
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00029936/00001
Material Information
 Title:
 Genetic search methods for multicriterion optimal design of viscoelastically damped structures
 Creator:
 Lin, ChyiYeu, 1957
 Publication Date:
 1991
 Language:
 English
 Physical Description:
 vii, 211 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Beams ( jstor )
Composite materials ( jstor ) Crossovers ( jstor ) Damping ( jstor ) Design optimization ( jstor ) Genetic algorithms ( jstor ) Integers ( jstor ) Objective functions ( jstor ) Term weighting ( jstor ) Trusses ( jstor ) Aerospace Engineering, Mechanics and Engineering Science thesis Ph. D Composite construction ( lcsh ) Dissertations, Academic  Aerospace Engineering, Mechanics and Engineering Science  UF Structural optimization ( lcsh ) Viscoelasticity ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1991.
 Bibliography:
 Includes bibliographical references (leaves 201210)
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by ChyiYeu Lin.
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 University of Florida
 Holding Location:
 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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25605195 ( OCLC )

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GENETIC SEARCH METHODS FOR MULTICRITERION OPTIMAL DESIGN
OF VISCOELASTICALLY DAMPED STRUCTURES
By
CHYIYEU LIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991
To my parents
ACKNOWLEDGEMENTS
The author would like to express his greatest gratitude to his committee chairman, Dr. Prabhat Hajela, for his continued guidance, instruction and friendship. He would also like to thank Professors Lawrence E. Malvern, ChangTsan Sun, Bhavani V. Sankar, and Ronald A. Cook for their careful review of this dissertation, and for their service on the committee. Appreciation is also extended to Dr. David C. Zimmerman for his volunteer review of this dissertation.
The author would like to express his appreciation to the Army Research Office and the Department of Aerospace Engineering, Mechanics and Engineering Science at the University of Florida, for providing financial support during the course of his graduate studies. The author would also like to say thanks to all his friends in the department for making his study a lovable memory. Special thanks are given to Dr. Vidish S. Rao for his help with his finite element codes, and to William J. Leath, Jr., for his proofreading. Finally, the author would like to express his deepest thanks to his wife DaiLih for her sacrifice, understanding, support and encouragement during his time of study at the University of Florida.
Ill
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS.........................................................................................iii
A B STR A CT......... ...........................................................................................................vi
CHAPTERS
1. INTRODUCTION .................................................. 1
1.1 Literature Survey......................................................... ........................... 1
1.2 Scope of Present Research ...........................................6
2. DAMPING TREATMENT AND ANALYSIS ................................................... 9
2.1 Background ..................................................................................................... 9
2.2 Measures of Damping . ................................................................... 17
2.3 D am ping A nalysis ................................................................................... 22
2.4 Design Optimization Formulation................................26
3. TRADITIONAL METHODS OF OPTIMIZATION ............................30
3.1 Introduction.......................... ................................................................. 30
3.2 FeasibleUsable Search Direction Method............................................31
3.3 Modified BranchandBound Search................................ ....36
3.4 MinMax Global Criterion Approach.................................38
3.5 Num erical Implem entation.....................................................................43
4. GENETIC ALGORITHMS ..................................................................................60
4.1 Introduction .................................................................................................. 60
4.2 G enetic Algorithm s................................................................................ 63
4.2.1 R eproduction.............................................................................. 68
4.2.2 C rossover... ................................................................................ 70
4.2.3 M utation................................................. ................................... 72
4.2.4 Im plicit Parallelism ..................... .................................................. . 73
4.3 Representation Schemes for Design Space.............................................77
iv
4.3.1 Continuous Design Variables.............................. ....77
4.3.2 Integer Design Variables........................78
4.3.3 Discrete Design Variables............................... .....81
4.3.4 Performance Evaluations of Representation Schemes............82
4.4 Multicriterion Design.............................................. .. ........ 85
4.4.1 Sharing Function Approach...................................87
4.4.2 VectorEvaluated Approach.............................. .....91
5. NUMERICAL IMPLEMENTATION OF GENETIC ALGORITHMS.....95
5.1 SingleCriterion Optimization..........................95
5.1.1 Riveted Lap Joint...................................... .......95
5.1.2 Series Stacked Belleville Spring.............................. ....101
5.1.3 ElevenBar Planar Truss............................. 105
5.1.4 Laminated Composite Plate Design.........................................109
5.2 Multicriterion Optimization .......................... 122
5.2.1 Statically Loaded TenBar Truss.....................................122
5.2.2 Dynamically Loaded TenBar Truss................................... 145
5.2.3 WingBox Structures ........................... 148
5.2.4 Viscoelastically Damped Laminated Composite Beam........163
6. APPROXIMATION CONCEPTS IN GENETIC ALGORITHM BASED
OPTIMIZATION ..................................................................................171
6.1 Introduction................................................................................................ 171
6.2 N eural N etworks................................................................................... 172
6.3 BackPropagation Training Algorithm........................... .....177
6.4 Multicriterion Optimal Design of ThreeDimensional
Viscoelastically Damped Laminated Composite Beam.................181
7. CONCLUDING REMARKS ......................................................................195
7.1 Conclusions................................ .... .......................................... 195
7.2 Recommendations for Future Work....................198
REFERENCES...........................................................................................................201
BIOGRAPHICAL SKETCH...........................................................................................211
v
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
GENETIC SEARCH METHODS FOR MULTICRITERION OPTIMAL
DESIGN OF VISCOELASTICALLY DAMPED STRUCTURES By
ChyiYeu Lin
December, 1991
Chairman: Dr. Prabhat Hajela
Major Department: Aerospace Engineering, Mechanics and Engineering Science
This study examines the use of genetic search techniques in the
multicriterion optimal design of structural systems. The global nature of these search techniques makes them better suited for design spaces that are known to be disjoint or nonconvex. The study was motivated by an application involving the optimal design of viscoelastically damped composite structures. In addition to the multicriterion nature of this design problem, its design space is a mix of continuous, integer, and discrete design variables. The more traditional mathematical programming based methods of optimization are known to generate suboptimal designs for this class of problems.
Genetic algorithms belong to a general category of stochastic search
methods. An optimal design is located by combining favorable characteristics of vi
several designs in such a way as to maximize a defined fitness function. The transformation operators that facilitate this combination have their philosophical basis in biological principles of evolution. As shown in this work, the method can easily accommodate the presence of integer and discrete variables in the design space. An extension of genetic search to the multicriterion optimal design problem is presented in this work. This design problem requires that an optimal design be generated for which each of the criteria are simultaneously optimized. Two distinct strategies for generating such designs are presented in this work. A distinct feature of these strategies is that they allow for a simultaneous generation of optimal designs in problems where the individual criterion are weighted differently.
The design of viscoelastically damped composite beam structures
considered in this work was based on a threedimensional finite element solution. This analysis is computationally demanding, and in order for it to be used for optimization, approximations to the exact analysis were required. A multilayer perceptron model of an artificial neural network was used to map the inputoutput relationships between the design variables and the output response, and this approximation was used in the optimal design in lieu of the exact analysis. Optimal designs of composite beams for minimum weight and maximum damping were obtained using this approach.
vii
CHAPTER 1
INTRODUCTION
1.1 Literature Survey
The earliest theoretical developments in the field of structural optimization may be traced back to a publication by Maxwell [1] in 1869. In 1904, Michell [2] proposed a theory which provided valuable insights into determining the optimum configuration of truss structures under a single load; however, the approach had limited practical applications. In the 1940's there was a new development in the theory for optimal design of structural componentsthe simultaneous failure mode theory. As the name suggests, the structural components were sized in a manner such that failure would occur simultaneously in several failure modes. In the area of plastic collapse design, linear programming methodology was used in the early 1950's to obtain the optimum configuration of truss and frame structures.
Since 1960, optimality criteria methods and nonlinear programming
algorithms (NLP) have become two of the most popular structural optimization methods. The optimality criterion methods belong to a class of indirect methods for optimal design. A structure with given loading conditions and prescribed constraints is deemed optimal when its derived optimality criterion is satisfied via
1
2
numerical algorithms. This optimality criterion technique [35] is not general, as the optimality criterion definition is problem dependent, and quite often difficult to obtain. The use of nonlinear programming techniques in optimum structural design affords a more direct and flexible solution procedure. Professor L. A. Schmit pioneered this approach, now being widely recognized as a viable design tool in the industry. A number of key review publications and illustrations of applications are available in References 617.
Although NLP methods have been developed to the point where they can be used almost routinely, they have several drawbacks. Such drawbacks may result from having an optimization problem with a large number of design variables and constraints. Furthermore, the design space may be disjoint or nonconvex. Nonlinear programming methods are not suitable for an optimization problem involving one or more design variables which are of discrete/integer type because of the difficulties in applying the onedimensional search technique, and in the determination of gradients of objective function and constraints. A combinatorial programming strategy has been proposed to overcome the discreteness of the design space. This technique demands significant investment of computational resources which worsens as the dimensionality of the noncontinuous design variables increases. Since the traditional NLP based methods essentially represent a local search strategy, they can only determine the relative optimum located nearest to the starting design. Further, as the more efficient methods belonging to the category of NLP require at least the first order gradient information of the
3
objective function and constraints, they will very likely fail as the search passes the disconnected region of a disjoint design space.
Decisions in engineering design typically involve multiple and frequently conflicting requirements. Perhaps the first effort in the field of multicriterion optimization may be attributed to an Italian economist, Pareto, who in 1896 introduced the concept within the framework of welfare economics [18]. The implications of his work to optimization theory, operations research, and control theory were only recognized in the late 1960's. Multicriterion optimization in engineering design has been examined, and numerous examples of the design of mechanical structures have been presented [1927]. Despite the multicriterion nature of most design problems, the principal focus of most research efforts has been on pursuing an efficient algorithm for the scalar objective function optimization problem. One common approach to solving multicriterion optimization problems is to select one of the criteria as the objective function, and to trade off among the remaining criteria using appropriate design constraints. The apparent simplicity afforded by this method is attractive; however, it should be noted that the treatment of criteria as constraints does not yield the same optimum design as would be obtained when solving the optimum design problem as one possessing multiple objectives [28].
The goal of this research was to develop efficient methods for
multicriterion optimal design. The approach was then applied to optimally design viscoelastically damped structures with maximum damping and minimum weight.
4
This exactly characterizes a multicriterion structural optimization problem. In this problem, however, the multicriterion optimization problem was complicated by the presence of discontinuity in the design space. The design variables were a mix of integer, discrete, and continuous types. Integer and discrete design variables were necessary, because the number of composite laminae forming a laminate is characterized by an integer variable, and the thicknesses of the constraining and viscoelastic layers were restricted to a manufacturerprovided list of discrete values. This multicriterion optimal design problem comprises a complicated design space which is discontinuous, and nonconvex.
A finite element based analysis of the structural damping ability of a constrainedlayer treated composite beam was investigated by Sun and his coworkers [2933]. The loss factor of a composite laminate of arbitrary stackingsequence, and with a discrete type of addon viscoelastic damping treatment can be effectively and accurately predicted. This work was used in the present research as an analysis tool for determining the characteristics and the dynamic/static response of viscoelastically damped composite structures.
In order to solve the multicriterion optimization problem with a
discontinuous and/or nonconvex design space effectively, two methods were explored. The first approach was an NLP based procedure with a branchandbound type algorithm used to account for the integer and discrete design variables. The second approach was the use of genetic algorithms, and this comprises a principal focus of the present study.
5
Genetic algorithms were first proposed by John Holland in 1975 [34], and have since been adapted for a large number of applications in game theory, induction systems, and other aspects of human cognition such as pattern recognition and natural language processing [3538]. These algorithms are generally regarded to be in the same category of stochastic search methods as simulated annealing [39]. Both approaches have their basis in natural processes (simulated annealing is derived from the principles of statistical mechanics).
Genetic search methods are based on Darwin's theory of survival of the fittest [40]. A set of design alternatives, (analogous to a population in a given generation) are allowed to reproduce and cross among themselves, with bias allocated to the most fit members of the population. Combinations of the most desirable characteristics of the mating pairs of the population result in progenies that are more fit than either of the parents. If a measure which indicates the fitness of a generation is also the desired goal of a design process, successive generations produce better values of the objective function. An obvious advantage in this approach is that the search is not based on gradient information, and has no requirements on the continuity or convexity of the design space. Another advantage is that the solution space in genetic algorithms is a discrete mapping of the design space; thereby a design space with a mix of integer and discrete design variables can be treated efficiently.
While the potential of genetic algorithms as a function optimizer has been a subject of more recent interest, the first use of genetic search methods in the
6
context of structural optimization was done by Hajela [41,42]. In that work [42], globally optimal structural designs were generated using genetic search methods, despite the existence of a nonconvex and disjoint design space.
The present research proposes and evaluates a number of distinct coding strategies for accommodating integer, discrete, and continuous design variables in genetic algorithms. It also extends the genetic search method to simultaneously obtain multicriterion optimal designs for several cases where the individual criterion are weighted differently relative to the remaining criteria.
1.2 Scope of Present Research
The scope of the present work and its organization can be summarized as follows. Damping in composite structural elements is the subject of chapter two. Both the inherent damping characteristics of composite structures and addon types of passive damping are discussed. The influences of constituent material properties and the geometrical configuration of the composite, on its damping characteristics, are presented. Three optimal design problems for damping enhancement are formulated, and a detailed implementation of these design problems is presented.
Chapter three provides an introduction to the traditional nonlinear
programming techniques, with a focus on the feasibleusable directions method. This method was used in conjunction with a modified branchandbound strategy for solving discrete/integer design variable problems. The minmax global
7
criterion strategy for multicriteria optimization is introduced. A twodimensional idealization of a cantilever beam with addon viscoelastic damping layers, is optimally configured as an example of an NLP based method for multicriterion optimal design of structural sandwich beam.
In chapter four, the theoretical background and the basic elements of genetic search methods are reviewed. The binary implementation of integer, discrete, and continuous design variables is discussed. Two distinct approaches for generating multicriterion optimal designs are presented in this chapter. These methods are proposed as an alternative to the more computationally intensive NLP based schemes for multicriterion design.
Chapter five describes the numerical implementation of genetic search
methods described in chapter four. To demonstrate the use of genetic algorithms in structural/mechanical optimization problems with a mix of integer, discrete, and continuous design variables, a lap rivet efficiency maximization problem, a Belleville spring sizing and stacking problem, and an optimal structural sizing problem were investigated. The optimal stackingsequence of square and rectangular, simplysupported laminated composite plates, for maximum transverse stiffness, was also considered. Genetic search methods for simultaneously generating multicriterion optimal designs were investigated on a number of structural optimization problems with multiple design criteria. Included in this study was a viscoelastically damped, laminated composite beam problem which used maximum damping and minimum weight as optimization criteria.
8
In chapter six, a neuralnetworkbased approximation of structural analysis is proposed for use in conjunction with genetic algorithms. This significantly alleviates the required computational effort in the optimal design. A number of multilayer perceptron models of neural networks and the backpropagation algorithm used in the training of such networks, are also discussed. This approximation strategy is examined in the context of a multicriterion optimization problem where a 3dimensional cantilever composite beam with addon viscoelastic damping tape is optimally configured for maximum damping. The influence of such damping mechanisms on the viscoelastically damped cantilever beam is studied in this implementation.
Conclusions drawn from this study and recommendations for future research are presented in the final chapter.
CHAPTER 2
DAMPING TREATMENT AND ANALYSIS
2.1 Background
The need for efficient lightweight structures in the aerospace industry has dramatically increased the use of composite materials in the last decade. The application of composite materials in structures that are subjected to dynamic loads has introduced stringent requirements on the dynamic characteristics of these materials. Vibrationinduced damage in structures has long been one of the leading causes for structural failure. Scientists and engineers have dedicated tremendous efforts to controlling and reducing vibration.
Active and passive control are among the most popular approaches
currently used to improve the damping ability of structures. Active control utilizes external devices referred to as sensors and actuators to shape the dynamic characteristics of the system. On the other hand, passive control mechanisms emphasize the integrated optimal design of inherent structural characteristics and external damping devices to maximize damping under expected dynamic loads. More specifically, for a laminated structural composite, an optimal passive damping can be achieved by a combination of configuring the layup and material selection of the base structure (the laminated composite itself), and by the
9
10
addition of external damping materials such as viscoelastic tapes to the surface of the laminated composite. The optimization problem associated with tailoring the passive damping ability of viscoelastically damped composite structural elements was the main focus of this study.
Damping is the ability of a structure to suppress the vibration through the dissipation of vibratory energy. In earthbound structures, there are a number of damping mechanisms in which vibratory energy is converted to other forms of energy such as sound or heat [43]. These mechanisms are classified under the class of nonmaterial damping. Energy can also be dissipated as the material deforms, and in such cases the damping mechanism is active within the material volume [4345]. Material damping is also referred to as either internal damping or structural damping. This damping is caused by the energy dissipated due to mechanisms such as viscoelastic behavior in polymeric material or the interfacial slip in metallic materials.
The principal mechanism for damping in glass and graphite reinforced
polymer matrix composites is the viscoelastic energy dissipation that occurs in the polymer matrix. Under a normal stress level, the composite exhibits linear viscoelastic behavior, and its dynamic stiffness and damping are independent of vibration amplitude.
The first attempt to tailor the damping ability of basic composite structures can be traced to an effort due to Plunkett and Lee [46]. Several models have been proposed for predicting the damping ability in short fiber reinforced composites.
Sun and his coworkers [47] have developed theoretical relationships for the material damping of aligned short fiberreinforced polymer matrix composites under offaxis loading. The complex modulus, Ex', of the shortfiber composite can be expressed as,
1 1 cos40 sin4 1 2VLT s (2.1)
 _ + +   smnOcos26
E; E + iE E E; G E
where Ex' and Ex" are the storage and loss moduli, respectively; E L, E"T and G',, represents the complex form of the elastic modulus in the longitudinal and transverse directions, and the shear modulus, respectively; v'LT represents the inplane Poisson's ratio. The damping or loss factor can be obtained as follows
r7 x (2.2)
The model based on Cox's shear lag theory, as shown in Figure 2.1, allows us to express the loss factor (a measure of the structural damping) as a function of fiber and matrix properties and the geometric parameters shown in Figure 2.1.
E.1
 =rix(E, E,,, G,,,, p, s, d, D, 0) (2.3)
where E'f, E'n,, and G'n, represent the complex form of the modulus of the short
12
X
7 \/ \  M a t r i X j e
F/ I1 r trix
I I
Figure 2.1 Representative volume element for offaxis loading.
13
fiber, the modulus of matrix, and the shear modulus of matrix, respectively. A detailed expression for this is presented by Gibson et al. [48]. Experiments [49] indicate that the damping ability in composites reinforced by discontinuous fibers is generally greater than that in continuous fiber composites. Sun and Wu [50] attributed this to the stress concentration effects present at the fiber ends, which facilitate the transfer and dissipation of energy in the viscoelastic polymer matrix. The analysis [48,50] clearly indicated the dependence of internal material damping in composites on the fiber and matrix geometry. Parameters such as loading angle, fiber aspect ratio, volume fraction, fiber length, fiber diameter, etc. were all found to influence the dynamic performance of the composite. The analysis also showed that the increase in damping ability was usually associated with a corresponding stiffness loss. A parametric study can be conducted by varying each of these variables, one at a time, to examine their influence on the damping. However, parametric studies of this form have been shown to yield suboptimal results. Hajela and Shih [51] proposed a design synthesis approach based on formal multicriterion optimization. Two distinct analysis models, Cox's shearlag theory and an advanced shearlag model, were used in their work. With this approach, the extensional loss factor of a representative volume element (RVE) was maximized subject to constraints on the element mass and stiffness characteristics.
Damping in laminated fiber composites can be determined through the laminated plate theory approach. The inplane material damping, 7ij, of the laminated composite is defined as follows,
14
77 = E i = 1,2,6 (2.4) Eo/
where E"j, and E'j represent the real and imaginary part of the inplane complex moduli E'ij, which can be expressed as
1 I .
E = E. + Eij (2.5)
where (Aij)1 is the element in the it" row and jth column of the inverse of matrix [A'ij], which is the complex form of the reduced inplane stiffness matrix, [Ai,]. Damping in the laminated fiber composites can also be determined through an energy approach in conjunction with a 3dimensional finite element method. This model represents a more realistic approach by including the energy dissipated at the interfaces. Detailed derivation of damping with both approaches is available in [52].
Although the internal damping capability in laminated composites can be adjusted, the extent to which it can be increased is limited by other design requirements. To further increase the damping ability, viscoelastic surface damping treatments can be used. Addon viscoelastic treatments can be used to increase the damping ability of the laminated composites without decreasing the stiffness of the laminated composites. Extensional damping treatment, a commonly used surface damping treatment, is also referred to as unconstrained or freelayer
15
damping treatment. The viscoelastic material is coated on one or both sides of a structure; whenever the structure is subjected to cyclic bending, the damping material will be subjected to tensioncompression deformation. The synchronized tensioncompression deformation of the viscoelastic layer dissipates the energy away from the vibrating base structure.
The shear type of damping treatment provides a more efficient damping enhancement ability than the unconstrained layer treatment on a constant weight basis [53]. This is largely due to the fact that the energy dissipation in the viscoelastic material is entirely due to the shear deformation and is independent of dilatational deformation [54]. The shear damping treatment is similar to the unconstrainedlayer type, except that the viscoelastic material is constrained by a stiff constraining layer. Therefore, whenever the structure is subjected to cyclic bending, the constraining layer will force the viscoelastic material to deform in shear. This shear deformation is the mechanism by which energy is dissipated. Figure 2.2 shows a sketch of both free and constrained layer damping treatments. It has been noted that the overall mass, stiffness, and damping characteristics of a structure will determine its damping ability. All of the above three parameters, and not just the damping material, need to be optimally configured in order to obtained the highest damping ability under some design constraints. Typically, in laminated composite structures, several factors, such as the stacking sequence in the laminate, the location, amount, and type of treatment, strongly influence the damping response.
16
FREE LAYER CONSTRAINED LAYER DAMPING TREATMENT I DAMPING TREATMENT
Damping material
Undefomted Composite Eleme Fol ccnsra~~Longhilayer den Shear defomiation
Composite Element Deformed
Figure 2.2 Illustration of mechanisms of damping treatments.
17
2.2 Measures of Damping There are various damping measures which are commonly used in different areas. Three common definitions of damping are introduced in this section.
For a single degree of freedom system shown in Figure 2.3, with viscous damping and an external excitation F(t), the differential equation of motion is, mI + ci + kx = F(t) (2.6)
where m represents the mass of the system, c represents the viscous damping, and k represents the spring constant. For viscouslydamped free vibration, the general solution to the homogeneous equation is
x = e"" (cle" + cze m (.
where
a c k ) k (2.8)
c1 and c2 are constants to be determined by initial conditions. An important characteristic of a structural system, the damping ratio or damping factor, E, is defined as
c = (2.9) Cc
18
k 1I c
m
F(t)
Figure 2.3 Sketch of viscous damping model.
19
where
cc = 2 mk (2.10)
A value cc for c determines the system to be overdamped (c > ce), underdamped (c < ce), or critically damped (c = ce). Depending whether it is overdamped, underdamped, or critically damped, the system exhibits an oscillatory motion, nonoscillatory motion, or critically damped motion, respectively.
A convenient way to determine the amount of damping present in a system is to measure the rate of decay of free oscillation. The logarithmic decrement, 6, is defined as the natural logarithm of the ratio of any two successive amplitudes of a free vibration, as shown in Figure 2.4. A relation exists between the damping ratio, 1, and the logarithmic decrement, 6, under the assumption of low damping; the relation is E = 6/7r.
A commonly used definition of structural damping, also referred to as
hysteretic damping was originally formulated as a basis for describing the internal damping properties of solids [55]. If the hysteresis loop for a material has a certain shape, the damping can be considered by using a complex stiffness, K' (= K' + iK"), where the real and imaginary part are referred to as the storage modulus and loss modulus, respectively. The loss factor is defined as the ratio of the imaginary part to the real part of moduli. The complex modulus is only defined for harmonic excitation. However, this complex modulus approach can be
20
F = In
X X2
Figure 2.4 Sketch of logarithmic decrement.
21
used for the analysis of a structural system under arbitrary excitation as any real signal can be represented as a Fourier series.
For a structure exhibiting steady state oscillation, the loss factor rl can be defined in terms of energy quantities as follows,
S D (2.11) (2r W)
where D is the energy dissipated per cycle and W is defined as the maximum energy stored during a cycle. Ungar and Kerwin [56] showed that this definition of W is unambiguous only in lightly damped systems. It was also shown that for an arbitrary seriesparallel network of viscoelastic springs, the composite loss factor reduced to the following simple expression
r  r?, V, (2.12)
where r7, and W, are the loss factors and the strain energy stored in the n"' spring, respectively.
Hysteresis damping in a continuous composite system can be modeled by using the correspondence principle of viscoelasticity [57]. According to this principle, the elastic constants of the materials are simply replaced by the corresponding viscoelastic counterparts. For example, The dynamic behavior of such materials can be defined in terms of the complex modulus
22
E* = E' + iE" = E'(1 + in) (2.13)
where
E' = storage modulus
E" = loss modulus
77 = loss factor
2.3 Damping Analysis
The analysis of a structure with constrainedlayer treatment is different from other situations because the addition of the constraining and viscoelastic layers on the base structure will change, in addition to its mass and stiffness characteristics, the geometry of the structure. The stiffness changes significantly through the thickness of the configuration, varying from a high value corresponding to the base structure to a low value associated with the viscoelastic layer. Ross, Ungar and Kerwin [58] developed what is probably the most widely used analysis to describe the behavior of different type of surface damping treatments. The fundamental work in the area of the analysis of sandwich viscoelastic beams was done by Kerwin [59]. Other literature on this topic may be found in [60].
