values of t at the points where the
transitions intercept the bounds on u^ and u^; since t represents a
normalized position x/L, these quantities can be thought of as an
intercept location expressed as percent of the beam's length. They
89
are defined with reference to the five distinct regions of the beam
previously given, where
r^ designates u^(t) intercept with lower bound
designates u^(t) intercept with upper bound
r^ designates u^Ct) intercept with lower bound
r^ designates u^Ct) intercept with upper bound
such that the five regions are:
(i)
0
<
t
<
r
c
control on upper bounds
(ii)
r
<
t
<
r
uâ€ž transition
c
a
2
(iii)
r
a
<
t
<
rd
control on upper/lower bounds
(iv)
rd
<
t
<
rb
u^ transition
(v)
rb
<
t
<
1.0
control on lower bounds
4.5 Inequality Stress Constraints
This section treats an inequality limit to allowable normal
and shear stresses associated with bending. Using the ordinary
strength of materials formulations it can be shown that for the recÂ¬
tangular cross section these constraints take the form
1 Vx)
2 I (x) h(x)  Â°MAX
Â£>
and
i Vx) . 2..
SOrf â€ (x)  TMAX
Jd
When dimensionless quantities are introduced
6 kr X3',â€œiU2  â€œmax
2 YL x4/ulu2 â€œMAX
90
the inequalities may be written in the required form for mixed conÂ¬
straints, i.e., as a function of both control and state components:
^(Xju) = x3/u^u2  < 0 (4.5.1)
4>2 ^ = â€œ x4y/uiu2 " tq  0 (4.5.2)
where
1 gMAX ,d
6 yL
2 TMAX
3 yL
The two stress constraints place restrictions upon the minimum
crosssectional dimensions to keep the normal and shear stresses less
than prescribed values. Specifically, from the constraints (4.5.1)
and (4.5.2), two control inequalities must be satisfied at each station
t, and these inequalities depend upon the state of the structural sysÂ¬
tem. The inequalities are:
ulu2 : X3/o0
uiu2;
 VT0
x4(t) < 0
from which can be derived boundary arcs in uspace:
%
ula(u2} = (x3/O0U2)2  0
U1t(u2) = " X4/T0U2  Â°
(A.5.3)
Both of these boundary arcs are hyperbolas restricted to the first
quadrant of uspace. Depending upon the location of the arcs, visavis
the rectangular 3U, inclusion of stress constraints has one of three
91
effects in determining what _u is admissible. At any station t in some
structural state x(t),
(i)if Â°q5tq is too small the stress boundary arc lies entirely
above rectangular 3U; all geometrically admissible u violate
the stress constraints.
(ii)if Gq,Tq is too large the stress boundary arc lies entirely
below rectangular 3U; all geometrically admissible ju satisfy
the stress constraints.
(iii)for some range of Oq,Tq the stress boundary arc divides the
rectangular 3U into two regions: the upper region consists of
geometrically admissible ju that satisfy the stress constraint,
u in the lower region are geometrically admissible but violate
the stress constraint.
Inclusion of stress constraints alters the admissible control
space from the rectangular shape previously considered to a shape that
may contain a stress boundary arc as part of its boundary. Consider the
normal stress boundary arc specified by (4.5.3) to be a part of OU.
Then to find u^p in the manner outlined in Section 4.3, 0(ii) must be
evaluated along u^ = u^C^). If a minimum exists along the orthogonal
projection of u^Cup) on t^ae Â®(n) surface, then
Min
U1
(30(u)
320(_u)
^ 3u2
0
0
92
Along the normal stress boundary arc U2a^u2^â€™
U1U2 = X3/O0 " U2 = X3/0OUl
Substituting for U2 in'$(_u) it follows from (4.3.3) that
$(u)
(x3/oQ + F2(t)a())/u;L
Since the expression in parentheses is positive, it is readily observed
that the value u^ = + â€œ minimizes $(u) along u^^iu^). From (4.5.3), the
value of U2 at this point in jaspace is zero.
The same result exists for (_u) evaluated along u^_; on the
shear stress boundary arc
ulu2 â€œ  VT0 * u2 "  VVl
Substituting for ^ in (11) it follows from (4.3.3)
$(u)
Ul = (_X4/t0) + (_x4/t0)_1 p2(t)/ui
lx
that
Since both the expression in parentheses and Fz(t) are positive, it can
be argued as above that the point of minimum Â®(_u) along occurs
at the point (u, = + 00, uâ€ž = 0) .
At any cross section there exists a single minimum of 0(u) along
either stress constraint boundary arc. Since these minima occur at the
point u: (+Â°Â°,0), as one proceeds along either boundary in the direction
of increasing u^,
reason, along the stress constraint boundary arc, it is necessary to
evaluate $(u) at only one point. That point is the intersection of the
stress boundary constraint arc with the rectangular boundary having the
93
larger value of u^. It is possible that the two stress boundaries
intersect, with the coordinates of that point given by
Even though this adds a complication to the boundary of what is admisÂ¬
sible _u, the monotonically decreasing property of both stress boundary
arcs still results in the necessity of evaluating $(u) at only a single
point along the stress boundary arcs. To illustrate this, Figure 4.7
depicts an admissible control region determined by a combination of
geometric constraints and a composite stress constraint boundary,
together with the point at which $(u_) is a minimum along the composite
stress boundary. With continued reference to the figure, u^ is that
point along line ABC which minimizes Â§(u_).
4.6 Results: Stress Constraints Included
For the purpose of comparison, the parameter values used in the
case with only geometric constraints were also used in the cases where
stress inequality constraints are present. The results are quite simply
stated: neither stress constraint is ever active. In general it was
found that the critical section occurs at the root. At that section
the greatest portion of geometrically admissible control space is proÂ¬
hibited by the stress constraints. The further along the beam towards
the tip, the less the prohibited geometrically admissible control space.
a/c 1
U2
Figure 4.7 Behavior of $(jj) Along Composite Stress Boundary
VO
95
Both and x^ decrease monotonically to zero at t = 0, and is the
reason for this phenomenon.
Each of the parameters and thus has three important
critical values. For the first pair of values the maximum allowable
stresses are sufficiently large that all geometrically admissible
control contained by the rectangular 3U satisfy the stress constraints.
In this case
oQ = 9.128
tq = 8.658
and the stress constraints intersect the lower lefthand corner of 9U
at t = 0 and lie below 9U for all t > 0. For the range of values
.5705 < aQ < 9.128
2.1645 < t0 < 8.658
the stress boundary arcs intercept the u^ = .2 portion of rectangular
9U. If they are exactly equal to the lower values, respectively, both
arcs pass through the point (1.,.2) at t = 0, and lie below that point
for all t > 0. However, these values are not as important as either
the first pair or the next. For the range of values
.1141 < oQ < .5705
.4329 < tq < 2.1645
both arcs intercept the u^ = 1 portion of rectangular 9U. When equal to
the lower values both arcs pass through the upper righthand corner of
3U at t = 0. If the values of
96
oQ < .1141
tq < .4329
then at t = 0 there is no control which satisfies both geometric
constraints and stress constraints. The problem has inconsistent
constraints such that no solution is possible.
A sequence of plots showing the admissible control space is
shown in Figure 4.8 for six separate stations along the beam. The
value of Oq is .1141 for this case such that in section t = 0 the set
of admissible controls is the single point jj: (1, 1). That portion of
the geometrically admissible control space disallowed by the stress
constraint is indicated by dashed lines. Additionally, the point corÂ¬
responding to is included for each section. By observing the
sequence of plots as t increases, both the stress boundary and ate
seen to move closer to the origin of _uspace. However, never overÂ¬
takes the stress boundary, thus the normal stress boundary never
becomes active. Figure 4.9 depicts the same case for the shearing
stress constraint.
Two additional figures are included with the intermediate
critical values of and t . The normal stress constraint case
Oq = 0.5705 is shown in Figure 4.10; shear stress constraint case
Tq = 2.1645, in Figure 4.11. No illustrations are provided for the
cases associated with the largest critical values since for t > 0 all
geometrically admissible controls satisfy the stress constraints.
t
.3
t = .5
Figure 4.8 Admissible Control: o =
o
â€¢^OPT
J
.1141
t = .1
I
L_
t = .7
t = .3
= .4329
Figure 4.10 Admissible Control: a =
o
VO
VO
.5705
Figure 4.11 Admissible Control: tq
U2
= 2.1645
100
CHAPTER V
CONSTRAINED DESIGN FOR AN OPTIMAL
EIGENVALUE PROBLEM
5.0 Introduction
A column is to be designed for maximum buckling load in order
to demonstrate (i) an application of the theory developed in Chapter III,
and (ii) an optimal eigenvalue problem. Difficulties pertinent to this
class of problems are discussed as well as a method to obtain a soluÂ¬
tion by using a theorem concerning selfadjoint problems. Use of this
theorem with the maximum principle overcomes the difficulty associated
with the simultaneous dual optimization required for problems where the
cost functional is an eigenvalue expressed as a Rayleigh quotient.
Numerical solutions to the combined set of state and adjoint variables
are obtained by the method of quasilinearization. Characteristics of
the solutions for a variety of control bounds are presented, along with
a discussion of attempts to also include a mixed, inequality constraint
related to stress.
5.1 Problem Statement
A vertical column, fixed at the base and free at the tip, is to
be designed such that the buckling load is maximized. Length L is
a specified constant as is the total weight W. The dependence of cross
sectional area A(x), material modulus E(x), and density p(x), upon
101
102
position x along the column is to be such that the vertical end load P
at buckling is maximized. Weight is not to be neglected and all cross
sectional shapes are similar.
In more specific terms, a given weight of material is to be
arranged with variable properties and geometry such that the resultÂ¬
ing mass distribution M(x) and stiffness distribution S(x) maximizes
the vertical tip load the column can support when the effect of weight
is included. Upper and lower bounds to all design variables
< A(x) < Ay
El 1 E(*) < Eu (5.1.1)
\  P(x) < RU
are also specified. Governing equations for the structural system are
derived in the next section.
5.2 Structural System
In the subsequent analysis, small deflection bending theory is
assumed: the result is a linear differential equation with variable
coefficients. Without this assumption, the governing equation is that
of the nonlinear elÃ¡stica problem. Additionally, axial compression
effects are neglected as second order. Let Y(x) denote the deflection
of the centerline at position x along the beam as depicted in Figure 5.1.
Since the system is conservative, the governing equations can be
derived quite simply by energy techniques. In terms of stiffness S(x)
and mass per unit length, the equation and boundary conditions are:
103
Vx)
Figure 5.1 Structural Conventions
104
(S(x)Y")" + PY" + g(Y' / M(OdÂ£)' = 0 (5.2.1)
C=x
Y(x) = Y'(x) = 0 x = 0
(S(x)Y") = (S(x)Y") + PY' = 0 x = L
Stiffness and mass distributions for the column are
S(x) = E(x)I(x)
M(x) = p(x)A(x)
where I(x) is the second moment of area about the neutral axis.
It can be shown that assuming similar cross sections provides
a relationship between I(x) and A(x). If f(x) represents the depenÂ¬
dence of one dimension of the cross section with the position along the
beam, then for some reference area A^,
A(x) = Aq f2(x) (5.2.2)
Choosing a reference area moment Iq related to A^ through a radius of
gyration for the crosssectional shape such that
ro  koAo
it can easily be shown that
ko
I(x) = â€” A2(x)
A dimensionless form of the differential equation is obtained
by introducing the following dimensionless quantities
105
t =
x/L
n =
Y/L
a(x) =
A(x)/Aq
e(x) =
E(x)/Eq
P(x) =
R(x)/Rq
m(x) =
M(x)/MQ
s(x) =
S(x)/SQ
where Ag, E^, designate some reference value and
Mo =
poAo
II
O
CO
Vo
and
m(x) =
a(x)r(x)
s(x) =
a2(x)e(x)
Design variables and the given geometric constraints (5.1.1) are also
converted to dimensionless form:
V 1
a(x) < ay
eL 
e(x) < e
rL; p
where the transformed bounds have been divided by the appropriate referÂ¬
ence value. Replacing variables in (5.2.1) by their dimensionless
equivalent gives
106
dt"
(s(t)n) + An + k ^f / m(Â£)dÂ£;j= 0
n = Ã± = 0
(s(t)n) = â€” (s(t)n) + xfi = 0
t = 0
t = 1
where
(5.2.3)
X =
PLZ
*0*0
k =
E0I0
Integrating by parts and using the given boundary conditions,
equivalent formulations to (5.2.3) are obtained which are used to
convert the governing equation to a state component representation.
Equivalent formulations are:
1
(s(t)h) = X[n(l)  n(t)] + k / m(0[n(0  n(t)]dÂ£
S=t
, 1
â€” (s(t)h) =  Xf)  kfi / m(C)d^ (5.2.4)
Â£=t
â€7 (Ã± 1^ (s(t)h)j = km(t)
Let
x = n Deflection
y*2 = Ã“ Slope
= s(t)q Moment
= Ã± ^ (s(t)h)
Shear * Slope
107
then, from this choice of state components and the last of the
equations (5.2.4) it follows that
Â¿2 = x3/s(t)
X3 = X2X4
x, = km(t)
4
To complete the transformation of the structural problem to an optimal
control problem, the design parameters are treated as the components
of control u(t) where
u1(t) = a(x(t))
u2(t) = e(x(t))
u3(t) = r(x(t))
Hence with the definition of control ja(t) and the representation of the
systemâ€™s state by a fourdimensional vector, the governing equations are
X1 X2
x0 = xj
u?u.
3 1 2
xâ€ž = xâ€žx.
3 2 4
x. = ku.,uâ€ž
4 13
x^CO) = 0
x2(0) = 0
x3(l) = 0
x^(l) = A
(5.2.5)
To complete the conversion of the structural problem to an
optimal control problem, the specified weight condition must be conÂ¬
sidered. In terms of the specific mass distribution M(x), that condiÂ¬
tion is
108
or
L
W = g / M(x)dx
0
1
w = P A gL / m(t)dt
This introduces yet another dimensionless parameter to the system in
that
1
1 = y / u (t)u (t)dt
0
where
y
poAo
w
gL
With this last development, the buckling problem can be stated
with more mathematical precision. Find
uQpT = ARGMAX [x4(l)] , x4(l) = A
U E Ã!
subject to:
(i) differential constraints
(ii) kinematic boundary conditions
natural boundary conditions
(iii) hard geometric constraint
(iv) hard material constraints
(v) subsidiary constraint
x = f(x,u)
x1(0)
= 0
x2(0)
= 0
x3(l)
= 0
x4d)
= A
aL 
UpCt) <
a
U
V I
CD
u2(t) <
e
U
r, <
u0(t) <
r
L 
3
U
1
n /
0
Ulu3dt .
= 1
109
A cursory examination of this problem statement reveals the inherent
difficulty of optimal eigenvalue problems. If the eigenvalue X is the
cost functional, the cost functional appears in the governing equation
and/or the boundary conditions. For the state components selected,
the eigenvalue X appears only as a boundary condition. However, as
stated in (5.2.3), the eigenvalue appears in both the equation and
boundary conditions.
5.3 Analysis of the Problem
In the preceding section the governing equation for column
buckling is expressed as either a second, third, or fourth order equaÂ¬
tion with appropriate boundary conditions. State components are then
defined from structural quantities, one of which is the shear divided by
the slope. Since the slope vanishes at t = 0, it must be shown that
this state variable, x^(t), is not indeterminate at the point in quesÂ¬
tion. Moreover, in order to use the theoretical techniques developed
in preceding chapters the first eigenvalue (cost) must be expressed as
some functional of the state and design parameters.
According to Bolotin (1963, p. 22), conservative systems are
described by selfadjoint boundary value problems. Classical elastic
stability theory is conventionally restricted to conservative problems
in which the buckling load is the fundamental eigenvalue. It is shown
in what follows that the column buckling problem is selfadjoint and
that the eigenvalue may be obtained via the total potential energy of
the system from the Rayleigh quotient. Sufficiency for x^(t) to be deterÂ¬
minate at t = 0 is that the problem be a selfadjoint eigenvalue problem.
110
Consider a dependent variable in the thirdorder representaÂ¬
tion of the governing equation (5.2.4) and appropriate boundary conÂ¬
dition. If
z = n
another valid representation of the system is
_d_
dt
(sz) + (A+k f m(5)dÂ£)z = 0
0
z = 0 ,
s(t)z = 0 ,
t = 0
t = 1
Using the notation of Lovitt (1924), who treats the general problem
(pu') + (q 4 Ar)u = 0
(puu')
0
= 0
the corresponding quantities are
u nÂ» z
x ^ t
p(x) n, s ( t)
1
q(x) ^ k / m(5)d?
t
r(x) ^ 1
When written as a linear differential operator L( ) and its adjoint
4. 4.
A A
L ( ), the operator L( ) is said to be selfadjoint if L ( ) H L( ).
If the TPBVP is to be selfadjoint for any admissible function
cp and ip, then by Green's Formula
Ill
b j, b
/ {ipL(Â»i/j)
a a
0
where B(4>,^) is the bilinear concomitant of L( ). In terms of a general
second order equation
L() = Pj^" + P2
l W = (Pj'i')"  Cp2^)' + (q + Ar)^
The necessary and sufficient conditions that L( ) be selfadjoint is
that p2 = p. If L( ) is a selfadjoint operator and the bilinear
concomitant vanishes, the problem is said to be selfadjoint. For the
general operator,
b
B(
= p1((J)> 
a
or in terms of Lovitt's equation,
operator,
which obviously has a selfadjoint
= p(<}>'il;  (pip')
1
0
it is seen that the problem is selfadjoint if the boundary conditions
of the adjoint and given systems are identical.
There is a welldeveloped theory associated with selfadjoint
eigenvalue problems, where the eigenvalue problem itself is the result
of minimizing an energy functional. Along this line, Lovitt shows that
under the requirements that p(x) > 0 and is piecewise smooth in (0,1),
and that q(x) is a wellbehaved function which is finite everywhere on
(0,1), there are an infinite number of real positive eigenvalues which
may be arranged as
112
0 < A1 < A2 <
< A <
n
If p(x) and q(x) are specified functions which satisfy the above conÂ¬
ditions, the fundamental eigenvalue is given by
A = Min [D(u) ]
u(x)
1
D(u) = / {p(x)(u')2  q(x)u2} dx
0
where
u e C2
puu' = 0
I 0
and
1
/ r(x)u2dx = 1
0
2
Lovitt shows for u e G and p(x), q(x), r(x) bounded, that A^ is finite.
Higher ordered eigenvalues are obtained by using the orthoÂ¬
gonality property of eigenfunctions as additional subsidiary condi
For example,
A
n
Min
â€žn(x)
[D(u )1
n
1
D(iO = / (p(x)(u^)2  q (x) u2 } dx
tions.