Since much of the difficulty in designing constrainedlayer damping
treatment is due to complicated geometries, it is therefore necessary to look at
23
finite element techniques for solutions to the problem. Since optimization techniques usually require a large number of function evaluations, efficient and accurate finite elements are desirable to model the composite configuration. Threedimensional finite element models of sandwich structures often result in highdimensionality stiffness matrices and associated significant computational requirements.
Sun, Sankar, and Rao [2933] have developed elements that are specially suited to modeling sandwich structures. The finite element implementation of the modal strain energy method [56,61] and the direct frequency response method [61] are used to analyze the free and forced vibration characteristics of the beam. Damping in the system is represented by using the complex stiffness approach which derives from the elasticviscoelastic correspondence principle. In the direct frequency response method, a forced vibration over a range of frequencies is considered and a theoretical response spectrum is generated. The loss factor, or the overall damping of the system, may then be calculated by using the halfpowerbandwidth technique. The technique is illustrated in Figure 2.5. The direct frequency response is computationally expensive, and it is mostly used to verify the predictions of other methods since it needs no assumptions about the level of damping. A more frequently used method to calculate damping is the modal strain energy method. The method is based on the following eigenvalue problem, [K] + [KG]]{U11 = o2[M]{U} (2.14)
24
Logarithmic Scale : At 3dB Below the Peak Value (AT
RESONANCE FREQUENCY) Linear Scale : At 0.707 the Peak Value
AFN
Loss Factor = Tj =
N
Z Half  Power Points
O
z
LL
FN
LL ()
z
FN FREQUENCY
Figure 2.5 Halfpowerbandwidth technique.
25
where
[K] = global conventional stiffness matrix
[KG] = global geometric stiffness matrix
[M] = global mass matrix
{ U = global displacement vector
The natural frequencies and mode shapes are calculated about the initially stressed state for the undamped system. The analysis can be based on the associated complex eigenvalue problem or the simplified real eigenvalue problem. When the analysis is performed based on the simplified real eigenvalue problem, the stiffness matrix is real and the corresponding nodal displacements are also real. Modal damping is then calculated by this technique using the modal strain energy method. The loss factor is calculated as
n
7 i (2.15) i=1
where
n = total number of elements
ri = loss factor of the i'h element
Ei = elastic shear energy; equal to 2 {u}T[k]{u}
{u} = displacement vector of the ith element
[k] = stiffness matrix of the i'h element
26
This technique is valid only for systems with relatively small levels of damping where the mode shapes and frequencies of the damped and the undamped structure are similar. The analytical predictions obtained by this method were in good agreement with the experimental results of Mantena [62] for several damped laminated beam configurations. This analysis [2933] was used in conjunction with optimization algorithms for solutions of optimization problems associated with viscoelastically damped composite elements in the present research.
2.4 Design Optimization Formulation The problem of optimization of a laminated composite beam for enhanced damping characteristics can be formulated as one or all of the following subproblems:
a) optimal design of the base structure.
b) design with addon viscoelastic damping treatment.
c) integral design of sandwich beam consisting of base structure and
addon damping treatments.
Damping ability of a laminated composite beam can be enhanced by
optimally configuring the internal structural parameters of the composites. These structural parameters include ply thicknesses, laminate thickness, material properties of each laminae, fiber orientation, etc. By using formal optimization
27
techniques, the damping ability can be maximized with the satisfaction of prescribed design constraints.
The use of addon viscoelastic damping is convenient and efficient, because the configuration of the base structure need not be altered. Given a laminated composite beam, the treatment of viscoelastic damping can be done by attaching the viscoelastic damping layers to the surface of the laminated beam. Only one viscoelastic layer is required for unconstrainedlayer treatments. One viscoelastic layer and a thin constraining layer are needed for constrainedlayer treatments. For a twodimensional cantilever beam, an arbitrary length of selected addon damping tape can be placed on the composite beam. Issues such as the precise location of damping material and its quantity are resolved by expressing them as decision variables in the dampingenhancement optimal design problem.
Placement of the tape in a certain form may create some problems in the analysis. As the tape moves from one point to another point, a new finite element model will be required, and a new finite element mesh of the structure is necessary. An innovative technique which assures a smooth variation in node locations during redesign is proposed in this work. For a noncontinuous placement of the damping tapes, a number of unconnected damping tapes can be placed along the length of a beam. Several different ways of placing damping tapes on a cantilever beam are shown in Figure 2.6. The problems associated with the discontinuity of damping tapes manifest themselves as discontinuities in the node numbers, and the requirement of an adaptive preprocessor which is able to
28
CONSTRAINING LAYER VISCOELASTIC LAYER
CANTILEVER BEAM
CONTINUOUS
/ I
CANTILEVER BEAM
CONTINUOUS
CANTILEVER BEAM
DISCRETE
Figure 2.6 Examples of damping treatment on cantilever beam.
29
create a new finite element model for each new structure created by the optimizer. For a 3dimensional laminated composite beam, the aforementioned difficulties were overcome again with a more sophisticated adaptive preprocessor. The mathematical representation of the presence or absence of damping treatment on a predetermined section may be resolved by assigning it a 0/1 type of variable. In this representation, the design problem contains discrete variables, and optimization algorithms which can account for discontinuous design variables in an efficient manner, would be required.
The integral design of the entire structure is most effective for the
enhancement of the damping ability of the structure. This method is basically the combination of the design of the base structure and the damping treatments at the same time; here, design constraints can be more easily satisfied due to the increased flexibility in changing the design.
CHAPTER 3
NONLINEAR PROGRAMMING BASED OPTIMIZATION METHODS
3.1 Introduction
Most practical engineering optimization problems involve linear or nonlinear equality/inequality constraints. Mathematical programming based solutions to such problems can be classified into three categories. The first category is that of linear programming for which there are relatively efficient solution algorithms available. However, as the name linear programming suggests, these optimization problems represent a limited number of physical problems, because the objective functions and constraints must be strictly linear. The more general optimization problems involving nonlinear objective functions and constraints are approached in two ways. The first entails converting the constrained problems into an unconstrained optimization problem, by appending a measure of constraint violation to the objective function. This unconstrained problem can be solved by one of the many nonlinear programming algorithms. To prevent this approach from becoming numerically illconditioned, the optimal solution is actually obtained by solving a sequence of unconstrained problems, corresponding to increasing values of the penalty on constraint violations. This type of method is referred to as a sequential unconstrained minimization 30
31
technique or SUMT. The third category consists of methods which are commonly used in practical engineering optimization problems. This approach deals with constraints directly in searching for an optimal solution, and is classified as a direct method. Such an approach was adopted extensively in the present work, primarily to obtain baseline optimal solutions to test problems. These solutions were then used for comparison with solutions obtained through other schemes proposed in this work. The direct method used here is referred to as the method of feasibleusable search directions, and is summarized in the following sections. A review of mathematical programming applications in structural optimization is available in [63].
3.2 FeasibleUsable Search Direction Method
An overview of the optimization problem and its formal mathematical
statement are introduced next. The method of feasibleusable search directions is then discussed.
In an engineering optimal design problem, the quantities to be determined are the design variables. For a given optimal design problem, there usually exist some restrictions dictated by the environment, processes, or resources, which have to be satisfied in order to produce an acceptable solution. These requirements are referred to as constraints, and describe dependencies among design variables and other parameters. Constraints are generally expressed mathematically by either equalities or inequalities. Among the designs satisfying all constraints, one has to
32
be selected as the best or optimal design based on the value of some merit or criterion function. This criterion is an integral part of the optimization problem formulation. It is generally a computable function of design variables and other parameters, and is called the objective function.
A general constrained optimization problem can be mathematically formulated as follows:
Minimize: F(X) objective function (3.1)
subject to
g (X) _ 0 j=1,m inequality constraints (3.2) /rk(X) = 0 k=l,n equality constraints (3.3)
xi < x, < x1 i=l1,p side constraints (3.4)
where X = [x1, x2, ..., Xp]T is a vector of design variables; xi' and xiU represent the lower and upper bounds of design variable xi, respectively.
If equality constraints are explicit in X, the number of design variables can often be reduced. Although side constraints (Eqn. 3.4) can be included in the inequality constraint set (Eqn. 3.2), it is usually convenient to keep them separate, as they define the region of search for the optimum.
33
Most mathematical programming algorithms require an initial design vector Xo, where initial values of each design variable x,0 20, x..., xP, are specified. From this starting point, the design is updated iteratively to an improved solution. The most generally used iterative update equation is as follow, Xq = Xq1 + aq Sq (3.5)
where superscript q is the iteration number, X is the vector of design variables, S is the search direction, and ",' is a scalar multiplier determining the amount of change in X for each iteration. The search direction, S, is first determined in each iteration, and the search moves in this direction to update the X vector according to Equation 3.5.
The method of feasible directions, or method of feasibleusable search directions [64], is introduced by an example with two design variables and two inequalities constraints with reference to Figure 3.1. The figure shows the contour lines of the constant objective function F(X), and also shows the constraints.
The goal of this method is to determine a search direction which rapidly reduces the objective function while maintaining a feasible design. Consider a design Xo on the constraint boundary gl(X)= 0. Gradient vectors of the objective function, VF(Xo), and the active constraint, VgI(XO), are calculated. The lines (hyperplanes in n dimensions) tangent to the contour of constant objective and tangent to the constraint boundary are now the linear approximation to the problem at this point. The next task is to determine a search direction S which
34
X2 Feasi
sector F(X)=constant
g1(X)=O X,
Figure 3.1 Illustration of the feasibleusable search direction approach.
35
reduces the objective function without violating the active constraint for some finite move. An S vector is called a usable direction if the objective function is reduced or maintained at its present value for a finite move along that direction. The usability requirement is mathematically defined as follows, VF(Xo) S 5 0 (3.6)
A direction S is called feasible if, for some finite move in that direction, the active constraint is not violated. The feasibility requirement can also be mathematically defined as follows:
Vgl(X) S 0 (3.7)
A region is called feasibleusable if any direction, S, inside that region satisfies both Equations 3.7 and 3.8. The search direction S is chosen to be located in the usablefeasible region such that the objective function, for a small move step, is reduced while the feasibility is maintained.
Since a method such as the feasibleusable direction approach is essentially a local search technique, only the relative optimum will be attained if the design space is nonconvex. A design can be treated as the globally optimal design with a higher degree of confidence only if a large number of optimization processes with different starting values of vector X converge to the same final design. If a selected number of optimization processes results in different "optimal" designs, it
36
usually indicates that the design space is multimodal or nonconvex. This problem can generally be solved by either switching to more sophisticated algorithms or by considering more initial designs to increase the probability of locating the global optimal design. It should be noted that any optimization method, except the exhaustive search technique, provides no guarantee of locating the global optimum.
To improve computational efficiency in nonlinear optimization problems, a piecewise linearization of the design space is frequently adopted. In this approach, the nonlinear programming problem (both objective function and constraints) is first linearized about the starting design. The solution to this approximate linear problem is obtained within some prescribed move limits so as to maintain the integrity after linear approximation. The problem is next linearized about the new solution, and the process repeated till convergence.
3.3 Modified BranchandBound Search The discrete nature of the design variables in most structural optimization problems has only received limited attention. Solutions to such problems are relatively more difficult to obtain than for optimization problems with only continuous design variables. Discrete design variables in structural optimization have often been treated as continuous design variables. Upon obtaining the optimal design for this simplified problem, design variables which represented discrete design variables were then adjusted to the nearest discrete value. This
37
simplified approach worked well for optimization problems with discrete design variables whose values were spaced reasonably close to one another. However, this simple rounding procedure often failed in locating the true optimal solution, or resulted in infeasible solutions [65]. One branch of mathematical programming, referred to as integer programming, offers a formal solution for optimization problems using integer/discrete design variables. Early work in obtaining systematic solutions to the integer linear programming problem are described by Gomory [66]. The branchandbound algorithms that emerged later are based on the enumeration of the space of the feasible integer solutions. A more detailed explanation of the branchandbound approach is available in [67].
The general framework for solving an integer programming problem involves decomposing the original problem into subproblems, modifying constraints to enlarge feasible domains, and finally a process referred to as fathoming. Fathoming involves checking a solution for feasibility, and establishing optimality.
A modified mixed integer and discrete programming algorithm was
proposed by Hajela and Shih [68]. The algorithm employs a strategy where a systematic search of continuous solutions is made, in which the discrete and integer variables are successively forced to assume specific values. The basic solution strategy used in this method for nonlinear mixed integer programming problem is a variant of the approach proposed by Garfinkel and Nemhauser [69]. The logical structure of the set of solutions was constructed as a binary tree. A
38
modified feasible directions algorithm [70] was used in the solution of the continuous nonlinear programming problem, with piecewise linear representation of the objective function and constraints.
3.4 MinMax Global Criterion Approach Decisions in engineering design typically require allocation of resources to satisfy multiple, and frequently conflicting, requirements. For instance, in the design of a viscoelastically damped cantilever beam, minimum weight and maximum damping are two criteria which always conflict with each other. Nevertheless, these two criteria are both important characteristics. Adding on viscoelastic damping tape would increase the damping ability of the beam while, at the same time, increase the weight. One approach to solving this multicriterion optimization problem is to select weight as the objective function, and to use damping as a design constraint, or vice versa. The apparent simplicity afforded by this method is attractive; however, it should be noted that the treatment of criteria as constraints does not yield the same optimal design as would be obtained when solving the optimum design problem as one possessing multiple objectives [28]. In the above optimization example, the influence of the relative importance of structural weight or modal damping on the distribution of viscoelastic damping material can be obtained only by formally solving a twocriterion problem with different weighting coefficient combinations on criteria.
39
In order to help understand the solutions to a multicriterion optimization problem, a frequently used expression, "Pareto optimal solution", needs to be defined. Given two designs Xi(xl,X,,...Xm) and Xj(xl,x,,...Xm) in an ncriterion minimization problem, the design Xi is referred to as being partially less than the design X,, if each element of the criterion vector of the design Xi is lessthanorequal to the corresponding element of design Xi, and, for at least one element, the relation is strictly lessthan. If Xi is partially less than Xj, X, is said to dominate Xj, or XJ is inferior to Xi. In a given set of designs, a population of designs which are not dominated by any other design in the given set is said to be nondominated or noninferior. The set of nondominated or noninferior designs are referred to as the Paretooptimal designs which are referred to as Paretooptimal solutions in an optimization problem.
An effective minmax variant of the global criterion approach for
multicriterion optimization problems was introduced by Hajela and Shih [71]. It is referred to as a global criterion method [72], and is broadly classified as belonging to a category of solution methods with no articulation of preference. In such an approach, a metric function is formulated to represent the distance between the ideal solution and the optimum solution, and a minimization of this function results in the true optimum. The ideal solution is an ndimensional vector for an optimization problem with n design criteria. An optimal value for each individual criterion defines a component of the ndimensional vector. These 'n' components of the ideal solution vector are obtained by considering the optimization problem
40
for each criterion separately. An application of the global criterion method is illustrated by Figure 3.2. This figure shows the feasible and infeasible sectors of the space of two criterion functions. If the ideal solution, fld(X), was located inside the feasible region, no further effort would be necessary. However, the ideal solution is typically infeasible, and the task becomes one of finding the feasible solution which is closest to the ideal solution. The Pareto optimal solution is obtained in such a fashion that any change of the optimal design vector will cause adverse effects on at least one of criteria.
One can define a vector objective function f(X) dependent on the design variable vector X in the following fashion:
f(X) = f,(X)f(X),...fi(X),...fk(X) ]
(3.8)
X = [x,,...]
where fi(X) is the i'" objective function or criterion. Likewise, f,(Xid) is the ideal or optimal value if the i'" objective function is considered as the sole criterion of a singlecriterion optimization problem, and Xid is the optimal design vector for such a criterion.
Minimization of the metric distance between the infeasible ideal solution and the candidate feasible solution by a minmax variant of the global criterion approach leads to the optimization problem described by the following mathematical statement.
41
f1(X) Pareto solutions
Min f2
:d Domain of feasible dn solutions
f id(X) Min f, f2(X)
Figure 3.2 Graphical representation of Pareto (noninferior) solutions in twocriterion
minimization problem.
42
Min Ma X)(X) =,2,...,k (3.9) fi(xid)
Solution of the above optimization problem yields the best compromise solution in which all criteria are considered equally important. Use of weighting coefficients can be introduced in conjunction with this method to rank the importance of each criterion. The minmax problem can be restated as follows:
Mmin Max fi() fi(i= ,2,...,k (3.10) fi(Xid)
where w, is the weighting coefficient representing the relative importance of the ih criterion.
A scalar variable p is introduced to transform the minmax problem of (3.10) into an equivalent scalar optimization problem. This singlecriterion optimization problem is as follows: Minimize 4 (3.11)
subject to the following additional constraints:
43
f,(X) f(Xid) 0 (3.12) f1(X id)
In a situation where the design variable set is a mixture of integer, discrete, and continuous design variables, it can be shown [71] that the ideal solution should be selected as the one obtained by treating all design variables as continuous.
3.5 Numerical Implementation
The modified branchandbound method, in conjunction with the minmax variant of the global criterion approach, was implemented in determining the multicriterion optimal design of a sandwich cantilever beam to enhance its damping ability. Damping ability in a structural composite or metal beam can be enhanced passively, via an external viscoelastic damping layer treatment. The design space for this problem consists of both continuous and discrete/integer design variables. An aluminum cantilever beam serves as a base structure of the structural system for which maximum damping in several modes is desired. Constrained viscoelastic layers were added to increase the damping ability of the structure. The structural system and two methods of adding damping layers are shown in Figures 3.3 and 3.4. Material properties for this structural system are given in Table 3.1. In the first case, the addon viscoelastic tape was applied starting from the root of the cantilever beam with the total length of the tape
44
constraining layer
viscoelastic layer
cantilever beam
0. 40m
S bea tcte
45
constraining layer
viscoelastic layer
cantilever beam
O. 40m
3.4 SaT ich bea st
46
Table 3.1 Material properties of the structural system.
constraining layer, and Viscoelastic layer base structure
Young's modulus 69.0 0.0021
(GN/m2)
Poisson's ratio 0.3 0.499 Damping factor 0.0 0.10
Density 2700.0 970.0
(kg/m3)
47
indicated by X5 (X4=0). In the other case, the addon damping tape was applied starting from an arbitrary point on the beam. The distance between the root of the beam and the starting point is indicated by X4, and the total length of the tape is indicated by X5.
One integrated approach to the optimal design of the viscoelastically
damped structure is to configure both the basic structure and addon damping layers. The optimal configuration is then determined by the thickness of the viscoelastic layer, the thickness of the constraining layer, and the thickness of the base structure. They are denoted as X1, X2, and X3, respectively. The remaining two design variables, X4 and Xs, result from how the damping layers were applied.
Finite element analysis was used to evaluate damping and natural
frequencies, as well as the static displacement in the structure resulting from the different applied lengths of the constrained viscoelastic layer. The structure of the finite element model representing the three layer sandwich is shown in Figure 3.5.
The base structure and constraining layer were modeled by using a
specially developed threenode, sevendegreeoffreedom, offset beam element as shown in Figure 3.6. The element contains sheardeformable characteristics, which are important in modeling fiberreinforced composites. A key feature of this model was its accountability of the coupling between the stretching and bending deformations. This allowed the nodes to be offset to one surface of the beam, coincident with the nodes of the adjoining element. The viscoelastic core was modeled using a rectangular plane stress element that is compatible with the
48
. X4 1 X 5 F
0.40m > Figure 3.5 Typical finite element model of the sandwich beam.
49
U,w,Y' w U,W,Y
Figure 3.6 Offset beam element.
50
offset beam element. The stiffness of the offset element was obtained from a strain energy formulation which was based on the following assumed displacement field:
u(x,z)=uo +(z )r(x) (3.13)
2
O(XZ) = W(x) (3.14)
* (x,) = *(x) (3.15)
where uo and r were obtained by linear interpolation and a quadratic interpolation was used to compute o. A consistent mass matrix, evaluated from the kinetic energy, was used in this analysis. A detailed description of the offset beam element is given in [31]. The loss factor of the system was determined by the strain energy based approach discussed in the previous chapter.
The vector of design objectives included minimum structural weight and maximum loss factors in the first three modes of vibration. The constraint set comprised lower and upper bounds on the first three natural frequencies, and the maximum displacement when a static load of 1N was applied at the tip of the beam. These are summarized in Table 3.2. Design variables X1 through X5 are shown in Figures 3.3 and 3.4. Thicknesses of the constraining and viscoelastic layers, X, and X,, were considered discrete variables, with admissible values selected from the set shown in Table 3.2. All other design variables were assumed
51
Table 3.2 Design constraints and allowable discrete values of design variables X1
and X,.
0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0010 0.0012 Xx, X2 0.0013 0.0014 0.0015 0.0016 0.0017 0.0018
0.0019 0.0020 (m)
20.0 < freq, _ 50.0 (Hz) Design 130.0 5 freq, < 200.0 (Hz)
Constraints 400.0 5 freq3 5 700.0 (Hz) tip displacement < 0.001 m
52
continuous. The mathematical statement of the optimum design problem can be written as follows:
Minimize (F,, F3, F4)
(3.16)
Subject to g < 0, j=1,2,...m
Here F1, F, and F3 represent the loss factors of the first three vibration modes, F4 is the structural weight, and g,
are the m inequality constraints for the problem as specified in Table 3.2. Two distinct cases were considered by setting X4 to zero, and by allowing X4 to assume finite values, respectively.
The locations and lengths of the damping tape were adjusted using an
adaptive preprocessor. Ten elements along the length of the base beam were used in the finite element analysis throughout the optimization process. A uniform node spacing was used in the situation that X4 and X5 are both multiples of onetenth of the length of the beam. When the length of an untreated free or fixed end is small, an unsmooth node spacing may result when an equallengthelement representation of the treated length is used. To avoid this situation, stretching techniques were utilized to assure a smooth variation in node spacing. A minimum node spacing ratio for any two consecutive finite elements so obtained resulted in an improved level of accuracy on the numerical prediction. Figure 3.7 shows a beam illustrating, from top to bottom, uniform node spacing, unstretched
53
 X4 I X5
0.4m
UNIFORM NODES
X4 X5
4. 0.4m
UNSTRETCHED NODES
X X4 X5
S0.4 m
STRETCHED NODES
Figure 3.7 Illustrations of node spacing arrangements.
54
node spacing, and stretched spacing. The adaptive preprocessor also possesses automatic node numbering ability to automatically create a finite element model based on the given set of design variables.
The minmax global criterion approach was used to transform the optimization statement (3.16) as follows: Minimize p (3.17)
subject to m original constraints plus the following four additional constraints:
fl(x) fl(Xid)
w f  id)
f(X id)
Sf(X) f3(Xid) 0id w2   0 (3.18)
f3(X id)
f (X) fX id)
wz  4<0 f,(x id)
where Wi, i= 1,4, are weighting parameters and each weight is used to emphasize the relative importance of the ith criterion. The nonlinear optimization problem presented in (3.17 and 3.18) was solved by the feasibledirection method with the modified branchandbound technique [68].
55
For case 1, the separately attainable optimal designs of each component of the objective function are shown in Table 3.3. The design variables were all assumed continuous in finding the ideal solutions. Assigning different weighting coefficients to the components of the objective function permitted the generation of a series of noninferior optimal solutions, as presented in Table 3.4. These noninferior solutions were such that no component of the objective vector could be improved without adversely affecting another component. Similar solutions for case 2 are presented in Tables 3.5 and 3.6. The inclusion of the X4 variable results in no improvement in F1d, F3id, and F4id. However, F2id (loss factor in second mode) increased by 2% and the length of the damping layer Xs decreased by 18%. Permitting X4 to assume finite values yields a noticeable improvement in the objective function, only when the weighting coefficients associated with F, outweigh those of F, and F3. This was because the middle region of the beam undergoes the most shear deformation for a displacement corresponding to the second mode of vibration. With X4 allowed to vary, and for weighting coefficients set as (0.0, 0.5, 0.0, 0.5), the damping factor of the second mode was increased by 5%c, while the weight of the structure was reduced by 1%.
56
Table 3.3 Ideal solutions for case 1.
F1ideal = 0.01825 F,ideal = 0.02003 F3ideal = 0.01833 F4ideal = 1.5860
xideal(F1) = (0.0020, 0.0001, 0.00454, 0.262) X'deal(F2) = ( 0.0020, 0.0001, 0.00446, 0.358) ideal (F3) = (0.0020, 0.0001, 0.00445, 0.400) X'deal(F4) = (0.0001, 0.0001, 0.00499, 0.000)
57
Table 3.4 Optimal solutions for problem in case 1.
Weighting Design Variables Loss Factor Weight Combination XI,X2,X3,Xs Mode 1
WzW4 (m) Mode 2 Mode 3
.20 0.0014 0.0142 1.8086
.20 0.0001 0.0154 .20 0.0044 0.0137
.40 0.359
.40 0.0019 0.0172 1.9350
.30 0.0001 0.0174 .10 0.0046 0.0098
.20 0.309
.20 0.0018 0.0167 1.9198
.40 0.0001 0.0187 .20 0.0044 0.0156
.20 0.357
.00 0.0015 0.0149 1.8397
.25 0.0001 0.0163 .25 0.0044 0.0140
.50 0.357
.00 0.0015 0.0149 1.8382
.50 0.0001 0.0163 .00 0.0044 0.0139
.50 0.355
58
Table 3.5 Ideal solutions for case 2.
F1ideal = 0.01825 F2ideal = 0.02037 F3ideal = 0.01833 F4ideal = 1.5829
X'dea'(F1) = ( 0.0020, 0.0001, 0.00454, 0.000, 0.262 ) Xideal(F2) = ( 0.0020, 0.0001, 0.00446, 0.060, 0.294 ) xideal(F3) = ( 0.0020, 0.0001, 0.00445, 0.000, 0.400) X'deal(F4) = ( 0.0001, 0.0001, 0.00498, 0.054, 0.000 )
59
Table 3.6 Optimal solutions for problem in case 2.
Weighting Design Variables Loss Factor Weight Combination X1,X,,X3,X4,X5 Mode 1
W,W4 (m) Mode 2 Mode 3
.20 0.0017 0.0115 1.8765
.20 0.0001 0.0172 .20 0.0045 0.0129
.40 0.022 0.319
.40 0.0019 0.0172 1.9350
.30 0.0001 0.0174 .10 0.0046 0.0098
.20 0.000 0.309
.20 0.0018 0.0167 1.9198
.40 0.0001 0.0187 .20 0.0044 0.0156
.20 0.000 0.357
.00 0.0015 0.0083 1.8433
.25 0.0001 0.0155 .25 0.0045 0.0144
.50 0.038 0.333
.00 0.0017 0.0066 1.8288
.50 0.0001 0.0171 .00 0.0046 0.0098
.50 0.059 0.262
CHAPTER 4
GENETIC SEARCH METHODS
4.1 Introduction
Mathematical nonlinear programming algorithms have emerged as the method of choice for applications in engineering optimization problems. They provide a general approach for obtaining solutions to both single and multiobjective design problems with a mix of equality and inequality constraints. The more efficient of this class of methods are generally gradient based, and require at least the firstorder derivatives of both objective and constraint functions with respect to the design variables. With this "slopetracking" ability, gradientbased methods can easily identify a relative optimum closest to the initial guess of the optimum design. There is no guarantee of locating the global optimum if the design space is known to be nonconvex [73]. These methods are also inadequate in problems where the design space is discontinuous, because the derivatives of both the objective function and constraints may become singular across the boundary of discontinuity.
In engineering design problems, the mix of continuous, discrete, and integer design variables has been approached by treating all variables as continuous, and then rounding specific variables either up or down to the nearest 60
61
integer or discrete variable. Branchandbound techniques based linear programming or combinatorial programming offers a formal solution to this class of optimization problems with a mix of integer, discrete and continuous design variables. This strategy consisted of a systematic search of continuous solutions in which the discrete and integer variables were successively forced to assume specific values. However, in doing so, the original optimization problem was undesirably expanded to a large number of suboptimization problems.
Exhaustive search and random search methods are among the simplest and most robust strategies for automated optimum design problems. These methods can work on almost all kinds of design spaces and without any restriction on types of design variables. An improvement on the simple enumerative techniques is possible with methods such as random walk and random walk with direction exploitation. The only drawback is that these methods often require thousands of function evaluations to achieve the optimum, even for the simplest of problems. It is hence crucial to examine alternative strategies for optimal structural design problems, which need less computational effort than required by the enumerative search techniques, and are also not susceptible to convergence to a local optimum as exhibited by gradientbased nonlinear programming algorithms. Genetic algorithms (GAs) as proposed by Holland [34], have the potential to successfully fill this gap.
Genetic algorithms belong to a category of stochastic search techniques, where only the most promising regions of the design space are enumerated to
62
locate the optimal design. These algorithms have their philosophical basis in Darwin's theory of survival of the fittest. Analogous to the natural process where a population of a given species adapts to a natural habitat, a population of designs is created and is then allowed to adapt to the design requirements. Designs that do not adapt in a favorable manner to the requirements are eliminated from consideration. The mechanism of adaptation borrows extensively from principles of biological evolution, in that basic characteristics of designs in one population are transferred to a population in another generation through gene transfer operators. Stated differently, design alternatives representing a population in a given generation are allowed to reproduce and cross among themselves with bias allocated to the most fit members of the population. Combination of the most favorable characteristics of the mating members of the population results in a progeny population that is more fit than the parent population. If the measure which indicates the fitness of a generation is also the desired goal of a design process, successive generations produce better values of the objective function.
The mechanics of genetic search, though simple to implement, encompass features that render the approach highly applicable to the problem of search in a nonconvex/disjoint design space with a mix of continuous, discrete, and integer design variables. These desirable characteristics are largely attributed to the fact that genetic search moves from a population of designs to another population of designs; this is in contrast to the point to point search available in traditional mathematical programming methods, and therefore offers a better possibility of
63
locating a global optimum. Furthermore, genetic algorithms work on a coding of the design variables rather than the variables themselves. This allows for an efficient treatment of integer and discrete variables. The terminology of genetic search and its principal components are summarized next.
4.2 Genetic Algorithms
The basic approach in genetic algorithms is to represent possible solutions to a given problem by a population of bit strings of finite lengths, and to subsequently use transformations analogous to biological reproduction and evolution to improve and vary the coded solutions. In natural populations, genetic information stored in chromosomal strings evolves over generations to adapt favorably to a static or changing environment. The chromosomal structure represents a generational memory, and is altered through chromosomal string inversions, occasional mutation, and a crossover of genetic information between reproducing members. In an elitist reproduction strategy, those members of the population that are deemed most fit are selected for reproduction, and are given the opportunity to strengthen the chromosomal makeup of the progeny generation. This approach is facilitated by defining a fitness function or a measure indicating the "goodness" of a member of the population in a given generation during the evolution process. For unconstrained maximization problems, the objective function could serve as the fitness function. The inverse of the objective function, or the difference between a large number (say the
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maximum value of the objective function in the population) and the objective function value for each member can be used as the fitness measure in a minimization problem. For constrained optimal design problems, an exterior penalty function formulation can be adapted to transform a constrained optimization problem into an unconstrained one.
In biological systems, generational memory is preserved and transferred to progeny populations in the form of chromosomes. A number of chromosome strings comprises a genotype, the total genetic makeup for an organism. In order to use genetic algorithms, an artificial chromosomelike string must be devised to represent the ndimensional solution space. One simple yet effective approach to accomplish this is to represent each design variable by a finite length binary string and then connect, headtotail, all n strings into a single binary string. Various simulations of genetic evolution and adaptation are conceivable. Three principal components of the genetransformation mechanism in this artificial evolution and adaptation simulation are reproduction, crossover, and mutation. These transformations are best described with reference to a specific optimization problem.
Consider the following function minimization problem:
3 2 (4.1) Minimize F(x,,2) = X1 +12 +X2 (4.1) Subject to: g,  x_ +xx, 5 0 (4.2) 2 = X XX2 < 0 (4.3)
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x in x xax i = 1,2 (4.4) xi < xi < xi i=1,2
To use genetic search, the constrained minimization problem is first converted into an unconstrained problem using the exterior penalty function formulation, resulting in the following problem:
Minimize F* = F + (4.5)
where P is the penalty parameter, and is typically given by,
p = r E 2 (4.6) j=1
where r is a penalty parameter of the form encountered in the exterior penalty function approach [74], and represents the violated constraints.
Careful consideration must be given to the selection of the penalty function P. Numerical experiments have shown that genetic search with a penalty function formulated as in Eqn. (4.6), exhibits slow convergence. This was largely due to the fact that a severely violated constraint yields a value of P which overwhelms the objective function value F. A more detailed explanation of this slow convergence will be given in a later section. In order to overcome this situation, the following bounding strategy was adopted. If the average fitness of feasible designs is Fa,., then a limiter value of the penalty L is selected as,
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L = k*F (4.7)
where k is of order 2; the penalty that is appended to an infeasible design is then obtained as follows:
2 2
r,if ri2E < L (4.8)
j=1 j=1
P=
L + (B L ,otherwise
The effect of this scaling is to prevent radical departures in the value of the penalty term from the specified value. If the slope parameter, a, is 0.0, penalty for all violated designs with P _ L is set to L. If instead, a is assigned a small value of the order 0.1, then the extent of constraint violation due to severely violated and less violated designs varies linearly from L, albeit with a small slope. To convert this function minimization into a fitness maximization as required by GA's, the following fitness function is created,
fi = Fmax  Fi (4.9)
where f, is the fitness of the ith design, and F 'ma is the maximum value of F'. The
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F'ma, can be chosen as the maximum F' in the population of that given generation, or of several previous generations. A fixed large value of F'ma can be used throughout the entire genetic search provided that there is not any F' which is larger than F'max. However, genetic search with a variable value of F'max selected to be slightly larger than the largest F" in the present or including past few generations, usually converges faster than with a fixed F'maxTo obtain a bit string representation of the design variables in this problem, each x, can be converted into a binary string of O's and l's. For purposes of discussion, we choose a 5 digit binary number, with the minimum and maximum values of xi denoted as x min and ximax, respectively, represented by the following binary numbers.
Xin,n = 00000 ximx = 11111
A linear scaling can be introduced to convert intermediate values of the binary number into design variable values. The binary string representations for x, and x2 can be placed headtotail to create a 10digit number, also referred to as a schema, which represents a solution to the problem. Several such 10digit binary strings are defined to constitute a population of designs, which includes a mix of feasible and infeasible designs. The fitness fi corresponding to each member of the population is computed before invoking the genetic transformations.
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4.2.1 Reproduction
The simulation of genetic evolution here is contrived in that the population size is forced to remain unchanged during the evolution process; two mating parents create only two progenies but are themselves eliminated. The reproduction process is one that biases the search toward producing more fit members in the population and eliminating the less fit ones. One simplistic approach to selecting members from an initial population to participate in the reproduction is to assign each member a probability of selection on the basis of its fitness. If f1 is the fitness measure of the i'h member, the probability for the ith member to participate in reproduction for each of m selections can be assigned as:
P(fi) = i=l1,m
m (4.10) k=1
where m is the population size. After m selections, a new population pool is created. The new pool is the same size as the original pool, but has a higher average fitness value. There is no new 'geneticmaterial' generated in the selection process.
Equation (4.10) shows that the probability of a design being selected for participation in the reproduction process depends not only on its fitness, but also fitness of other designs in the population. Furthermore, the value returned by a fitness function is not always an "exact" measure of fitness. The exact value
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returned can vary greatly depending on how the fitness function is implemented. For example, let a population in a minimization problem consist of four designs which have objective function values as 3, 5, 7, and 9. If the largest value 9 is chosen as F'm,,ax (Eqn. 4.9), then the fitness, fi, for these four designs will be 6, 4, 2, and 0, respectively. The probability of selection of these four designs are 6/12, 4/12, 2/12, and 0/12, respectively (Eqn. 4.10). The design with the lowest fitness will not be selected for participation in the reproduction process. The probability of the design with the best fitness is 3 times that of the design with the second worst fitness. Therefore, a higher fitness average is likely to occur after the reproduction process. This gives genetic algorithms power to guide the search towards more promising design areas. However, if the above designs have the following objective functions 3, 5, 7, and 85. The outstanding high objective function, 85, could result from multiple causes, for instance, it may be due to a severely violated constraint. If the value 85 were selected as F max, then, the fitness for these four designs will be 82, 80, 78, and 0. In each selection, the probabilities of the four designs for participation in the reproduction process would be 82/240, 80/240, 78/240, and 0/240. Almost no differences exist between the designs when considering the probabilities of participation with the exception of the worst case. This phenomenon makes it difficult for good designs to stand out from the population, and consequently the genetic search would not concentrate in the more promising areas. This illustrates a drawback of induced bad fitness as caused
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by violated constraints, and the advantage of using the modified penalty P, which was formulated in Equation (4.8).
Fitness proportionate reproduction (Eqn. 4.10) has been used almost
exclusively for allocating reproductive trials of genetic algorithms. Reproduction strategies that differed from the traditional, proportionate reproduction plan were implemented [75,76] in an effort to obtain improved performance of genetic searches. Baker [75] reported experiments where reproductive trials were allocated according to the rank of individual strings in the population rather than by individual fitness values as related to the population average. The rankbased allocation of reproductive trials slowed convergence. The method gave a slow, but more accurate optimization. Whitley [76] presented new evidence and arguments which suggested that allocating reproductive trials according to rank can be used to speed up a genetic search if appropriately implemented.
4.2.2 Crossover
The process of reproduction assures that more copies of dominant or fit designs will be present in a population. The crossover process allows for an exchange of design characteristics, among members of the population pool, with the intent of improving the fitness of the next generation. This is similar to the transfer of genetic material in biological reproduction processes as facilitated by DNA and RNA strings. Crossover is executed by selecting chromosomal strings of two mating parents, randomly choosing sites on the strings, and swapping strings
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of O's and 1's between these chosen sites. There are a number of crossover operations, such as a onepoint crossover, twopoint crossover, multipoint crossover, segmented crossover, uniform crossover, and shuffle crossover. The twopoint crossover operator was used in this research. An illustration of the twopoint crossover process between mating parents represented by 20 digit binary strings is as follows:
Parents = 11010100100110011010 Parent2 = 01001110001011001000
Child = 11001110001010011010 Child2 = 01010100100111001000
The crossover sites on the parent strings are indicated by an understrike. A probability of crossover p, is defined to determine if crossover should be implemented. In the case of the more fundamental onepoint crossover process, only one randomly generated site is created, and strings after the selected site are then swapped between two partners. More information about the implementation, nature, and evaluation of different crossover operators is presented in [77].
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4.2.3 Mutation
Mutation safeguards the genetic search process from a premature loss of valuable genetic material during reproduction and crossover. The process of mutation is simply to choose few members from the population pool according to the probability of mutation p,,, and to switch a 0 to 1 or vice versa at a randomly selected mutation site on the selected string. Most genetic search methods give a constant probability of mutation, Pm. A method employing variable probability of mutation in the genetic search was proposed by Fogarty [78]. The study measured the performance of genetic searches in which the p,,'s were a) exponentially decreased with successive generations, b) increased over the bit representation of each integer, and c) was a combination of the increasing and decreasing probability methods. The variable pn, methods were compared with the performance of a genetic search with a fixed pm. The varied probability Pm was shown to significantly improve the performance of the genetic search, but only for cases in which the initial populations contained all O's or all l's. Randomly generated initial population cases showed no improvement in performance.
Running a genetic algorithm entails setting a number of initial parameter values, such as the size of population, probabilities of crossover, p, mutation probability, pm, etc. Determining initial conditions which suit one's problem is not a trivial task. If poor initial settings are used, the performance of a genetic search can be severely impacted. Two useful strategies for determining effective crossover and mutation probabilities have been developed [79,80]. A strategy to
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adapt the probabilities of crossover and mutation during the course of the genetic search was implemented by Davis [81]. This method showed limited success, and more efforts were needed to make it robust and effective. The performance of genetic search methods in obtaining the optimum ply stacking sequence of a laminated composite plate under a concentrated load was examined for a number of combinations of p, and pm. This experiment is detailed in Chapter 5.
4.2.4 Implicit Parallelism
It is important to emphasize that genetic algorithms use probabilistic transition rules only to guide a highly exploitative search. To this extent, they should not be considered a variant of the random walk approach. The implicit parallelism available in these methods is significant from a computational standpoint, and has been explored [82]. If a binary code is used in an ndigit string to represent a design, then a total of 2' variations of the design are available in the representation scheme. However, the exploitative power of GA's extends beyond this if one takes into consideration the fact that the presence of a
0 or 1 at some key location along the string may be significant in itself. As an example, consider the design of a twobar truss structure for minimum weight, with each crosssectional area mapped into a 5digit binary string. The maximum and minimum areas would be denoted as follows.
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Amax = 11111
A i = 00000
If the bit string representation of the design is obtained by stacking the binary representation of area A2 after A1, then a design represented by 1001000010 indicates a value of A1 close to its maximum and A2 closer to its minimum value. Further, if we use a '*' to indicate that a location along the string can occupy either a 1 or a 0, then a string of the form 1***0000** conveys the same information. In this manner, one can think of the following schemata:
H1 = *0*1****1*
H2 = 1***00***0
H3 = 1*01***0**
Our design of areas A1 and A2 could belong to each one of these schemata. In such a 10digit representation, there are a total of 310 (59049) schemata among 210 (1024) possible unique designs. A decrease in the number of *'s in a schema makes it more specific. In this context one defines the order of a schema denoted as O(H) as the number of O's or 1's in a schema. The order of H1 is 3 and that of H2 and H3 is 4. Another item of interest is the defining length of a schema d(H), which is taken as the distance between the first and last specific digit on the string; d(H1)=7, while d(H2)=9. Schemata with higher defining lengths are more
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likely to be disrupted during crossover. If we denote the number of schemata of type H at a given generation t as m(H,t), then, due to reproduction alone, at generation t+ 1 this number would be [82],
m(H,t +1) = m(H,t)f(H) (4.11) where f(H) is the fitness of schema H and fv is the average fitness of the population. Clearly, schema with a fitness higher than the average fitness of the population are increased exponentially in successive generations. The affect of crossover and mutation is disruptive. At the same time, however, these processes contribute to the evolution of new characteristics in the population. In terms of the probabilities of crossover and mutation, pc and p,, respectively, and the string length L, the growth or decay of a schema in a population is expressed as follows:
nm(H,t+1) _ m(H,t) 1 Pc L1 O(H)p, (4.12)
Here, the predicted growth due to reproduction alone has been modified by considering the probability of surviving the crossover and mutation transforms. As is clear from this expression, loworder, shorter defininglength schemata, which are also fit, increase rapidly in the population. Holland [34] estimated that the actual processing of n strings corresponding to the population size results in the processing of O(n3) schemata, giving the approach a tremendous computational advantage.
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The choice of population size, n, for a genetic search represents a
fundamental decision in GA implementation. A population with too few members can result in a premature convergence of the genetic search, where not enough genetic combinations are processed to converge to the ideal solution. Premature convergence resulted from an effect referred to as genetic drift [79,83] or a preferential selection pressure. An overly large population size results in increased number of function evaluations before significant improvement is observed in the serial evaluation of individuals. Goldberg [84,85] provides strategies for determining the optimal initial population sizes of binarycoded serial and parallel genetic algorithms. The performances of genetic search methods were computed for a number of structural and mechanical optimal design problems using different population sizes. The results are presented in Chapter 5.
Premature convergence can be defined as an event where a population of designs achieves a high level of uniformity at all loci, without containing nearoptimal design characteristics. This undesirable convergence situation usually results from stochastic errors associated with a small population size. Strings representing important schemata are not usually represented sufficiently when a small population size is selected. Finite populations eventually converge to nonoptimal solutions without the presence of selective pressure. This phenomenon is given the name genetic drift. A large population size or a high probability of mutation decreases the occurrence of genetic drift. Detailed descriptions of genetic drift and of the Markov analysis of genetic drift are presented in [83].
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4.3 Representation Schemes for Design Space
4.3.1 Continuous Design Variables
An mdigit binary number representation of a continuous design variable allows for 2"' distinct variations of that design variable to be considered. If the description of a design variable is required to a precision of Ac, then the number of digits in the binary string may be estimated from the following relationship:
' XU, XL 1 (4.13) 2"' +
AC I
Here, XL and xu are the lower and upper bounds of a continuous variable x. As an example, if one considers representing a variable x to within a precision 0.1, and the lower and upper bounds on x are 0.0 and 0.7, then 2m >_ 8. This gives the result that m=3, and indeed the eight 3digit combinations of 0 and 1, enlisted as 000, 001, 010, ..., 111, can be assigned to numbers 0.0, 0.1,...0.7. Note that an integer value of m will not yield a precision of 0.14, and a 3digit string would still be necessary to represent the six numbers 0.0, 0.14, 0.28, ..., 0.7. There would be two excessive binary representations which would be digested by distributing them to partial discrete design space as described in later sections. It is important to recognize that even when dealing with continuous variables, GA's work on a discrete representative set of those variables; the method is therefore ideally suited for applications to problems with a mix of continuous, integer, and discrete variables.
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The binary representation of population characteristics has dominated GA research. Advantages of binary representation include: analytical simplicity of binary vectors, the elegance of genetic operators in binary strings, requirements for computational speed, and most importantly, the theoretically motivated desires to maximize the number of schemata sampled by a given set of individuals in a population [34,82]. Binary representation is not an exclusive model for genetic algorithms, and higherlevel representations in both theory and applications are worthy of investigation [86].
4.3.2 Integer Design Variables
Due to the discrete nature of the binary representation schemes, integer design variables can be simply regarded as continuous design variables with a fixed accuracy Ac equal to 1. If m could be found to meet (xU  XL) = 2'1, a onetoone correspondence could be readily established. In most cases, however, this is not possible, and the excessive binary strings must be assigned in an appropriate manner. There are a number of ways in which this is done, and these are described as follows.
(i) Penalty approach: In this approach, the smallest number m which meets the inequality
2'" > (x  XL) + 1 (4.14)
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is computed. Of the 2m possible mdigit binary strings, a unique string is assigned to each of the N integer variables. The remaining (2m  N) strings are assigned to outofbound integers. As an example, in representing six integers between 0 and 5, the computed value of m satisfying the above inequality is 3. The remaining two binary strings are assigned to outofbound variables as follows,
[0, 1, 2, 3, 4, 5, 6*, 7*]
[000, 001, 010, 011, 100, 101, 110*, 111*]
where, '*' indicates an outofbound variable. A penalty measure is then allocated to the fitness function of a design which includes the outofbound value of an integer variable. While this approach yields a onetoone correspondence between the integer variables and their binary representations, careful consideration must be given to the magnitude of penalty assigned to the fitness function due to the presence of an outofbound variable. Large penalties on the fitness will adversely affect the genetic search, making it difficult to distinguish between good and average designs.
(ii). ExcessiveDistribution Method: In this approach, m is first computed on the basis of Eqn. (4.14). The excessive binary representations are then assigned to integers in the admissible range, whereby one or more integers may have more
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than one binary representation. For the case described above, one of the two binary strings could be assigned to integer 4 and the other to integer 5. Although this method does not create any additional design constraints, it does have the effect of an uneven expansion in the design space. The affect of this partial expansion is hard to predict without apriori information of the objective function space. Clearly, the expansion results in a larger number of schemata in the design space. If the schemata happen to belong to the poorer regions of the design space, the convergence of genetic search would be adversely influenced. One method of avoiding this problem is to distribute the excess binary strings evenly along the extent of the feasible integer space. As an example, if twenty excessive binary strings are to be distributed evenly among one hundred integers, one excess binary representation can be assigned to every fifth consecutive integer.
The uneven distribution described above may be avoided by creating excess strings on purpose to obtain a similar number of binary representations for each integer. For the example described above, two integers (33%) have twice the number of binary representations (100% more) than the other four (67%) of the integers. If m=4, we would have a total of 16 binary strings to be assigned to six integers. In this case, each integer could be assigned 2 strings, and four integers (67%) could be assigned one additional string (50% more) than the remaining two (33%) integers. With increasing value of m, the disparity in distribution can be removed at the price of increasing the number of schemata to be explored.
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4.3.3 Discrete Design Variables
Discrete type design variables are characterized by an uneven spacing between two consecutive values. With the unique nature of genetic search, wherein gradient information is not required, these variables can be handled in the same manner as continuous or integer variables. Mapping of these variables is a twostage process. In the first stage, discrete variables are mapped to an equivalent number of integer variables. Then, techniques of mapping integer variables into binary strings described in the preceding sections are applicable with no additional manipulation. An illustration of this process is the mapping of a set of eight discrete variables into binary strings.
[2.4, 3.76, 5.96, 8.25, 9.37, 13.70, 20.55, 24.0]
[1 2 3 4 5 6 7 8]
[000 001 010 011 100 101 110 111 ]
A design space with a mix of continuous, discrete, and integer variables can be represented as required in genetic search by connecting, headtotail, binary string equivalents of these variables as described above.
In genetic search, the average fitness of a population is increased over generations of evolution. In an unconstrained maximization, the objective function can be chosen as the fitness function. If an unconstrained minimum is the objective, the fitness must be revised to be the inverse of the objective function.
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As stated earlier, another alternative is to choose the maximum objective function value of all members of the population, and subtract individual objective function values from this quantity to obtain the corresponding fitness values. In constrained optimization, designs with violated constraints are considered less fit. Hence in constrained minimization, a penalty corresponding to the constraint violations is appended to the objective function much as in the exterior penalty function approach. In the present work, the traditional penalty function approach is compared to a revised formulation discussed earlier, which is characterized by the imposition of an upper bound on the penalty. Without such a bound, a highly penalized infeasible design and a good feasible design would be difficult to separate, and the genetic search would deteriorate into a random search.
4.3.4 Performance Evaluation of Representation Schemes
DeJong [79] defines online and offline performance measures to gage the efficiency of a genetic search. While online performance is an average of all evaluations of fitness, and is therefore indicative of how well the entire population adapts, the offline parameter is an average of the current best evaluations of fitness. These two parameters, in addition to the average fitness of a population at each generation of evolution, were used to measure the performance of genetic search with the different representation schemes described in preceding sections.
A test problem for this purpose was chosen as follows:
Minimize F,(x) = x + + x (4.15)
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where x1, xz, and x3 could assume continuous values ranging between 512 and 511. A onetoone correspondence can be obtained by using a 10digit binary string (210 distinct representations). String lengths of 12, 14, 16, 18 and 20 were also used, yielding 4, 16, 64, 256 and 1024 binary representations per integer variable, respectively. A fixed population size of 50 was used in conjunction with probabilities of crossover and mutation set to 0.6 and 0.005, respectively. Both online and offline performance of each of these cases was obtained. The results show improvement in both performance measures with an increase in the number of representations for each admissible value of the integer variables (20digit string performed the best). This was attributed to two possible factors.
a. More schemata are available in such larger string representations, thereby
increasing the probability of having a better initial population.
b. In populations where there is exactly one representation for each
admissible value of the integer variable, there is the risk of losing that
variable very easily during genetic transformations. Note that each 0 or 1
along the string very specifically denotes an integer variable, and is
susceptible to elimination by even a simple mutation transform. These
advantages must be weighed against the slower improvement in the
population, a direct consequence of schema disruption with longer binary
strings.
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A modification of this problem was a change in the design variable upper bound to 238. For string length of 10, there were excess binary representations, which were treated by the methods discussed earlier. In the penalty approach, a penalty (0(106)) was imposed on any design containing design variable values between 239 and 511. This choice of penalty parameter is not advised because high penalties on the objective function resulted from the design variable range violations will slow the convergence of the genetic search. The magnitude of the penalty for the design variable range violation should be appropriately selected in accordance to the average magnitude of fitness functions. An arbitrary assignment of penalty parameters should be avoided. Another genetic search with an initial penalty (0(10')), and which was linearly decreased in every generation, was also used with more satisfactory results.