113
for
u e C2
n
.1
pu u' =0
n nlo
1
/ r(x)u2 dx = 1
0 n
and
1
I r(x)u u.dx = 0 i = 1,...,nl
0 n 1
The immediate use of the selfadjoint property of the column
buckling problem is that the fundamental eigenvalue is finite. This
is used to prove that x^(0) is determinate. Recall the definition
x
4
.1
n
From the thirdorder formulation of the governing equation (5.2.4)
Id 1
H â€œjj: (sri) =  A  k / m(Â£)dÂ£
t
such that
1
x, (0) =  A  k / m(Â£;)dÂ£; =  3
0
where g is a positive constant to be determined. By substituting the
specified weight subsidiary condition for the integral
x^(0) =  (A + k/p) =  6
Both k and p are specified, finite parameters of the system. If A is
finite then x^(0) is also. In fact, the buckling load \ is the
114
fundamental eigenvalue A^ which is shown by Lovitt to be finite for
the conditions prescribed. Hence, x^(0) is determinate.
In the literature of classical elastic stability problems,
the eigenvalue to be optimized is expressed as a Rayleigh quotient.
This quotient can be found from the total potential energy. For the
general equation of Lovitt, the quotient is
1
/ (p(u')2  qu2} dx
/ ru2dx
0
or in terms of the column buckling parameters and components (5.2.5)
1 1
/ (x2/s(t)  k / m(Â£)dÂ£ x2} dt
, .. 0 t
A____ 
/ x2 dt
0
On reversing the order of integraion for the double integral contained
in the numerator, and introducing design parameters, the Rayleigh
quotient becomes
1 t
/ (x2/u2u  ku u / x2(Â£)dÂ£} dt
0 1J0
A = 1
/ x2 dt
0
The normalization condition of Lovitt requires that the
quotient's denominator equals unity for all admissible solutions, or
1
/ x2(t)dt = 1
0
115
This particular normalization is selected in order to obtain the
simplest expression for the functional A. Also, recall that
x^(0) =  (A + k/p)
and
x^(l) = A
It is seen from these boundary conditions and the definition
x^ = ku^u^ that the subsidiary weight condition can be written as a
mixed boundary condition on x^(t):
x^(l)  x^(0) = k/p
This, together with converting the normalization condition to another
state component with boundary conditions specified at both ends,
transforms the problem to:
\vIax
Max {J[u]}
u e Q
1
Jtu] = / {x^/u^u2  ku^u^x^} dt
subject to
Â¿1
= X2
x1(0)
= 0
Â¿2
= x3/u2u2
x2(0)
= 0
*3
X2X4
x3(1)
= 0
= kulu3
x4(i)
 x4(0)
X5
= *2
x5(0)
= 0
x5(l)
= 1
k/p
116
where _u e means that _u Â£ C2 and satisfies the geometric control
inequalities. For reasons as yet undetermined, the quasilinearizaÂ¬
tion algorithm did not converge when the latter two state boundary
conditions were used, even though they belong to the class of problems
to which the algorithm is theoretically applicable. This might be due
solely to the mixed boundary condition being dependent on the value of
the cost functional being minimized.
On failing to obtain a solution by minimization of the Rayleigh
quotient, the next attempt was suggested by the state boundary condiÂ¬
tions. From the condition
x4(l) =  A
and definition of A, whereby A > 0, it follows that
Min [x4(l)] = Max [A]
u u
Thus, the problem can be cast as a Mayer type of optimal control problem
Vx = Mi" [x4(1)1
u e Q
X1 = X2
X2 = X3/U1U2
X3 = X2X4
X4 = kulU3
x1(0) = 0
x2(0) = 0
x3(l) = 0
X4 (0) = Â£
where B is a positive constant to be determined. The quasilinearizaÂ¬
tion algorithm did not converge for this formulation of the problem
also, again for undetermined reasons.
117
Consider from the choice of state variables that
x. = km(t) and x,(1) =  A
4 4
In an earlier derivation, g was defined such that
1
x,(0) =  A  k  m(t) dt =  g < 0
0
In another form,
1
 B + k j m(t) dt =  A
0
(Constant) + (Weight) =  (Load)
or in terms of the weight
A = B  kW
Therefore, if the weight W is minimized subject to constraints, this
is the same as maximizing A, the load, subject to the same constraints.
Notice that constant 3 is an undetermined boundary condition, i.e.,
a free condition, so that the corresponding adjoint variable's boundary
condition is known.
At this point the results of a doctoral dissertationâ€”Salinas
(1968)â€”were used in another formulation of the same problem. Salinas
considered the application of the energy method to selfadjoint probÂ¬
lems. He showed that the following problems are identical:
(i) Maximum load, specified weight
u0PT = ARGMAX {Jp[u]>
U Â£ ft
Jp = the buckling load cost functional
J = a constant, the weight
w
118
(ii) Minimum weight, specified load
UqPT = ARGMIN {Jw [u]}
li e Ã2
Jw = the weight cost functional
Jp = a constant, the load
Both problems are subject to the same differential constraints,
boundary conditions, and control admissibility constraints. A concise
way of stating the equivalence is
u = ARGMAX {J [u] : J specified}
â€”OPT _ r â€” W
u e Ã2
= ARGMIN (J [u] : Jp specified}
_u e
The equivalence of these two problems is used to state the
column buckling problem as
u^ = ARGMIN [Jw]
UEÃ1
JW = U1U3 dt
0
X1 " x2
Xl(0) = 0
x2 x3' ul"*2
,/u?u,
x2(0) = 0
X3 X2X4
.3(1)  0
x, = ku,u.
4 13
x.(1) =  A
4 s
where the subscript s on the eigenvalue indicates a specified load.
With reference to the statement of the problem as a Mayer type, notice
that
119
JW = k  x4^0^ Min tx^(l)]
u Â£ ft
with x^(0) =  3, a free boundary condition. Furthermore, since
x^(l) =  X and X =
observe that
 J(r = 3  JD
y W P
Admissibility of control (u e ft) requires that u.(t) is piecewise
continuous through the second derivative, and that the control inequalÂ¬
ity constraints are not violated. For every value Xg assigned a priori
to the eigenvalue X there is a solution (in , ) that minimizes
OPT ^OPT
the weight with respect to u e ft and satisfies the eigenvalue problem.
However, an arbitrary Xg need not satisfy the specified weight condiÂ¬
tion and in general does not. To fulfill the given total weight requireÂ¬
ment the value of X^ was adjusted such that
1
Jw 1% 3 = n/ uiu3dt = 1
OPT 0
When the cost is not only a minimum but is also equal to unity, all
conditions of the buckling problem as stated in section 5.2 are satisÂ¬
fied. By Salinas' theorem the Xg causing this satisfaction is the
maximum lowest eigenvalue, i.e., the maximum buckling load. This effect
is illustrated in Figure 5.2 and is mathematically stated by
Max
{JP[yy ]
OPT
V^W ] = 1}
OPT
= ARGMIN ÃJtI[u] : X = X }
W â€” s
UÂ£il
where
Figure 5.2 Relationship Between Optimal Eigenvalue
and Minimum Weight Cost Functional
120
121
5.4 Application of the Maximum Principle
In the preceding analysis difficulties associated with the
optimal eigenvalue problem are discussed. Circumvention of such difÂ¬
ficulties is described for those problems that are selfadjoint.
It is shown that the column buckling belongs to that class of probÂ¬
lems. However, before the theoretical developments presented in the
third chapter may be applied, the particular forms of parameters and
functions must be identified. Accordingly, for the optimization of
the column buckling load
x ^ (x, xâ€ž xâ€ž x.)
â€” 1 2 j 4
T
J ^ JW = / V*>u)dt
L0(x,u) = puiu3
%(0) = 0
xâ€ž (0)
Boundary
Conditions
s
122
(auu1)(u1aL)
Â£(2Â£>u) % \ ~ ^eU~u2^ul_eL)
(rU~U3) (VrL)
where the maximum value of the As is that value such that
1
JW = p / u u dt = 1
0
(5.4.1)
Based upon these particular forms for the general parameters,
the variational Hamiltonian is
H =  Ãp^^2 + P2X3^UÃU2 + P3X2X4 + P4kulu3^ + pulu3
k
Adjoint variables p/t) are defined from the Hamiltonian H which con
tains the constraint functions. Since the constraint functions ^ are
independent of x for the case of only geometric constraints
JL.
p =  h'â€™ = H
x x
From this result and from the state boundary conditions it follows that
P1 = 0 Pj^l) = o
P2 = ~pix2 " P3X4 p2(1) = 0
P3  p2/xu2 P3(0) = 0
P^ = P3X2 P4(Â°) = 0
Variable p^(t) may be integrated by inspection:
p^(t) = 0 for all t > 0
which immediately reduces the number of adjoint variables to be deterÂ¬
mined and simplifies the Hamiltonian as well. Thus,
123
H =  {p3x2x4 + p2x3/uâ€œUl + (kp^y^u }
where
p2 =  p3x4 P9(l) = 0
p3 =  p2/u2u2 P3(0) = 0
P4 =  P3x2 P4(Â°) = 0
Optimal control as determined by the PMP stated in (3.1.8) is
UqPT = ARGMIN [H(x0pT,u,p)] (5.4.2)
U Â£ Q
where the admissibility restriction in Pontryagin's derivation
requires only that _u(t) be piecewise continuous. Applying Salinas'
theorem further restricts the control to _u e C2. Considering for the
moment that no further constraints on the control exist, the unconÂ¬
strained UQPp is that choice of (up(t), u7(t), u3(t)) which minimizes
the variational Hamiltonian H. Recognizing that the physical quantities
represented by u have meaning only if they are positive, assume this to
be the case. With nonnegative controls, the state equations and
boundary conditions jointly imply that
x.(t) > 0 i = 1,2,3
x4(t) < 0
Furthermore, by means of (5.4.2) and the expression for H
(P2X3) > 0 > u2 = 0
(p2x3) <0 * u2 = <*>
124
and
(kp^y) >0 * u^ = 00
(kp^y) < 0 > = 0
Control up remains to be determined.
Assuming that u2>u^ are known quantities, then u
independently to minimize the variational Hamiltonian.
Hu   {2p2x3/u^u2 + (kp^y)u^} = 0
H =  {6p xZu'iu } > 0
upup 23 12
The unconstrained optimal control variable up is defined
first condition; the second condition indicates that p2(
tive for H(2copT,uopT,Â£) to be a minimum. So,
ui â€˜
2p2x3
,1/3
Â»p4v)u2u3_
and
P9(t) < 0
which leads to the following argument on the signs of pp
p2
>
0 e
P2(t)
< 0
and
P2(l)
= 0
P3(t)
>
0 Â«
P2
= "P3X4
and
x4(t)
< 0
h
<
0
p4
= 'P3X2
and
x2(t)
> 0
p4(t)
<
0
K
< 0
and
P4(Â°)
= 0
p must be chosen
In that context
to satisfy the
t) must be nega
(t):
125
It is noted that with these signs if (kp^y) < 0 the optimal unconÂ¬
strained control is
r
ui
â– ^OPT ^
V.0 y
k
where the existence of u^ is discussed in the next section. What is
important is that the solution requires a material of infinite strength
and zero density. Some structural analysts commonly refer to this
hypothetical material as Bolognium.
In the course of an actual design process, control bounds correÂ¬
sponding to the constraint functions are specified. On the basis of
the preceding analysis with given bounds on controls, optimal u^(t) and
u^(t) are bangbang controls defined by
u2(t) = %(eJ+eu) + %(eLeu) SGN (p2x3) (5.4.3)
u3(t) = %(rI+ru)  %(rL~ru> SGN (kp^y) (5.4.4)
where for an arbitrary argument z of function SGN ( )
r+l
SGN (z) = < 0
z > 0
z = 0
1
z < 0
k
Depending upon the magnitude of u visÃ¡vis a and a , there are three
.Li U
possibilities for u at each point (t,jx(t) ,Â£(t)) :
126
^OPT
â– â€œ2
^OPT "\U2 /
 â€œ2
That class of problems having no mixed constraints requires computing
the Lagrangian multiplier functions jj(t) of (3.3.4) only as a check.
Each individual multiplier function must be nonnegative for all
0 < t < 1. A positive value in a mathematical programming application
indicates that the usable, feasible direction is attempting to leave
an active constraint surface. With the similarity between this and
PMP demonstrated in Chapter III, the existence of a negative multiplier
function indicates a similar inconsistency in active constraints and
the constrained direction of steepest descent of H with respect to _u.
In the usual application of PMP to a problem, the constraints
are adjoined to the cost function and another functional defined:
H
127
Each constraint 4>Â£(x>_y) which is active such that
f (x,u) = 0
is solved for a component of control. The resulting nonzero multiÂ¬
plier function p? (t) must be determined from the necessary condition
*
H =0
_u
This p^(t) is used in turn to determine the effect upon the system
from the solution's lying on a mixed constraint surface which by
definition depends upon the state. This effect is manifested by the
JL
A
adjoint differential equation Â¿ which can be written as
Â¿ =  + ^0
If constraints ^ depend only upon the control, the second term is
T
unnecessary because ^ = 0.
Such is the case of the column buckling problem possessing only
control bounds. For this problem, whenever the constraints are active
the multiplier functions determined by the above procedure are
P1(t)
2P2x3/uu2 + (kp4y)u,
2â€ž1  CaL + O
P2(t)
2P2X3/U!U2
2u2 
P3(t)
(kp^u)u1
2u  (r +r )
3 u L
(5.4.5)
(5.4.6)
(5.4.7)
128
These same parameters will now be determined from the explicit formulaÂ¬
tion of jj(t) derived in Chapter III.
Recall that where I, denotes the set of active constraints which
A
are r in number:
IA = (a^, o^,. . . .a^}
= { ...
a, aâ€ž a
12 r
4= [v
3$,
8u.
i
i = 1,...,m
j = 1,â€¢â€¢,r
and that in terms of and H, jj(t) is defined as
1
y(t) i i Hu
\ U â€” J u â€”
(3.3.4)
For the column buckling problem subject to constraints (5.4.1),
order to calculate _p the following quantities must be defined:
r c ~\ r ~\
*1,1
0
0
*l,u= <
0
> *2,u <
'**2,2
>
(
*3,u" <
0 Ã
0
J
0
^ J
fu
ui
/'2p2K3/u3â€ž2
+
(kp/y)u3*N
H =
u 1
hu2
>  <
'2P2X3/U1U2
+
0
^HU3y
^ o
+
(kp4y)u1^
in
129
with
, = 2u 
(a +aT)
1,1 1
U L
= 2uâ€ž 
W
2,2 2
â€ž = 2uâ€ž 
(rU+rL)
3,3 3
Consider first the case where u^ is not on a control boundary.
Only the latter two constraints are active since ^ and are bang
bang controls, thus = 0 and both ^ anc^ Ã3 are determined by (3.3.4)
From the preceding paragraph's definitions
r = 2 > IA = = {2,3}
(i To
U â€”
0 *2,2 Â°
0 0
2,2
3,3.
(1 xO"1
u â€”
2
h,2
jj =
^3
0 (f>
2,2
0 0
0
0 4>
2
3,3
0
4>
3,3.
r \
H
U1
H
u2
H
Vu3y
\\ h*
2
H /
^u3 3,3
130
Hence,
_ ~P2X3/U1U2
"2 " 2ul ' (eu+eL)
(kp^y)u1
2u3 â€ (rU+rL)
which are identical to the expressions (5.4.6) and (5.4.7) obtained by
the usual method.
Whenever the constraint is active, u^ is prescribed and y^ is
nonzero. If the derivation in the preceding paragraph is expanded to
include this additional constraint, the resulting expression for y^ is
identical to (5.4.5).
The result of this analysis is a nonlinear TPBVP whose solution
is the desired optimal solution:
Â¿1=
X2
x]_(0)
= 0
X2 =
x3/ulâ€œ2
x2(0)
= 0
X3 =
X2X4
x3d)
= 0
*â€¢
ii
kulu3
x4(d
= A
s
p2 =
P3x4
p2(1)
= 0
p3 =
"P2/UÃU2
p3(o)
= 0
II
â€¢ a
P3X2
p4(Â°)
= 0
^OPT
= APvGMIN
ue!!
H =  (p^x^ + P2x3/fuiu2 +
A
Max
^JP^OPT* : JW^OPT^
1}
131
where ui e requires that ui satisfy constraints (5.4.1) and Lagrangian
multiplier functions given by (5.4.5) through (5.4.7) must be nonÂ¬
negative. A solution to this problem is obtained by the method of
quasilinearization; a subroutine listing of the equations required by
the IBM program is given in Appendix C.
One additional preliminary item must be attended to before preÂ¬
senting results. In order to provide solutions subject to comparison
standards, an eigenfunction normalization must be prescribed. Most
Rayleigh quotient problems in the open literature normalize the eigenÂ¬
function such that the denominator of the quotient is always equal to
unity. For the column buckling problem this is equivalent to
1
/ x2(t) dt = 1 (5.4.8)
0
The unmodified results of solution via quasilinearization do not
satisfy this condition; instead,
1
/ x2(t) dt = A2
0
where A is some constant. A transformation which satisfies normalizaÂ¬
tion (5.4.8) such that the differential equations, boundary conditions,
and optimal control are unaffected by the transformation, is given by
* x^/A
x2 > x2/A p2 * Ap2
x3 * VA p3 + Ap3
x4 x4 p4 P4
132
With a specified normalization of the eigenfunction, the state soluÂ¬
tion Xqp^, may be compared to any other similar solution for which the
normalization is known. All subsequent results are normalized as above
to satisfy (5.4.8).
5.5 Results: Geometric Control Constraints
Three dimensionless parameters of the system are derived in
section 5.3. The first is A, the eigenvalue to be maximized. The
second is k, which is a relative measure of the weight of a uniform
column to its stiffnessâ€”how strongly inclusion of the weight affects
the solution. The third parameter, p, is a measure of the weight of
a uniform column described by the reference conditions relative to the
weight of the column having distributed geometry and properties.
Values of k and y describe a particular problem statement and must be
specified a priori.
Since the present discussion is directed towards technique, as
opposed to results for particular geometries, loading conditions, and
materials, a single value of both y and k is used for all cases. Most
of the literature compares the optimal eigenvalue to that obtained for
a uniform cylinder of identical weight. This corresponds to a value
of unity for y, selected so that the results may be compared. ParamÂ¬
eter k involves both material and geometric properties. For convenience
the material selected is structural steel and the Euler buckling slimÂ¬
ness ratio (L/k^) value of 60 is assumed. With these assumptions the
values used throughout are
y = 1
k = .012
133
At this point a comment upon the effect that the magnitude of
k has upon the nature of Ug (t) is appropriate. In the last section
it was shown that u^ and u^ are bangbang controls dependent upon the
sign of switching functions (?2X3^ an<^ (kp^y) , respectively. IntuiÂ¬
tively one expects the strongest, least dense material to be optimal,
corresponding to
(p2x3) < 0  u2
e
U
(kp^y) <0 Â»â– u^ = rL
Also shown in the last section is that for Hamiltonian H to be minimized
JL.
A
by an unconstrained value of u^ = u^ at some time, it is necessary that
P2(t) < 0. Note that if this does not occur somewhere in 0 < t < 1
the control is uniform throughout, and the system described is a uniform
column; unless = tt2/4 the solution of the state is the trivial soluÂ¬
tion.^ A direct consequence of p2 being nonpositive is that p^(t)
is also. Hence any positive value of k is sufficient to make (kp^y) <0,
so that the minimum density is required.
This insignificant requirement that a nontrivial, optimal soluÂ¬
tion to the buckling problem exist provides an unusual result. When
the variable signs required for a nontrivial solution are used in the
PMP expressions (5.4.3) and (5.4.4), the intuitive solution is shown
mathematically to be the optimal solution. It is for this reason that
all subsequent cases were run with control bound values
^Coding errors produced precisely this case, and indeed,to
within the numerical limits of the computer the algorithm converged
to the trivial solution.
134
eu = 1Â°
e = 0.8
JLj
ru = 1'1
r = 1.0
J_i
Unity values were selected to simplify results; nonunity control bound
values were included to reveal unforeseen switching in and u^. None
were encountered. Effectively, the problem as stated has a single
control component u^(t) = a(x/L).
To illustrate the operation of the quasilinearization algorithm
detailed results are presented for one case. Data for this case are:
a = 0.9
â– Li
(5.5.1)
Just as in the cantilever beam example, the initial
from the uniform solution and must satisfy only the
Initial guesses are in terms of
C(t) = cos (%7Tt)
0 < t < 1
S(t) = sin (%rrt)
guess is
boundary
taken
conditions
such that
x1(t)
x2(t)
x3(t)
x4(t)
x5(t)
x6(t)
x7(t)
= ~ (lC(t))
7T
= 2S(t)
= C(t)
=  X  (lt)k/p
S
â–  Ã cÂ«
=  s(t)
TT
135
Given this initial guess, the value of A^ which satisfies the
weight condition is A._ =2.75. This is demonstrated by the follow
Max
ing values of A^ and which represent specific points on a curve
like that of Figure 5.2:
A JT7
s w
2.74108 0.998S19
2.74396 0.999210
2.75000 1.000000
For A>r = A such
Max s
that J__ = 1
W
the convergence
is measured
in terms of
ERROR as defined in Section 4.4
, and shown in the following tabulation:
Iteration
ERROR
Component
(Position)
Cos
Jw
â€¢f
JP
0
.696*10"2
*2(o)
1.00000
2.75
1
.151*10~2
x2(.630)
1.00017
2.75
2
.128*10"4
x^(.555)
1.00003
2.75
3
.250*10â€œ12
x2(.550)
1.00000
2.75
Deflection of the
centerline is
shown in Figure
5.3 which
contains
three curves: the initial guess, a plot of subsequent iterates (coinÂ¬
cident for the scale used), and the normalized solution. The optimal
control is displayed in Figure 5.4. Control functions for iterations
subsequent to the initial guess plot coincident to a single curve.
All nonzero components of the state and adjoint variables are shown
in Figure 5.5. Shear (represented by the product x^^) a*so included
to give a complete set of structural variables.
13b
tte*
ates
Soluâ€˜
ciÂ°e.
Guess
* 0.9
53
T5/Ã>ic
al
tlec
tlÂ°a
o5
CeÃ1
tet
Figure 5.4 Typical Optimal Control Function
137
138
Figure 5.5 Typical State and Adjoint Variable Soltuions
139
Notice that for the case where density and modulus are equal
to unity, the condition on specified weight simplifies to
1
JTT = / u, (t)dt = 1 (5.5.2)
w 1
where implicitly y = 1. As the value of increases, it reaches an
upper limit of unity: values of a^ larger than unity violate the
weight condition (5.5.2). Thus, for no upper bound on u^ there are two
limiting cases associated with lower bound values,
a = 0 and a = 1.0
L Ji
Values of a not contained in this range automatically violate some
Jj
given condition. Each value of 0 < a < 1.0 has a corresponding optimal
eigenvalue A^ arid design profile u., (t) a, a(x/L) . Several of these
1
profiles are shown in Figure 5.6. In all but the uniform column limitÂ¬
ing case for a =1.0, the optimal profile has a transition region
* *
ere = up The point where decreasing u^(t) intersects the lower
wh
bound a occurs at a time designated t . Optimal control is on the
Li iJ
lower bound for t > t .
Li
A comparative measure for the effectiveness of the optimizaÂ¬
tion is the ratio of A., to the eigenvalue for a uniform column of
Max
equal weight. This latter eigenvalue is the classical Euler buckling
load denoted A,, and corresponds to the case a = 1.0. Thus,
h Li
> Vl ii
E" Vo ' 4
The dependence of the effectiveness ratio A^^/A^, and the lower bound
intercept tT upon aT is shown in Figure 5.7. As is typical of simple
Li Li
Figure 5.6 Optimal Area Distributions Parametric in Lower Bound:
No Upper Bound
140
141
\,a /A*
Max n
CINT
Figure 5.7 Effectiveness Ratio and Lower Bound Intercept:
No Upper Bound
142
problems of this type, the most improvement occurs for the smallest
excursions from the uniform case (.6 < a < 1.0). To allow a to be
"" Li ~ L
reduced from .6 to 0 provides much less improvement in the cost.
Observe that the maximum value for a(x) occurs at t = 0 for
a^ = 0, and is 1.333. Thus for a^ > 1.333 no optimal profile in
Figure 5.6 is constrained by the upper bound on u^(t). For a^ = 1.333,
at one point t = 0,
a(0) = a^
and
a(t) < au t > 0
Therefore, instead of being unbounded in (5.5.1), it is only necessary
that a^ > 1.333. An a^ this large is essentially unbounded for the
given weight constraint.
Results similar to those above have been obtained for similar
problems by other methods. The next examples are for a case not yet
reported in the open literature. These cases could not be obtained
without specifying both upper and lower control bounds. The first
example has a lower bound specified with a range of various upper
bounds, decreasing to the limiting case of a uniform solution. The
first example of this case is
1.0 < afJ < 1.333
Since the nature of the solution methods differ only in details, no
solution methods are presented. Optimal profiles are presented for
various values of a^ given a^ = 0. In Figure 5.8 the limiting cases
Figure 5.8 Optimal Area Distributions Parametric in Upper Bound:
Lower Bound a =0.0
iu
143
144
are shown with three intermediate cases as well. One case is displayed
as a dashed line, since it is obtained as an approximation. As a^*1.0
the numerical procedure becomes unstable for reasons discussed in
Chapter VII; the case for a^ = 1.05 must approach the uniform cylinder
associated with a^ = 1.0. Because the transition region where
a(x/L) < a^ is approaching a vertical straight line in the limit as
a^ >â– 1.0, the approximation is taken to be:
a(x/L) = ay 0 < x/L < ty
a(x/L)
aV j
t^ < X/L < l
That value of t^ which satisfies the specified weight condition under
the assumed form of the approximation is given by
t
U
1
No further use is made of this approximation; its sole reason for
existence is to provide an additional curve in Figure 5.8 to more effecÂ¬
tively demonstrate the transition from the case with no upper bound to
the uniform beam case.
The characteristic form of solutions is indicated by a plot of
tjj versus a^ shown in Figure 5.9 together with the effectiveness ratio
A., /A^. Just as observed in the case possessing only a lower bound,
Max E
when only an upper bound is present the first excursions from a uniÂ¬
form column (1.0 < a^ < 1.2) provides the greatest cost improvement.
In the preceding case, although only t^ is presented, the
implication that tT does not exist is incorrect. By observation of
Li
145
AMax/AE'
Figure 5.9 Effectiveness Ratio and Upper Bound Intercept:
Lower Bound a. =0.0
146
Figure 5.8 and the definition of t it is obvious that for all cases
l_i
t = 1.0. This occurs as a consequence of selecting a = 0.0; in order
Li Lj
to demonstrate a more general behavior of the solution process, a nonÂ¬
zero lower bound is selected in order to provide a nonconstant t^.
The third configuration of the column buckling problem to be
considered is for the control bounds
aL = 0.3
1.0 < ajj < 1.327
where the upper limit to a^ is such that for a^ > 1.327 no upper bounds
are ever active as u^ is essentially unbounded from above. Optimal
profiles for the limiting and three intermediate cases are shown in
Figure 5.10. The dashed curve is an approximation composed of three
straight line segments (numerical instability was again encountered as
the uniform column limiting case was approached). Mathematically, the
approximation is
a(t)
U
< Ã cu
(t tu)
' aÂ« â– (vl>
t <
t
"L
such that the case which satisfies the specified weight condition
requires the control bound intercepts of the approximation to satisfy
'U
+ fcL =
1 
aTT 
a
a
L
L
2
Figure 5.10 Optimal Area Distributions Parametric in Upper Bound:
Lower Bound a =0.3
Li
147
148
From an extrapolation of intercept values associated with higher values
of a in Figure 5.10, t is found to be about 0.985. The corresponding
U ii
value of t is thus 0.882. Just as in the preceding case it must be
emphasized that this approximation is included only to more clearly
illustrate the transition of a(x/L) from one limiting case to the other
as a^ decreases from 1.327 to 1.0.
Effectiveness of design optimization is revealed in the plot
of A,, /Aâ€ž versus a_T (see Figure 5.11). Characteristic solution forms
Max E U
are also depicted there in the plot of t and t^ dependence upon a^.
It is to be noted that tT deviates little from the value of unity but
very definitely has a functional dependence upon a^.
5.6 Inequality Stress Constraints
Having obtained solutions for various control constraints, the
next problem configuration attempted involves a mixed constraint.
Initially it was decided to restrict the maximum allowable normal
stress due to bending:
K(X)I < aMax 0 < x < L
Recognizing that for any cross section the maximum value of Og(x) occurs
at the "outermost fibres," the distance d(x) of these fibres from the
bending axis must be determined. On the basis of assuming similar cross
sections, the function f(x) appearing in the A(x) expression (5.2.1)
also determines d(x), i.e.,
d (x) = dQf(x) = dQaLi(x)
149
Figure 5.11 Effectiveness Ratio and Control Bound Intercepts:
Lower Bound a^ = 0.3
150
where d^ is some reference value. In terms of BernoulliEuler bending
equation, the inequality constraint on bending normal stress is
laB(x)
(x)d(x)
Vâ€
<
 o Max
(EIY")d(x)
Vx)
<
 a Max
E(x) d(x)Y"
<
 a Max
By introducing dimensionless variables and squaring to remove the
absolute value signs,
e2 (t)a(t)ri
<
a
2
Max
which is manipulated into the desired mixed constraint form when written
in terms of state and control variables:
0
where
(5.6.1)
L o
Max
Eodo
Besides satisfying the geometric constraints, the u^ control
component must also satisfy a state dependent constraint
ul(t) > (x3/o0)2/3 > 0
Results for this case of mixed constraints are not presented. To be
a valid demonstration, the optimal control ought to be constrained by
the mixed constraint (5.6.1) on at least part of the interval 0 < t < 1.
151
In the present example, for small values no solution exists that can
fulfill both the stress and geometric constraints. Large values of
result in all geometrically admissible control simultaneously satisfyÂ¬
ing the mixed constraint. Intermediate values produce solution iterÂ¬
ates in the quasilinearization process that were constrained by (5.6.1)
over some interval. However, the final iterate u^^Ct) is not conÂ¬
strained by the mixed constraint at any point 0 < t < 1. For this reaÂ¬
son no mixed constraint examples are presented. Theoretical considerÂ¬
ations of other possible stress constraints are given in the following
paragraphs.
A more realistic stress constraint is to limit the stress
component o(x) normal to a cross section, composed of a term Og(x) due
to bending and a term Op(x) attributed to the end load P. With referÂ¬
ence to the definition of 0 shown in Figure 5.1, a(x) and its constituÂ¬
ents are:
o(x) = oB(x) + cr (x)
Og (x)
PL (x)d(x)
"Vr
ap(x)
P cos 6(x)
A(x)
Differential calculus and trigonometry provide two identities that are
used to evaluate cos (0) as a function of x in terms of the slope of
the centerline. They are
dY
= tan 0
dx
tan20 + 1 = sec20
152
From this,
cos 0 = < 1 +
(gj
2
Thus, the normal stress inequality becomes
M (x)d(x) r
a(x) =  + f h +
IB(x) A(x) ^ \dx /
<
Max
By introducing dimensionless state and control variables, after
much manipulation the following mixed constraint inequality is obtained
, k,3 , a
=  (up + 
0
"<(u?> + x3/a0
Three parameters exist for this derivation,
L a.
an =
Max
0 Eodo
the Euler buckling slimness ratio
â€” a, a geometrical slimness ratio
JL
A simpler constraint function can be obtained by taking Op(x) as merely
Â°p(x) "iw
for which
(x, u)
UJ + x3/o0
153
This function can be analytically solved for whenever
%
= 0* What must be solved is a cubic equation in u^.
(up3  A (up  B = 0
* â– t ft' ft
(5.6.2)
B = x3/a0
A reduced cubic equation with real coefficients
y3 + py + q = 0
has a discriminant
A =  108R, R = (Â¿ p)3 + (~ q)2
that determines the nature of the roots to the equation. For,
A < 0 > one real, two complex roots
A = 0 *â– two real, equal roots
A > 0 > three distinct real roots
Consider a typical problem where the cross section is circular, and
having typical parameter values:
k.
1
2
oâ€ž =
.06
it follows from
154
and
A =  4p3  27q3
/11.664 o ) x 10
X3 x10 5/7
Thus for typical parameter values, there is a single real u^ value
whenever
x3 > X3 x 10â€œ5/7 = 4 x 10~5
or
x3 > (X3 x10~5/7)^ * 6.2 x l(f3
In other words, near the tip where both t and x^ approach zero, three
real distinct roots to (5.6.2) exist. Unless this possibility is
excluded by a sufficiently large minimum bound to u^, the question of
which root is appropriate must be answered. Having demonstrated how
complicated the analysis becomes for this particular stress constraint,
no further considerations are presented. The developments are included
to show that even simple stress constraints can be too complicated for
ordinary methods of analysis.
CHAPTER VI
FINITE ELEMENT METHODS IN STRUCTURAL
OPTIMIZATION: AN EXAMPLE
6.0 Introduction
In order to provide a contrasting comparison to the foregoing
PMP examples the cantilever beam problem is solved by using finite
elements in conjunction with mathematical programming techniques. It
is assumed the beam has a constant specified width in order to better
exemplify the concepts contained in this chapter. Numerical solutions
obtained by a feasible directions algorithm are compared to the PMP
results. This example serves to illustrate the principles used in the
theoretical derivations.
6.1 Finite Element Problem Statement
The same cantilever beam problem solved in Chapter IV is treated
here with different techniques. In what follows the structural system
is modeled by and solved with finite elements. A direct consequence
of this mathematical representation is that the optimization problem is
transferred from a vector space of continuous functions to a space of
finite dimension. Since no contributions to finite element theory are
made, the method itself is not discussed. The reader is referred to
Zienkiewicz (1971) for details of the method. BernoulliEuler Bending
Theory is applied to a fixed length cantilever beam of constant width
155
156
and material properties, which is symmetric about its central axis.
The problem is to find the variable height h(x) that gives the minimum
tip deflection due to its own height, subject to hard constraints
of maximum and minimum allowable heights.
A higher ordered element is used to obtain better accuracy;
both the displacement and its derivative at the nodes are prescribed
to be the independent variables. For this simple problem, n nodes divide
the beam into (n1) elements of equal length.
Hence, the displacement and its derivative are the deflection
and rotation of the beam's centerline at the locations of the nodes.
It is also assumed that each element is linearly tapered, whereby the
BernoulliEuler bending equation for element "i" takes the form
EI (x) = M(x)
dx2
I(x) = J2 wh3(x)
Position along the beam x, height h(x), and displacement of the beam
y(x), and slope 9(x) of the centerline, are approximated by vectors
composed of discrete elements:
T
x = [x x . .. x ]
â€” 1 z n
h(x) = [h(x ) h(x ) ... h(x Ã]1
1 Z n
z(x) = [yix^ y(x2> ... y(xn)]T
0(x) = [0(x ) 6(x ) ... e(x )]T
1 z n
Whenever a height distribution is specified, a finite element solution
for the beam's slope and deflection can be obtained.
157
6.2 Mathematical Programming:
Gradient Projection Method
With the structural analysis given in the form of a finite
element solution, the mathematical statement of the optimization problem
is
Given: (i) material properties p,E
(ii) dimensions L, w
(iii) number of nodes n
Find: Min [y(x )]
h(x) n
subject to b < h(x^) < d i = l,...,n
This may be stated as a nonlinear mathematical programming problem,
whose optimality conditions are prescribed by the KuhnTucker Theorem
given in section 3.2. For this particular beam problem corresponding
quantities of the theorem are
x ^ h_
F(x) ^ v(x)
x=L
rh(x )  d i = 1,...,n j = 1
g (x) ^ <
J v_b  h(>:^) i = l,...,n j = i+n
The numerical algorithm used to obtain the optimal solution
satisfying the KuhnTucker conditions is a modified gradient projecÂ¬
tion method described by Kowalik (1970). Briefly speaking, it uses
projections of the cost function gradient into a subspace satisfying
currently active constraints. Starting from a feasible design point
h^, a sequence of feasible designs is generated such that
158
where
â– V! â€™2â€™ â€¢ â€¢ * â€¢ â€¢ â€¢ â€™^OPTIMAL
h t = h + a S
q+1 ~q q
S =  PVF
a >0
q
T 1 T
P = I  N(N N) N
A derivation of projection matrix P is found in Kowalik (1970).
Columns of N are the gradient vectors of the active constraints; thus,
N is an (n * r) matrix where r is the number of active constraints and
0
reduces to the identity matrix I. With P so defined, those components
of VF that lead to a constraint violation are subtracted from the
direction of steepest descent. In this manner S is the direction which
best improves the cost function without leaving the region of feasible
designs where g^ (x) < 0. From the form of g^(x) in this particular
problem, the typical element N of N is defined as:
1K.
N., =
ik 9x
i = 1,...,n k=l,...,r
where a, are those indices .1 < j < 2n associated with the r active
k  J 
constraints for which g^. (x) = 0. Hence,
N., = 0
xk
i ^ a
r i
i = a.
N
ik
ll i = a.
and
and
a, < n
k 
a, > n
k
159
There is no difficulty in calculating the gradients required
by N; since the g^(x) are specified functions the gradients can be
determined explicitly. Derivatives required to evaluate VF are not
so easily obtained. To circumvent the lack of an explicit formulation,
the derivatives are obtained as follows. The i*"^ component of VF is
found numerically by
3F
9x.
i
F(x + Ax)  F(x)
Ax
where the only nonzero component of Ax is
Ax.
i
JL_
10
x.
i
To avoid instability as the solution is approached, the increment is
modified to
10
kx.
x
where k is the maximum percent change of all components of from the
previous iteration. No numerical instabilities were encountered with
this scheme; however, if too small a value of numerical tolerance is
chosen, the convergence criteria are never satisfied and the algorithm
begins to "hunt" in the neighborhood of the optimal solution. A simÂ¬
plified flowchart is shown in Figure 6.1.