In the excess distribution method, two schemes were implemented. In the first, the 273 excess representations were distributed evenly over the range 512 ,238; in the second, they were all distributed to integers on the side of 238. Four other experiments were conducted, with string lengths of 11, 12, 13 and 14, and all excessive representations were distributed evenly for the entire range of integer variables. It can be shown that a string length of 13 yields the most even distribution with 91% of the integer variables assigned only about 10% more representations than the remaining 9% of the integer variables.
Some general conclusions emerged from these numerical experiments. Although conventional wisdom in genetic search advocates the use of smaller
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length strings, for a discrete design space it appears advantageous to have more than a onetoone correspondence between the integer/discrete design variables and their binary representations. If a onetoone correspondence is chosen with only a few excess representations, then the design variable representation can be based on either the penalty concept or the excessive distribution approach. The penalty approach is highly sensitive to the choice of the penalty parameter, and is generally not recommended unless information about the magnitude of the objective function is available. Of the excessive distribution method, the even distribution schemes perform better than the onesided distribution methods. This conclusion would not be valid only if excessive distribution is done in those regions of the design space where the optimal point is located. When an excessive distribution method is used for a design variable, the design space corresponding to this variable will be partially expanded. With larger length binary numbers, the excessive strings can be more evenly distributed and result in an even expansion over all variables. This approach also makes available a larger number of schemata for each design variable.
4.4 Multicriterion Design
A relatively simple approach to account for the multicriterion nature of the problem in optimal design is through the utility function method [74]. A utility function is defined by assigning a relative importance to an individual criterion in relation to other criteria. For an mcriterion optimization problem, a practical
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definition of the utility function can be shown to be,
U = WiFi(x) (4.16) i=1
where Wi serves as the weighting factor for each objective function Fi(x). Since the magnitude of each objective criterion can be quite different, a scaling strategy must be implemented. Here, the following scaled form of the utility function was used,
U = i Fi (4.17) i=1 Fi
where F," are the scaling parameters for each candidate criterion.
With this definition, the multicriterion optimal design problem is changed into a single objective optimal design problem, and can be solved in a relatively straightforward manner. In a situation where the multicriterion optimal design is required for a number of different weighting combinations, a genetic search would have to be performed for each of these cases separately. One of the principal contributions of the present research is an exploration of methods wherein single runs of genetic search can be used to simultaneously generate a family of Pareto optimal designs corresponding to different weighting coefficients of individual candidate criterion. Two distinct strategies of achieving this objective are proposed next.
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4.4.1 Sharing Function Approach
Goldberg and Richardson [87] introduced the concept of sharing functions into genetic algorithms to simultaneously locate relative optima in a multimodal design space. As the name suggests, the approach is based on a concept of shared resources among distinct sets of population. Each such set converges to one relative optimum, and in doing so, maximizes its payoff. The principle of sharing is implemented by degrading the fitness of each design in proportion to the number of designs located in its neighborhood through the use of sharing functions. The extent of sharing is controlled by a sharing parameter ash, in terms of which the sharing function is defined as follows:
1 ,d ,
(d i) = ,d as, (4.18)
0 ,otherwise
Here di is a metric indicative of the distance between designs i and j. If two designs i and j are very close, dij is almost zero, and O(dj) = 1. If dj > og, o(dj)
0. The fitness of a design i is modified as,
4 m (4.19) E (dj)
j=1
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where m is the number of designs located in vicinity of the i'h design. Note that the raw fitness value of each design is decreased due to the presence of a large number of designs in its vicinity.
If the distance metric dij is evaluated in the decoded design space, the
sharing is called a phenotypic sharing. The distance metric dij can be computed as follows:
d, = (xk Xkj)2 (4.20) Sk=1
In order to adapt the sharing principle into an approach where genetic
search can simultaneously locate the optima corresponding to different weighting combinations, the weighting variables were included in the set of design variables. As in mathematical programming based multicriterion design, the criterion weights W are required to meet the following requirement.
EWi = 1 (4.21) i=1
As an illustration, if 9 different weighting combinations [W1 =0.1, W2=0.9], [W1=0.2, W,=0.8], ..., [W1=0.9, W,=0.1] are considered for a twocriterion optimal design problem, the weighting variable can be defined as follows:
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x, = (1,2 ....., 9)
= ([0.1,0.9], [0.2,0.8], ...[0.9,0.1])
This weighting variable x, is then regarded as one additional design
variable taking on values between 1 and 9, with a step size of 1. This variable is of integer type, and representation schemes discussed earlier can be used in a routine manner.
For the above twocriterion problem, one weighting design variable, xw, was needed to represent the nine weighting sets. If the number of criteria is greater than two, then two approaches can be taken to represent the weighting variable sets. The first approach introduces m1 integer variables, x, into the design variable set of an mcriterion problem. Consider the following nine different weighting combinations for a threecriterion problem:
[W, =0.2, W2=0.3, W3=0.5], [W1 = 0.2, W2= 0.4, W3= 0.4], [WI = 0.2, W2 = 0.5, W3= 0.3], [W1 = 0.3, W2= 0.3, W3= 0.4], [WI= 0.3, W2= 0.4, W3=0.3], [WI = 0.3, W2= 0.5, W3= 0.2], [W1 = 0.4, W2 = 0.3, W3= 0.3], [W1 = 0.4, W2 = 0.4, W3= 0.2],
[W1= 0.4, W, = 0.5, W3=0.1].
Two weighting variables xwl and x2, were used to represent W, and W,, respectively. The value of W3 in each weighting set was then determined by Eqn.
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(4.21). The second approach utilizes a single weighting variable which represents the possible combinations of the optimization problem. Weighting combinations are arranged in a series, and the weighting variable components map to each combination of the problem criteria. For the above ninecriterion example, the following variable map was generated:
xw = (1, 2, ..., 9)
([0.2,0.3,0.5], [0.2,0.4,0.4], ..., [0.4,0.5,0.1])
The binary coded weighting variable and binary representations of true
design variables are connected headtotail to form a binary string representing a design with a specific weighting combination. In order to generate optimal designs corresponding to all weighting combinations, the sharing principle is applied selectively on the weighting variable. The distance metric d, between two designs xi = [xw,i, xl.i, x., ..., Xpi] and x = [xWj, x1,j x2,j ..., xp,] is based only on the variable representing weight combinations as follows.
di= (xwi  X)2 (4.22)
With appropriate choices of sharing parameter osh and population size, sharing directs the genetic search toward locating optimal designs corresponding to most of weighting combinations. In order both to speed convergence and to impart stability to the genetic search, it was found to be necessary to introduce
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restrictions in mating between members belonging to different subsets of the population [88]. A parameter of mating restriction, ama, was defined. This confined the mating process among those members for which the metric distance as measured along the weighting variables was within the radius amat. Some caution is necessary in this approach as it is highly reminiscent of inbreeding, and sources of diversity in the population must be introduced. It is important to include in future work, mating restrictions other than those based entirely on the weighting coefficient combinations. Since the population of designs in each specific weighting group is determined on the basis of the penalized fitness of the entire population, a better average fitness for a specific weighting group will eventually result in that group attracting more members in successive generations. The fitness or the overall utility function has to be carefully scaled so as to distribute designs evenly over each weighting combination, and for each weighting group to possess roughly similar exploitative abilities in the genetic search.
4.4.2 VectorEvaluated Approach
This approach has its parallel in a method proposed by Schaffer [89], referred to as the vector evaluated genetic algorithm (VEGA) approach, and intended for multiobjective optimization. Given the fixed population size as P and the number of optimization criteria as N, each member of the population was evaluated for its fitness as it relates to each of the N criteria. For each of the N criteria, subpopulations of size P/N were extracted from the pool on the basis of
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the fitness function for the criterion under consideration. This sorting was done in a rotational sequence, where one member was selected for each criterion, followed by a second round of selection. The selection was performed using the traditional genetic selection (reproduction) method, which ensured the preservation of individual designs which performed above average on any criterion. Subpopulations were then shuffled and grouped into a population which was subjected to the usual genetic transfer processes.
The method proposed in the present research for simultaneously generating a family of Pareto optimal designs draws upon such a vector evaluated approach, and is summarized as follows. A utility function is first defined to obtain a scalar measure of fitness for a given set of weighting parameters Wk as ni
Uk = Wik Fi (4.23) i=1
where the superscript k denotes the kth weight combination. Each set of weight combination for which an optimal design is needed constitutes a component in the Nvector of utility functions. The entire population of P designs was evaluated for each of the weight combinations. A subpopulation of size P/N for each weighting combination was formed as described above. Two distinct strategies were adopted in this context. In the first approach, designs were selected from the pool as those that were deemed most fit for that weight combination. Selection of a design for one subpopulation did not exclude its subsequent selection in another
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subpopulation. In the second approach, an exclusionary principle was applied wherein a design could contribute its genetic makeup to only one subpopulation and was not considered for mating in another subpopulation. Both of these processes result in N subpopulations, each containing P/N designs with superior measures of fitness corresponding to a specific weight combination. For both methods, in general, this step simply represents a regrouping with no genetic transformations invoked on the designs. In particular, in the exclusionary scheme, the entire genetic pool is preserved and simply divided up in the most opportunistic manner to contribute towards evolution of designs for specific requirements. After the selection of each subpopulation is completed, genetic transformations such as reproduction, crossover and mutation are invoked separately in each subpopulation. The process of population classification is repeated for the next generation in the same manner. In the discussion of sample results, these two strategies are denoted as VEinc and VEexc, respectively.
Due to the possible small size of each subpopulation, genetic drift might result, with associated undesirable characteristics such as convergence to a suboptimal design. A possible approach to prevent such a premature convergence is to introduce a more moderate sized population for genetic evolution. The underlying assumption here is that a larger population is likely to be more diverse, and hence offers a higher possibility of locating the true optimum. This can be achieved in a strategy where the crossover transformation is a twostage process. In the first stage, a crossover is performed between a

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GENETIC SEARCH METHODS FOR MULTICRITERION OPTIMAL DESIGN OF VISCOELASTICALLY DAMPED STRUCTURES By CHYIYEU LIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991
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To my parents
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ACKNOWLEDGEMENTS The author would like to express his greatest gratitude to his committee chairman, Dr. Prabhat Hajela, for his continued guidance, instruction and friendship. He would also like to thank Professors Lawrence E. Malvern, ChangTsan Sun, Bhavani V. Sankar, and Ronald A. Cook for their careful review of this dissertation, and for their service on the committee. Appreciation is also extended to Dr. David C. Zimmerman for his volunteer review of this dissertation. The author would like to express his appreciation to the Army Research Office and the Department of Aerospace Engineering, Mechanics and Engineering Science at the University of Florida, for providing financial support during the course of his graduate studies. The author would also like to say thanks to all his friends in the department for making his study a lovable memory. Special thanks are given to Dr. Vidish S. Rao for his help with his finite element codes, and to William J. Leath, Jr., for his proofreading. Finally, the author would Hke to express his deepest thanks to his wife DaiLih for her sacrifice, understanding, support and encouragement during his time of study at the University of Florida. iii
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1 TABLE OF CONTENTS page ACKNOWLEDGEMENTS iii ABSTRACT vi CHAPTERS 1. INTRODUCTION 1 LI Literature Survey 1 1.2 Scope of Present Research 6 2. DAMPING TREATMENT AND ANALYSIS 9 2.1 Background 9 2.2 Measures of Damping 17 2.3 Damping Analysis 22 2.4 Design Optimization Formulation 26 3. TRADITIONAL METHODS OF OPTIMIZATION 30 3.1 Introduction 30 3.2 FeasibleUsable Search Direction Method 31 3.3 Modified BranchandBound Search 36 3.4 MinMax Global Criterion Approach 38 3.5 Numerical Implementation 43 4. GENETIC ALGORITHMS 60 4.1 Introduction 60 4.2 Genetic Algorithms 63 4.2.1 Reproduction 68 4.2.2 Crossover 70 4.2.3 Mutation 72 4.2.4 Implicit Parallelism 73 4.3 Representation Schemes for Design Space 77 iv I I
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page 4.3.1 Continuous Design Variables 77 4.3.2 Integer Design Variables 78 4.3.3 Discrete Design Variables 81 4.3.4 Performance Evaluations of Representation Schemes 82 4.4 Multicriterion Design 85 4.4.1 Sharing Function Approach 87 4.4.2 VectorEvaluated Approach 91 5. NUMERICAL IMPLEMENTATION OF GENETIC ALGORITHMS 95 5.1 SingleCriterion Optimization 95 5.1.1 Riveted Lap Joint 95 5.1.2 Series Stacked Belleville Spring 101 5.1.3 ElevenBar Planar Truss 105 5.1.4 Laminated Composite Plate Design 109 5.2 Multicriterion Optimization 122 5.2.1 Statically Loaded TenBar Truss 122 5.2.2 Dynamically Loaded TenBar Truss 145 5.2.3 WingBox Structures 148 5.2.4 Viscoelastically Damped Laminated Composite Beam 163 6. APPROXIMATION CONCEPTS IN GENETIC ALGORITHM BASED OPTIMIZATION 171 6.1 Introduction 171 6.2 Neural Networks 172 6.3 BackPropagation Training Algorithm 177 6.4 Multicriterion Optimal Design of ThreeDimensional Viscoelastically Damped Laminated Composite Beam 181 7. CONCLUDING REMARKS 195 7.1 Conclusions I95 7.2 Recommendations for Future Work 198 REFERENCES 201 BIOGRAPHICAL SKETCH 211
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GENETIC SEARCH METHODS FOR MULTICRITERION OPTIMAL DESIGN OF VISCOELASTICALLY DAMPED STRUCTURES By ChyiYeu Lin December, 1991 Chairman: Dr. Prabhat Hajela Major Department: Aerospace Engineering, Mechanics and Engineering Science This study examines the use of genetic search techniques in the multicriterion optimal design of structural systems. The global nature of these search techniques makes them better suited for design spaces that are known to be disjoint or nonconvex. The study was motivated by an application involving the optimal design of viscoelastically damped composite structures. In addition to the multicriterion nature of this design problem, its design space is a mix of continuous, integer, and discrete design variables. The more traditional mathematical programming based methods of optimization are known to generate suboptimal designs for this class of problems. Genetic algorithms belong to a general category of stochastic search methods. An optimal design is located by combining favorable characteristics of vi
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several designs in such a way as to maximize a defined fitness function. The transformation operators that facilitate this combination have their philosophical basis in biological principles of evolution. As shown in this work, the method can easily accommodate the presence of integer and discrete variables in the design space. An extension of genetic search to the multicriterion optimal design problem is presented in this work. This design problem requires that an optimal design be generated for which each of the criteria are simultaneously optimized. Two distinct strategies for generating such designs are presented in this work. A distinct feature of these strategies is that they allow for a simultaneous generation of optimal designs in problems where the individual criterion are weighted differently. The design of viscoelastically damped composite beam structures considered in this work was based on a threedimensional finite element solution. This analysis is computationally demanding, and in order for it to be used for optimization, approximations to the exact analysis were required. A multilayer perceptron model of an artificial neural network was used to map the inputoutput relationships between the design variables and the output response, and this approximation was used in the optimal design in lieu of the exact analysis. Optima] designs of composite beams for minimum weight and maximum damping were obtained using this approach. vii
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CHAPTER 1 INTRODUCTION 1.1 Literature Survey The earliest theoretical developments in the field of structural optimization may be traced back to a publication by Maxwell [1] in 1869. In 1904, Michell [2] proposed a theory which provided valuable insights into determining the optimum configuration of truss structures under a single load; however, the approach had limited practical applications. In the 1940's there was a new development in the theory for optimal design of structural componentsthe simultaneous failure mode theory. As the name suggests, the structural components were sized in a manner such that failure would occur simultaneously in several failure modes. In the area of plastic collapse design, linear programming methodology was used in the early 1950's to obtain the optimum configuration of truss and frame structures. Since 1960, optimality criteria methods and nonlinear programming algorithms (NLP) have become two of the most popular structural optimization methods. The optimality criterion methods belong to a class of indirect methods for optimal design. A structure with given loading conditions and prescribed constraints is deemed optimal when its derived optimality criterion is satisfied via 1
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2 numerical algorithms. This optimality criterion technique [35] is not general, as the optimality criterion definition is problem dependent, and quite often difficult to obtain. The use of nonlinear programming techniques in optimum structural design affords a more direct and flexible solution procedure. Professor L. A. Schmit pioneered this approach, now being widely recognized as a viable design tool in the industry. A number of key review publications and illustrations of applications are available in References 617. Although NLP methods have been developed to the point where they can be used almost routinely, they have several drawbacks. Such drawbacks may result from having an optimization problem with a large number of design variables and constraints. Furthermore, the design space may be disjoint or nonconvex. Nonlinear programming methods are not suitable for an optimization problem involving one or more design variables which are of discrete/integer type because of the difficulties in applying the onedimensional search technique, and in the determination of gradients of objective function and constraints. A combinatorial programming strategy has been proposed to overcome the discreteness of the design space. This technique demands significant investment of computational resources which worsens as the dimensionality of the noncontinuous design variables increases. Since the traditional NLP based methods essentially represent a local search strategy, they can only determine the relative optimum located nearest to the starting design. Further, as the more efficient methods belonging to the category of NLP require at least the first order gradient information of the
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3 objective function and constraints, they will very likely fail as the search passes the disconnected region of a disjoint design space. Decisions in engineering design typically involve multiple and frequently conflicting requirements. Perhaps the first effort in the field of multicriterion optimization may be attributed to an Italian economist, Pareto, who in 1896 introduced the concept within the framework of welfare economics [18]. The implications of his work to optimization theory, operations research, and control theory were only recognized in the late 1960's. Multicriterion optimization in engineering design has been examined, and numerous examples of the design of mechanical structures have been presented [1927]. Despite the multicriterion nature of most design problems, the principal focus of most research efforts has been on pursuing an efficient algorithm for the scalar objective function optimization problem. One common approach to solving multicriterion optimization problems is to select one of the criteria as the objective function, and to trade off among the remaining criteria using appropriate design constraints. The apparent simplicity afforded by this method is attractive; however, it should be noted that the treatment of criteria as constraints does not yield the same optimum design as would be obtained when solving the optimum design problem as one possessing multiple objectives [28]. The goal of this research was to develop efficient methods for multicriterion optimal design. The approach was then applied to optimally design viscoelastically damped structures with maximum damping and minimum weight.
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4 This exactly characterizes a multicriterion structural optimization problem. In this problem, however, the multicriterion optimization problem was complicated by the presence of discontinuity in the design space. The design variables were a mix of integer, discrete, and continuous types. Integer and discrete design variables were necessary, because the number of composite laminae forming a laminate is characterized by an integer variable, and the thicknesses of the constraining and viscoelastic layers were restricted to a manufacturerprovided list of discrete values. This multicriterion optimal design problem comprises a complicated design space which is discontinuous, and nonconvex. A finite element based analysis of the structural damping ability of a constrainedlayer treated composite beam was investigated by Sun and his coworkers [2933]. The loss factor of a composite laminate of arbitrary stackingsequence, and with a discrete type of addon viscoelastic damping treatment can be effectively and accurately predicted. This work was used in the present research as an analysis tool for determining the characteristics and the dynamic/static response of viscoelastically damped composite structures. In order to solve the multicriterion optimization problem with a discontinuous and/or nonconvex design space effectively, two methods were explored. The first approach was an NLP based procedure with a branchandbound type algorithm used to account for the integer and discrete design variables. The second approach was the use of genetic algorithms, and this comprises a principal focus of the present study.
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5 Genetic algorithms were first proposed by John Holland in 1975 [34], and have since been adapted for a large number of applications in game theory, induction systems, and other aspects of human cognition such as pattern recognition and natural language processing [3538]. These algorithms are generally regarded to be in the same category of stochastic search methods as simulated annealing [39]. Both approaches have their basis in natural processes (simulated annealing is derived from the principles of statistical mechanics). Genetic search methods are based on Darwin's theory of survival of the fittest [40]. A set of design alternatives, (analogous to a population in a given generation) are allowed to reproduce and cross among themselves, with bias allocated to the most fit members of the population. Combinations of the most desirable characteristics of the mating pairs of the population result in progenies that are more fit than either of the parents. If a measure which indicates the fitness of a generation is also the desired goal of a design process, successive generations produce better values of the objective function. An obvious advantage in this approach is that the search is not based on gradient information, and has no requirements on the continuity or convexity of the design space. Another advantage is that the solution space in genetic algorithms is a discrete mapping of the design space; thereby a design space with a mk of integer and discrete design variables can be treated efficiently. While the potential of genetic algorithms as a function optimizer has been a subject of more recent interest, the first use of genetic search methods in the
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context of structural optimization was done by Hajela [41,42]. In that work [42], globally optimal structural designs were generated using genetic search methods, despite the existence of a nonconvex and disjoint design space. The present research proposes and evaluates a number of distinct coding strategies for accommodating integer, discrete, and continuous design variables in genetic algorithms. It also extends the genetic search method to simultaneously obtain multicriterion optimal designs for several cases where the individual criterion are weighted differently relative to the remaining criteria. 1.2 Scope of Present Research The scope of the present work and its organization can be summarized as follows. Damping in composite structural elements is the subject of chapter two. Both the inherent damping characteristics of composite structures and addon types of passive damping are discussed. The influences of constituent material properties and the geometrical configuration of the composite, on its damping characteristics, are presented. Three optimal design problems for damping enhancement are formulated, and a detailed implementation of these design problems is presented. Chapter three provides an introduction to the traditional nonlinear programming techniques, with a focus on the feasibleusable directions method. This method was used in conjunction with a modified branchandbound strategy for solving discrete/integer design variable problems. The minmax global
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7 criterion strategy for multicriteria optimization is introduced. A twodimensional idealization of a cantilever beam with addon viscoelastic damping layers, is optimally configured as an example of an NLP based method for multicriterion optimal design of structural sandwich beam. In chapter four, the theoretical background and the basic elements of genetic search methods are reviewed. The binary implementation of integer, discrete, and continuous design variables is discussed. Two distinct approaches for generating multicriterion optimal designs are presented in this chapter. These methods are proposed as an alternative to the more computationally intensive NLP based schemes for multicriterion design. Chapter five describes the numerical implementation of genetic search methods described in chapter four. To demonstrate the use of genetic algorithms in structural/mechanical optimization problems with a mix of integer, discrete, and continuous design variables, a lap rivet efficiency maximization problem, a Belleville spring sizing and stacking problem, and an optimal structural sizing problem were investigated. The optimal stackingsequence of square and rectangular, simplysupported laminated composite plates, for maximum transverse stiffness, was also considered. Genetic search methods for simuhaneously generating multicriterion optimal designs were investigated on a number of structural optimization problems with multiple design criteria. Included in this study was a viscoelastically damped, laminated composite beam problem which used maximum damping and minimum weight as optimization criteria.
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8 In chapter six, a neuralnetworkbased approximation of structural analysis is proposed for use in conjunction with genetic algorithms. This significantly alleviates the required computational effort in the optimal design. A number of multilayer perceptron models of neural networks and the backpropagation algorithm used in the training of such networks, are also discussed. This approximation strategy is examined in the context of a multicriterion optimization problem where a 3dimensional cantilever composite beam with addon viscoelastic damping tape is optimally configured for maximum damping. The influence of such damping mechanisms on the viscoelastically damped cantilever beam is studied in this implementation. Conclusions drawn from this study and recommendations for future research are presented in the final chapter.
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CHAPTER 2 DAMPING TREATMENT AND ANALYSIS 2.1 Background The need for efficient lightweight structures in the aerospace industry has dramatically increased the use of composite materials in the last decade. The application of composite materials in structures that are subjected to dynamic loads has introduced stringent requirements on the dynamic characteristics of these materials. Vibrationinduced damage in structures has long been one of the leading causes for structural failure. Scientists and engineers have dedicated tremendous efforts to controlling and reducing vibration. Active and passive control are among the most popular approaches currently used to improve the damping ability of structures. Active control utilizes external devices referred to as sensors and actuators to shape the dynamic characteristics of the system. On the other hand, passive control mechanisms emphasize the integrated optimal design of inherent structural characteristics and external damping devices to maximize damping under expected dynamic loads. More specifically, for a laminated structural composite, an optimal passive damping can be achieved by a combination of configuring the layup and material selection of the base structure (the laminated composite itself), and by the 9
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10 addition of external damping materials such as viscoelastic tapes to the surface of the laminated composite. The optimization problem associated with tailoring the passive damping ability of viscoelastically damped composite structural elements was the main focus of this study. Damping is the ability of a structure to suppress the vibration through the dissipation of vibratory energy. In earthbound structures, there are a number of damping mechanisms in which vibratory energy is converted to other forms of energy such as sound or heat [43]. These mechanisms are classified under the class of nonmaterial damping. Energy can also be dissipated as the material deforms, and in such cases the damping mechanism is active within the material volume [4345]. Material damping is also referred to as either internal damping or structural damping. This damping is caused by the energy dissipated due to mechanisms such as viscoelastic behavior in polymeric material or the interfacial slip in metallic materials. The principal mechanism for damping in glass and graphite reinforced polymer matrk composites is the viscoelastic energy dissipation that occurs in the polymer matrix. Under a normal stress level, the composite exhibits linear viscoelastic behavior, and its dynamic stiffness and damping are independent of vibration amplitude. The first attempt to tailor the damping ability of basic composite structures can be traced to an effort due to Plunkett and Lee [46]. Several models have been proposed for predicting the damping ability in short fiber reinforced composites.