To furnish a comparison standard, the BernoulliEuler equation
for a beam with a linearly tapered center section was integrated analÂ¬
ytically. This revealed some interesting qualitative results which are
discussed below, with quantitative results given in the following secÂ¬
tion. First, the expression for the second derivative of y(x) is
160
Figure 6.1 Simplified Flowchart of Algorithm
161
only piecewise smooth due to I(x). To integrate the equations for the
linearly tapered beam required dividing the range of x into three
reqions. In the region 0 < x < the height h(x) = d and y(x) is
a quartic polynomial in x. Variable height complicates the equation in
the region x^ < x < x^; however, the centerline deflection is still
a function of a single variable, u(x), defined as
u(x) = 1  ( l ~ L
xi
X 1
where
A = 2 â€” tan a, a constant
d
tan a =
1 /LV1 (1 "
2 Id
X2 X1
such that
u(x2) = Â¿
One of the complicated terms of the expression for y(x) in this region
of the beam is
12 In (u (x) )  u (x) ]
which is undefined at x = when b/d = 0. Within the third region
x^ < x < L, the height h(x) = b and the centerline deflection is a
quartic polynomial in (x  x^) â€¢
It is shown in Hornbuckle et al. (1974) that in all three
regions the deflection could be written in nondimensionalized form
162
iM=fct
, b.
(xâ€™d}
y
o
(6.2.1)
This result supports Dixon's statement that d/L is not a parameter of
the dimensionless system. These equations indicate that dimensionless
results indeed depend only upon b/d. However, for a particular beam of
given dimensions, the actual tip deflection does increase as the square
of the slimness ratio L/d.
6.3 Results
Before any attempts were made to obtain results, certain basic
questions had to be answered. First, the integration inherent to the
finite element solution as described in Zienkiewicz (1971) is done
numerically, thereby requiring determination of the proper step size.
It was found that with the range of parameters used in this problem,
ten integration intervals per element gave the desired accuracy without
excessive computation. Secondly, there was a question of how many nodes
are required to give a valid representation of the beam. Because the
system is so simple, as few as two finite elements produce good strucÂ¬
tural results visavis tip deflection. However, unless eight or sixÂ¬
teen elements were used, the approximation of h(x) by h(x) was not
acceptable. The results for eight or sixteen elements were almost idenÂ¬
tical. (For the latter two cases the execution times on an IBM370/165
were 15 seconds and 110 seconds, respectively.)
163
Basic operation of the algorithm and convergence to the optimal
solution is illustrated in Figure 6.2 for a typical value of b/d. The
profile is displayed at each stage in the design iteration process,
including the final design. A "stage" in the design process occurs
whenever another constraint is encountered, requiring the recomputation
of VF, P, and S. To simplify the figure only nine nodes were used; the
optimal profile for the beam having linear taper is shown for comparison
The listing below indicates how constraints are encountered as the numer
ical process approaches the optimal design. Tip deflections for each
design iterate are included. Thus, for the design process depicted by
Figure 6.2,
No. of Active
Cons traints
0
1
2
3
4
5
Comparison
Tip Deflection
.108879
.027556
.020673
.017103
.016646
.015351
.015332
Initial Design
Final Design
Linear Taper Design
In the earlier studies, the problem was found to have a single
parameter b/d; however, two different beam profiles h(x) were obtained
for this problem by various authors. It was found with the finite
element method that for numerical precision corresponding to slide
rule accuracy the optimal profile is an almost linearly tapered beam.
Greater numerical accuracy provides a distinct curvature in the tapered
portion of the beam. This is shown in Figure 6.3 by the two curves
generated with the finite element method. The profile depicted by the
Intermediate Feasible Design
Constrained Optimal Design
Figure 6.2 Design Iteration to Optimal Profile
164
b/d = .25
L/d = 20
h(x/L)
d
Figure 6.3 Optimal Profile: Comparison of Methods and Numerical Tolerance
165
166
dots is taken from the work by Boykin and Sierakowski (1972), and is
presented for corroboration. For this latter curve the absolute magniÂ¬
tude of the error in satisfying the differential equations is less
, __â€”15
than 10
Since tip deflection is the cost function, the tip deflection
from the finite element algorithm was compared to that from the linearly
tapered beam (Figure 6.4). Differences were less than .5% except near
b/d = 0. This is attributed to the natural logarithm term in the linear
taper solution discussed previously. No computer results were obtained
for the linear taper solution with b/d almost zero because the natural
logarithm term became excessively large. This may explain why all
earlier studies using this example problem contain no results for
b/d < 0.1. However, the finite element algorithm converged to a soluÂ¬
tion for b/d = 0 with no difficulty.
Further analysis centered on a statement in which Bellamy
and West (1969), claim: ". . .as b/d increases, the midsection of
the beam profile reduces in length and increases in slope." Having
found quite close agreement with the linear tapered tip deflection,
we expected at least qualitative agreement for the angle of taper a.
Angle of taper for the linearly tapered beam and the finite element
results are shown in Figure 6.5. Notice that both curves approach zero
in the limit as b/d approaches one. Of greater importance is the fact
that the angle of slope for both methods is reasonably close only for
b/d < 0.5. Figure 6.6 shows that for b/d > 0.5, if the beam is divided
into sixteen finite elements, the tapered section contains only three
Y
TIP
(feet)
Figure 6.4 Tip Deflection Versus b/d
167
Figure 6.5 Angle of Taper Versus b/d
168
h(x/L)
d
Figure 6.6 Optimal Beam Profiles via Finite Elements
169
170
nodes because it spans such a short section of beam. This is believed
to be insufficient for accurate representation. The same shortcoming
is also apparent in Figure 6.7, which shows the optimal profile dimenÂ¬
sions reported in the various studies. For b/d > 0.5, the value of r.
d
obtained by the finite element method diverges from the other solution
method results.
To verify independence of the solution to L/d, a series of runs
were made with fixed b/d and the slimness ratio L/d ranging from five
to eighty. Resulting profiles at first suggested that L/d is a paramÂ¬
eter, but closer analysis revealed there was an oversight in the use of
numerical tolerances. Initially a single tolerance was applied to both
geometrical constraints and the KuhnTucker test. The former is indeÂ¬
pendent of relative beam shape, but the latter is not; this is implied
by the form of the nondimensionalization coefficient given by (6.2.1).
Basically, the projection matrix P deletes those components
from VF which are normal to an active geometrical constraint boundary.
Since all numerical schemes contain inherent accuracy limits, the Kuhn
Tucker test is considered to be satisfied when at those points not on
a boundary
3y(xn)
3x. < CG
i
Yet as the slimness ratio L/d increases, for any point on the beam the
magnitude of the component of the gradient also increases. By requirÂ¬
ing the KuhnTucker test to satisfy the same tolerance for all values
of L/d, as L/d increases the solution is effectively forced to become
more accurate. In fact, more accuracy than is possible from the
171
Digital (Dixon)
â€¢â€” Analog  Hybrid
Quasilinearization
Figure 6.7 Dependence of Intercept Location upon b/d
172
algorithm was required and the system began hunting around the solution,
failing to converge. By numerical experimentation and curve fitting
it was foundâ€”Hornbuckle et al. (1974)â€”that uniform accuracy could be
obtained for this particular problem when the tolerance used in the
KuhnTucker test is
N0
Â£g = Â£o + ko(L " V
^U5
d
Nq = 2.77825
kQ = 1.184045 x 106
Subscript zero indicates values associated with reference length correÂ¬
sponding to L^/d; the value of five was chosen because it represents an
approximate lower bound for the validity of BernoulliEuler bending
theory. In uniform beams whenever L/d < 5, the effect of shear is no
longer negligible. As an example of why care should be exercised in
selecting the range of parameters to generate data, Dixon (1967, 1968)
obtained results for 0.2 < b/d < 4; having assumed the effect of shear
negligible, he considered only the range of L/d where it is significant.
CHAPTER VII
COMMENTS ON NUMERICAL INSTABILITY IN
THE OUASILINEARIZATION ALGORITHM
7.0 Introduction
Quasilinearization is an indirect type of numerical scheme in
which a solution is obtained for the differential equations that must
be satisfied by the optimal solution. Since a standard IBM SHARE proÂ¬
gram is used without modification, the method is not discussed. InterÂ¬
ested readers are referred to any standard textbook on numerical techÂ¬
niques, or to Ghosh (1973).
This type of algorithm requires an initial guess of the state
and then iterates from that guess to the solution. Control functions
associated with the final iterate are the optimal control functions
actually sought. If the process converges at all, the convergence is
rapid and typically requires fewer than ten iterates. However, the
paramount difficulty of using this method is to select an initial guess
which is "good" enough to yield convergence. This chapter contains some
numerical data related to specific cases of the example problems where
convergence difficulties are encountered.
7.1 Computer Program Convergence Features
As explained in section 4.4, convergence is measured by whether
or not the iterate satisfies the differential equation. In terms of
173
174
the quantity ERROR defined by (4.4.2), in subroutine QUASI the test for
numerical instability involves three program parameters:
E1MAX the value of ERROR for the initial guess
E2MAX the value of ERROR for some subsequent iteration
BLO a specified instability parameter
Whenever the ratio of E2MAX to E1MAX exceeds BLO the program terminates
on an error exit. Typical values of BLO are 10 to 20. The primary mode
of convergence is for E2MAX, which decreases with each iteration under
normal circumstances, to become smaller than a specified value (assumed
0
to be 0.5 x 10 ). Other special modes of convergence contained in
QUASI were never used by any of the cases run for either example problem.
The purpose of this test is twofold. In some instances the
initial guess is not "good" enough and the first few iterations possess
E.2MAX values greater than E1MAX, but is sufficiently good for the algoÂ¬
rithm to ultimately converge. The other purpose is to prevent excessive
computation for those cases where the initial guess is so bad that the
numerical process is divergent.
Another related convergence test, is required by the subroutine
RECIP which performs required matrix inversion. This test involves
a program parameter SUM which is defined in terms of a matrix to be
inverted, and the inverse C 1 obtained numerically by RECIP. If C .
and C denote appropriate elements respectively, then
SUM = E T. ICT,1 C, . I
. ... ' xk kj 1
k i/j
175
If the matrix has been inverted exactly the value of SUM is zero.
Whenever SUM is less than program parameter OFFDG the inversion is conÂ¬
sidered to be sufficiently accurate. The value selected for all cases
15
for both example problems is 5 x 10 . Whenever the inverted matrix
fails to satisfy the test on SUM, the subprogram iterates upon the
inverse to improve it. If the improved inverse satisfies the accuracy
requirement, the result is returned to QUASI. After five unsuccessful
improvement attempts, results of the fifth attempt are returned to QUASI
A singular matrix results in immediate termination of all calculations
via an error exit.
During the operation of the quasilinearization algorithm two
different matrices must be inverted. One of these, D., exists for every
â€”i
element i and each separate _D^ must be inverted. From all of the Ih
another matrix is defined which also requires inversion. The specific
form of these matrices is not important; however, their definitions may
be found in Ghosh (1973, pp. 5258). It is noteworthy that in the cases
run for the example problems no difficulties were experienced in the
inversion of any I) matrix. All matrix inversion problems were assoÂ¬
ciated with the J3 matrix.
7.2 Numerical Instabilities for
Cantilever Beam Example
An initial guess for the beam problem was obtained from the
solution for a uniform beam, that is, for parameter values a/c = b/d = 1.
Cases were then run for a range on both parameters from 0.1 to 1. As
would be expected, no convergence difficulties were encountered with
176
combinations of parameter values corresponding to a nearly uniform
beam. In fact, with the initial guess based upon the uniform beam,
only three cases failed to converge. For this initial guess given by
(4.4.3) the following three cases exhibited numerical instability.
a/c = 0.1 b/d = 0.3
= 0.2
= 0.1
(7.2.1)
All other combinations of parameter values resulted in convergence by
the quasilinearization algorithm.
Since all of those cases which did converge showed very little
change in structural state components x^.(t), i = 1,...,4, the uniform
beam initial guess was replaced by a guess based upon the case where
a/c = 0.1 and b/d = 0.4. It was suspected that the adjoint variable
guesses were not sufficiently "good." Hence, the equations (4.4.3)
were replaced by:
Xi(t) = .025 * t2
x2(t) = .040 * t
x (t) = .100 * (1t)
x,(t) = .400 * (1t)2
P3(t) = 5.00 * t3
P4(t) = 1.25 * t3
0 < t < 1
(7.2.2)
With this initial guess only one case corresponding to the three sets
of parameter values (7.2.1) failed due to numerical instability; the
first two cases of these cases converged.
177
Having succeeded in forcing two additional cases to converge
by improving the initial guess, this same approach was tried with the
third case as well. The initial guess was based upon the converged
solution for the case with a/c = 0.1 and b/d = 0.2. Equations (7.2.2)
were replaced by
x (t) = .020 * t2
x2(t) = .040 * t
x3(t) = .080 * (1t)3
x.(t) = .350 * (1t)3
4
P3(t) = 13.0 * t4
p.(t) = 2.00 * t4
4
0 < t < 1
(7.2.3)
With this initial guess the case for a/c = .1 and b/d = .1 converged
also. This last case gave a complete data set for parameter values
.1 < a/c < 1
.1 < b/d < 1
in increments of .1. No cases for parameter values equal to zero were
run since the parameters appear in the denominator of various expressions.
As mentioned earlier, indirect methods such as quasilinearizaÂ¬
tion generally converge rapidly if the initial guess is sufficiently
good. To try to determine why the three cases failed to converge, all
of the cases were closely examined. Two characteristics were observed:
(i) There is little change in the maximum value of each separate
state component, regardless of the parameter values. CurvaÂ¬
ture of x3(t) and x^(t) did change for lower parameter values;
178
this is evident in different expressions used for the
initial guess of these two variables. However, the varÂ¬
iable that actually caused the instability termination
was p^(t) and not a state component.
(ii) The maximum value of both adjoint variables increases with
decreasing parameter values, with an associated increase
in curvature.
With p^(t) isolated as the source of convergence difficulties for cases
with small parameter values, p^(t) was plotted for several combinations
of small parameter values.
The increased curvature mentioned above is shown in Figure 7.1.
All cases that failed to converge exhibit a "numerical instability
termination" for which p^(t) becomes divergent. Maximum error in satisÂ¬
fying the p^ equation occurs in the region .755 < t < .965 for these
divergent runs. From the figure it can be seen that this is the region
of greatest change in the curvature of the solution p^(t)  This sugÂ¬
gests that the linearization gradients in the quasilinearization algoÂ¬
rithm may not be valid. If that is the case, then the linear expansion
about the known iterate cannot be expected to result in an improvement.
A modification developed by Ghosh (1973) to inhibit instabilities assoÂ¬
ciated with toolarge improvements has no effect upon these numerically
divergent cases. Removal of this possible source of numerical instabilÂ¬
ity further implies the difficulty is related to the linearity assumptions.
An explanation for the increasing slope is available from the
differential equation for the adjoint variables, where
179
Figure 7.1 Solutions for Adjoint Variable p^(t)
for Certain Cases of Interest
180
P3 = (lt)/u13u2
P3(0) = 0
(7.2.4)
P4(0) = 0
Since both and u^ are equal to their minimum allowable values near
t = 1, p>3 increases with decreasing parameter values in this region.
However, the term (1t) forces j>3 to zero at t = 1. For unknown reaÂ¬
sons, the large curvature associated with this effect seemed to pose
no difficulties for the numerical algorithm. As seen in equations
(7.2.4), if P3(t) can be determined, then p^(t) is readily obtained.
Curves for p4(t), corresponding to cases presented in Figure 7.1, are
given in Figure 7.2. All six cases exhibit positive curvature and
possess none of the inflection points seen in their counterparts of
Figure 7.1. The increase in amplitude of these too variables for
decreasing values of b/d is illustrated in Figure 7.3. In this figure
it is seen that with respect to increasing amplitude as b/d Â» 0, P3(t)
is the critical variable.
The major disadvantage of the indirect solution method is that
little is known about the connection between convergence and requireÂ¬
ments upon the initial guess. For this specific example, only one of
the seven variables causes convergence difficulties, and the resultÂ¬
ing numerical instability occurs in a region of large, rapidly changing
curvature. No convergence problems are encountered when the curvature
of the initial guess more closely resembles the final solution. All of
this suggests that the question of whether or not an initial guess
allows the quasilinearization algorithm to converge, may depend upon the
respective curvatures of the initial guess and the solution.
181
Figure 7.2 Solutions for Adjoint Variable p^(t)
for Certain Cases of Interest
P3(t)
10 p^(t)
â€¢2 .4 .6 .8 1.0
b/d
Figure 7.3 Terminal Point Amplitude of p^(t) and p^(t):
Dependence upon Parameter b/d Value
182
183
7.3 Numerical Instabilities for
Column Buckling Example
Another type of convergence failure was experienced with the
column buckling problem. In these cases which did not converge, each
of the variables was divergent, increasing with each iteration beginÂ¬
ning with the initial guess. All of these numerically unstable cases
were for parameter values which resulted in a solution that approached
the uniform column limiting solution. Attempts to force convergence
were unsuccessful.
The first attempt was to improve the initial guess used as a
starting point by the algorithm. Although this technique succeeded
with the beam problem, it failed with the column problem. The next
attempt to obtain convergence was to employ the instability inhibiting
modification developed by Ghosh. The only result was to prolong the
numerical divergence. At this point it was suspected that perhaps the
source of difficulty was related to normalization of the eigenfunction.
However, when each iterate was normalized as indicated in section 5.4,
the sole result was that the state variables maintained reasonable
amplitudes while the amplitudes of the adjoint variables increased more
than without normalization. Doubling the number of intervals in the
finite difference approximation (from 100 to 200) was also ineffective.
A possible source of the numerical difficulty was the matrix
inversion. For those cases failing to converge each of the JD matrices
was inverted accurately but the inversion of C was questionable, as
indicated by the value of SUM. Since each iterate depends directly
184
upon the inverse of JC, a degradation in would cause a correspondÂ¬
ing degradation in the "solution" at each iteration.
To illustrate what happens in these cases consider a specific
example: a = 0, a =1.1. A plot of values of SUM versus t , the
ij U u
intercept of the upper bound, is shown in Figure 7.4. These data points
represent the C_ ^ accuracy at different iterations for various initial
guessesâ€”acceptable accuracy is indicated by the straight line. BasiÂ¬
cally, if an initial guess yielded a t of approximately onehalf, each
iteration gives an increased value of t^ with a correspondingly less
accurate C) As t^ approaches one, the solution approaches the limitÂ¬
ing uniform column.
The result of this is that may become illconditioned if the
parameters are chosen such that the solution is similar to a uniform
column. There are no explicit data or derivations to indicate this is
the situation, only the implicit suggestion of the data in Figure 7.4.