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11 Sun and his coworkers [47] have developed theoretical relationships for the material damping of aligned short fiberreinforced polymer matrix composites under offaxis loading. The complex modulus, E^', of the shortfiber composite can be expressed as, 1 ^ 1 ^ cos'^e ^ sin^e ^ '^x where E^' and are the storage and loss moduli, respectively; E*l, E'jand G'lrepresents the complex form of the elastic modulus in the longitudinal and transverse directions, and the shear modulus, respectively; v^jrepresents the inplane Poisson's ratio. The damping or loss factor can be obtained as follows (2.2) sin^0cos0 (2.1) The model based on Cox's shear lag theory, as shown in Figure 2.1, allows us to express the loss factor (a measure of the structural damping) as a function of fiber and matrix properties and the geometric parameters shown in Figure 2.1. E" = vA^n El Gl p, s, d, D, 6) (2.3) where E'f, E\^, and G'Â„, represent the complex form of the modulus of the short
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12 X Figure 2.1 Representative volume element for offaxis loading.
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13 fiber, the modulus of matrix, and the shear modulus of matrix, respectively. A detailed expression for this is presented by Gibson et al. [48]. Experiments [49] indicate that the damping ability in composites reinforced by discontinuous fibers is generally greater than that in continuous fiber composites. Sun and Wu [50] attributed this to the stress concentration effects present at the fiber ends, which facilitate the transfer and dissipation of energy in the viscoelastic polymer matrix. The analysis [48,50] clearly indicated the dependence of internal material damping in composites on the fiber and matrix geometry. Parameters such as loading angle, fiber aspect ratio, volume fraction, fiber length, fiber diameter, etc. were all found to influence the dynamic performance of the composite. The analysis also showed that the increase in damping ability was usually associated with a corresponding stiffness loss. A parametric study can be conducted by varying each of these variables, one at a time, to examine their influence on the damping. However, parametric studies of this form have been shown to yield suboptimal results. Hajela and Shih [51] proposed a design synthesis approach based on formal multicriterion optimization. Two distinct analysis models. Cox's shearlag theory and an advanced shearlag model, were used in their work. With this approach, the extensional loss factor of a representative volume element (RVE) was maximized subject to constraints on the element mass and stiffness characteristics. Damping in laminated fiber composites can be determined through the laminated plate theory approach. The inplane material damping, r]^, of the laminated composite is defined as follows.
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14 E" ri, = ^ ij = 1,2,6 (2.4) F where E",j, and E'jj represent the real and imaginary part of the inplane complex moduli E'ij, which can be expressed as ^(z = Â—r^, = ^.j ^ ^^iJ (2.5) (4) where (A j)"^ is the element in the i'" row and j"' column of the inverse of matrix [A'ij], which is the complex form of the reduced inplane stiffness matrix, [Ajj]. Damping in the laminated fiber composites can also be determined through an energy approach in conjunction with a 3dimensional finite element method. This model represents a more realistic approach by including the energy dissipated at the interfaces. Detailed derivation of damping with both approaches is available in [52]. Although the internal damping capabiHty in laminated composites can be adjusted, the extent to which it can be increased is limited by other design requirements. To further increase the damping ability, viscoelastic surface damping treatments can be used. Addon viscoelastic treatments can be used to increase the damping ability of the laminated composites without decreasing the stiffness of the laminated composites. Extensional damping treatment, a commonly used surface damping treatment, is also referred to as unconstrainedor freelayer
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15 damping treatment. The viscoelastic material is coated on one or both sides of a structure; whenever the structure is subjected to cyclic bending, the damping material will be subjected to tensioncompression deformation. The synchronized tensioncompression deformation of the viscoelastic layer dissipates the energy away from the vibrating base structure. The shear type of damping treatment provides a more efficient damping enhancement ability than the unconstrained layer treatment on a constant weight basis [53]. This is largely due to the fact that the energy dissipation in the viscoelastic material is entirely due to the shear deformation and is independent of dilatational deformation [54]. The shear damping treatment is similar to the unconstrainedlayer type, except that the viscoelastic material is constrained by a stiff constraining layer. Therefore, whenever the structure is subjected to cyclic bending, the constraining layer will force the viscoelastic material to deform in shear. This shear deformation is the mechanism by which energy is dissipated. Figure 2.2 shows a sketch of both free and constrained layer damping treatments. It has been noted that the overall mass, stiffness, and damping characteristics of a structure will determine its damping abihty. All of the above three parameters, and not just the damping material, need to be optimally configured in order to obtained the highest damping ability under some design constraints. Typically, in laminated composite structures, several factors, such as the stacking sequence in the laminate, the location, amount, and type of treatment, strongly influence the damping response. i
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16 FREE LAYER DAMPING TREATMENT Damping material Undefomied Longfcjdinai deformation CONSTRAINED LAYER DAMPING TREATMENT Defomied Foa constraining layer Shear deformation Figure 2.2 Illustration of mechanisms of damping treatments.
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17 2.2 Measures of Damping There are various damping measures which are commonly used in different areas. Three common definitions of damping are introduced in this section. For a single degree of freedom system shown in Figure 2.3, with viscous damping and an external excitation F(t), the differential equation of motion is, where m represents the mass of the system, c represents the viscous damping, and k represents the spring constant. For viscouslydamped free vibration, the general solution to the homogeneous equation is mx + cx + kx = F{t) (2.6) (2.7) where 11 a = c k (2.8) m Cj and c, are constants to be determined by initial conditions. An important characteristic of a structural system, the damping ratio or damping factor, is defined as
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18 xwwwwww Figure 2.3 Sketch of viscous damping model.
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19 where = 2 sf^ (2.10) A value c^, for c determines the system to be overdamped (c > cj, underdamped (c < c^), or critically damped (c = cj. Depending whether it is overdamped, imderdamped, or critically damped, the system exhibits an oscillatory motion, nonoscillatory motion, or critically damped motion, respectively. A convenient way to determine the amount of damping present in a system is to measure the rate of decay of free oscillation. The logarithmic decrement, 6, is defined as the natural logarithm of the ratio of any two successive amplitudes of a free vibration, as shown in Figure 2.4. A relation exists between the damping ratio, I, and the logarithmic decrement,
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I 20 Figure 2.4 Sketch of logarithmic decrement.
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21 used for the analysis of a structural system under arbitrary excitation as any real signal can be represented as a Fourier series. For a structure exhibiting steady state oscillation, the loss factor r? can be defined in terms of energy quantities as follows, ri = (2.11) (InW) where D is the energy dissipated per cycle and W is defined as the maximum energy stored during a cycle. Ungar and Kerwin [56] showed that this definition of W is unambiguous only in lightly damped systems. It was also shown that for an arbitrary seriesparallel network of viscoelastic springs, the composite loss factor reduced to the following simple expression T] = ^ (2.12) E K where ry,, and WÂ„ are the loss factors and the strain energy stored in the n"' spring, respectively. Hysteresis damping in a continuous composite system can be modeled by using the correspondence principle of viscoelasticity [57]. According to this principle, the elastic constants of the materials are simply replaced by the corresponding viscoelastic counterparts. For example. The dynamic behavior of such materials can be defined in terms of the complex modulus
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22 E' = E' + iE" = Â£'(1 + it]) (2.13) where E' = storage modulus E" = loss modulus t] = loss factor 2.3 Damping Analysis The analysis of a structure with constrainedlayer treatment is different from other situations because the addition of the constraining and viscoelastic layers on the base structure will change, in addition to its mass and stiffness characteristics, the geometry of the structure. The stiffness changes significantly through the thickness of the configuration, varying from a high value corresponding to the base structure to a low value associated with the viscoelastic layer. Ross, Ungar and Kerwin [58] developed what is probably the most widely used analysis to describe the behavior of different type of surface damping treatments. The fundamental work in the area of the analysis of sandwich viscoelastic beams was done by Kerwin [59]. Other literature on this topic may be found in [60]. Since much of the difficulty in designing constrainedlayer damping treatment is due to complicated geometries, it is therefore necessary to look at
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23 finite element techniques for solutions to the problem. Since optimization techniques usually require a large number of function evaluations, efficient and accurate finite elements are desirable to model the composite configuration. Threedimensional finite element models of sandwich structures often result in highdimensionality stiffness matrices and associated significant computational requirements. Sun, Sankar, and Rao [2933] have developed elements that are specially suited to modeling sandwich structures. The finite element implementation of the modal strain energy method [56,61] and the direct frequency response method [61] are used to analyze the free and forced vibration characteristics of the beam. Damping in the system is represented by using the complex stiffness approach which derives from the elasticviscoelastic correspondence principle. In the direct frequency response method, a forced vibration over a range of frequencies is considered and a theoretical response spectrum is generated. The loss factor, or the overall damping of the system, may then be calculated by using the halfpowerbandwidth technique. The technique is illustrated in Figure 2.5. The direct frequency response is computationally expensive, and it is mostly used to verify the predictions of other methods since it needs no assumptions about the level of damping. A more frequently used method to calculate damping is the modal strain energy method. The method is based on the following eigenvalue problem, [[^ + [Kg]W) = oy'[Af\{U} (2.14)
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24 Logarithmic Scale : At 3dB Below the Peak Value (AT RESONANCE FREQUENCY) Linear Scale : At 0.707 the Peak Value Loss Factor = t\
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25 where [K] = global conventional stiffness matrix [K^] = global geometric stiffness matrix [M] = global mass matrix iU) = global displacement vector The natural frequencies and mode shapes are calculated about the initially stressed state for the undamped system. The analysis can be based on the associated complex eigenvalue problem or the simphfied real eigenvalue problem. When the analysis is performed based on the simplified real eigenvalue problem, the stiffness matrix is real and the corresponding nodal displacements are also real. Modal damping is then calculated by this technique using the modal strain energy method. The loss factor is calculated as It rt = i:^ (2.15) i1 where n = total number of elements r?i = loss factor of the i"" element Ej = elastic shear energy; equal to V2{u}'^[k]{u} {u} = displacement vector of the i"' element [k] = stiffness matrbc of the i"' element
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26 This technique is vah'd only for systems with relatively small levels of damping where the mode shapes and frequencies of the damped and the undamped structure are similar. The analytical predictions obtained by this method were in good agreement with the experimental results of Mantena [62] for several damped laminated beam configurations. This analysis [2933] was used in conjunction with optimization algorithms for solutions of optimization problems associated with viscoelastically damped composite elements in the present research. 2.4 Design Optimization Formulation The problem of optimization of a laminated composite beam for enhanced damping characteristics can be formulated as one or all of the following subproblems: a) optimal design of the base structure. b) design with addon viscoelastic damping treatment. c) integral design of sandwich beam consisting of base structure and addon damping treatments. Damping ability of a laminated composite beam can be enhanced by optimally configuring the internal structural parameters of the composites. These structural parameters include ply thicknesses, laminate thickness, material properties of each laminae, fiber orientation, etc. By using formal optimization
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27 techniques, the damping abihty can be maximized with the satisfaction of prescribed design constraints. The use of addon viscoelastic damping is convenient and efficient, because the configuration of the base structure need not be altered. Given a laminated composite beam, the treatment of viscoelastic damping can be done by attaching the viscoelastic damping layers to the surface of the laminated beam. Only one viscoelastic layer is required for unconstrainedlayer treatments. One viscoelastic layer and a thin constraining layer are needed for constrainedlayer treatments. For a twodimensional cantilever beam, an arbitrary length of selected addon damping tape can be placed on the composite beam. Issues such as the precise location of damping material and its quantity are resolved by expressing them as decision variables in the dampingenhancement optimal design problem. Placement of the tape in a certain form may create some problems in the analysis. As the tape moves from one point to another point, a new finite element model will be required, and a new finite element mesh of the structure is necessary. An innovative technique which assures a smooth variation in node locations during redesign is proposed in this work. For a noncontinuous placement of the damping tapes, a number of unconnected damping tapes can be placed along the length of a beam. Several different ways of placing damping tapes on a cantilever beam are shown in Figure 2.6. The problems associated with the discontinuity of damping tapes manifest themselves as discontinuities in the node numbers, and the requirement of an adaptive preprocessor which is able to
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28 CONSTRAINING LAYER VISCOELASTIC LAYER I / / / / / // / / // CANTILEVER BEAM CONTINUOUS ////////// / / / // / / //, CANTILEVER BEAM CONTINUOUS / / // CANTILEVER BEAM DISCRETE Figure 2.6 Examples of damping treatment on cantilever beam.
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29 create a new finite element model for each new structure created by the optimizer. For a 3dimensional laminated composite beam, the aforementioned difficulties were overcome again with a more sophisticated adaptive preprocessor. The mathematical representation of the presence or absence of damping treatment on a predetermined section may be resolved by assigning it a 0/1 type of variable. In this representation, the design problem contains discrete variables, and optimization algorithms which can account for discontinuous design variables in an efficient manner, would be required. The integral design of the entire structure is most effective for the enhancement of the damping ability of the structure. This method is basically the combination of the design of the base structure and the damping treatments at the same time; here, design constraints can be more easily satisfied due to the increased flexibility in changing the design.
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CHAPTER 3 NONLINEAR PROGRAMMING BASED OPTIMIZATION METHODS 3.1 Introduction Most practical engineering optimization problems involve linear or nonlinear equality/inequality constraints. Mathematical programming based solutions to such problems can be classified into three categories. The first category is that of linear programming for which there are relatively efficient solution algorithms available. However, as the name linear programming suggests, these optimization problems represent a limited number of physical problems, because the objective functions and constraints must be strictly linear. The more general optimization problems involving nonlinear objective functions and constraints are approached in two ways. The first entails converting the constrained problems into an unconstrained optimization problem, by appending a measure of constraint violation to the objective function. This unconstrained problem can be solved by one of the many nonlinear programming algorithms. To prevent this approach from becoming numerically illconditioned, the optimal solution is actually obtained by solving a sequence of unconstrained problems, corresponding to increasing values of the penalty on constraint violations. This type of method is referred to as a sequential unconstrained minimization 30
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31 technique or SUMT. The third category consists of methods which are commonly used in practical engineering optimization problems. This approach deals with constraints directly in searching for an optimal solution, and is classified as a direct method. Such an approach was adopted extensively in the present work, primarily to obtain baseline optimal solutions to test problems. These solutions were then used for comparison with solutions obtained through other schemes proposed in this work. The direct method used here is referred to as the method of feasibleusable search directions, and is summarized in the following sections. A review of mathematical programming applications in structural optimization is available in [63]. 3.2 FeasibleUsable Search Direction Method An overview of the optimization problem and its formal mathematical statement are introduced next. The method of feasibleusable search directions is then discussed. In an engineering optimal design problem, the quantities to be determined are the design variables. For a given optimal design problem, there usually exist some restrictions dictated by the environment, processes, or resources, which have to be satisfied in order to produce an acceptable solution. These requirements are referred to as constraints, and describe dependencies among design variables and other parameters. Constraints are generally expressed mathematically by either equalities or inequalities. Among the designs satisfying all constraints, one has to
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32 be selected as the best or optimal design based on the value of some merit or criterion function. This criterion is an integral part of the optimization problem formulation. It is generally a computable function of design variables and other parameters, and is called the objective function. A general constrained optimization problem can be mathematically formulated as follows: Minimize: F{X) objective function (31) subject to gj{X) < 0 i=\,m inequality constraints (3.2) ^'a (^ " 0 ^=1,^ equality constraints (3.3) x' < < x" i=l,p side constraints (34) where X = [xj, x,, Xp]'^ is a vector of design variables; xand x" represent the lower and upper bounds of design variable x,, respectively. If equality constraints are explicit in X, the number of design variables can often be reduced. Although side constraints (Eqn. 3.4) can be included in the inequality constraint set (Eqn. 3.2), it is usually convenient to keep them separate, as they define the region of search for the optimum.
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33 Most mathematical programming algorithms require an initial design vector X^, where initial values of each design variable XjÂ°, X2Â°, XpÂ°, are specified. From this starting point, the design is updated iteratively to an improved solution. The most generally used iterative update equation is as follow, A"? = X^'^ + a^' 5^ (35) where superscript q is the iteration number, X is the vector of design variables, S is the search direction, and a^' is a scalar multiplier determining the amount of change in X for each iteration. The search direction, S, is first determined in each iteration, and the search moves in this direction to update the X vector according to Equation 3.5. The method of feasible directions, or method of feasibleusable search directions [64], is introduced by an example with two design variables and two inequalities constraints with reference to Figure 3.1. The figure shows the contour lines of the constant objective function F(X), and also shows the constraints. The goal of this method is to determine a search direction which rapidly reduces the objective function while maintaining a feasible design. Consider a design XÂ° on the constraint boundary gi(X) = 0. Gradient vectors of the objective function, VF(XÂ°), and the active constraint, Vgi(XÂ°), are calculated. The lines (hyperplanes in n dimensions) tangent to the contour of constant objective and tangent to the constraint boundary are now the linear approximation to the problem at this point. The next task is to determine a search direction S which
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34 Figure 3.1 Illustration of the feasibleusable search direction approach.
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35 reduces the objective function without violating the active constraint for some finite move. An S vector is called a usable direction if the objective function is reduced or maintained at its present value for a finite move along that direction. The usability requirement is mathematically defined as follows, VF{X^)S < 0 (36) A direction S is called feasible if, for some finite move in that direction, the active constraint is not violated. The feasibility requirement can also be mathematically defined as follows: Vg,(ZÂ°)S<0 (3.7) A region is called feasibleusable if any direction, S, inside that region satisfies both Equations 3.7 and 3.8. The search direction S is chosen to be located in the usablefeasible region such that the objective function, for a small move step, is reduced while the feasibility is maintained. Since a method such as the feasibleusable direction approach is essentially a local search technique, only the relative optimum will be attained if the design space is nonconvex. A design can be treated as the globally optimal design with a higher degree of confidence only if a large number of optimization processes with different starting values of vector X converge to the same final design. If a selected number of optimization processes results in different "optimal" designs, it
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36 usually indicates that the design space is multimodal or nonconvex. This problem can generally be solved by either switching to more sophisticated algorithms or by considering more initial designs to increase the probability of locating the global optimal design. It should be noted that any optimization method, except the exhaustive search technique, provides no guarantee of locating the global optimum. To improve computational efficiency in nonlinear optimization problems, a piecewise linearization of the design space is frequently adopted. In this approach, the nonlinear programming problem (both objective function and constraints) is first linearized about the starting design. The solution to this approximate linear problem is obtained within some prescribed move limits so as to maintain the integrity after linear approximation. The problem is next linearized about the new solution, and the process repeated till convergence. 3.3 Modified BranchandBound Search The discrete nature of the design variables in most structural optimization problems has only received limited attention. Solutions to such problems are relatively more difficult to obtain than for optimization problems with only continuous design variables. Discrete design variables in structural optimization have often been treated as continuous design variables. Upon obtaining the optimal design for this simplified problem, design variables which represented discrete design variables were then adjusted to the nearest discrete value. This
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37 simplified approach worked well for optimization problems with discrete design variables whose values were spaced reasonably close to one another. However, this simple rounding procedure often failed in locating the true optimal solution, or resulted in infeasible solutions [65]. One branch of mathematical programming, referred to as integer programming, offers a formal solution for optimization problems using integer/discrete design variables. Early work in obtaining systematic solutions to the integer linear programming problem are described by Gomory [66]. The branchandbound algorithms that emerged later are based on the enumeration of the space of the feasible integer solutions. A more detailed explanation of the branchandbound approach is available in [67]. The general framework for solving an integer programming problem involves decomposing the original problem into subproblems, modifying constraints to enlarge feasible domains, and finally a process referred to as fathoming. Fathoming involves checking a solution for feasibility, and establishing optimality. A modified mixed integer and discrete programming algorithm was proposed by Hajela and Shih [68]. The algorithm employs a strategy where a systematic search of continuous solutions is made, in which the discrete and integer variables are successively forced to assume specific values. The basic solution strategy used in this method for nonlinear mixed integer programming problem is a variant of the approach proposed by Garfinkel and Nemhauser [69]. The logical structure of the set of solutions was constructed as a binary tree. A
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38 modified feasible directions algorithm [70] was used in the solution of the continuous nonlinear programming problem, with piecewise linear representation of the objective function and constraints. 3.4 MinMax Global Criterion Approach Decisions in engineering design typically require allocation of resources to satisfy multiple, and frequently conflicting, requirements. For instance, in the design of a viscoelastically damped cantilever beam, minimum weight and maximum damping are two criteria which always conflict with each other. Nevertheless, these two criteria are both important characteristics. Adding on viscoelastic damping tape would increase the damping ability of the beam while, at the same time, increase the weight. One approach to solving this multicriterion optimization problem is to select weight as the objective function, and to use damping as a design constraint, or vice versa. The apparent simplicity afforded by this method is attractive; however, it should be noted that the treatment of criteria as constraints does not yield the same optimal design as would be obtained when solving the optimum design problem as one possessing multiple objectives [28]. In the above optimization example, the influence of the relative importance of structural weight or modal damping on the distribution of viscoelastic damping material can be obtained only by formally solving a twocriterion problem with different weighting coefficient combinations on criteria.
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39 In order to help understand the solutions to a multicriterion optimization problem, a frequently used expression, "Pareto optimal solution", needs to be defined. Given two designs Xi(xi,X2,...xÂ„) and Xj(xi,X2,...Xn,) in an ncriterion minimization problem, the design Xj is referred to as being partially less than the design X^ if each element of the criterion vector of the design Xj is lessthanorequal to the corresponding element of design Xj, and, for at least one element, the relation is strictly lessthan. If X is partially less than X^, X, is said to dominate X^, or Xj is inferior to Xj. In a given set of designs, a population of designs which are not dominated by any other design in the given set is said to be nondominated or noninferior. The set of nondominated or noninferior designs are referred to as the Paretooptimal designs which are referred to as Paretooptimal solutions in an optimization problem. An effective minmax variant of the global criterion approach for multicriterion optimization problems was introduced by Hajela and Shih [71]. It is referred to as a global criterion method [72], and is broadly classified as belonging to a category of solution methods with no articulation of preference. In such an approach, a metric function is formulated to represent the distance between the ideal solution and the optimum solution, and a minimization of this function results in the true optimum. The ideal solution is an ndimensional vector for an optimization problem with n design criteria. An optimal value for each individual criterion defines a component of the ndimensional vector. These 'n' components of the ideal solution vector are obtained by considering the optimization problem
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40 for each criterion separately. An application of the global criterion method is illustrated by Figure 3.2. This figure shows the feasible and infeasible sectors of the space of two criterion functions. If the ideaJ solution, f^(X), was located inside the feasible region, no further effort would be necessary. However, the ideal solution is typically infeasible, and the task becomes one of finding the feasible solution which is closest to the ideal solution. The Pareto optimal solution is obtained in such a fashion that any change of the optimal design vector will cause adverse effects on at least one of criteria. One can define a vector objective function f(X) dependent on the design variable vector X in the following fashion: m = [f,(X}/,{X),...f^{x),.../,(x)f (3.8) where f,(X) is the i"' objective function or criterion. Likewise, {^(X"^) is the ideal or optimal value if the i"" objective function is considered as the sole criterion of a singlecriterion optimization problem, and X''^ is the optimal design vector for such a criterion. Minimization of the metric distance between the infeasible ideal solution and the candidate feasible solution by a minmax variant of the global criterion approach leads to the optimization problem described by the following mathematical statement.
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41 Figure 3.2 Graphical representation of Pareto (noninferior) solutions in twocriterion minimization problem.
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42 Min Max (3.9) Solution of the above optimization problem yields the best compromise solution in which all criteria are considered equally important. Use of weighting coefficients can be introduced in conjunction with this method to rank the importance of each criterion. The minmax problem can be restated as follows: where W; is the weighting coefficient representing the relative importance of the i"" criterion. A scalar variable y3 is introduced to transform the minmax problem of (3.10) into an equivalent scalar optimization problem. This singlecriterion optimization problem is as follows: Min Max wi=l,2,...,k (3.10) Minimize (3.11) subject to the following additional constraints:
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43 f,iX)f^{X") ax') p < 0 i=\,2,,k (3.12) In a situation where the design variable set is a mixture of integer, discrete, and continuous design variables, it can be shown [71] that the ideal solution should be selected as the one obtained by treating all design variables as continuous. 3.5 Numerical Implementation The modified branchandbound method, in conjunction with the minmax variant of the global criterion approach, was implemented in determining the multicriterion optimal design of a sandwich cantilever beam to enhance its damping ability. Damping ability in a structural composite or metal beam can be enhanced passively, via an external viscoelastic damping layer treatment. The design space for this problem consists of both continuous and discrete/integer design variables. An aluminum cantilever beam serves as a base structure of the structural system for which maximum damping in several modes is desired. Constrained viscoelastic layers were added to increase the damping ability of the structure. The structural system and two methods of adding damping layers are shown in Figures 3.3 and 3.4. Material properties for this structural system are given in Table 3.1. In the first case, the addon viscoelastic tape was applied starting from the root of the cantilever beam with the total length of the tape
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44 Figure 3.3 Sandwich beam structure.
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45 Figure 3.4 Sandwich beam structure.
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46 Table 3.1 Material properties of the structural system. cuiisudining layer, ano base structure viscoeiastic layer Young's modulus (GN/m^) 69.0 0.0021 Poisson's ratio 0.3 0.499 Damping factor 0.0 0.10 Density (kg/m^) 2700.0 970.0
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47 indicated by X5 (X4 = 0). In the other case, the addon damping tape was applied starting from an arbitrary point on the beam. The distance between the root of the beam and the starting point is indicated by X4, and the total length of the tape is indicated by X5. One integrated approach to the optimal design of the viscoelastically damped structure is to configure both the basic structure and addon damping layers. The optimal configuration is then determined by the thickness of the viscoelastic layer, the thickness of the constraining layer, and the thickness of the base structure. They are denoted as Xj, X2, and X3, respectively. The remaining two design variables, X^ and X5, result from how the damping layers were applied. Finite element analysis was used to evaluate damping and natural frequencies, as well as the static displacement in the structure resulting from the different applied lengths of the constrained viscoelastic layer. The structure of the finite element model representing the three layer sandwich is shown in Figure 3.5. The base structure and constraining layer were modeled by using a specially developed threenode, sevendegreeoffreedom, offset beam element as shown in Figure 3.6. The element contains sheardeformable characteristics, which are important in modeling fiberreinforced composites. A key feature of this model was its accountability of the coupling between the stretching and bending deformations. This allowed the nodes to be offset to one surface of the beam, coincident with the nodes of the adjoining element. The viscoelastic core was modeled using a rectangular plane stress element that is compatible with the
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48 X4 X5 i 0.40m Figure 3.5 Typical finite element model of the sandwich beam.
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49 A w Figure 3.6 Offset beam element.
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50 offset beam element. The stiffness of the offset element was obtained from a strain energy formulation which was based on the following assumed displacement field: u{x^)=u,^(z!L)y\f{x) (3.13) o)(x^) = w{x) (314) i^{x^) = ^^{x) (3.15) where Uq and ir were obtained by linear interpolation and a quadratic interpolation was used to compute o). A consistent mass matrix, evaluated from the kinetic energy, was used in this analysis. A detailed description of the offset beam element is given in [31]. The loss factor of the system was determined by the strain energy based approach discussed in the previous chapter. The vector of design objectives included minimum structural weight and maximum loss factors in the first three modes of vibration. The constraint set comprised lower and upper bounds on the first three natural frequencies, and the maximum displacement when a static load of 1N was applied at the tip of the beam. These are summarized in Table 3.2. Design variables Xj through X5 are shown in Figures 3.3 and 3.4. Thicknesses of the constraining and viscoelastic layers, and X,, were considered discrete variables, with admissible values selected from the set shown in Table 3.2. All other design variables were assumed
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51 Table 3.2 Design constraints and allowable discrete values of design variables Xi and X,. Xj, X2 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0010 0.0012 0.0013 0.0014 0.0015 0.0016 0.0017 0.0018 0.0019 0.0020 (m) Design Constraints 20.0 < freqi < 50.0 (Hz) 130.0 < freq, < 200.0 (Hz) 400.0 < freqj < 700.0 (Hz) tip displacement < 0.001 m
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52 continuous. The mathematical statement of the optimum design problem can be written as follows: Minimize (F^,F,,FÂ„F^) (3.16) Subject to g. < 0, j=l,2,...m Here Fj, F,, and F3 represent the loss factors of the first three vibration modes, F^ is the structural weight, and g^ are the m inequality constraints for the problem as specified in Table 3.2. Two distinct cases were considered by setting X4 to zero, and by allowing X4 to assume finite values, respectively. The locations and lengths of the damping tape were adjusted using an adaptive preprocessor. Ten elements along the length of the base beam were used in the finite element analysis throughout the optimization process. A uniform node spacing was used in the situation that X4 and X5 are both multiples of onetenth of the length of the beam. When the length of an untreated free or fixed end is small, an unsmooth node spacing may result when an equallengthelement representation of the treated length is used. To avoid this situation, stretching techniques were utilized to assure a smooth variation in node spacing. A minimum node spacing ratio for any two consecutive finite elements so obtained resulted in an improved level of accuracy on the numerical prediction. Figure 3.7 shows a beam illustrating, from top to bottom, uniform node spacing, unstretched
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X4 53 X5 0.4 m H X4 I UNIFORM NODES X5 0 .4 mUNSTRETCHED NODES X4^ r Â• Â— Â• Â—
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54 node spacing, and stretched spacing. The adaptive preprocessor also possesses automatic node numbering ability to automatically create a finite element model based on the given set of design variables. The minmax global criterion approach was used to transform the optimization statement (3.16) as follows: Minimize (3.17) subject to m original constraints plus the following four additional constraints: w, f,{X)f,{X'') f,iX)f,{X") f,{X)f^{X'') mux^') 13 < 0 13 < 0 (3 < 0 13 < 0 (3.18) where W^, i = l,4, are weighting parameters and each weight is used to emphasize the relative importance of the i"^ criterion. The nonlinear optimization problem presented in (3.17 and 3.18) was solved by the feasibledirection method with the modified branchandbound technique [68].
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55 For case 1, the separately attainable optimal designs of each component of the objective function are shown in Table 3.3. The design variables were all assumed continuous in finding the ideal solutions. Assigning different weighting coefficients to the components of the objective function permitted the generation of a series of noninferior optimal solutions, as presented in Table 3.4. These noninferior solutions were such that no component of the objective vector could be improved without adversely affecting another component. Similar solutions for case 2 are presented in Tables 3.5 and 3.6. The inclusion of the X4 variable results in no improvement in F,'", Fj'", and F^^ However, F,*'' (loss factor in second mode) increased by 2% and the length of the damping layer X5 decreased by 18%. Permitting to assume finite values yields a noticeable improvement in the objective function, only when the weighting coefficients associated with F3 outweigh those of Fj and F3. This was because the middle region of the beam undergoes the most shear deformation for a displacement corresponding to the second mode of vibration. With X4 allowed to vary, and for weighting coefficients set as (0.0, 0.5, 0.0, 0.5), the damping factor of the second mode was increased by 5%, while the weight of the structure was reduced by 1%.
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56 Table 3.3 Ideal solutions for case 1. Fi"*^^' = 0.01825 F,""^" = 0.02003 Fj"'^^' = 0.01833 F;'"=^' = 1.5860 X"'"'(Fj) ( 0.0020, 0.0001, 0.00454, 0.262 ) X"^"'(F2) = ( 0.0020, 0.0001, 0.00446, 0.358 ) = ( 0.0020, 0.0001, 0.00445, 0.400 ) = ( 0.0001, 0.0001, 0.00499, 0.000 )
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57 Table 3.4 Optimal solutions for problem in case 1. \A/ j^i rthti'nrT vv cigjiiing uesign vanaDies Loss Factor weignt ^^omomdiion V V V V Mode 1 W JW4 (m) Mode 2 MnHp 3 70 n nni/i n MAO U.UUUl U.Ui!)4 .zu U.UU44 0.013 / .40 0.359 .40 0.0019 0.0172 1.9350 .30 0.0001 0.0174 .10 0.0046 0.0098 .20 0.309 .20 0.0018 0.0167 1.9198 .40 0.0001 0.0187 .20 0.0044 0.0156 .20 0.357 .00 0.0015 0.0149 1.8397 .25 0.0001 0.0163 .25 0.0044 0.0140 .50 0.357 .00 0.0015 0.0149 1.8382 .50 0.0001 0.0163 .00 0.0044 0.0139 .50 0.355
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58 Table 3.5 Ideal solutions for case 2. Fj"^"' = 0.01825 = 0.02037 F/^"' = 0.01833 F,"""'^ = 1.5829 X.deai^p^^ = ( 0.0020, 0.0001, 0.00454, 0.000, 0.262 ) X"''=^'(F2) = ( 0.0020, 0.0001, 0.00446, 0.060, 0.294 ) X"'"'(F3) = ( 0.0020, 0.0001, 0.00445, 0.000, 0.400 ) X""^^'(F,) = ( 0.0001, 0.0001, 0.00498, 0.054, 0.000 )
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59 Table 3.6 Optimal solutions for problem in case 2. Weighting Design Variables Loss Factor Weight Combmation Xj,X,,X3,X4,X5 Mode 1 W1W4 (m) Mode 2 Mode 3 .20 0.0017 0.0115 1.8765 .20 0.0001 0.0172 .20 0.0045 0.0129 .40 0.022 0.319 .40 0.0019 0.0172 1.9350 .30 0.0001 0.0174 1 A .10 0.0046 0.0098 .20 0.000 0.309 .20 0.0018 0.0167 1.9198 .40 0.0001 0.0187 .20 0.0044 0.0156 .20 0.000 0.357 .00 0.0015 0.0083 1.8433 .25 0.0001 0.0155 .25 0.0045 0.0144 .50 0.038 0.333 .00 0.0017 0.0066 1.8288 .50 0.0001 0.0171 .00 0.0046 0.0098 .50 0.059 0.262
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CHAPTER 4 GENETIC SEARCH METHODS 4.1 Introduction Mathematical nonlinear programming algorithms have emerged as the method of choice for applications in engineering optimization problems. They provide a general approach for obtaining solutions to both single and multiobjective design problems with a mix of equality and inequality constraints. The more efficient of this class of methods are generally gradient based, and require at least the firstorder derivatives of both objective and constraint functions with respect to the design variables. With this "slopetracking" ability, gradientbased methods can easily identify a relative optimum closest to the initial guess of the optimum design. There is no guarantee of locating the global optimum if the design space is known to be nonconvex [73]. These methods are also inadequate in problems where the design space is discontinuous, because the derivatives of both the objective function and constraints may become singular across the boundary of discontinuity. In engineering design problems, the mix of continuous, discrete, and integer design variables has been approached by treating all variables as continuous, and then rounding specific variables either up or down to the nearest 60
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61 integer or discrete variable. Branchandbound techniques based linear programming or combinatorial programming offers a formal solution to this class of optimization problems with a mix of integer, discrete and continuous design variables. This strategy consisted of a systematic search of continuous solutions in which the discrete and integer variables were successively forced to assume specific values. However, in doing so, the original optimization problem was undesirably expanded to a large number of suboptimization problems. Exhaustive search and random search methods are among the simplest and most robust strategies for automated optimum design problems. These methods can work on almost all kinds of design spaces and without any restriction on types of design variables. An improvement on the simple enumerative techniques is possible with methods such as random walk and random walk with direction exploitation. The only drawback is that these methods often require thousands of function evaluations to achieve the optimum, even for the simplest of problems. It is hence crucial to examine alternative strategies for optimal structural design problems, which need less computational effort than required by the enumerative search techniques, and are also not susceptible to convergence to a local optimum as exhibited by gradientbased nonlinear programming algorithms. Genetic algorithms (GAs) as proposed by Holland [34], have the potential to successfully fill this gap. Genetic algorithms belong to a category of stochastic search techniques, where only the most promising regions of the design space are enumerated to
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62 locate the optimal design. These algorithms have their philosophical basis in Darwin's theory of survival of the fittest. Analogous to the natural process where a population of a given species adapts to a natural habitat, a population of designs is created and is then allowed to adapt to the design requirements. Designs that do not adapt in a favorable manner to the requirements are eliminated from consideration. The mechanism of adaptation borrows extensively from principles of biological evolution, in that basic characteristics of designs in one population are transferred to a population in another generation through gene transfer operators. Stated differently, design alternatives representing a population in a given generation are allowed to reproduce and cross among themselves with bias allocated to the most fit members of the population. Combination of the most favorable characteristics of the mating members of the population results in a progeny population that is more fit than the parent population. If the measure which indicates the fitness of a generation is also the desired goal of a design process, successive generations produce better values of the objective function. The mechanics of genetic search, though simple to implement, encompass features that render the approach highly applicable to the problem of search in a nonconvex/disjoint design space with a mix of continuous, discrete, and integer design variables. These desirable characteristics are largely attributed to the fact that genetic search moves from a population of designs to another population of designs; this is in contrast to the point to point search available in traditional mathematical programming methods, and therefore offers a better possibility of
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63 locating a global optimum. Furthermore, genetic algorithms work on a coding of the design variables rather than the variables themselves. This allows for an efficient treatment of integer and discrete variables. The terminology of genetic search and its principal components are summarized next. 4.2 Genetic Algorithms The basic approach in genetic algorithms is to represent possible solutions to a given problem by a population of bit strings of finite lengths, and to subsequently use transformations analogous to biological reproduction and evolution to improve and vary the coded solutions. In natural populations, genetic information stored in chromosomal strings evolves over generations to adapt favorably to a static or changing environment. The chromosomal structure represents a generational memory, and is altered through chromosomal string inversions, occasional mutation, and a crossover of genetic information between reproducing members. In an elitist reproduction strategy, those members of the population that are deemed most fit are selected for reproduction, and are given the opportunity to strengthen the chromosomal makeup of the progeny generation. This approach is facilitated by defining a fitness function or a measure indicating the "goodness" of a member of the population in a given generation during the evolution process. For unconstrained maximization problems, the objective function could serve as the fitness function. The inverse of the objective function, or the difference between a large number (say the
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64 maximum value of the objective function in the population) and the objective function value for each member can be used as the fitness measure in a minimization problem. For constrained optimal design problems, an exterior penalty function formulation can be adapted to transform a constrained optimization problem into an unconstrained one. In biological systems, generational memory is preserved and transferred to progeny populations in the form of chromosomes. A number of chromosome strings comprises a genotype, the total genetic makeup for an organism. In order to use genetic algorithms, an artificial chromosomelike string must be devised to represent the ndimensional solution space. One simple yet effective approach to accomplish this is to represent each design variable by a finite length binary string and then connect, headtotail, all n strings into a single binar>' string. Various simulations of genetic evolution and adaptation are conceivable. Three principal components of the genetransformation mechanism in this artificial evolution and adaptation simulation are reproduction, crossover, and mutation. These transformations are best described with reference to a specific optimization problem. Consider the following function minimization problem: Minimize F{x^^^) = xf +X^X2 +X2 (4.1) Subject to: = x^+xfx^ < 0 (4.2) (4.3)
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65 xr ^ X, < xr i = 1,2 (4.4) To use genetic search, the constrained minimization problem is first converted into an unconstrained problem using the exterior penalty function formulation, resulting in the following problem: Minimize F' = F + p (45) where P is the penalty parameter, and is typically given by, P=rj:^ (4.6) where r is a penalty parameter of the form encountered in the exterior penalty function approach [74], and represents the violated constraints. Careful consideration must be given to the selection of the penalty function P. Numerical experiments have shown that genetic search with a penalty function formulated as in Eqn. (4.6), exhibits slow convergence. This was largely due to the fact that a severely violated constraint yields a value of P which overwhelms the objective function value F. A more detailed explanation of this slow convergence will be given in a later section. In order to overcome this situation, the following bounding strategy was adopted. If the average fitness of feasible designs is F. . O civ ' then a limiter value of the penalty L is selected as.
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66 L =k*FÂ„ (4.7) where k is of order 2; the penalty that is appended to an infeasible design is then obtained as follows: otherwise (4.8) The effect of this scaling is to prevent radical departures in the value of the penalty term from the specified value. If the slope parameter, a, is 0.0, penalty for all violated designs with P > L is set to L If instead, a is assigned a small value of the order 0.1, then the extent of constraint violation due to severely violated and less violated designs varies linearly from L, albeit with a small slope. To convert this function minimization into a fitness maximization as required by GA's, the following fitness function is created. f. = F* F* J I max ^ ( (4.9) where f, is the fitness of the i"^ design, and F'Â„,,^ is the maximum value of F*. The
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67 F can be chosen as the maximum F' in the population of that given generation, or of several previous generations. A fixed large value of F\^^^ can be used throughout the entire genetic search provided that there is not any F* which is larger than F'^^^^ However, genetic search with a variable value of F'^^^. selected to be slightly larger than the largest F' in the present or including past few generations, usually converges faster than with a fixed F'Â„,3^.. To obtain a bit string representation of the design variables in this problem, each x, can be converted into a binary string of O's and I's. For purposes of discussion, we choose a 5 digit binary number, with the minimum and maximum values of x^ denoted as x,"^'" and x,"''^ respectively, represented by the following binary numbers. x,""" = 00000 x,"'"' 11111 A linear scaling can be introduced to convert intermediate values of the binary number into design variable values. The binary string representations for Xj and X, can be placed headtotail to create a 10digit number, also referred to as a schema, which represents a solution to the problem. Several such 10digit binary strings are defined to constitute a population of designs, which includes a mix of feasible and infeasible designs. The fitness f, corresponding to each member of the population is computed before invoking the genetic transformations.
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68 4.2.1 Reproduction The simulation of genetic evolution here is contrived in that the population size is forced to remain unchanged during the evolution process; two mating parents create only two progenies but are themselves eliminated. The reproduction process is one that biases the search toward producing more fit members in the population and ehminating the less fit ones. One simplistic approach to selecting members from an initial population to participate in the reproduction is to assign each member a probability of selection on the basis of its fitness. If fj is the fitness measure of the i*" member, the probabiHty for the i"^ member to participate in reproduction for each of m selections can be assigned as: m (4.10) where m is the population size. After m selections, a new population pool is created. The new pool is the same size as the original pool, but has a higher average fitness value. There is no new 'geneticmaterial' generated in the selection process. Equation (4.10) shows that the probability of a design being selected for participation in the reproduction process depends not only on its fitness, but also fitness of other designs in the population. Furthermore, the value returned by a fitness function is not always an "exact" measure of fitness. The exact value
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69 returned can vary greatly depending on how the fitness function is implemented. For example, let a population in a minimization problem consist of four designs which have objective function values as 3, 5, 7, and 9. If the largest value 9 is chosen as F\^^ (Eqn. 4.9), then the fitness, fj, for these four designs will be 6, 4, 2, and 0, respectively. The probability of selection of these four designs are 6/12, 4/12, 2/12, and 0/12, respectively (Eqn. 4.10). The design with the lowest fitness will not be selected for participation in the reproduction process. The probability of the design with the best fitness is 3 times that of the design with the second worst fitness. Therefore, a higher fitness average is likely to occur after the reproduction process. This gives genetic algorithms power to guide the search towards more promising design areas. However, if the above designs have the following objective functions 3, 5, 7, and 85. The outstanding high objective function, 85, could result from multiple causes, for instance, it may be due to a severely violated constraint. If the value 85 were selected as F*Â„,3^, then, the fitness for these four designs will be 82, 80, 78, and 0. In each selection, the probabilities of the four designs for participation in the reproduction process would be 82/240, 80/240, 78/240, and 0/240. Almost no differences exist between the designs when considering the probabilities of participation with the exception of the worst case. This phenomenon makes it difficult for good designs to stand out from the population, and consequently the genetic search would not concentrate in the more promising areas. This illustrates a drawback of induced bad fitness as caused
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70 by violated constraints, and the advantage of using the modified penalty P, which was formulated in Equation (4.8). Fitness proportionate reproduction (Eqn. 4.10) has been used almost exclusively for allocating reproductive trials of genetic algorithms. Reproduction strategies that differed from the traditional, proportionate reproduction plan were implemented [75,76] in an effort to obtain improved performance of genetic searches. Baker [75] reported experiments where reproductive trials were allocated according to the rank of individual strings in the population rather than by individual fitness values as related to the population average. The rankbased allocation of reproductive trials slowed convergence. The method gave a slow, but more accurate optimization. Whitley [76] presented new evidence and arguments which suggested that allocating reproductive trials according to rank can be used to speed up a genetic search if appropriately implemented. 4.2.2 Crossover The process of reproduction assures that more copies of dominant or fit designs will be present in a population. The crossover process allows for an exchange of design characteristics, among members of the population pool, with the intent of improving the fitness of the next generation. This is similar to the transfer of genetic material in biological reproduction processes as facilitated by DNA and RNA strings. Crossover is executed by selecting chromosomal strings of two mating parents, randomly choosing sites on the strings, and swapping strings
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71 of O's and I's between these chosen sites. There are a number of crossover operations, such as a onepoint crossover, twopoint crossover, multipoint crossover, segmented crossover, uniform crossover, and shuffle crossover. The twopoint crossover operator was used in this research. An illustration of the twopoint crossover process between mating parents represented by 20 digit binary strings is as follows: Parentl = 11010100100110011010 Parent2 = 01001110001011001000 Childl = 11001110001010011010 Child2 = 01010100100111001000 The crossover sites on the parent strings are indicated by an understrike. A probability of crossover p, is defined to determine if crossover should be implemented. In the case of the more fundamental onepoint crossover process, only one randomly generated site is created, and strings after the selected site are then swapped between two partners. More information about the implementation, nature, and evaluation of different crossover operators is presented in [77].
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72 4.2.3 Mutation Mutation safeguards the genetic search process from a premature loss of valuable genetic material during reproduction and crossover. The process of mutation is simply to choose few members from the population pool according to the probability of mutation pÂ„Â„ and to switch a 0 to 1 or vice versa at a randomly selected mutation site on the selected string. Most genetic search methods give a constant probability of mutation, p^. A method employing variable probability of mutation in the genetic search was proposed by Fogarty [78]. The study measured the performance of genetic searches in which the p,Â„'s were a) exponentially decreased with successive generations, b) increased over the bit representation of each integer, and c) was a combination of the increasing and decreasing probability methods. The variable pÂ„, methods were compared with the performance of a genetic search with a fixed pÂ„. The varied probability p^ was shown to significantly improve the performance of the genetic search, but only for cases in which the initial populations contained all O's or all I's. Randomly generated initial population cases showed no improvement in performance. Running a genetic algorithm entails setting a number of initial parameter values, such as the size of population, probabilities of crossover, p^, mutation probability, p^, etc. Determining initial conditions which suit one's problem is not a trivial task. If poor initial settings are used, the performance of a genetic search can be severely impacted. Two useful strategies for determining effective crossover and mutation probabilities have been developed [79,80]. A strategy to
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73 adapt the probabilities of crossover and mutation during the course of the genetic search was implemented by Davis [81]. This method showed Hmited success, and more efforts were needed to make it robust and effective. The performance of genetic search methods in obtaining the optimum ply stacking sequence of a laminated composite plate under a concentrated load was examined for a number of combinations of p^ and pÂ„,. This experiment is detailed in Chapter 5. 4.2.4 Implicit Parallelism It is important to emphasize that genetic algorithms use probabilistic transition rules only to guide a highly exploitative search. To this extent, they should not be considered a variant of the random walk approach. The implicit parallelism available in these methods is significant from a computational standpoint, and has been explored [82]. If a binary code is used in an ndigit string to represent a design, then a total of 2" variations of the design are available in the representation scheme. However, the exploitative power of OA's extends beyond this if one takes into consideration the fact that the presence of a 0 or 1 at some key location along the string may be significant in itself. As an example, consider the design of a twobar truss structure for minimum weight, with each crosssectional area mapped into a 5digit binary string. The maximum and minimum areas would be denoted as follows.
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74 = 11111 = 00000 If the bit string representation of the design is obtained by stacking the binary representation of area A, after Aj, then a design represented by 1001000010 indicates a value of Aj close to its maximum and A, closer to its minimum value. Further, if we use a '*' to indicate that a location along the string can occupy either a 1 or a 0, then a string of the form 1***0000** conveys the same information. In this manner, one can think of the following schemata: J_J2 _ Â«QÂ«***Â«Â« H2 = 1***00***0 H3 = 1*01***0** Our design of areas A^ and Aj could belong to each one of these schemata. In such a 10digit representation, there are a total of 3^Â° (59049) schemata among 2'Â° (1024) possible unique designs. A decrease in the number of *'s in a schema makes it more specific. In this context one defines the order of a schema denoted as 0(H) as the number of O's or I's in a schema. The order of Hi is 3 and that of H2 and H3 is 4. Another item of interest is the defining length of a schema d(H), which is taken as the distance between the first and last specific digit on the string; d(Hl) = 7, while d(H2) = 9. Schemata with higher defining lengths are more
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75 likely to be disrupted during crossover. If we denote the number of schemata of type H at a given generation t as m(H,t), then, due to reproduction alone, at generation t+ 1 this number would be [82], f a\' where f(H) is the fitness of schema H and is the average fitness of the population. Clearly, schema with a fitness higher than the average fitness of the population are increased exponentially in successive generations. The affect of crossover and mutation is disruptive. At the same time, however, these processes contribute to the evolution of new characteristics in the population. In terms of the probabilities of crossover and mutation, p^ and p^, respectively, and the string length L, the growth or decay of a schema in a population is expressed as follows: m{H,t^l) > miH,t)l^(lp^^0{H)p\ (4.12) f L1 Here, the predicted growth due to reproduction alone has been modified by considering the probability of surviving the crossover and mutation transforms. As IS dear from this expression, loworder, shorter defininglength schemata, which are also fit, increase rapidly in the population. Holland [34] estimated that the actual processing of n strings corresponding to the population size results in the processing of O(n^) schemata, giving the approach a tremendous computational advantage.
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76 The choice of population size, n, for a genetic search represents a fundamental decision in GA implementation. A population with too few members can result in a premature convergence of the genetic search, where not enough genetic combinations are processed to converge to the ideal solution. Premature convergence resulted from an effect referred to as genetic drift [79,83] or a preferential selection pressure. An overly large population size results in increased number of function evaluations before significant improvement is observed in the serial evaluation of individuals. Goldberg [84,85] provides strategies for determining the optimal initial population sizes of binarycoded serial and parallel genetic algorithms. The performances of genetic search methods were computed for a number of structural and mechanical optimal design problems using different population sizes. The results are presented in Chapter 5. Premature convergence can be defined as an event where a population of designs achieves a high level of uniformity at all loci, without containing nearoptimal design characteristics. This undesirable convergence situation usually results from stochastic errors associated with a small population size. Strings representing important schemata are not usually represented sufficiently when a small population size is selected. Finite populations eventually converge to nonoptimal solutions without the presence of selective pressure. This phenomenon is given the name genetic drift. A large population size or a high probability of mutation decreases the occurrence of genetic drift. Detailed descriptions of genetic drift and of the Markov analysis of genetic drift are presented in [83].
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77 4.3 Representation Schemes for Design Space 4.3.1 Continuous Design Variables An mdigit binary number representation of a continuous design variable allows for 2"" distinct variations of that design variable to be considered. If the description of a design variable is required to a precision of A^,, then the number of digits in the binary string may be estimated from the following relationship: 2"' > (4.13) Here, Xl and x^^ are the lower and upper bounds of a continuous variable x. As an example, if one considers representing a variable x to within a precision 0.1, and the lower and upper bounds on x are 0.0 and 0.7, then 2'" > 8. This gives the result that m = 3, and indeed the eight 3digit combinations of 0 and 1, enlisted as 000, 001, 010, Ill, can be assigned to numbers 0.0, 0.1,...0.7. Note that an integer value of m will not yield a precision of 0.14, and a 3digit string would still be necessary to represent the six numbers 0.0, 0.14, 0.28, 0.7. There would be two excessive binary representations which would be digested by distributing them to partial discrete design space as described in later sections. It is important to recognize that even when dealing with continuous variables, GA's work on a discrete representative set of those variables; the method is therefore ideally suited for applications to problems with a mix of continuous, integer, and discrete variables.