However, as stated in the preceding section, there is little known
about why certain initial guesses fail to converge in the indirect
solution algorithms.
SUM
1 x 10
12
5
2
1 x 10
13
5
2
1
a = 0.0
!j
Â©
Â©
o
au = 1,1
Â©
o
Â©
Â©
Â©
/
/ / /
/
/ /
/
/
/ / / /
/
0
\
Â©
Acceptable Matric
Inversion
0
.4
.6
!
.8
1.
t
U
Figure 7.4
Degradation of C ^ Accuracy as Iterate Approaches a Solution
Similar to the Uniform Column Limiting Solution
185
CHAPTER VIII
CONCLUSIONS AND RECOMMENDATIONS
8.0 Summary and Conclusions
This dissertation treats the optimal design of elastic structures
subject to both hard inequality constraints and subsidiary conditions.
It is shown that structural problems can be classified into two general
types depending upon whether or not the cost functional is an eigenvalue.
For the conservative systems considered, which are described by self
adjoint differential equations, it is shown that the minimum weight
problem is identical to the maximum buckling load problem.
Pontryagin's Maximum Principle is analyzed as a nonlinear
programming problem in Chapter III. Specifically, the theory of the
gradient projection method is applied to the maximum principle, with
the control subject to inequality constraints. From this an explicit
formulation is obtained for the Lagrangian multipliers that adjoin the
constraints to the variational Hamiltonian.
Additional characteristics of the maximum principle result from
the gradient projection method analogy. In the latter, the projected
gradient is that specific direction which minimizes the cost without
violating the constraints in effect at any point in question. As
applied to the maximum principle, the projected gradient is
* â– p
H =  (H + _Â¿)Ã¼ which in uspace denotes the direction of constrained
186
187
maximum descent whenever jj is given by equation (3.3.4). While this
attribute is not pursued further in the dissertation, it may prove useÂ¬
ful in subsequent applications of the maximum principle. To further
illustrate the nonlinear programming principles used in the theoretÂ¬
ical development, the example of Chapter IV is approximated by finite
elements, and solved by a gradient projection algorithm. Operation of
the algorithm is illustrated and the results compared to those obtained
with the maximum principle. When a sufficient number of finite eleÂ¬
ments are used to adequately describe the structure, the results of the
two methods are identical.
A problem concerning the minimum deflection of a beam is optiÂ¬
mized by using classical maximum principle techniques. It is shown
that no finite solution is obtained for unconstrained control; this also
holds for the column buckling problem treated in Chapter V. FurtherÂ¬
more, it is shown that for the conservative system corresponding to the
buckling problem, the minimum weight and maximum buckling problems are
identical. Another result of using the maximum principle is that for
the problem described, in order to have a nontrivial solution the moduÂ¬
lus and density must be the maximum and minimum allowable constant
values, respectively. This is a mathematical verification of an intuiÂ¬
tive result.
Numerical instabilities experienced by the quasilinearization
algorithm are detailed in Chapter VII. While no concrete results are
obtained, from the data presented it appears that the question of conÂ¬
vergence is related to the respective curvatures of the initial guess
and the solution.
188
8.1 Recommendations
Further work suggested in the course of dissertation research
falls into three categories: structural considerations, mathematical
theory, and numerical solution techniques. Under the first category
the most obvious extension is to not invoke the linear bending assumpÂ¬
tion. Since the quasilinearization algorithm handles nonlinear TPBVP,
theoretically there is no need for the assumption. In contrast to the
first of equations (5.2.4) the complete, nonlinear bending problem is
described by
(s(t)h) = M(t)[l + Ãš2]3/2 , 0 < t < 1
1
M(t) = A[n(l)  u(t)] + k / m(Â£)[n(Â£)u(t)]dÂ£
t
n (0) = Ã±(0) =0
Another interesting possibility is to include the effect of shear.
These structural aspects pertain to specific problems. A more
general result, and hence more interesting, is related to expanding the
class of problems that can be treated by energy techniques. It is sugÂ¬
gested that the mutual potential energy method and the application of
adjoint systems to nonconservative problems be investigated to see if
a "general energy method" exists. Such a development would expand the
class of problems now considered to be amenable to the energy method
and its welldeveloped theory.
In terms of mathematical theory, the double optimization
problem associated with extremization of eigenvalues for selfadjoint
systems is an inviting prospect. No significant contributions have been
189
published in recent years. Any new discovery of a general nature would
be a significant development. Another useful area of mathematical
research would be the further study of the nonlinear programming aspect
of the maximum principle. Beyond this, it might also prove fruitful to
combine the latter with a comparison to dynamic programming.
Finally, the discussion of numerical instabilities of Chapter VII
indicates the possibility that the respective curvatures of the initial
guess and the solution determine convergence. Any development that
could indicate convergence, or the lack thereof, for an initial guess
required by an indirect solution technique, would be a significant
development. One possibility is a study of the sensitivity of the
solution increment function, je(t), to error sources such as initial
guess, step size, etc., for a case with a known solution. Little is
known about this subject.
APPENDIXES
APPENDIX A
HISTORICAL DEVELOPMENTS
The dissertation subject matter originated from two papers:
a short survey of structural optimization problems by Prager and Taylor
(1968) and a technical note by Boykin and Sierakowski (1972) .
A survey of the literature began with the former and revealed that what
were presumed to be recent developments were in fact two hundred years
old. Besides engendering a real sense of humility, my discovery aroused
a fear that some fundamental part of my work might be a duplication of
a much earlier study. I also found the historical development to be
quite intriguing; for that reason this appendix has been included to
give the reader a brief history of the separate development of mechanics
and the calculus of variations. Neither is intended to be comprehensive
however, the reader may enjoy being able to associate some recognizable
equation or method with a specific person and period of time.
The appendix consists of three sections, where the first is an
anecdotal discussion of the major early contributors to the calculus
of variations and a table listing major developments in chronological
order. This is follox^ed by a similar account for mechanics, also having
a table of chronologically ordered developments. Concluding the appenÂ¬
dix is a short biographical table that lists the life span and nationalÂ¬
ity of most of the people mentioned.
191
192
Newton posed a problem in Book II of the Principia
which requires the techniques of the calculus of variations for solution.
His goal was to find the solid of revolution which has the minimum
resistance in axial flow. No apparent significance was attached to the
problem of either Newton or his contemporaries.
The calculus of variations actually begins with two Swiss
mathematicians, James Bernoulli and his younger brother John. Until
1690 the latter was a student of James, but soon became a rival.
In June, 1696, John Bernoulli posed the brachistochrone problem in
Acta Eruditorum:
A New Problem, to the Solution of Which
Mathematicians Are Invited
Given two points A and B in a vertical plane, to find for
the movable (particle) M, the path AMB descending along which
by its own gravity, and beginning to be urged from the point A,
it may in the shortest time reach the other point B.
According to various sources, solutions were offered by Newton,
Leibnetz, de l'Hopital, and the Bernoullis. At Leibnetz' request
the solution was withheld to encourage others to consider the problem.
In January, 1697, the problem was reannounced. In the following May
issue of Acta Eruditorum, the solutions of the Bernoullis and the
Marquis de l'Hopital were published. John's is alleged to be the most
readable but since it is solved as an analog to an optics problem, the
method is restricted to a small class of problems. The solution of
James is quite geometrical, treating the curve as a polygon with sides
of infinitesimal length. It was also assumed that whatever optimal
193
property the entire curve possessed, was also possessed by each part.
The solution of de l'Hopital was given, but without proof.
At the conclusion of his paper, James posed two more difficult
problems:
First: find the curve of quickest descent from a fixed point A
to a given vertical line. This version of the brachistochrone
problem involves conditions for minimizing a definite integral
where variations at the limits are permitted. Such conditions
lead to natural boundary conditions, or transversality condiÂ¬
tions, and were not obtained until some years later by Lagrange.
Second: among all curves with given length and base, find the
the curve such that a second curve, whose coordinates are
some function of the first, contains a maximum or minimum area.
This subsidiary condition is called an isoperimetrical constraint.
Besides inviting all mathematicians to attempt solutions, James specifÂ¬
ically offered his brother John a prize of fifty ducats for the correct
solutionâ€”according to Bliss (1925, p. 12). John claimed the award but
James refused, showing that the assumption of a uniform optimality propÂ¬
erty for each infinitesimal element was not valid. James published the
correct solutions in the May, 1701, issue of Acta Eruditorum; to include
the effect of the constraint, he used three adjacent polygon elements of
infinitesimal length instead of two that sufficed for the brachistochrone,
and no fewer than four solution techniques:
(1) Equilibrium of forces on an element
(2) Equilibrium of moments on an element
(3) Principle of virtual work
(4) Principle of minimum potential energy
Whether or not the prize existed in immaterial; it is true the problem
embroiled the brothers in a bitter quarrel, and together with the brachÂ¬
istochrone problem marks the beginning of the calculus of variations.
194
All of this work by the Bernoullis was based upon complicated
geometrical results. A student of theirs, Leonhard Euler, expanded and
generalized their initial developments. His first major contribution
to the calculus of variations was the differential equation bearing his
name. Euler suggested that for continuous functions a definite integral
vanishes identically if the integrand vanishes. When the integrand is
the product of some function M(x,x,t) and an arbitrary variation Sx(t)
the function must vanish if the variation is to be arbitrary. Euler's
equation is M(x,x,t) = 0; up to the first part of this century this
vanishing integrand was referred to as "Euler's Fundamental Lemma"â€”
"Euler's equation" is the more recent appellation. He also was the first
to indicate the two general classes of optimal solutions obtained from
his technique: absolute and relative. After reading a memoir by Lagrange
which introduced the symbol "6," Euler adopted its use for his own
studies and named it in an article entitled "Elementia Calculi Varia
tionum." Other investigations related to this include combining the calÂ¬
culus techniques of Newton and Leibnetz via differential elements, and
what is apparently the earliest study of integrability. Euler's list of
886 books and articles also inclucbs the introduction of "Â£" to denote
summation, and an intuitive description of continuous functionsâ€”all of
which are necessary elements of the calculus of variations.
As mentioned in the previous paragraph, Lagrange introduced the
concept and symbol "6"; using it enabled him to derive Euler's equation
195
without resorting to the infinitelysided polygon and the accompanying
complicated geometry. It also facilitated the evaluation of problems
having definite integrals with unspecified conditions at the limitsâ€”
these were called "the definite equations," and later "terms at the
limits," but are currently referred to as "natural boundary conditions."
This represented a generalization on the type of problem originally
posed by James Bernoulli. Lagrange introduced yet another generalizaÂ¬
tion by using "indeterminate multipliers" to enforce implicit constraints,
e.g., differential equations. The final major contribution by Lagrange
to be cited here was the extension to problems having double integrals
with fixed conditions at the limits.
Following a study of the major works of Newton, Laplace, and
Lagrange, William Rowan Hamilton applied the analytical method of
Lagrange to his own study of optics. In the "Theory of Systems of Rays"
(1828) and its supplements, Hamilton deduced all optical properties
from a single "Characteristic Function" using the "Law of Varying Action,"
a generalization of the "Law of Least Action" due to P. L. M. de Maupertis.
When applied to dynamical systems in "On General Methods of Dynamics"
(1834) this method is recognized as Hamilton's Principle. The canonical
equations of motion derived from this method are called Hamilton's equaÂ¬
tions, and were used by Jacobi to later derive the HamiltonJacobi equaÂ¬
tion.
Euler derived the first necessary condition for an optimal soluÂ¬
tion in the calculus of variations: in the manner developed by Lagrange,
the first variation must vanish. But this may be satisfied by either
196
a minimum, maximum, or an "inflection point." Legendre made the first
attempt to distinguish what the case may be by investigating the second
variation in 1786. Using a transformation of variables, he arrived at
what was believed to be a sufficient condition (for either a maximum or
a minimum) for fifty years. However, in 1837 Jacobi discovered that
Legendre's condition failed in some cases. The "conjugate points" assoÂ¬
ciated with these cases were found from derivatives of the solution to
Euler's equation with respect to the constants of integration. As a
result, Legendre's condition was determined to be only necessary, along
with the newer Jacobi's condition, and not sufficient as originally
thought.
Derivation of a true sufficient condition was done by Weierstrass
in 1879. It occurred as a result of his reexamination of earlier works,
formulating the problems very precisely. Both his condition and his
"corner condition" for a discontinuous solution are stated in terms of
the "Weierstrass function."
Further elaborations, involving geodesics and the existence of
integrals, are attributed to Kneser and Hilbert about 1900. Twenty years
later Bliss applied the "adjoint system" to the solution of a variational
problem. The "maximum principle" was suggested in 1937 by Valentine but
is generally attributed to Pontryagin in 1956.
All of these basic developments are presented in Table A.l,
together with less pertinent details. According to Bliss (1925, p. 181)
the classical memoirs of the Bernoullis, Euler, Lagrange, Legendre, and
Jacobi are contained in Ostwald's "Klassiker der exacten Wissenschaften,"
197
TABLE A.1
MAJOR DEVELOPMENTS
IN THE CALCULUS OF VARIATIONS
1686 Newton
first to pose a problem of a type requirÂ¬
ing the calculus of variations; not
treated as significant at the time.
1696 John Bernoulli
posed the brachistochrone problem. SoluÂ¬
tions were obtained by James Bernoulli,
Leibnetz, and Newton, as well as by John
Bernoulli and de l'Hopital. AccompanyÂ¬
ing his solution were two additional
problems proposed by James Bernoulli,
one involving natural boundary conditions,
another with an isoperimetrical constraint.
1701 James Bernoulli
solved the isoperimetrical constraint
problem posed in his solution to the
brachistochrone.
1722 Euler
extended the work of the Bernoulli brothers
to several classes of problems with varÂ¬
ious constraints.
1740 Euler
introduced the technique that setting
coefficient of 6Y in integrand to zero
satisfies necessary condition. The
result is called Euler's Equation.
1744 Euler
divided optimization problems into two
general classes: absolute and relative.
This is the first systematic exposition;
all preceding work represented ad hoc
solutions to specific problems.
1766 Euler
published one tract giving the name
"calculus of variations," and another
containing the first study of conditions
of integrability.
* Lagrange
introduces the symbol "6" to distinguish
between variation and derivative.
* Lagrange
deduced Euler's Equation without using adjaÂ¬
cent elements of an infinitelysided polygon
No specific dates are available; two memoirs on the general subject from
which these might originate were published in 1762 and 1770.
198
TABLE A.l (Continued)
* Lagrange
determined the method to treat unspecÂ¬
ified end conditions; the results are
now called natural boundary conditions.
* Lagrange
... introduced enforcement of implicit conÂ¬
straints by the use of indeterminate
multipliers.
* Lagrange
... extended the calculus of variations to
include double integrals with fixed
boundary conditions.
1786 Legendre
... investigated the second variation of
the cost function to find a criterion
with which maxima and minima could be
distinguished. First results were not
really a sufficient condition, because of
nonunique solutions demonstrated by
Lagrange
1810 Brunaci
... extended Legendreâ€™s work to double
integrals, retaining the same general
flaw.
1834 Hamilton
... developed the canonical equations and
Hamilton's Principle based upon a generalÂ¬
ization of the Law of Least Action.
1837 Jacobi
... studied the transformation of Legendre,
discovering how to determine when the
method failed. The resulting conjugate
points are determined from derivatives
of solutions to Euler's equation with
respect to the constants of integration.
1842 Sarrus
... won the French Academy of Science prize
for mathematics by obtaining the natural
boundary conditions for a general multiple
integral having variable limits. His
technique used a new substitution sign to
designate a particular value for general
variable.
1844 Cauchy
... simplified the work of Sarrus to the form
used currently.
No specific dates are available; two memoirs on the general subject
from which these might originate were published in 1762 and 1770.
199
TABLE A.l (Concluded)
1852 Brioschi
1858 Clebsch
1871 Todhunter
1877 Erdmann
1879 Weierstrass
1919 Bliss
1937 Valentine
was the first to apply determinants to
the investigation of second order terms.
generalized Jacobi's Theorem to several
dependent variables, with or without
connecting equations, and for multiple
integrals.
suggested that variations might be of
restricted sign, allowing the possibility
of discontinuous solutions.
derived the conditions that must be
satisfied by a discontinuous solution,
called the WeierstrassErdmann conditions
helped to make the theory of the calculus
of variations more precise, in general.
Specifically, he found a new necessary
condition in terms of his function
E(X,Y,P,Y') and very clearly determined
those conditions which are sufficient.
was perhaps the first to use the adjoint
system of variational equations.
derived the earliest equivalent to a
strong minimum principle, using the
Weierstrass condition and slack variables
1956 Pontryagin
together with colleagues, derived the
Pontryagin maximum principle.
200
Nos. 46 and 47. Historical developments are treated by Carll (1881),
Todhunter and Pearson (1893), Bolza (1904), and Bliss (1925, 1946).
Biographical data are found in Truesdell (1968), Bergamini (1963),
Ireland (1962), and Asimov (1960).
Analytical mechanics became a viable method with the introducÂ¬
tion of calculus by Leibnetz and Newton. For example, Galileo Galilei
observed in 1585 that the free fall of various objects of different size
and weight is given by the formula Y = 16t2. Newton is reported to have
differentiated twice to discover that all objects fall with an accelerÂ¬
ation of thirtytwo feet per second squared. Yet there were differences
between the calculus of Newton and Leibnetz; the former designated a
derivative by a dot above the variable, and the latter used the form
"dY/dt" to indicate the derivative of "Y" with respect to "t." Newton
also deduced that maxima/minima occur at points where the rate of change
is zero, whereas Leibnetz insisted such points occurred where the tangent
to the curve has zero slope.
To show that the work of these two was equivalent was one of
the many accomplishments of Euler. In using this discovery he
demonstrated that much of what was then considered to be pure physics
was transformed quite simply into problems of mathematics. Such work led
him to sufficiently important developments that specific equations bear
his name in the separate fields of the calculus of variations, differÂ¬
ential equations, solid mechanics, and rigid body dynamics. According to
Truesdell (1968), Euler is the major source of rational analysis. CerÂ¬
tainly he is one of the most prolific, having published 886 books and
articles, and fathering thirteen children.
201
A more detailed list of the basic developments of mechanics
is given in Table A.2; a few biographical data are presented in
Table A.3.
TABLE A.2
MAJOR DEVELOPMENTS IN ANALYTICAL MECHANICS
1694
James
Bernoulli
1704
James
Bernoulli
James Bernoulli
1704 James Bernoulli
Leibnetz
1727 Euler
published the very first paper on the
mathematical theory of elasticity.
concluded a thirteen year study which
included an investigation of the
catenary (results were published in 1744).
Problem was also solved by John Bernoulli
using a theorem concerning the center of
gravity of an arc.
declared elasticity is generally nonÂ¬
linear, with linear elasticity a special
case. He also derived the differential
equation for the elÃ¡stica.
first characterized a material with stress
as a function of strain instead of the
then accepted load versus displacement
for a particular specimen. He avoided
the E of linear elasticity, suggesting
E^ by implication.
performed the first analysis of a loaded
beam, assuming that some fibres were in
tension. That the tension varied linÂ¬
early gave the result that the moment
was proportional to the second area moÂ¬
ment of the cross section.
derived the BernoulliEuler equation for
bending of beams while a student of John
Bernoulli. The work was not published
until 1862.