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78 The binary representation of population characteristics has dominated GA research. Advantages of binary representation include: analytical simplicity of binary vectors, the elegance of genetic operators in binary strings, requirements for computational speed, and most importantly, the theoretically motivated desires to maximize the number of schemata sampled by a given set of individuals in a population [34,82]. Binary representation is not an exclusive model for genetic algorithms, and higherlevel representations in both theory and applications are v^'orthy of investigation [86]. 4.3.2 Integer Design Variables Due to the discrete nature of the binary representation schemes, integer design variables can be simply regarded as continuous design variables with a fixed accuracy A, equal to 1. If m could be found to meet (x,j Xl) = 2"'l, a onetoone correspondence could be readily established. In most cases, however, this is not possible, and the excessive binary strings must be assigned in an appropriate manner. There are a number of ways in which this is done, and these are described as follows. (i) Penalty approach: In this approach, the smallest number m which meets the inequahty 2'" > {X, xj . 1 (4.14)
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79 is computed. Of the 2"' possible mdigit binary strings, a unique string is assigned to each of the N integer variables. The remaining (2"" N) strings are assigned to outofbound integers. As an example, in representing six integers between 0 and 5, the computed value of m satisfying the above inequality is 3. The remaining two binary strings are assigned to outofbound variables as follows, [0, 1, 2, 3, 4, 5, 6*, 7*] [000, 001, 010, Oil, 100, 101, 110*, 111*] where, '*' indicates an outofbound variable. A penalty measure is then allocated to the fitness function of a design which includes the outofbound value of an integer variable. While this approach yields a onetoone correspondence between the integer variables and their binary representations, careful consideration must be given to the magnitude of penalty assigned to the fitness function due to the presence of an outofbound variable. Large penalties on the fitness will adversely affect the genetic search, making it difficult to distinguish between good and average designs. (ii). ExcessiveDistribution Method: In this approach, m is first computed on the basis of Eqn. (4.14). The excessive binary representations are then assigned to integers in the admissible range, whereby one or more integers may have more
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80 than one binary representation. For the case described above, one of the two binary strings could be assigned to integer 4 and the other to integer 5. Although this method does not create any additional design constraints, it does have the effect of an uneven expansion in the design space. The affect of this partial expansion is hard to predict without apriori information of the objective function space. Clearly, the expansion results in a larger number of schemata in the design space. If the schemata happen to belong to the poorer regions of the design space, the convergence of genetic search would be adversely influenced. One method of avoiding this problem is to distribute the excess binary strings evenly along the extent of the feasible integer space. As an example, if twenty excessive binary strings are to be distributed evenly among one hundred integers, one excess binary representation can be assigned to every fifth consecutive integer. The uneven distribution described above may be avoided by creating excess strings on purpose to obtain a similar number of binary representations for each integer. For the example described above, two integers (33%) have twice the number of binary representations (100% more) than the other four (67%) of the integers. If m = 4, we would have a total of 16 binary strings to be assigned to six integers. In this case, each integer could be assigned 2 strings, and four integers (67%) could be assigned one additional string (50% more) than the remaining two (33%) integers. With increasing value of m, the disparity in distribution can be removed at the price of increasing the number of schemata to be explored.
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81 4.3.3 Discrete Design Variables Discrete type design variables are characterized by an uneven spacing between two consecutive values. With the unique nature of genetic search, wherein gradient information is not required, these variables can be handled in the same manner as continuous or integer variables. Mapping of these variables is a twostage process. In the first stage, discrete variables are mapped to an equivalent number of integer variables. Then, techniques of mapping integer variables into binary strings described in the preceding sections are applicable with no additional manipulation. An illustration of this process is the mapping of a set of eight discrete variables into binary strings. [2.4, 3.76, 5.96, 8.25, 9.37, 13.70, 20.55, 24.0] [ 1 2 3 4 5 6 7 8 ] [000 001 010 Oil 100 101 110 111 ] A design space with a mix of continuous, discrete, and integer variables can be represented as required in genetic search by connecting, headtotail, binary string equivalents of these variables as described above. In genetic search, the average fitness of a population is increased over generations of evolution. In an unconstrained maximization, the objective function can be chosen as the fitness function. If an unconstrained minimum is the objective, the fitness must be revised to be the inverse of the objective function.
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82 As stated earlier, another alternative is to choose the maximum objective function value of all members of the population, and subtract individual objective function values from this quantity to obtain the corresponding fitness values. In constrained optimization, designs with violated constraints are considered less fit. Hence in constrained minimization, a penalty corresponding to the constraint violations is appended to the objective function much as in the exterior penalty function approach. In the present work, the traditional penalty function approach is compared to a revised formulation discussed earlier, which is characterized by the imposition of an upper bound on the penalty. Without such a bound, a highly penalized infeasible design and a good feasible design would be difficult to separate, and the genetic search would deteriorate into a random search. 43.4 Performance Evaluation of Representation Schemes DeJong [79] defines online and offline performance measures to gage the efficiency of a genetic search. While online performance is an average of all evaluations of fitness, and is therefore indicative of how well the entire population adapts, the offline parameter is an average of the current best evaluations of fitness. These two parameters, in addition to the average fitness of a population at each generation of evolution, were used to measure the performance of genetic search with the different representation schemes described in preceding sections. A test problem for this purpose was chosen as follows: Minimize F^(x) = x'^ + jc,' + x," (4.15)
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83 where Xj, x,, and X3 could assume continuous values ranging between 512 and 511. A onetoone correspondence can be obtained by using a 10digit binary string (2'Â° distinct representations). String lengths of 12, 14, 16, 18 and 20 were also used, yielding 4, 16, 64, 256 and 1024 binary representations per integer variable, respectively. A fixed population size of 50 was used in conjunction with probabilities of crossover and mutation set to 0.6 and 0.005, respectively. Both online and offline performance of each of these cases was obtained. The results show improvement in both performance measures with an increase in the number of representations for each admissible value of the integer variables (20digit string performed the best). This was attributed to two possible factors. a. More schemata are available in such larger string representations, thereby increasing the probability of having a better initial population. b. In populations where there is exactly one representation for each admissible value of the integer variable, there is the risk of losing that variable very easily during genetic transformations. Note that each 0 or 1 along the string very specifically denotes an integer variable, and is susceptible to elimination by even a simple mutation transform. These advantages must be weighed against the slower improvement in the population, a direct consequence of schema disruption with longer binary strings.
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84 A modification of this problem was a change in the design variable upper bound to 238. For string length of 10, there were excess binary representations, which were treated by the methods discussed earlier. In the penalty approach, a penalty (O(IO^)) was imposed on any design containing design variable values between 239 and 511. This choice of penalty parameter is not advised because high penalties on the objective function resulted from the design variable range violations will slow the convergence of the genetic search. The magnitude of the penalty for the design variable range violation should be appropriately selected in accordance to the average magnitude of fitness functions. An arbitrary assignment of penalty parameters should be avoided. Another genetic search with an initial penalty (0(10"*)), and which was linearly decreased in every generation, was also used with more satisfactory results. In the excess distribution method, two schemes were implemented. In the first, the 273 excess representations were distributed evenly over the range 512 Â« 238; in the second, they were all distributed to integers on the side of 238. Four other experiments were conducted, with string lengths of 11, 12, 13 and 14, and all excessive representations were distributed evenly for the entire range of integer variables. It can be shown that a string length of 13 yields the most even distribution with 91% of the integer variables assigned only about 10% more representations than the remaining 9% of the integer variables. Some general conclusions emerged from these numerical experiments. Although conventional wisdom in genetic search advocates the use of smaller
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85 length strings, for a discrete design space it appears advantageous to have more than a onetoone correspondence between the integer/discrete design variables and their binary representations. If a onetoone correspondence is chosen with only a few excess representations, then the design variable representation can be based on either the penalty concept or the excessive distribution approach. The penalty approach is highly sensitive to the choice of the penalty parameter, and is generally not recommended unless information about the magnitude of the objective function is available. Of the excessive distribution method, the even distribution schemes perform better than the onesided distribution methods. This conclusion would not be valid only if excessive distribution is done in those regions of the design space where the optimal point is located. When an excessive distribution method is used for a design variable, the design space corresponding to this variable will be partially expanded. With larger length binary numbers, the excessive strings can be more evenly distributed and result in an even expansion over all variables. This approach also makes available a larger number of schemata for each design variable. 4.4 Multicriterion Desig n A relatively simple approach to account for the multicriterion nature of the problem in optimal design is through the utility function method [74]. A utility function is defined by assigning a relative importance to an individual criterion in relation to other criteria. For an mcriterion optimization problem, a practical
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86 definition of the utility function can be shown to be, U = YlW^.(x) (4.16) 1=1 where serves as the weighting factor for each objective function Fj(x). Since the magnitude of each objective criterion can be quite different, a scaling strategy must be implemented. Here, the following scaled form of the utility function was used, m p U = Y: W~ (417) '=1 f: where F ' are the scaling parameters for each candidate criterion. With this definition, the multicriterion optimal design problem is changed into a single objective optimal design problem, and can be solved in a relatively straightforward manner. In a situation where the multicriterion optimal design is required for a number of different weighting combinations, a genetic search would have to be performed for each of these cases separately. One of the principal contributions of the present research is an exploration of methods wherein single runs of genetic search can be used to simultaneously generate a family of Pareto optimal designs corresponding to different weighting coefficients of individual candidate criterion. Two distinct strategies of achieving this objective are proposed next.
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87 4.4.1 Sharing Function Approach Goldberg and Richardson [87] introduced the concept of sharing functions into genetic algorithms to simultaneously locate relative optima in a multimodal design space. As the name suggests, the approach is based on a concept of shared resources among distinct sets of population. Each such set converges to one relative optimum, and in doing so, maximizes its payoff. The principle of sharing is implemented by degrading the fitness of each design in proportion to the number of designs located in its neighborhood through the use of sharing functions. The extent of sharing is controlled by a sharing parameter o^^^, in terms of which the sharing function is defined as follows: sh ^ 0 4ij < o sh (4.18) otherwise Here djj is a metric indicative of the distance between designs i and j. If two designs i and j are very close, djj is almost zero, and 0(d,j) Â« 1. If d^ > a.^,
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88 where m is the number of designs located in vicinity of the i"" design. Note that the raw fitness value of each design is decreased due to the presence of a large number of designs in its vicinity. If the distance metric d^ is evaluated in the decoded design space, the sharing is called a phenotypic sharing. The distance metric djj can be computed as follows: In order to adapt the sharing principle into an approach where genetic search can simultaneously locate the optima corresponding to different weighting combinations, the weighting variables were included in the set of design variables. As in mathematical programming based multicriterion design, the criterion weights W. are required to meet the following requirement. m E = 1 (4.21) (=1 As an illustration, if 9 different weighting combinations [Wj = 0.1, W, = 0.9], [Wi = 0.2, W2 = 0.8], [Wi = 0.9, W, = 0.1] are considered for a twocriterion optimal design problem, the weighting variable can be defined as follows:
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89 xw = (1,2, , 9) = ([0.1,0.9], [0.2,0.8], ...[0.9,0.1]) This weighting variable is then regarded as one additional design variable taking on values between 1 and 9, with a step size of 1. This variable is of integer type, and representation schemes discussed earlier can be used in a routine manner. For the above twocriterion problem, one weighting design variable, Xy^,, was needed to represent the nine weighting sets. If the number of criteria is greater than two, then two approaches can be taken to represent the weighting variable sets. The first approach introduces m1 integer variables, x^, into the design variable set of an mcriterion problem. Consider the following nine different weighting combinations for a threecriterion problem: [Wi=0.2, \V2 = 0.3, W3 = 0.5], [Wj = 0.2, W, = 0.4, W3 = 0.4], [W,=0.2, W.0.5, W3 = 0.3], [Wi = 0.3, W2 = 0.3, W3 = 0.4], [Wj = 0.3, W, = 0.4, W3 = 0.3], [W, = 0.3, W, = 0.5, W3 = 0.2], [Wj = 0.4, W, = 0.3, W3 = 0.3], [Wi = 0.4, W, = 0.4, W3 = 0.2], [W,=0.4, W2 = 0.5, W3 = 0.1]. Two weighting variables x^i and x^, were used to represent W, and WÂ„ respectively. The value of W3 in each weighting set was then determined by Eqn.
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90 (4.21). The second approach utilizes a single weighting variable which represents the possible combinations of the optimization problem. Weighting combinations are arranged in a series, and the weighting variable components map to each combination of the problem criteria. For the above ninecriterion example, the following variable map was generated: xw = (1, 2, 9) = ([0.2,0.3,0.5], [0.2,0.4,0.4], [0.4,0.5,0.1]) The binary coded weighting variable and binary representations of true design variables are connected headtotail to form a binary string representing a design with a specific weighting combination. In order to generate optimal designs corresponding to all weighting combinations, the sharing principle is applied selectively on the weighting variable. The distance metric d,j between two designs Xj = [x\v,i, Xj j, x, I, Xp j] and Xj = [x^j, x^j, X2j, XpJ is based only on the variable representing weight combinations as follows. With appropriate choices of sharing parameter o^^ and population size, sharing directs the genetic search toward locating optimal designs corresponding to most of weighting combinations. In order both to speed convergence and to impart stability to the genetic search, it was found to be necessary to introduce
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91 restrictions in mating between members belonging to different subsets of the population [88]. A parameter of mating restriction, a^^,, was defined. This confined the mating process among those members for which the metric distance as measured along the weighting variables was within the radius a^^,. Some caution is necessary in this approach as it is highly reminiscent of inbreeding, and sources of diversity in the population must be introduced. It is important to include in future work, mating restrictions other than those based entirely on the weighting coefficient combinations. Since the population of designs in each specific weighting group is determined on the basis of the penalized fitness of the entire population, a better average fitness for a specific weighting group will eventually result in that group attracting more members in successive generations. The fitness or the overall utility function has to be carefully scaled so as to distribute designs evenly over each weighting combination, and for each weighting group to possess roughly similar exploitative abilities in the genetic search. 4.4.2 VectorEvaluated Approach This approach has its parallel in a method proposed by Schaffer [89], referred to as the vector evaluated genetic algorithm (VEGA) approach, and intended for multiobjective optimization. Given the fuced population size as P and the number of optimization criteria as N, each member of the population was evaluated for its fitness as it relates to each of the N criteria. For each of the N criteria, subpopulations of size P/N were extracted from the pool on the basis of
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92 the fitness function for the criterion under consideration. This sorting was done in a rotational sequence, where one member was selected for each criterion, followed by a second round of selection. The selection was performed using the traditional genetic selection (reproduction) method, which ensured the preservation of individual designs which performed above average on any criterion. Subpopulations were then shuffled and grouped into a population which was subjected to the usual genetic transfer processes. The method proposed in the present research for simultaneously generating a family of Pareto optimal designs draws upon such a vector evaluated approach, and is summarized as follows. A utility function is first defined to obtain a scalar measure of fitness for a given set of weighting parameters W,"" as m U, = E F, (4.23) where the superscript k denotes the kth weight combination. Each set of weight combination for which an optimal design is needed constitutes a component in the Nvector of utility functions. The entire population of P designs was evaluated for each of the weight combinations. A subpopulation of size P/N for each weighting combination was formed as described above. Two distinct strategies were adopted in this context. In the first approach, designs were selected from the pool as those that were deemed most fit for that weight combination. Selection of a design for one subpopulation did not exclude its subsequent selection in another
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93 subpopulation. In the second approach, an exclusionary principle was applied wherein a design could contribute its genetic makeup to only one subpopulation and was not considered for mating in another subpopulation. Both of these processes result in N subpopulations, each containing P/N designs with superior measures of fitness corresponding to a specific weight combination. For both methods, in general, this step simply represents a regrouping with no genetic transformations invoked on the designs. In particular, in the exclusionary scheme, the entire genetic pool is preserved and simply divided up in the most opportunistic manner to contribute towards evolution of designs for specific requirements. After the selection of each subpopulation is completed, genetic transformations such as reproduction, crossover and mutation are invoked separately in each subpopulation. The process of population classification is repeated for the next generation in the same manner. In the discussion of sample results, these two strategies are denoted as VEj^, and VE^^Â„ respectively. Due to the possible small size of each subpopulation, genetic drift might result, with associated undesirable characteristics such as convergence to a suboptimal design. A possible approach to prevent such a premature convergence is to introduce a more moderate sized population for genetic evolution. The underlying assumption here is that a larger population is likely to be more diverse, and hence offers a higher possibility of locating the true optimum. This can be achieved in a strategy where the crossover transformation is a twostage process. In the first stage, a crossover is performed between a
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94 subset of members within the same subpopulation (akin to inbreeding). The remaining members from each subset forms a larger gene pool where a second stage of crossover is conducted. In this second stage, a mating restriction must be applied on the variable denoting the weighting combination associated with a design, so as to prevent crossovers between designs corresponding to radically different weight combinations.
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CHAPTER 5 NUMERICAL IMPLEMENTATION OF GENETIC ALGORITHMS This chapter describes the numerical implementation of genetic algorithms in single and multicriterion optimization problems. Several singlecriterion optimization problems in structural and mechanical system design are presented in the first part of the chapter. The second section describes a number of multicriterion optimization problems in representative structural systems. These single and multicriterion problems were formulated such that they contained a mix of integer, discrete and continuous design variables. The two variations of genetic search described in the previous chapter were used to simultaneously obtain families of Pareto optimal solutions. 5.1 SingleCriterion Optimization 5.1.1 Riveted Lap Joint The first problem involved the design of a lap joint between two steel plates in which the rivet size and the number and arrangement of the rivet pattern were considered as design variables. The configuration of the plates and the rivets is shown in Figure 5.1. The number of rows parallel to side AB is 95
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96 ^600mmÂ« ^4 ^4 p 4Â— EZZZZ 15mm 15mm Figure 5.1 Geometry of riveted lap joint.
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97 represented by an integer variable Xj with permissible values between 1 and 32. The number of the rivets in each row x, was an integer variable and was allowed to assume values between 1 and 128. The diameter X3 of all rivets was assumed to be the same, and is chosen from a conmiercially available set: X3 = [6,8,10,12,14,16,18,20,22,24,27,30,33,36,40,45] (mm) This variable is therefore of the discrete type. The objective of this optimization was to maximize the efficiency of the joint, defined as the ratio of the strength of joint to the strength of the plate. The strength of the joint is obtained as the minimum of the shearing failure strength P,, tension failure strength PÂ„ and compression or bearing failure P^. These can be formulated as follows [90,91]: 2 T TTX
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98 where. T = 80 MPa a, = 90 MPa = 120 MPa Stress concentrations due to the close placement of any two rivets were avoided by the imposition of the following linear constraints [90]: SxjXj + < 500 (5.4) SxjX, + < 2000 Further, the strength of the plate was specified as 2700 kN in the present analysis. For the genetic search, population sizes of 30 and 60 were considered, with two distinct penalty function formulations. The total number of allowed values for design variables Xj, x,, and X3 were 32 (2^), 128 (2''), and 16 (2^), respectively. Design variables Xj, x,, and X3 were then represented as binary strings of 5, 7 and 4, digits, respectively. Results of the genetic search simulation using a maximum bound on the constraint penalty are shown in Figures 5.2 and 5.3; they are marked with an asterisk to distinguish them from those obtained with an unbounded penalty formulation. A population size of 30, when used with a bounded constraint penalty, performed most favorably. An optimum design of Xj* = 5, x,' = 13, and X3' = 27mm was obtained at the 58th generation, and resulted in a joint efficiency of 82.45%. The average efficiency of the joint for both population sizes, and using no bounds
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99 Figure 5.2 Iteration history of lap joint efficiencyonline and offline measures. (* indicates use of bounded penalty).
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100 1 0.9 0 8 0 7 30 60 30* 60* TRIALS (xlOOO) re 5.3 Iteration history of lap joint efficiencyaverage value. (* indicates use bounded penalty).
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101 on the constraint penalty, actually decreased over generations of evolution. This underscores the need for a careful selection of penalty functions in the presence of constraints. 5.1.2 Series Stacked Belleville Spring A second example deals with the sizing and selection of the number of the belleville springs to be stacked in series to maximize the load carrying capacity with a minimum attendant deformation. A Belleville spring is shown in Figure 5.4, and the configuration for a seriesstacked Belleville spring is shown in Figure 5.5. The compressive stress at the convex innerdiameter corner would be the critical consideration in such a statically loaded situation. The load deflection relation is obtained as follows [92,93], where d is the deflection under load P. The expression for maximum stress is obtained as, P = [ihf){h0.5f)t . P] (5.5) (lv)Mfl[C,(h 0.5/) . C,r] (5.6) (1 v)Mawhere.
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Figure 5.5 Series stacked Belleville spring.
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103 C, = M 6 R 1 nlnR 6 \R 1 nlnR 2 6 {R 1)2 nlnR (5.7) (5.8) (5.9) a = 0.5 O.D. R = O.D./I.D. f = vertical deflection of the spring H = h + t cos q, the height of a single spring N = number of springs in stack Hj = N*H, total height of the stacked springs In the above equations, a value of Poisson's ratio of v =0.3 and Young's modulus E = 2.06 MN/mmwere used. It is important to note that the loaddeflection relations are only valid for d < 0.85h. Consequently, the objective of the optimization is to find the maximum load at d = 0.8h, and subject to other constraints. When an h/t ratio greater than 1.3 is used in a stack, the load deflection relation tends to be erratic, as some of the springs have a tendency to snap through the flat position. To avoid this problem, a constraint of h/t < 1.3
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104 was imposed. The maximum allowable working stress was set to 1400 MPA and a restriction on R to be at least 1.5 was also included. The design variables Xj, x,, X5 are the thickness t, outer diameter O.D., inner diameter I.D., number of springs N, and disk cone angle 6, respectively. The first three were treated as discrete variables selected from the following sets. 0. D. s [200, 201, 202,...263] (mm) 1. D. s [112, 113, 114,...175] (mm) t = [4.8, 5.5, 6.05, 6.5, 7.1, 7.49, 9.4, 11.25] (mm) The number of springs was an integer variable ranging from ] to 16; the cone angle 0 was a continuous variable bounded between 1 deg and 20 deg. It was represented to an accuracy of 0.1 in the binary scheme. Population sizes of 50 and 100 were considered in the numerical simulation, with constraints treated both by a general penalty and a bounded penalty formulation. Binary strings of lengths 3, 6, 6, 4, and 10 were chosen for design variables Xj through x^. The choices of binaiy strings of lengths 3, 6, 6, and 4 for x^, x^, X3, and x, resulted in onetoone mapping between the actual design variable space and its corresponding binary representation space. The choice of a binary string of length 10 resulted in an even distribution where 1024 binary representations were assigned to 200 candidate Xj values. This meant that 5 representations were almost uniformly distributed to each design value. The results obtained from the simulation are
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105 shown in Figures 5.6 and 5.7, where, as before, results corresponding to the bounded penalty formulation are denoted with an asterisk. This bounded penalty formulation yielded the most stable genetic search when working with a population size of 50. At the 317th generation, an optimal design X* = [9.4 (mm), 232 (mm), 130 (mm), 7, 8.1 (deg)] was obtained, corresponding to a maximum load of 142,211 lbs. 5.1.3 Eleven Bar Planar Truss A minimum weight truss design with constraints on displacements comprised the third test problem. The structure and the loading are shown in Figure 5.8. The crosssectional areas of each of the eleven members of the truss are discrete variables chosen from the following available sections: A' = [0.111, 0.141, 0.196, 0.250, 0.307, 0.391, 0.442, 0.563, 0.602, 0.766, 0.785, 0.994, 1.000, 1.228, 1.266, 1.563] (in). Material properties of steel with specific weight p= 0.283 Ib/in^ and Young's modulus E= 30x10^ psi were prescribed for this application. Binary string length of four was used to represent the design variables; this results in exactly one binary string for each variable. The displacement constraints were defined as follows:
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106 Figure 5.6 Iteration history of maximum spring Ioadingonline and offline measures (* indicates use of bounded penalty).
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107 z o o o < o 150 140 130 120 110 100 CO 1j 1'*' J W *' Â•*^" Â• v::: : : :l: i; : Â• '\' : ^' i V::! * v'^ : :Â• . '* ",' \'' ^wVlr.''' / Â•''<Â•' \ , 60* 100* Â• 50 8 10 12 TRIALS (xlOOO) 14 16 18 20 jure 5.7 Iteration history of maximum spring loadingaverage value (* indicates use of bounded penalty).
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108 4000 i 8000 u3 u1 u2 5000 7000 Y Figure 5.8 Eleven bar planar truss.
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109 Ui < 0.5 in. u, < 0.5 in. (5.10) U3 < 0.2 in. The traditional penalty function formulation was used with population sizes of 60 and 120; for the bounded penalty function approach, sizes of 50 and 100 were considered. These results are summarized in Figures 5.9 and 5.10, respectively. As in previous problems, the bounded penalty formulation yielded better performance. An optimal weight of 82.61 lbs was obtained with the corresponding design variable vector as follows: X* = [0.442, 0.602, 0.307, 0.442, 0.602, 0.111, 0.111, 0.602, 0.602, 0.442] (in") 5.1.4 Laminated Composite Plate Desig n A twentyply fiberreinforced laminated composite plate subjected to a central concentrated load was chosen as the fourth test problem. The optimum stacking sequence of the twenty unidirectional laminae was determined to minimize the central displacement. A simply supported laminated composite plate under a concentrated force is shown in Figure 5.11. Two laminated composite plates, one square and the other rectangular, were considered in this problem. Each side of the square plate was 0.5m. The rectangular plate was 0.6m by 0.3m. The 0.6m side of the rectangular plate was in the xdirection.
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110 Figure 5.9 Iteration history of structural weightonline and offline measures (* indicates use of bounded penalty).