â€¢k
No specific dates are available from ordinary sources; items are
located with respect to related entries of the table.
202
TABLE A.2 (Continued)
1729 Musschenbroek
... performed the first comprehensive experÂ¬
iments on the strength of materials. He
observed that beams and columns bend
before breaking and that breaking strength
varies inversely with the square of the
length and also depends upon depth and
breadth.
1736 Euler
... introduced the study of mechanics, specifÂ¬
ically the concepts point mass, accelerÂ¬
ation, and vectors (which he called
"geometrical quantity").
1739 Daniel Bernoulli
... developed the principle of minimum potenÂ¬
tial energy, although he is best known
for his work in fluid mechanics.
1740 _ Euler
... learned how to solve analytically, linear
ordinary differential equations with conÂ¬
stant coefficients. Prior to that time
only series solutions were available.
* Euler
... after making some calculations on an
actual beam deflecting under its load,
discovered that most real applications
result in small strain. All previous
work had presumed that deformations were
finite.
1743 Euler
... determined that the equation of the
elastic curve could be obtained from the
minimum potential energy.
1743 Euler
... derived the column buckling formula; the
results were compared to Musschenbroek's
experiments in 1765.
1743 John Bernoulli
... obtained the first differential equation
for the motion of a system by studying
the dynamics of a weighted cord using
lumped masses.
No specific dates are available from ordinary sources; items are
located with respect to related entries of the table.
203
TABLE A.2 (Continued)
1743 D'Alembert
... introduced the first partial differential
equations. He also was the first to
separate constraint forces from external
forces, but never stated "D'Alembert's
Principle," which was given later by Euler
and Lagrange.
1744 Euler
published the earliest example of what
came to be called the "Newtonian Method."
Also defined were linear momentum and
kinetic energy derived by integration.
1747 Euler
... was the first to derive "Newton's EquaÂ¬
tions" for a discrete system. By applyÂ¬
ing them to each part of the system, all
the required equations were obtained.
The example used was the threebody
problem of celestial mechanics.
1750 Euler
... extended the preceding study to "mechanÂ¬
ical systems of all kinds," i.e., to conÂ¬
tinuous systems. Examples used included
rigid body motion and transverse vibraÂ¬
tions of a rod.
1754 Riccati
... suggested that the work done to deform
an elastic body was recoverable in part.
1760 Euler
... developed an entire theory of rigid body
motion. He defined center of mass and
distinguished it from center of gravity.
Additionally, he obtained the inertia
tensor and described its elements with
respect to the mass.
* Lagrange
determined the infinite sequence of theoÂ¬
retical buckling loads.
1773 Lagrange
... attempted to find the column shape which
would have the largest buckling load for
given height and volume. His analysis was
incorrect; this results, a circular
cylinder.
No specific dates are available from ordinary sources; items are
located with respect to related entries of the table.
204
TABLE A.2 (Concluded)
1778
Euler
â€¢ â€¢ â€¢
determined the height at which a uniformly
heavy column buckles.
1788
Lagrange
â€”
published one of the most important works
in mechanics: "MÃ©chanique Analitique."
In this book he systematized mechanics,
using the calculus of variations to
derive general equations which apply to all
problems of mechanics. Generally speakÂ¬
ing, he obviated the use of complicated
geometry, replacing it with pure algebra.
More specifically, ha introduced the
misnomer "D'Alembert's Principle," genÂ¬
eralized coordinates, and Lagrange's
equations.
1877 Gauss ... proved the fundamental theorem of algebra
in his doctoral dissertation, requiring
the introduction of complex numbers. He
later recognized their vector character
and applied them to the solution of
complicated mechanics problems.
205
TABLE A.3
MAJOR CONTRIBUTORS IN THE DEVELOPMENT OF
THE CALCULUS OF VARIATIONS AND ANALYTICAL MECHANICS
NEWTON, Sir Isaac
(16421727)
English mathematician
LEIBNETZ, Baron von
Gottfried Wilhelm
(16461716)
German mathematician
BERNOULLI, John (Jean,
Johan)
(16671748)
Swiss mathematician
BERNOULLI, James (Jacques,
Jakob)
(16541705)
Swiss mathematician
MUSSCHENBROEK, Pieter van
(16921761)
Dutch physicist and
mathematician
BERNOULLI, Daniel
(17001782)
Swiss mathematician
EULER, Leonhard
(17071783)
Swiss mathematician
ALEMBERT, Jean Le Rond D'
(17171783)
French astronomer and
mathematician
LAGRANGE, Comte Joseph
Louis
(17361813)
French mathematician
LEGENDRE, Adrien Marie
(17521833)
French mathematician
GAUSS, Karl Friedrich
(17771855)
German astronomer and
mathematician
HAMILTON, Sir William
Rowan
(18051865)
Irish mathematician
JACOBI, Karl Gustav Jacob
(18041851)
German mathematician
ERIOSCHI, Francesco
(18241897)
Italian mathematician
CLEBSCH, Alfred Rudolf
Friedrich
(18331872)
German mathematician
TODHUNTER, Isaac
(18201884)
English mathematician .
historian
WEIERSTRASS, Karl Theodor
(18151897)
German mathematician
BLISS, Gilbert Ames
(18761951)
American mathematician
PONTRYAGIN, Lev S.
(1908 )
Russian mathematician
APPENDIX B
A SIMPLE PROOF OF THE KUHNTUCKER THEOREM
Consider the minimization of a general unspecified functional
with respect to an ndimensional vector x, subject to m < n inequalÂ¬
ity constraints,
functional:
Min [F(x)]
x
inequality constraints: g^(x) < 0
j 1, .. . , m
where
x = [x, xâ€ž ... x ]
â€” 12 n
Inequality constraints are converted to equality constraints by introÂ¬
ducing "slack" variables s^ 00 defined such that
gj(x) + s2(x)= 0 j = l,...,m
(B.l)
or
^2
sj (x) = [_8j Cx) ] 2  0
(B .2)
Since for all j the lefthand side of equation (B.l) sums to zero,
for arbitrary values of A.
J
m
Â£ A.[g.(x) + s?(x)] = 0
3=1 2 2 2
(B.3)
it follows that:
m
$(x,Â£,A_) = F00 + Â£ A. [g . (x) + s2. (x) ] = F 00
3=1 2 2 2 "
206
207
Necessary conditions for the adjoined cost function $ to be
a minimum are derived from the requirement that the first variation
vanish.
s$
. T W , . T 9$ , ,,T 9$
6x â€” + Ã³s â€” + SX â€”
â€” 9x â€” 9s â€” 9A
0
If all the variations are to be independent, then
V F(x) + A. V g.(x) =0 j summed
x  J x J ~
2A. s.(x) = 0 j = 1,.. . ,m
J J ~
gj(x) + Sj(x) = 0 j = l,...,m
where repeated indices in a product imply summation. The independence
of j3 and _A can be argued heuristically. Slack variables are an artiÂ¬
ficial contrivance used to force a desired condition without altering
the original problem. Any change in s^. (x) which satisfies the given
constraint must be accompanied by a change in (x) such that, the net
effect on $ is zero; thus $Cx,_s,_A) is stationary with respect to j3.
Furthermore, since the sum in (B.3) is equal to zero for any arbitrary
values of A^, any arbitrary variation of $ with respect to A^ must also
be zero.
Given the definition of s.(x) in (B.2), the three necessary
J â€œ
conditions may be expressed as
V [F(x) + A.g.(x)] = 0
x â€” j J ~
AjgjM = 0
Sj(x) < 0
(B. 4)
(B .5)
208
The first condition will be used with the concept of a feasible, usable
direction to derive the final condition remaining before the KuhnTucker
condition may be stated. Any direction in xspace is feasible if an
increment Ax in that direction reduces the cost function F(_x) . DirecÂ¬
tion d is said to also be usable if an increment Ax: in that direction
causes no constraint violation.
Consider first the concept of a feasible direction; for any
direction d the directional derivative is defined to be
dF
f = d â€¢ V F(x)
ds x â€”
where
lid  = 1
In addition to this restriction on the norm of direction Ã¡, let the
increment dx be prescribed by
dx = (ds)d (B.6)
and the vector scalar product be written as the equivalent inner product
such that
d â€¢ V F E dT V F H (V F, d)
X X X
In terms of an inner product notation
dF a
dl â–
For an increment in direction d of length ds, the change in the cost
dF = (V F, d) ds
x
function is
209
By the linearity property of the inner product
dF =  (V F, d) ds
Hence the cost F(x) is decreased by an increment in any direction d
which satisfies
(V F, d) > 0 (B.7)
Conversely, a direction d is feasible if (V^F, d) > 0, i.e., the angle
between d and the direction of steepest descent is acute.
For any direction to be usable, an increment Ax in that direcÂ¬
tion must cause no constraint violation. On the constraint boundaries
g. (x) = 0 for j Â£ I as defined in Chapter III. With any direction again
1 A
denoted by d, the change in g^. (x) with respect to an increment dx is
defined through the directional derivative:
where
such that
(Vv g, (x) 3)
Ãœ J
I!
= i
d) ds
Since ds denotes a vector norm by way of the definition of djc given in
equation (B.6), it is a positive quantity. Hence the sense of change
dg. is prescribed by the inner product (V g., d) for which there are
J 2Â£ J
three possibilities.
210
(i) (V g., d) <0 * dg. <0
2Â£ J J
It was assumed that active constraints are being discussed, for
which g_. (x) = 0. The constraints require all g.. (>c) to be nonÂ¬
positive. An increment dx in the direction d used in the inner
product satisfies the constraint with the inequality sign.
Because the direction does not lead to a constraint violation
it is by definition usable. Actually, for such a direction d,
even though point x lies on the boundary g^(x) = 0, the point
(x + dx) does not.
(ii) (V g , d) = 0 * dg = 0
J J
Under the assumption that g^. (x) is active where g^ (x) = 0, when
the increment die in direction d causes no change in g^ , d is
tangent to the constraint surface. With x + dx used in the
inner product, the point (x + dx) may still lie on the constraint
boundary. With such an increment the constraint may perhaps not
be violated, and if not, the direction is therefore usable.
Note that for linear constraints, the word "may" is deleted from
the previous sentence. It is included because nonlinear conÂ¬
straints require an argument calling for infinitesimal increÂ¬
ments; a finite increment leads to a point that is not on the
constraint boundary. A finite increment from a point on a nonÂ¬
linear constraint boundary results in that constraint either
becoming inactive on being violated, depending upon whether the
constraint surface is concave or convex, respectively.
211
(iii) (V g., d) > O > dg. > 0
A J J
That the constraint (x) is active requires C2S) = 0.
If an increment d_x in the direction d used in the inner
product results in a positive dg^, the constraint has been
violated. Therefore, d is not a usable direction.
The result of these possibilities is that for any point x on
a constraint boundary, for a direction d to be usable it must satisfy
(V g., d) < 0 j e I (B.8)
A J A
If the set of all directions d which are feasible contains no usable
directions at the point x in question, then x = x^pp and x is the
constrained minimum point.
Conditions for a usable, feasible direction given by (B.8) and
(B.7) are used with the gradient condition (B.4) to prove that the
Lagrangian multipliers of (B.5) must be nonnegative at the point where
x is a constrained optimum. Condition (B.4) states that at the conÂ¬
strained minimum point, the negative gradient of the cost function lies
in a subspace defined by the gradients of the active constraints, i.e.,
 yw I b b b
In the following, implied summation is replaced by the symbol E, and
the dependence of F and g. upon is implied. Resolving V F into com
J A
ponents along the V g. directions via the inner product gives
X .1
212
Thus
m (VXF, Vxg )
V F = Â£ = =J V g.
 3 = 1 V 8j  * J
A. =
J
<y. y.j)
II Vj II
3 e 1,
This condition relating the Lagrangian multipliers to an inner
product is used with the conditions necessary for the existence of a
constrained minimum to show that A. > 0. At a constrained optimal point,
for every feasible direction d, the cost decreases and
(V F, d) > 0 d is feasible
x
the constraints are violated (otherwise not optimal),
(V g,, d> > 0 < d is not usable
2Â£ 3
Assume that some j, A^ is negative and a feasible but not usable direcÂ¬
tion Ã¡ exists. If d is taken to be coincident with V F, this direc
x
tion is feasible and usable, contrary to the original assumption.
That A^ > 0 is proved by contradiction. Mathematically stated,
Given:
(V F, d) > 0
for all d
(vxgj, d) > 0
for all j e I
Assume: A . < 0
3
for some j e I
A
(i) A. < 0 (V F, V g.) < 0
3 x x J
213
(ii)
Consider d = V F
X
then
(_VxF, d) = (VxF, VxF) = VxF 2
>0 > d is feasible
(iii)
But
(V g., d) = (V g., V F)
X J X J X
<0 > d is usable
by virtue of (i). This says the direction of steepest descent
is a feasible direction which is also usable, contrary to the
given condition.
(iv)
Therefore,
X. > 0, j e T Q.E.D.
J A
This completes the proof of the KuhnTucker conditions for the
existence of constrained minimum of F(x) subject to g. (x) < 0
j = 1Â» â€¢
..,m. They are
(i)
g (x ) < 0
SjVÂ±0PT' â€œ
(ii)
X. > 0
J
(iii)
Ajgj(2k)PT) = Â°
in
+ 3, xj vxVW â– 0
 J=1 
APPENDIX C
COMPUTER SUBROUTINE LISTINGS
In order to use the IBM SHARE quasilinearization program, three
usersupplied subroutines must be combined with three SHARE subroutines
The latter three are written in terms of a general problem described in
section 3.5 and do not change from one problem to another. Their names
and functions are:
QUASI
 a quasilinearization algorithm
RECIP
 a matrix inversion subroutine
OUTPUT
 a subroutine that prints out different, specified
combinations of problem variables
In addition to
these IBM subroutines, the user must provide three subÂ¬
routines which
convert a specific problem into the general formulation
required by the above subroutines. Their names and functions are:
MAIN
 reads data, establishes initial guess, calls QUASI,
and terminates execution with additional output if
desired
DIFEQ
 is called by QUASI to evaluate the differential
equation and linearization gradients at every point
for each iteration
CORRC
 corrects specified boundary conditions after each
iteration
These six subroutines are combined into a single computer program for
solving the given TPBVP.
214
215
Listings of nine subroutines are contained in this appendix.
The first six represent a program for solving the minimum deflection
beam problem subject to stress constraints. The order of presentaÂ¬
tion is:
MAIN
DIFEQ
CORRC
QUASI
RECIP
OUTPUT
Following these are the three usesupplied subroutines required to
solve the maximum buckling load problem subject to only geometric and
material bounds. They are presented in the following order:
MAIN
DIFEQ
COPTIC
IOC
IMPLICIT REAL*8(AH,0Z )
CGMMCN/QUCOMM/X.ERRORsOFFOGjHjElMAX.tZMAXrBLQ
CGNMCN/DATA/8D,AC,SIG,TAU
Cl MENS ION GI6),M6),T(1C1I*P[1).AI6,6],B(6,6),BLI6,6) Â»
1ER(6,6),C(6,6),CG{6,6),R(101J6),Y{101,6),G16,6,1CC)
CIMENSION PH 118 )
I FLAG
= 0
BC
= .25
AC
= .20
S IG
= 8.75
TAU
= 2.31
N
= 6
M
= 100
NC
= 6
M
= 100
NP
= 101
NPAR
= 1
MT
= 15
IPRM
= 1
ERRCR
= 0.5C6
CFFCG
= 5.0C15
ELC
= 10.
CO 100 1=1,6
CC 100 J1,6
e l < i,
J) = 0.0
BR t 1 Â»
J ) =0.0
CONTINUE
ELI 1,
1 ) = 1.0
GL12,
2) = 1.0
EL 1 5,
5) = 1.0
DU 6 ,
6) = 1.0
ERl 3,
3) = 1.0
e R 1 A ,
A) = 1.0
CO 200 I=a,NP
216
Til) = ÃI1.J/M
Z = Til)
Y tIÂ» 1 ) = .0 20 *(Z * *2 )
Y(1,2) = .O 4 O * Z
Y( I , 3) = .080*1 (l.Z)**3)
Y( I ,4) =.035*1 (l.Z ) * * 3 i
YÃ 1,5) = 13â€¢O*(Z * *4 )
Y ( I j 6 ) = 2.00*(Z**4 )
200 CONTINUE
CALL CUASI ( N, M, NC, N I, N P ,.NP AR, N I T, IPRN T , G , W, T, P , A, B , B L, B R, C , DG Â» R,
1Y,D,+2000 )
IF(E2MAX.LE.ERROR) CO TO 9GC
800 CONTINUE
WRITE (6,810) E2MAX,ERROR
810 FOR Y AT ( * 1 â€¢ /*0â€¢,10X,â€¢DIC NOT CONVERGE, E2MAX =. â€¢ , DI 2.5 , ' . GT. ERROR
li* , C12.5,// )
IFLAC = 1
GO TC 920
900 WRITE (6,910) E2MAX,ERROR
910 FORMAT Ã â€¢11/* O', 10X,5 CONVERGED, E2MAX = â€™,012.5,*.LE. ERROR =â€™,D12
1.5,//)
920 CONTINUE
WRITE(6,701)
701 FORMAT ( â€¢O', 10X, 9 ... X AND YJ(X) AT EXIT QUASI ...',//)
WRITE(6,7010 )
7010 FORM AT ( 9X1HX, 8X 2HY 1,8X 2HY 2,8X2HY3,8X2H Y4,8X2H Y5,8X2H Y6,8X2HY7 ,
18X2FY8,8X2HY9,7X3HY10//)
WRITE(6,702) (T( I 5, (Yt IÂ»J),J=1,N)Â»I=1Â»NP)
70 2 FORMAT(5X,F7.4,6F10.4)
I F(IFLAG.LT.0) GO TO 30CG
W R I T E(6, 1000 )
1000 FORMAT ('1'/'0â€™, 2X , 5 CONTROL DEPENDENT PORTION OF HAMILTONICNâ€™Â»//)
WRITE (6,1100)
1100 FORMAT(T7,â€™Xâ€¢,T 1 5, *U1 *,T23, â€™U2=.20',T37, 902 = .30â€™,T51,'U2=.40â€™ , T65,
217
1 Â»U2=. 50'Â»T79,Â»U2=.60', T9 3Â» Â»U2 = .70*,T107Â» 'U2=.80â€¢,T121,Â»U2 = .90â€˜)
CC 1600 1=2,11
Z = ( 11. 1/10.
II = 10 *(I1) + 1
F2X =(l.Z)Â«Y( I I 3 ) / Y { 11,6)
CG 1600 J = 1, 16
U1 = .25 + .05 *(Jl.)
CO 1AGO K = 1,8
U2 = .20 + .1*(K1. )
1400 PHI1K) = U 1*U 2 + F2X/( (U1 * * 3)* U2)
WRITE (6,1500) Z,U1,(PHI(L),L=1,8)
1500 FORMAT l Â»0' ,2X,F6.3,3X, F6.3,8D14.5)
1600 CONTINUE
GO TC 3000
2000 CONTINUE
WRITE(6,2100)
2100 FORMAT('0Â»,1CX,'ERROR EXIT C STRESS CONSTRAINT EXCLUDES ALL GEOMET
1RICALLY ADMISSIBLE CONTROLÂ»)
3000 CONTINUE
STOP
ENC
SUBROUTINE DIFEG (NfNCfNPAR,IM,ITER,GÂ»W,P,DGt*)
IMPLICIT REALMS(AHÂ»QZ )
COMNON/QUCOMM/X,ERRORfOFFDGfHtE1MAX,E2MAX,8L0
COMMON/DATA/ 80, AC,.S IG, TAU
DIMENSION G(NC) ,W(NC ) ,P(NPAR ) â–ºDG(NC,NC)
DIMENSION S I GBl 10 ) Â» TAUB(10)
FUNCiUlÂ»U2Â»F2X) = U1*U2 * F2X / { (Ul** 3)*U2 )
F2X
= ( l.X )*W(3)/W(6)
100
FLL
= FUNCtBC,AC, F2X )
Ul
= BD
U2
= AC
PHI
= PLL
MPI
= 0
110
US2
= (F2X/(BDÂ«*4).}**.5
IFtUS2.LT.AC.OR.US2.CT.1. } GO TO 120
PH I S= FUNCtBC,US2,F2X )
I F{PHIS.GT.PHI ) GO TO 12C
U2 = US 2
PHI = PHIS
MPI = 2
120 FLU = FUNCtBC,l.Â»F2X)
IFtPLU.GT .PHI ) GO TO 130
0 2 * 1.
PHI = PLU
MPI = 0
130 CONTINUE
US1 = (3.*F2X)**.25
IFtUSl.LT .BD.CR.US1.GT . 1. ) CO TO 1400
P HI S = FUNCÃUS1Â»1.Â»F2X)
I F(PHIS.GT.PHI) GO TO 1400
Ul = US1
U 2 =1.
PHI = PHIS
MPI = 1
219
no no oooooo non
1400
1401
140 2
1403
CONTINUE
N 8 =0
INB = 0
U1BNSL = l S I G * W ( 3 ) /A C 1 * * â€¢ 5
U1BSSL = TAU*W(4)/AC
U10NSU = (SIG*W{3 ) ) **. 5
U1BSSU = T AU *W(4 )
IFÃU1BNSU.LE.1. .ANC.U1BSSU.LE.1.) GO TO 1402
STRESS CONSTRAINTS ACMIT M0 GEOMETRICALLY ADMISSIBLE
NB = 1
WRITE(6,1401) X,U1BNSU.U1BSSU
FORMAT('010X, 'X =â€¢ , F1C . 4, 'U1BNSU ='Â» D 1 5.6 Â»'U1BSSU *
GO TO 500
CONTINUE
STRESS CONSTRAINTS ACMIT GEOMETRICALLY ADMISSIBLE
IF(UIBNSL.GE.BD.OR.UIBSSL.GE.BD) GO TO 1403
STRESS CONSTRAINTS ACMIT ALL GEOMETRICALLY ADMISSIBLE
NB =0
GO TO 140
CONTINUE
U28NSU = S I G*W(3 )
U2BSSU = T AU*W(4)
IF1U2BSSU.GT .U2DNSU) GO TO 1404
IF(U2BNSUoLT.AC ) GO TO 1410
CONTROL
Â»D15.6)
CONTROLS
CONTROL
220
on oooo or> oooo
C LI UPPER BOUNDARY IS 0 U2LB .LE. U2 .LE. 1.0
C BOUNDARY IS DEFINED BY MAXIMUM NORMAL STRESS
U2LE = U2BNSU
INB = I
NB =2
GO TC 140
C
c
1404 CONTINUE
I F(U2BSSU.LT.AC ) GO TO 1410
U1 UPPER BOUNDARY IS 0 U2LB .LE. U2 .LE. 1.0
BOUNDARY IS DEFINED BY MAXIMUM SHEAR STRESS
U2LB = U2BSSU
INB = 2
NB = 2
GG TC 140
1410 CONTINUE
IFÃU1BSSL.GT oUlENSL ) GO TO 1415
U2 LOWER BOUNDARY IS C UlLB .LE. U1 .LE. 1.0
BOUNDARY IS DEFINED BY MAXIMUM NORMAL STRESS
L1LB = U1BNSL
INB = 1
NB = 1
GC TC 140
1415 CONTINUE
C
221
o o o
U2 LCWER BOUNDARY IS O UlLB .LE. U1 .LE. 1.0
BOUNDARY IS DEFINED DY MAXIMUM SHEAR STRESS
L1LB = U1BSSL
INB = 2
ND = 1
C
C
C
C
c
INB
= 1
... UKSIG, AC ) .GT.UK TAUt AC)
c
INB
= 2
... U1(SIG,AC).LT.UKTAU,AC)
c
NB
= l
... NC ADMISSIBLE CONTROL
c
NB
= 0
... NO ACTIVE STRESS CONSTRAINT
c
NB
= 1
... UlLB .LE. Ui .LE. 1 AND
U2 .EQ. AC
c
NB
= 2
... U2LB .LE. U 2 .LE. 1 AND
U1 .EQ. 1.0
C
C
c
c
140 PUU = FUNCIl.tl.jF2X)
I F(PLU . GT . PH I ) GO TO 15C
U1 =1.
U2 =1.
PHI = PUU
MPI = O
150 CONTINUE
LS2 = (F 2 X)**.5
IFÃNE.EQ.2) CO TO 155
IF(US2.LT.AC.OR.US2.GT. 1. ) GO TO 16C
153 PHIS= FUNC( 1Â»,US2Â». F2X)
I F(PHIS.GT.Phl ) GO TO 160
U1 = 1.
L2 = US 2
222
CÃœ TO 180
FHI = PHIS
NPI = 2
IFÃNB.EQ.2)
GC TC 16C
155 IFIU2LB.LT.US2.AND.US2.LT .1Â» ) GO TO 153
FU L B = FUNC (1.j U 2 L BÂ»F2X)
IF(PULB.GT.PHI) GO TO 180
U1 =1.
U2 = U2LB
FHI  PULB
PPI = 3
GO TC 180
160 PUL = FUNCÃ1.,ACÂ»F2X )
I F(FUL.GT.PHI ) GO TO 170
U1 = 1.
U2 = AC
FHI = PUL
KPI = 0
170 CONTINUE
LSI = (3.*F2X/(iC**2).)**.25
IF(NE.EQ.l) GO TO 175
IFtUSl.LT .BD.CR.US1.GT. 1. ) GO TO 180
173 PH IS= FUNC(US1,AC,F2X)
I F{PHIS.GT.PHI ) GO TO 180
LI = US1
L2 = AC
PHI = PHIS
FPI = 1
GO TC 180
175 IF(U1LB.LT.US1.AND.US1.LT.1 . ) GO TO 173
PUL 0= FUNCIU1LB,ACÂ»F2X)
IF(PLLB.GT.Phi ) GO TO 180
LI = U1L B
L2 = AC
223
PHI = PULO
y pi â€¢ 4
180 CONTINUE
200 G(1) = W(2)
G(2) = W(3)/Ã(U1**3)*U2}
G ( 3 ) = W ( 4 )
G(4 ! = U 1*U 2
G(5 ) = ( l.X )/( (U1**3)*U2)
G Ã 6 )
=WÂ¡5)
CO 220 1=1,6
CO 220 J = 1,6
2 20
CG ( I
jJ ) = 0.0
LIU
= 0.0
CG 11
,2) = 1.0
CG C 3
,4 ) = 1.0
CG ( 6
,5) =1.0
C
c
c
c
NPI
â€œ0 Â« Â« o
U1
ANC
U2 ARE CONSTANTS
c
y p i
â€œ 1 9 0 9
U1
IS
NOT CONSTANT
c
ppi
~ 2 * â€¢ â€¢
U 2
IS
NOT CONSTANT
c
y pi
= 3 tt 9 â€¢
U 2
IS
ON A STRESS CONSTRAINT
BOUNDARY
c
ypi
= 4
U1
IS
ON A STRESS CONSTRAINT
BOUNDARY
c
c
c
NPIG
= MPI + 1
GO TC (230,240*250Â»260,2705Â» MPIG
230 CG{2 Â» 3) = l./t(U1**3)*U2)
GO TC 290
240 CONTINUE
LIU = l./( (Ul**4 )*U2)
CU1W3 = .25*U1/W(3)
224
250
2 60
265
270
CU1W6
C G Ã 4 y 3 ) =
CG(4,6 ) =
CG(5 j 3 ) =
CG(5 Â» 6 ) =
CG(2,3 ) =
C G ( 2 Â» 6 ) =
CO TC 290
CONTINUE
UIU
CU2W3
CU2W6
C G (4 Â» 3 ) =
C G ( 4 Â» 6 ) =
CG(5,3 ) =
CG(5,6) =
CG(2 Â» 3) =
CG(2,6 ) =
GO TC 290
CONTINUE
IF{INB.EC
C G ( 2 Â» 3 ) =;
CG(4 Â» 3 ) =
CG Ã 5 Â» 3 ) =.
CC TC 290
CONTINUE
C G ( 2,3 J =
CG{2 t 4) =
CG(4 j 4) =
C G ( 5 j 4 ) =
GO TC 290
CONTINUE
IF ( INB.EC
CG { 2,3) =
.25*U1/W(6)
U2*DU1W3
U 2 * D L 1W 6
3. Â»( icX )*UIU*DU1W3
OG Ã5,3 )*CU1W6/DU1W3
3.*W(3 )*UIU*DU1W3 + l./l(U1**3)*U2)
3.Â«W(3 )*UIU*DU1W6
1v/{(UlÂ»*3)*(U2**2))
.5*U2/W(3 )
â™¦5Â»U2/W(6 )
U1*CU2W3
U1#CU2W6
(1 â€”X)*UIUÂ»DU 2W3
DG(5 T 3)Â»CU2W6/DU2W3
W(3)*UIU*DU2W3 + UIU*U2
â€œW(3 ) *UIU*DU2H6
2) GO TO 265
S IGÂ«G(2)/U2 + 1./U2
S IG
S IG*G(5)/U2
1 ,/U2
TAU*C( 2 ) /U2
TAU
TAU*G(5)/U2
2) GO TO 275
1.5*SIG*G(2)/(Ul*Ul*AC) + I./({Ul**3)*AC)
225
CG(4,3 > = O â€¢ 5*S IG/U1
CG(5Â» 3) = 1.5*SIG*GÃ51/(U1*U1*AC)
GC TG 290
275 CCMINUE
C G(2,3) = 1./{(U1*Â«3)*AC)
C G(2,4 ) = 3.*TAU*G(2 ) /Ã U1* AC )
CG{4 14) = TAU
CC(5 Â»4) = 3 .*TAU*G(5)/ÃU1*AC)
290 CONTINUE
XCS = SIG * W(3)
XCT = T AU*W{4 )
U1BI= XCS/XCT
U2B1= X CTÂ» * 2/X C S
CC 295 IX = 1, 10
XU2 = 0.1*IX
SIGE(IX) = CSQRT(XC5/XU2 )
TAUBIIX) = XCT/XU2
295 CCMINUE
Z = X + .5 *H
WRITEÃ6Â»3C0 ) ZÂ»U1,U2,F2X
30C FOR N AT( 2X, â€¢ X= ' , D13.6,6X , ' U 1=; â€™ , D15.6,6X , â€™ U2 = ' , D1 5.6,5 X, 'F2X= â€™ t D15 .
16 )
VvRI TEÃ6Â» 310 ) U1BUU2BI, INB,NB,MPI
310 FORMAT(26X,D15.6,9X,D15.6â€ž58X,3 I 3)
WRITEC6, 320) {S IGB( 1X5, IX = 1,10)
URITEÃ6,321) (TAUBÃ IX)Â» I X = 1Â»10)
320 FORMAT(2X,10C12.3 )
321 FORMAT(2X,10C12.3,// )
RETURN
500 CGNTINUE
RETURN 1
ENC
226
SUBROUTINE CCRRC (N,M,NC,NP,NPAR,I TER,T,P,8L,BR,Y)
IMPLICIT REALÂ»8(AH,0Z )
COI* NON/QUCOMM/XÂ» ERROR Â»OFFDG, H, E1MAX,E2M AX, BLO
CONNGN/DATA/.BD, AC,.$IG,TAU
Cl MENS ION T(NP) ,P ÃNPAR),BL(NC,NC),BR(NC,NC) ,Y(NP,NC)
V(1, 1 ) = 0.0
Y(1 * 2 ) = 0.0
Y(101,3) = 0.0
Yl101,4) = 0.0
Y(1 , 5 ) = 0.0
Y(1,6) = 0.0
RETURN
ENC
1
SU'd RCUT I N E QUAS 1 (N , M Â»NC , NIÂ»IMP ,.N PARÂ» NI T. I PRNT , G,W,T,P,A,Q,BL,BR,C,
1 C G Â» R Â» Y , D * * )
IMPLICIT REAl.Â«8(AH,QZ ) 3
CCMMCN/QUCOMM/X,ERROR,OFFDG,H,EIMAX,E2MAX,BLO 4
COMMCN/DATA/BC,ACÂ»SIG,TAU
DIMENSION G(NC) ,W(NC)Â»T(NP),P(NPAR),A(NC,NC),B(NC,NC), 6
IBL(NCÂ»NC).BR(NCjNC),C(NC Â»NC ) ,DG{NC,NC),R(NP,NC) , 7
2Y(NPfNC)*D(NCÂ»NCÂ»NI ) 8
M1 = M + I 9
CALL OVER FL{JJJ ) 10
ITÂ£R=0 II
1 IM = C 12
CALL CQRRC (N,M,NC,NP,NPAR,I TER,T,P,BL,BR,Y) 13
I F( I PRNT) 99,99,9B 14
98 CALL OUTPUT (N, M, NC ,NP, IMI PRNT, I TER,G, W , T,DG ,.Y ) 15
99 E2MAX = O.ODO 16
MPAS = 3 17
VvRITE (6,1200)
1200 FORMAT('1'/'0',10X,'CONTROL VECTOR COMPONENTS*,//)
DC 30 IM =1,M 18
IM1=IM +1 19
H=T( IM1)T(IM) 20
X=T(IM)+0.5*F 21
CO 2 1=1,N 22
2 W(I ) =0 â€¢ 5 *(Y( IM, I )+Y( IM1, I ) ) 23
CALL CIFEC (N,NC,NPAR,IM,ITER,G,W,P,DG,+9997)
KKK =1 25
CALL OVERFKJJJ ) 26
IF!JJJ2) 1000,1001,1001 27
1000 WRITE (6,110) KKK,.ITER,IM 28
110 FORMAT<17HOOVERFLOW OF TYPE 13, 12H ITERATION 14, 1CH INTERVAL 29
114///) 30
1001 DC 29 1=1,N 31
W(I)=Y(I Ml,I )Y( IM,I) 32
ro
S3
co
G(I)=h*G(I) 33
G = DABS(W(I)G{I)) 34
IF(E2MAXG) 6,29,29 35
6 E2f'AX=G 36
12 I = IM 37
J2J = I 33
29 CONTINUE 39
IFUPRNT1) 28,28,27 40
27 CALL CUT FUT (N,P,NCÂ»NPÂ»IM,IPRNT,I TER,GÂ»W,T,DG,Y) 41
28 H=0â€¢ 5 Â»H 42
CC 5 1=1,N 43
DO 5 J = 1Â»N 44
5 A(I,J)=H*CG(IÂ» J ) 45
CO AA 1=1,N 46
A( I , I ) = 1 .+M I, I ) 47
44 G(I )=K( I )G( I ) 48
CALL RECIP (N,M,NC,IM,IPRNT,ITER,MPAS,W,A,B,C,DG) 49
K K K = 2 50
CALL CVERFL(JJJ ) 51
IFUJJ2) 1002,1003,1003 52
1002 WRITE (6,110) Ki
1003 CO 30 1=1,N 54
R(IM,I)=0â€¢ 55
CO 8 J=1,N 56
R(IN,I) = R(IM, I ) + A( I,J ) *G(J ) 57
8 C(I,J,IM) = 2.*A(I,J) 58
30 C(I,I,IM) = 1.+ C(I,I,IM) 59
IF(IPRNT) 34,34,35 60
35 X = G.5*(T{ I 2 I) + T( 121 + 15 ) 61
WRITE (6,105) E2MAX,J2J,X 62
105 FOR R AT (16HOMAX * ABSâ€ž ERR .= E15.6,6H FOR V 12,6H AT X=,F7.4) 63
WRITE (6,1201) Y(N P,1 )
1201 FORPATt'O',10X,'TIP DEFLECTION *,020.12)
34 IF(ITER) 31,32,31 64
229
32 E1MAX = E2MAX 65
GC TC 37 66
31 IF ( E2MAX  BLC*E1MAX) 26,36,96 67
96 WRITE (6,106) ITER 68
106 F0RMAT125H0INSTABILITY IN ITERATION 13//) 69
WRITE(6,1311)
WRITE(6,1312) (T( I ) , l Y( I,J),J=1,N),I=1Â»NP)
1311 FORMAT (4X1HX,8X2HY1,8X2HY2,8X2HY3,8X2HY4,8X2HY5,8X2HY6,8X2HY7,
18X2HY8,8X2HY9,7X3HY10//)
1312
FORMAT (1X.F7.4
, 6F10.4 )
97
RETURN
70
36
IF ( E2MAX  ERROR) 57,97,37
71
37
IF ( ITER  NIT)
137,97,97
72
137
IF (E2MAX  .01
Â» E 1M A X ) 38,38,33
73
38
IF {E2MAX  PRERR) 33,32,97
74
33
ITER=ITER+1
75
FRERR = E2MAX
76
MPAS = 5
77
CO 111 1=1,N
78
G ( I ) =<0 â€¢
79
CC 1C J = 1,N
80
10
A(I,J)=C.
81
111
A ( I , I ) = 1 .
82
CO 12 IM= 1Â» M
83
IF (MOD( IN,2 ) )
212,312,212
84
212
X = 1.
85
GO TC 152
86
312
X= 1.
87
152
CO 52 1=1,N
88
CO 52 J= 1, N
89
52
G ( I ) = G ( I ) X * A ( I
,J)*R(IM,J)
90
CO 56 1=1,N
91
CO 56 J=1,N
92
8 ( I ,J)=0 .
93
230
CG 56 K=1j N 94
56 B(I,J)= B tI * J ) + A( I,K)*C(K,J, I M ) 95
CC 12 1=1, N 96
CC 12 J=UN 97
12 A{ I , J) = B( I,J ) 98
IF (F'OC( K,2 ) Â¡4 12,512,412 99
412 X = 1. 100
GO TO 157 101
512 X = 1. 102
157 CC 57 1=1,N 103
Rtf'l , I ) = 0. 104
CO 57 J=1,N 105
R(M1,I)=R(M1Â»I)BL( IJ J *G ( J ) 106
57 0(IÂ»J)=XÂ»A(I,J) 107
CO 67 1=1,N 108
CO 67 J=1,N 109
A(I,J)=0 â€¢ 110
CO 68 K=1,N 111
68 MI, J)=A( I, J )+BL( I,K)*B(K,J ) 112
67 A(I , J ) = A t I,J ) +BR( I,J ) 113
KKK=3 114
CALL CVERFL(JJJ) 115
IFUJJ2) 1004, 1005,1005 116
1004 WRITE (6,110) KKK,ITERÂ»M1 117
1005 CALL RECIP (N,M,NC,M1,IPRNT,ITER,MPASÂ»W,A,B,C,DG) 118
K K K = 4 119
CALL OVER FL(JJJ ) 120
IFUJJ2) 1006, 1007,1007 121
1006 WRITE (6,110) KKK, ITER,K1 122
1007 CO 14 1 = 1, N 123
G( I ) =0. 124
CO 14 J = 1 ,N 125
14 G( I )=G( I )+A( I, J )*R( M1,.J ) 126
CO 15 1=1,N 127
231
Y(Ml, I) = Y(Ml, I) + Gil)
15 R(M1, I ) = G( I )
CO 40 IM=1,M
ID=M1IM
ID1= ID + 1
CO 16 1=1,N
G(I)=0 .
CC 16 J = 1,N
16 G(I) = G(I)  Ã¼( I,J, ID) *R( IC1,J )
CO 40 1 = 1,N
R(IC 11> = R(ID, I )+G( I )
40 Y( IC, I ) = Y(ID, I ) + R( IC, I )
KKK = 5
CALL CVERFL(JJJ )
IFtJJJ2) 1008,1,1
1003 WRITE (6,110) KKK,ITER,M1
CO TC 1
9997 RETURN 1
ENC
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
232
SUBROUTINE RECIF ( N Â»M ,NC , IM , IPRNT, I TERt MPAS , W,.A , B,C ,DG ) 146
IMPLICIT REAL#8(AH Â» OZ) 147
COMNCN/QUCOMM/XÂ» ERROR,OFFOG,H,E 1MAX,E2MAX ,810 148
COMMCN/DATA/BD,AC,SIG,TAU
DIMENSION W{NC),A(NC,NC),B(NC,NC),C(NC,NC),DG(NC,NC) 150
CG 1 1=1,N 151
CO 1 J=1,N 152
1 C(I ,J) = A( I,J ) 153
CC 7C 1=1,N 154
FACT = A{ I , I ) 155
A(I,I)=1. 156
IF(FACT) 72,34,72 157
34 WRITE (6,100) I, IM 158
100 FORMAT(22H0ZER0 ON DIAGONAL ROW 13, 7HMATRIX 13///) 159
STCP 160
72 DO 71 J=1Â»N 161
71 A(I,J)=A(I,JJ/FACT 162
CO 70 K=1,N 163
IF(KI) 73,70,73 164
73 I F(A(K , I) ) 74,70,74 165
74 F ACT = A Ã K Â» I ) 166
A(K , I ) =0 . 167
CO 75 J=1,N 168
75 A{K,J)=A(K,J)FACT*A( I,J ) 169
70 CONTINUE 170
N P A S = 1 171
14 SUM =0. 172
CO 2 1=1,N â€¢ 173
CO 2 J = 1,N 174
CG(I,J)=0.0 175
CO 2 K=1,N 176
2 CG(I , J) = CG(I,J)+A(I,K)*C(K,J5 177
M = N1 178
CC 4 1=1,N1 179
233
11=1+1 180
CO 4 J = 11Â»N 181
4 SUM = SUM+ DABS(CG( IÂ» J ) ) + DABS(DG(J,I>) 182
IF{ IPRNT3) 31,32,32 183
32 WRITE (6,101) SUM 184
101 FORMAT(30H SUM OF ABS OF OFFDIAG TERMS E15.8) 185
31 IF (SUM  OF FOG ) 13,13,12 186
12 IF (NPAS 1) 9,5,15 187
15 PRERR = 2.*PRERR 183
IF (SUM  PRERR) 8,200,200 189
200 CO 201 1=1,N 190
CO 201 J= 1Â» N 191
201 A( I , J ) = B < I,J ) 192
IF ( IPRNT  3 ) 13,203,203 193
203 WRITE (6,102) 194
102 FORM AT ( 30FI USED NEXT TO LAST RECIPROCAL.) 195
GO TO 13 196
8 IF (NPASMPAS ) 9,13,13 197
9 PRERR = SUM 198
NPAS = NPAS +1 199
CO 202 I = 1,N 200
CO 2C2 J = 1,N 201
202 B(I , J) = A( I, J ) 202
CO 7 1=1,N 203
7 CG C I , I) = CG(I,I)1. 204
CO 1C 1=1,N 205
CG 11 J=1, N 206
W(J)=C. 207
CO 11 K=1,N 208
11 W(J)=W(J)+A(K,I )Â« DG(J Â»K ) 209
CO 10 J=1Â»N 210
10 A(J,I) = A(J, I )W(J ) 211
GO TC 14 212
13 RETURN 213
234
SUBROUTINE OUTPUT ( N, M, NO , NP , I Ml, IPRNT, I TER, G , W , T , DG , Y) 215
IMPLICIT RÂ£Al*8(AH, OZ ) 216
COMMCN/QUCOMM/X,ERROR,OFFDG,H,E1MAX,E2MAX.BLO 217
COMMCN7DATA/BC,AC,SIG,TAU
DIMENSION G(NC),W(NC)Â»T(NP)Â»DG(NC,NC)Â»Y(NP,NC) 219
IF (IM) 1,1,16 220
1 WRITE (6,100) ITER 221
WRITE (6,102) 222
Ml = N+l 223
CO 5 1=1,Ml 224
6 WRITE (6,104) T( I), (Y(IÂ»J),J = 1Â»N) 225
IF (IPRNT2) 21,11,11 226
11 WRITE (6,106) ITER 227
WRITE (6,108) 228
WRITE (6,110) 229
GO TO 21 230
16 WRITE (6,112) X,(C< J ) , J = 1 ,N ) 231
WRITE (6,112) X, (W(J ) ,J = 1,N ) 232
IF (IPRNT3) 21,21,18 233
13 CO 19 1=1,N 234
19 WRITE (6,114) (CG(I,J ) ,J = l,N) 235
21 RETURN 236
100 FORMAT (25H1S0LUT ION AFTER ITERATION 13//) 237
102 FORMAT (4X1HX,8X2HY1,8X2HY2,8X2HY3,8X2HY4,8X2HY5,8X2HY6,8X2HY7, 238
18X2HY8,8X2HY9Â»7X3HY10// ) 239
104 FORMAT ( 1XF7 .4 , 10F1C .4 ) 240
106 FGRMAT !19H1ERRORS IN ITERATE 14//) 241
108 FORMAT (4X1HX,6X4HH*G1,6X4HF*C2,6X4HH*G3,6X4HH*G4,6X4HH*G5, 242
16X4hF*G6,6X4HH*G7,6X4HH*G0,6X4HH*G9,5X5HH*G10) 243
110 FORMAT (4X1HX,7X3HDY1,7X3HDY2,7X3H0Y3,7X3HDY4,7X3HDY5,7X3HDY6, 244
17X3FCY7,7X3HCY8,7X3FCY9,6X4FDY10// / ) 245
112 FORMAT (1XF7.4,10F10.4) 246
114 FORMAT l1X10 F10.3 ) 247
END 248
235
IMPLICIT REAL*8(AH,QZ ) I
REAL*8 MU tKMÂ»ML Â»L AM 2
COMMCN/QUCOMM/X, ERROR, QF FOG, HÂ»E IMAX , E2MAX ,610 3
COMMCN/DATA/ AUÂ»AL,EUÂ»EL,RU,RLÂ»MU*KM,DUÂ»SUÂ»LAM 4
COM MCN/TCC/ Ul, US 1. ASF, XhT 5
DIMENSION G(7),W(7),T(2C1),P(10),A(7,7),B(7,7), 6
1 BL(7,7 ) ,BR(7,7 ) ,C(7,7 ) , DG (7,7 ) ,R(201,7) , 7
2 Y(201Â»7).D(7,7Â»20C) 8
CIMENSION DYS AV E(20 1,7 ) 9
C ... MUST SPECIFY ML = FCT(X) 10
ML = 0.0 11
Ul = 1.0 12
AU = 1.1
EU = 1.0 14
RU = 1.0 15
AL = 0.9 16
EL = 1.0 17
RL = 1.0 18
MU = 1.0
CU = 2.5 20
SU = 0.5 21
KM = .012 22
N =7 23
NC = 7 24
M = 200 25
M = 2C0 26
l\P =201 27
N PAR = 1 28
NIT = 5
IPRNT =1 30
ERROR = l.OD8 31
CFECC = 5.0D15 32
BLC = 10. 33
CG ICO 1=1,7 34
NJ
O'
CG ICO J = 1Â» 7 35
EL( ItJ) = 0.0 36
ORI I fJ) = 0.0 37
100 CONTINUE 38
E L ( 1 , 1 ) = 1.0 39
BL(2,2) = 1.0 40
8 R C 3 â™¦ 3 ) = 1.0 41
ER(4,4) = 1.0 42
D R ( 5 Â» 5 ) = 1.0 43
EL(6,6) = 1.0 44
0L{7 j 7 ) = 1.0 45
150 CONTINUE 46
FI = 3.141586 49
FI2 = P I * 2 50
P02 = .5 Â» PI 51
LAN = 2.74950
CO 2C0 1=1,NP 53
T(I) = (Il.J/M 54
Z = T(I ) 55
CT = DCCS(PG2#Z) 56
ST = DSIN(P02*Z ) 57
V( I , 1) = 1.0  CT 58
Y(1,2) = 1.75#ST 59
Yl I,3J = 2.50CT 60
Y( I , 4 I = LAN  ( 1. Z )* K M/MU 61
Y(I,5) = 5C0Â»CT 62
Y(1 ,6) = .350*ST 63
Yd, 7 ) = ~.275#ST 64
200 CONTINUE 69
CALL CUASI (N,M,NCtNI,NP,.NPAR,NIT,IPRNT,G,W,T,P,A,B,BLrBR,C,DG,R, 79
1 Y , C , OYSAV E) 80
IF{E2MAX .LE. ERROR ) GO TO 90C 81
000 CONTINUE 82
WRITE (6,810) E2NAX,ERROR 83
237
810 FORMAT (Â» 1*/'O' , 1CX, ' CIC NOT CONVERGE, E2MAX = ' ,D12Â« 5 , '.GT. ERROR 84
1=4,C12.5) 85
CC TC 920 86
900 WRITE (6,910) E2MAX,ERROR 87
910 FORMAT (* 1â€¢/*0â€¢,10X, 'CONVERGED, E2MAX = â€¢ ,012.5, â€¢ .LE. ERROR =*,012 88
1.5) 89
920 CONTINUE 90
CIF = ( LAM + Y(l,A) }*MU/KM 91
XHT = l./M 47
YI2 = 0.0 48
CC 930 1=1,NP
CYT = Y( I,2)*Y( I,2)*XHT 65
IFCNPI) 925,925,930
925 CYT = .5*DYT
930 Y I 2 = Y I 2 + CYT
ASF = DSGRT(Y I 2) 70
AIN = l./ASF 71
CC 935 J=1,N 72
CC 935 1=1,NP 73
GO TC (931,931,931,935,923,933,935), J 74
931 Y ( I , J ) = A IN *Y ( I,.J ) 75
GC TC 935 76
933 Y l I , J ) = ASF*Y(I,.J) 77
935 CONTINUE 78
WRITE (6,945) CIF 92
945 FORMAT ( *1*/* 0 * , 20X,* STATE VARIABLES ... COST FUNCTION =',D15.6) 93
WRITE (6,946) Y I 2 Â»â€¢ A S F 94
946 FORMAT!Â»0*,20X, 'NORM SQUARED ...*,D15.6,5X,'NORMALIZATION CONSTANT 95
1 ...',C15.6,//) 96
WRITE (6,947) 97
947 FORMAT (9X1HX,8X2hY1,8X2hY2,8X2HY3,8X2HY4,8X2HY5,8X2HY6,8X2HY7, 98
18X2FY8,8X2HY9,7X3HY1C//) 99
WRITE (6,950) ( T ( I ) , ( Y ( I ,. J ) , J = 1,7 ) , I = 1, N P ) 100
950 FORMAT(5 X,F7.4,7F10.4) 101
238
STGP
ENC
SUBROUTINE D I F EG (NÂ»NC,NPAR,IM,I TER,G,W,P,DG) 104
IMPLICIT REAL*8(A~H, GZ ) 105
REAL*8 MU,KM,ML,LAM 106
COM MCN/QUCOMM/X,ERROR,0FFDG,H,E1MAX,E2MAX,BLO 107
COM M CN/UAT A/ AU,AL,EU,ELÂ»RU,RL,MU,KM,DU,SU,LAM 108
COMMCN/TCO/ Ul,ÃœSlf ASFÂ»XIT 109
DIMENSION G(NC),W(NC),.P(NPAR),DG(NCÂ»NC) 110
C ... MUST SPECIFY ML = FCT(X) 111
ML = 0.0 112
EXU = 1./3. 113
US2 = 1.0 114
US3 = 1.0 115
XN = 2 â€¢ *W ( 5 ) * W ( 3 ) 116
XD = (KM*W(7)  MU)*US2*US3 117
ARG = XN/XD 118
Z1 = DSIGNIUS2,ARG) 119
US1 = ( CABS(ARG) )**EXU 120
US1 = Z1*US1 121
IF(US1.GT.AL) GC TO 60 122
50 U1 = AL 123
CUW 3 = 0.0 124
CUW 5 = 0.0 125
CUW7 = C.O 126
CAM 1 = 0.0 127
GC TC 100 128
60 IF(USl.GE.AU) GC TO 70 129
U1 = US1 130
CUW3 = US1/(3.Â»W(3 ) ) 131
CUW5 = US1/(3.*W(5 ) ) 132
CUW7 = EXU*USlÂ«KM/(KM*h(7)  MU) 133
GAM 1 = C.O 134
GC TC 100 135
70 CONTINUE 136
LI = AU 137
240
CUW3 =
0.0
CUW5 =
0.0
CUW7 =
0.0
GAÃ‘I =
0 .0
100
CONTINUE
U2
US2
U3
US3
GAM 2 =
0.0
GAM 3 =
0.0
G ( 1 ) =
W (2 )
G ( 2 ) =
W(3)/(U1*U1Â«U2)
G ( 3 ) *
W(2)*W(4 )
G (4 ) =
KM*(ML 4 U 1 * U 3 )
G { 5 ) =
~ W ( 6 ) * W ( 4 )
G ( 6 ) =
W(5 ) /{U1Â«U1Â«U2)
C ( 7 ) =
â€” to { 6 ) #W { 2 )
200
CONTINUE
CO 220
1 = 1,
7
CC 2 20
J = l,
7
220
CG( I , J)
=
0.0
CG(1,2)
=
1.0
CGU
= 
2.*G(2)/U1
C G ( 2 Â» 3 )
=
CGUÂ» CUW3 +
CG(2,5)
=
DGU*CUW5
CG(2,7 )
=
C G U * C U W 7
CG(3 Â» 2 )
=
W ( 4 )
CG{3,4 )
=
W ( 2 )
CGU
=
KM*U3
CG(4 t 3 )
=
DGU*CUW3
CG(4 Â» 5 )
=
DGUCUW5
CG{4i7 )
=
DGU*CUW7
CG(5,4)
= 
W ( 6 )
CG{5,6 )
= 
W ( 4 )
CGU
= 
2 .*G(6}/U1
1./(U1*U1*U2)
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
241
300
CGl6 Â»3 )
CG C 6 * 5 )
CG Ã 6,7)
CG(7,2 )
C G ( 7 Â» 6 )
CONTINUE
RETURN
ENC
DGU * CUW 3
DCUÂ»CUW5
DGU*CUW7
W ( 6 )
W ( 2 )
i. / (U1*U1*U2)
172
173
174
175
176
177
173
179
242
SUBROUTINE CCRRC (N,M,NC,NP,NPAR,ITER,T,P,BL,BR,Y) 180
IMPLICIT REAL *8( AH , 0â€œZ ) 181
R E A L * 8 MU,KM,ML,LAM 182
C0MMGN/QUC0MM/X,ERROR,ÃœFFDG,H,E1MAX,E2MAX,BL0 183
COMMCN/DATA/ AU , AL, EU , EL,RU,RL,MU,KM,DU,SU,LAM 184
DIMENSION T(NP),P(NPAR),BL(NC,NC ) ,BR(NC,NC),Y(NPÂ»NC) 135
Y<1, 1 ) = 0.0 186
Y(1 ,2) = 0.0 187
Y{N P , 3 ) = 0.0 188
Y ( N P Â»4 ) =LAM 189
Y(NP,5 ) =0.0 190
Y Ã1,6 ) = 0.0 191
Y(l, 7) =, 0.0 192
RETURN 193
ENC 194
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BIOGRAPHICAL SKETCH
Jerry C. Hornbuckle was born in Clearwater, Florida, in 1942.
On graduating from Stranahan High School in Fort Lauderdale, he attended
the Aerospace Engineering program of the University of Florida. Studies
there were interrupted by a year's employment at Honeywell's inertial
guidance division in St. Petersburg, Florida. He received his BSAE
from the University of Florida in 1965.
Following graduation he was employed by the Boeing Company in
Huntsville, Alabama. While there he worked on error analyses, the
ApolloSaturn V onboard computer flight program, and trajectory simuÂ¬
lation. During the evenings he was a parttime student at the UniverÂ¬
sity of Alabama in Huntsville, receiving an MSE in Engineering MechanÂ¬
ics in 1971.
In 1969 he began doctoral studies in the Engineering Sciences
department of the University of Florida. Although specializing in
dynamics and vibrations, his dissertation research was in the field of
structural optimization. He expects to graduate in August, 1974, with
a Ph.D. in Engineering Mechanics.
Mr. Hornbuckle is married to the former Carolyn A. Smith of
Fort Lauderdale and has two children. He enjoys golf, tennis, racquet
ball, music, and reading. In the area of reading, he is interested in
psychology and history, especially military and aviation history.
254
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
/ /
William H. Boykin, Jr., Chairman
Associate Professor of Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Robert L. Sierakowski, CoChairman
Professor of Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
j/l/
Gene W. Hemp
Associate Professor of Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Associate Professor of Mechanical Engineering
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted in partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
August, 1974
Dean, Graduate School
PAGE 2
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