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Ill Figure 5.10 Iteration history of structural weightaverage value (* indicates use bounded penalty).
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112 Figure 5.11 Simply supported laminated plate under concentrated load.
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113 which defined the zero degree direction of the fiber orientation. A vertical concentrated force of lOON was apphed at the center of both plates. The laminated plate was assumed to be symmetric with respect to its middle plane. This impHed that only plies above the plane of symmetry needed to be considered in the design variable set. These ten plies were further assumed to consist of two identical stacks. Hence, the stacking sequences of laminae in the plate could be defined by the fiber orientations of only the top five plies. One design variable was assigned to each of these five plies for the representation of the fiberorientation. The ply stacking sequence of the laminated plate was expressed as [xj, x,, X3, x^, Xj],/^. A layout of the ply stacking sequence of the laminated plate is shown in Figure 5.12. The design variables Xj through x^ were assumed to be discrete types, and each was selected exclusively from the following set: [90, 85, 80, 75, 0, 5, 10, 85] (deg.) A Fourier series solution was obtained to the midpoint displacement of the simply supported laminated plate under a concentrated load. A higher order plate theory, which included the effects of transverse shear deformation and rotatory inertia [94] was used in this analysis. Results were obtained for graphiteepoxy laminated plates (E, = 111. 9 GN/m", E, = 10.27 GN/m", G^, = 7.31 GN/m, V = 0.24, t = 0.000127 m). For the genetic search, a population size of 50 was considered. Each design variable, Xj through X5, was represented as a
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114 M.P Figure 5.12 Ply stacking sequence of laminated plate.
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115 binary string of 7 digits. Four genetic searches were initially performed on the square plate with different combinations of the probability of crossover P^, and the probability of mutation P^. A population size of 50 was selected and the genetic search was terminated after 100 generations. The iteration history of average central displacement for four combinations of P, and PÂ„, is shown in Figures 5.13. The iteration history of the smallest central displacement for corresponding value of P^ and P,Â„ is shown in Figure 5.14. The displacement shown in Figures 5.13 and 5.14 was nondimensionalized. This was obtained by dividing the true displacement by the displacement (0.00392m) of the laminated plate with the ply stacking sequence as [90, 54, 18, 18, 54],/,. The final designs and their displacements for these four cases are shown in Table 5.1. Table 5.1 Final designs and displacements. Genetic search parameters Plyorienlations Nondimensionalized midpoint displacement Pc = 0.6, PÂ„ = 0.01 P, = 0.6, PÂ„ = 0.001 P,=0.8, PÂ„, = 0.01 Pe=0.8, P,Â„ = 0.001 [45, 45,45, 45,40],/, [ 40, 50, 45, 45, 45],/, [45, 45,45, 45, 45],,, [45, 45, 45, 45, 45],;, 0.8101 0.8090 0.8073 0.8073 Another four genetic searches were performed on the rectangular plate (aspect ratio = 2) with the same combinations of the probability of crossover P^, and the probability of mutation PÂ„,. A population size of 50 was selected and the genetic search was terminated after 100 generations. The iteration history of average central displacement for four combinations of P, and P,Â„ is shown in
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116 LU IE LU O < CL Q GENERATION Figure 5.13 Iteration history of average displacement for the square plate.
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117 0.88 0.87 0.86 0.85 0 .84 0.83 0 .62 0.81 0.8 Pc0.6 PmÂ«0.01 Pc0.6 Pm0.001 Pc0.8 Pm0.01 PcÂ«:0.8 Pm = 0.001 GENERATION igure 5.14 Iteration history of smallest displacement for the square plate.
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118 Figures 5.15. Similar results for the minimum central displacement are shown in Figure 5.16. As before, the displacement shown in Figures 5.15 and 5.16 has been nondimensionalized. They were obtained by dividing the true displacement by the displacement (0.00183m) of a laminated plate with the ply stacking sequence as [90, 54, 18, 18, 54]2/3. The final designs which had the same values of 0.8808 for displacement in these four cases are shown in Table 5.2. Table 5.2 Final designs and displacements. Genetic search parameters Plyorientations Nondimensionalized midpoint displacement P, = 0.6, PÂ„ = 0.01 P,=0.6, PÂ„ = 0.001 P,=0.8, PÂ„, = 0.01 P, = 0.8, PÂ„, = 0.001 [ 60, 60, 65, 60, 60]2/, [60, 60, 60, 65, 60],/, [ 60, 60, 65, 60, 60]j/, [60, 60, 60, 65, 65]y, 0.8808 0.8808 0.8808 0.8808 From the trend of the numerical solution obtained in these two cases (aspect ratios equal to 1 and 2), the optimal ply orientation would be greater than 60 degrees for laminated plates of an aspect ratio greater than 2. As an extreme case, the optimum fiber orientation would be 90 degrees for all laminae as the plate's aspect ratio approaches infinity. In both these examples, it was observed that genetic searches with a higher crossover probability, P, = 0.8, located the best designs quicker than the search with a lower crossover probability of P^ = 0.6. This was attributed to the larger number of new designs which were created as a result of the higher probability of crossover. Genetic searches with the lower mutation probability, PÂ„, = 0.001, produced better average fitnesses than simulations with a
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119 1.1 ^ 1.08 1.06 1.04 1.02 1 1 A 0 .98 1 1 1 0.96 0 .94 0 .92 0.9 0 88 T Pc0.6 PmÂ»0.01 PcÂ»0.6 Pm0.001 PcÂ«0.8 PmÂ«0.01 Pc0.8 Pm0.001 ~r20 Â— r40 Â— rÂ— 60 80 100 GENERATION Figure 5.15 Iteration history of average displacement for the rectangular plate.
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I 120 0.893 0.892 0.891 0 .89 0.889 0.888 I I I 0.887 o 0.886 0.885 0.884 0.883 0 882 0.881 0.88 Pc0.6 Pm = 0.01 Pc0.6 Pm0.001 Pc0.8 Pm0.01 Pc = 0.8 Pm = 0.001 \ _ _ Â— 1Â— 20 40 60 GENERATION ;ure 5.16 Iteration history of smallest displacement for the rectangular plate.
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121 larger value of Pn, = 0.01. This was attributed to the lower degree of schema disruption induced by the lower probability of mutation. The real advantage to the use of genetic search methods to solve the ply stacking sequence problem becomes more evident when the approach is compared with proposed mathematical programming solutions to the problem. Haftka and Walsh [95] proposed a standout strategy for a stackingsequence design for both maximum buckling load for given total thickness, and minimum total thickness subject to a buckling constraint. The laminate was assumed to be composed exclusively of 0deg, 90deg, and Â±45deg plies. In order to define the orientation of each ply, four plyorientationidentity variables, x^, x^, X45, and x^5 were created. The variable, Xq, x^q, X45, or x^5 is equal to one if there is a 0deg, 90deg, 45deg, or 45deg ply, respectively. An additional constraint which assures a unique orientation for each ply is given as follows: + X90 + X45 + X_,5 < 1 A linear integerprogramming formulation of this problem was solved using a branchandbound algorithm. It was noted that a large number of subproblems would be created with the use of such an algorithm. The number of such subproblems often grows exponentially with the number of the integer/discrete variables. The computational expense would be extremely high if a large number of possible fiber orientations were allowed.
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122 However, in the genetic search method, only one discrete variable is needed to represent the fiber orientation of each ply regardless of the total number of allowed fiber orientations. Furthermore, the discrete variables were handled without any more extra cost than continuous variables in this approach. 5.2 Multicriterion Optimization The multicriterion optimization strategies described in preceding chapter were implemented for a class of representative structural design problems. All analysis was performed using the finiteelementbased structural analysis program ANSYS. The analysis was linked to the genetic search procedure via preand postprocessors. 5.2.1 Statically Loaded Ten Bar Truss A statically loaded ten bar truss shown in Figure 5.17, comprised the first test problem. A load P = 100 kips was applied at nodes 3 and 5 in the direction of displacements d, and dg. A truss bar length L = 360 in. was specified. As depicted in Table 5.3, three variations of this problem were constructed, with different dimensionality as represented by the number of design variables and constraints. The table also shows the mix of continuous and discrete design variables in each problem. The sizing of this truss was done with the dual goal of minimizing both the structural weight and the vertical displacement d^. In addition, displacement constraints were also imposed as specified in Table 5.3.
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123 100K to lOOKIb Figure 5.17 Ten bar truss structure. Element numbers are indicated by 0.
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124 Table 5.3 Description of the statically loaded ten bar truss problem. TEN BAR TRUSS (STATIC LOAD) CASE 1 Design Variable Element Type Constraint XI 1,3 Continuous < 3.0 in. X2 7,8,9,10 Discrete X3 2,4,5,6 Discrete CASE 2 Design Variable Element Type Constraint XI 1,3 Continuous do < 3.0 in. X2 2,4 Discrete dj < 1.5 in. X3 5,6 Discrete X4 X5 7,8 9,10 Discrete Discrete CASE 3 Design Variable Element Type Constraint XI 1 Continuous d(, < 3.0 in. X2 2 Discrete df, < 1.5 in. X3 3 Continuous dj < 0.8 in. X4X10 410 Discrete ds < 0.8 in. Elements of discrete set 0J5 0.50, 0.85, 1.25, 1.90, 2.20, 2.45, 2.60, 3.50, 3.90, 4.60, 4.90, 5.25, 5.75, 6.00 6 55 7 40 7.70, 8.10, 8.60, 9.00, 9.45, 9.80, 10.25, 10.50, 11.15, 11.65, 12.45, 13.1o,13.70, 14 00 M 45 14.8M4.90, 15.35, 15.50, 16.50, 17.50,18.75, 20.00 (in^)
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125 Different weights were assigned to each of these goals. In order to reduce the differences between the magnitudes of individual criteria, a scaling on each individual criterion was necessary. The scaled, overall fitness function for this problem was defined as follows, where the scaling constants were determined by a numerical assessment of the problem domain. F = F max ^3298.9 h.313 * (5.11) The term may be thought of as a penalty term introduced to account for the problem constraints, much in the manner of an exterior penalty function. is the weighting factor for the first criterion, the weight of the truss, and ranges from 0.0 through 1.0 with a step size 0.1. Similar variations were allowed for WÂ„ the weight parameter associated with displacement d^. As defined in Table 5.3, a population size of 100 was maintained at each generation, and the genetic evolution stopped after 40 generations. Crossover and mutation probabilities of 0.8 and 0.01, respectively, were used in this exercise. Results were obtained for each of the defined test cases and using both the sharing function and the vectorevaluated fitness strategies. Iteration histories showing the best fitness values corresponding to a given weighting combination for cases 1, 2, and 3 of Table 5.3, at five different generations of evolution are shown in Figures 5.18ac, 5.19ac, and 5.20ac, respectively. Three, five and ten design variables were specified for cases 13. A
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126 Fitness 1.1 0.9 0.8 0.7 0 .6 i 0.5 0.4 0.3 Staticcase 1 1 St generation + 10th generation O 20th generation A 30tti generation X 40th generation "1 I I \ 1 1 1 1 r 0.2 0.4 0.6 0.8 gure 5.18a Best fitness values obtained using the sharing function approach.
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129
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131 0 .6 , 0.5 0.4 0.3 ?ure 5.19c Best fitness values obtained using the VE^^, approach.
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135 comparison with results generated through conventional mathematical programming methods is also shown on Figures 5.19ac. The optimal designs for ten different weighting coefficient combinations using the sharing function, the VEiÂ„Â„ and the VE^^^ approaches for cases 1, 2, and 3, are summarized in Tables 5.4ac through 5.6ac. In general there was good agreement among the methods for cases 1 and 2 corresponding to 3 and 5 design variables; such agreement was not observed for the 10 design variable problem. Based on our earlier discussion on defining lengths of^ strings, it is not difficult to account for this trend. The string lengths in the 10 design variable case are considerably longer, allowing for greater possibilities of schema disruption, and hence also the need to carry out the iteration for a larger number of generations. Approaches where such longer strings may be decomposed into smaller defining length strings should be explored as possible remedies to the schema disruption problem. A threecriterion optimization problem was implemented on the statically loaded truss with design variables and constraints defined in case 2, as shown in Table 5.3. The scaled, overall fitness function for this problem was a variation of Eqn. (5.11) and is defined as follows, F = F weight (5.12) max 3298.9 1.313 10,000 where the new term, a^. represents the stress in the element 6 of the truss. One weighting variable which utilized a 3bit binary string was used to form a oneto
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136 Table 5.4a Optimal designs for the statically loaded ten bar truss (case 1), using sharing function approach. STATICCASE 1 [sharing] Weighting factors Design variables Objective Weight (ib) Displacement (in) Wl W2 XI X2 X3 0.0 1.0 19.53 18.75 15.35 0.529 5844.7 0.695 0.1 0.9 19.84 20.00 10.25 0.644 5485.0 0.697 0.2 0.8 19.84 18.75 11.65 0.760 5443.4 0.705 0.3 0.7 19.40 15.50 6.55 0.832 4320.9 0.824 0.4 0.6 15.92 11.65 5.25 0.885 3360.4 1.045 0.5 0.5 18.21 9.00 4.60 0.887 2992.2 1.139 0.6 0.4 14.26 6.00 3.50 0.876 2163.9 1.589 0.7 0.3 13.83 6.55 4.60 0.837 2352.1 1.478 0.8 0.2 9.95 4.60 1.90 0.711 1514.7 2.259 0.9 0.1 6.20 5.25 1.25 0.575 1332.9 2.772 1.0 0.0 8.96 5.75 2.60 0.522 1722.0 2.003
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137 Table 5.4b Optimal designs for the statically loaded ten bar truss (case 1), using the VEjn^ approach. STATICCASE 1 [VE J Weighting factors Design variables Objective Weight (lb) Displacement (in) Wl W2 XI X2 X3 0.0 1.0 19.95 20.00 20.00 0.501 6595.0 0.657 0.1 0.9 19.96 20.00 12.45 0.641 5740.9 0.682 0.2 0.8 19.95 18,75 8.60 0.753 5104.4 0.727 0.3 0.7 20.00 14.45 5.75 0.828 4096.2 0.854 0.4 0.6 19.84 11.15 5.25 0.870 3502.2 0.975 0.5 0.5 17.49 9.00 5.25 0.890 3025.0 1.134 0.6 0.4 14.79 8.10 3.50 0.871 2530.0 1.348 0.7 0.3 11.72 6.00 2.60 0.813 1918.2 1.777 0.8 0.2 8.69 4.60 1.90 0.710 1443.3 2.364 0.9 0.1 5.97 4.60 1.25 0.557 1215.8 2.956 1.0 0.0 5.93 4.60 1.25 0.368 1213.6 2.964
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138 Table 5.4c Optimal designs for the statically loaded ten bar truss (case 1), using the VE^j.^. approach. STATICCASE 1 [VE,J Weighting factors Design variables Objective Weight (lb) Displacement (in) Wl W2 XI X2 X3 0.0 1.0 19.91 20.00 20.00 0.501 6592.7 0.658 0.1 0.9 19.93 20.00 13.70 0.642 5880.7 0.676 0.2 0.8 19.89 18.75 9.45 0.754 5197.2 0.720 0.3 0.7 19.95 14.00 6.55 0.827 4111.9 0.849 0.4 0.6 19.95 11.65 5.25 0.871 3588.5 0.953 0.5 0.5 17.86 9.00 3.90 0.887 2893.2 1.178 0.6 0.4 15.16 7.70 3.50 0.869 2486.9 1.370 0.7 0.3 11.88 5.75 2.60 0.813 1887.2 1.806 0.8 0.2 10.05 4.60 2.20 0.711 1554.3 2.195 0.9 0.1 6.70 3.50 2.20 0.545 1188.6 2.905 1.0 0.0 5.51 3.90 2.20 0.359 1185.3 2.986
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139 Table 5.5a Optimal designs for the statically loaded ten bar truss (case 2), using the sharing function approach. STATICCASE 2 [sharing] Weighting factors Design variables Objective Weight (lb) Displacement (in) Wl W2 XI X2 X3 X4 X5 0.0 1.0 18.SS 14.85 6.55 17.50 14.00 0.584 4801.2 0.768 0.1 0.9 18.42 10.25 4.60 18.75 17.50 0.670 4784.7 0.766 0.2 0.8 19.49 9.45 4.60 13.10 20.0 0.772 4547.8 0.814 0.3 0.7 19.44 8.10 5.25 18.75 9.45 0.825 4113.2 0.847 0.4 0.6 18.81 5.75 2.20 13.10 9.00 0.862 3285.6 1.014 0.5 0.5 19.19 5.25 4.90 10.50 6.55 0.885 3025.4 1.119 0.6 0.4 15.24 5.25 4.90 10.50 5.75 0.880 2737.8 1.254 0.7 0.3 15.08 5.75 2.45 5.25 7.70 0.847 2354.2 1.523 0.8 0.2 13.68 4.90 4.60 6.00 6.00 0.787 2272.5 1.548 0.9 0.1 13.91 1.90 0.50 5.75 4.60 0.635 1751.6 2.066 1.0 0.0 4.50 5.25 0.85 5.75 6.55 0.480 1584.5 2.707
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140 Table 5.5b Optimal designs for the statically loaded ten bar truss (case 2), using the VEjnj approach. STATICCASE 2 [VE,J Weighting factors Design variables Objective Weight (lb) Displacement (in) Wl W2 XI X2 X3 X4 X5 0.0 1.0 19.96 20.00 20.00 20.00 20.00 0.501 6595.5 0.657 0.1 0.9 19.96 16.50 10.50 20.00 20.00 0.638 5859.7 0.672 0.2 0.8 19.93 11.65 4.60 20.00 13.70 0.743 4745.3 0.747 0.3 0.7 19.67 9.00 3.50 15.35 11.65 0.816 3982.0 0.851 0.4 0.6 19.67 9.00 2.60 14.00 9.00 0.863 3610.9 0.931 0.5 0.5 18.78 5.25 2.20 9.80 8.10 0.874 2917.4 1.135 0.6 0.4 14.46 4.60 1.90 9.00 6.00 0.856 2387.0 1.386 0.7 0.3 11.04 4.60 1.90 9.00 5.25 0.815 2133.4 1.584 0.8 0.2 9.16 1.90 0.85 4.60 4.60 0.708 1410.5 2.405 0.9 0.1 6.36 1.90 0.85 4.60 4.60 0.551 1252.0 2.750 1.0 0.0 6.43 1.90 0.15 4.60 4.60 0.369 1216.4 2.990
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141 5.5c Optimal designs for the statically loaded ten bar truss (case 2), using the VEg^^ approach. STATICCASE 2 fVE,J Wei fac ghting :lors Design variables Objective Weight (lb) Displacement (in) Wl W2 XI X2 X3 X4 X5 0.0 1.0 19.89 14.85 15.35 20.00 20.00 0.512 6036.9 0.672 0.1 0.9 19.89 14.85 9.00 20.00 20.00 0.638 5677.5 0.680 0.2 0.8 19.89 9.45 3.90 20.00 13.70 0.743 4584.0 0.764 0.3 0.7 19.63 9.45 3.50 15.50 12.45 0.817 4081.2 0.836 0.4 0.6 19.75 6.00 2.60 13.10 8.60 0.859 3341.6 0.992 0.5 0.5 17.46 5.25 1.90 10.25 8.10 0.874 2861.7 1.157 0.6 0.4 14.77 4.60 1.90 8.60 6.55 0.856 2416.6 1.367 0.7 0.3 12.30 3.90 0.85 6.00 5.75 0.807 1905.6 1.763 0.8 0.2 9.27 2.45 1.25 5.75 3.90 0.700 1506.5 2.200 0.9 0.1 7.12 1.90 0.85 3.90 3.50 0.534 1151.0 2.889 1.0 0.0 7.17 2.45 2.20 3.90 2.45 0.357 1177.3 2.920
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142 Table 5.6a Optimal designs for the statically loaded ten bar truss (case 3), using the sharing function approach. STATICCASE 3 (shanDgj Wc fa ghcing :lors Design vanables Obj. Weighl (lb) Disp. (m) VVI W2 Xl .\2 \4 .\f X6 X7 XS .X9 XIO 0.0 1.0 18.93 14.45 18.42 20.00 11.15 11.55 15J0 nso 14.90 5.75 0.606 4824.4 0.796 0.1 0.^ 1931 9.00 16.68 20.00 11.65 4.90 12.45 18.75 14.90 9.00 0.700 4512.8 0.821 0.2 0.8 1736 5.75 18.71 14.90 7.70 2.45 13.10 18.75 11.15 4.90 0.785 38093 0.909 OJ 0.7 17.13 2.tO 19.75 9.00 11.65 3.90 14.85 20.00 12.45 11.15 0.846 4151J 0.879 0.4 O.O 18.N 2.45 18.81 6.00 4.90 3.90 12.45 11.15 1630 12.45 0.881 3641J 0.961 OS OS 17.44 7.40 14.16 14.85 860 4.60 12.45 11.15 8.60 4.90 0.922 3382.3 1.076 0,0 OA W.Â« 3.90 12.07 13.70 0.15 6,55 13.10 8.60 11.15 4.60 0.890 3083.6 1.082 0.7 03 17.20 2.60 18.81 6.00 11.65 030 4.60 15 J5 9.00 oso 0.888 2785.0 1300 0.8 0.2 5.42 5.75 16.83 1.90 5.25 3S0 10.25 9.80 3.90 4.60 0.821 2236.4 1.827 0.9 0.1 5,29 5.25 13.25 3.50 4.60 1.90 5.75 6.00 8.10 3.90 0.674 1906.8 2.018 1.0 0.0 5.25 4.60 5.47 7.70 6S5 0.50 5.75 6.00 8.10 3.90 03)6 18013 2.408
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143 Table 5.6b Optimal designs for the statically loaded ten bar truss (case 3), using the VEj,,^ approach. STATICCASE 3 VEÂ„^1 Wc fa ghling Â•tors Design vanables Obj. Weighl (lb) Disp. (in) Wl W2 Xl X2 .\3 X4 X5 X6 X7 X8 X9 XIO 0.0 1.0 20.00 20.10 19.97 0.10 8.10 20.00 20.00 20.00 2.20 20.00 0.659 4987.4 0.866 O.I 0.9 20.00 20.10 19.97 0.10 8.10 20.00 20.00 20.00 2.20 20.00 0.744 4987.4 0.866 0.2 O.S 19.W 19.10 19.97 0.10 810 20.00 20.00 11.05 1.25 20.00 0.826 4582.4 0.900 0.1 0.7 19.84 14.10 19.97 0.10 5.75 13.70 20.00 9.45 0.15 20.00 0.889 406J.0 0.975 O.l 0,6 14.00 10.10 19.90 0.10 1.25 9.80 20.00 2.45 030 1630 0.922 3139.6 1.185 as OS 12J7 10.10 19.98 0.10 0.85 9.80 18.75 2.20 030 14.45 0.922 2948.0 1,247 0.6 0.4 10.00 8.10 HSS 0.10 0.85 8.10 14.45 1.90 0.15 11.65 0.901 2448.2 1.496 0.7 03 6.89 8.10 15.88 0.10 0.15 8.10 14.00 0.85 0.15 9.45 0.S40 20883 1.738 O.S 0.2 J. 41 3.10 15.18 0.10 0.15 3.90 10.25 030 0.15 9.45 0.74f. 1602.3 2J58 0.9 0,1 5.26 3.10 15.03 0.10 0.85 330 655 030 0.15 9.00 0.602 1436.2 2.758 1.0 0.0 6.40 3.10 15,03 0.10 0.85 3S0 4.60 1.25 0.15 9.00 0.431 14203 2.984
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144 Table 5.4c Optimal designs for the statically loaded tenbar truss (case 3), using the VEgj.^ approach. STATICCASE 3 [VE^,^ Wei fac ghlmg lors Design vanables Obj. (ID) Disp. (in) Wl W2 XI X2 X3 X4 X5 X6 X7 X8 X9 XIO 0.0 1.0 ISI.W 20.10 19.08 0.10 15J5 20.00 20.00 20.00 9.00 20.00 0.652 5427.9 0.857 0,1 0. 19.10 19.24 0.10 12.45 18.75 20.00 14.00 0.15 20.00 0.830 4692.4 0.896 0.3 0.7 ljJU 15.10 18.74 0.10 8.60 15J5 18.75 7.70 0.15 20.00 0.898 4056.3 o.^Â»;.' 0.4 0.6 IbAb 10.10 18.73 0.10 6^5 11.65 18.75 4.60 0.15 163 0.937 34003 1.149 OJ O.S 9.22 10.10 19.98 0.10 0.15 9.00 18.75 0.85 0.15 12.45 0.921 2Â«2.7 1J5S o.c 0.4 9.21 10.10 18.73 0.10 050 9.00 15 J5 0.50 0.15 12.45 0.897 2486.8 1.460 0.7 0.3 5J8 8.10 17^ 0.10 050 655 1535 0.85 0.15 9.45 0.846 2111.6 1.742 0.8 0.2 6.7] 0.1 0 10^5 0.10 2.20 6.00 9.00 1.90 0.15 9.45 0.751 1 71 6.4 2.220 0.9 0.1 2.S4 5.10 14.58 0.10 0.8S 7.70 9.80 030 0.15 4.60 0.611 14S7J 2.r*2 1.0 0.0 2.6S 6.10 9J9 0.10 2.45 2.60 14.00 030 0.15 5.75 0.44'< 1482.1 2.880
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145 one mapping between binary representations and the eight weighting combinations. The population size at each generation was 50, and the genetic search was allowed to proceed through 40 generations. The iteration histories of the best fitness value for eight different combinations of weighting factors are shown in Figure 5.21. An interesting feature observed in the use of the sharing function approach was the lack of a steady improvement in the fitness with increasing iterations. This is largely attributed to the fact that the binary bits representing the weighting variable were altered during the crossover process, resulting in an undesirable vectorelement switch and a resultant degradation in fitness value. An approach where the weighting combination variable is protected from the crossover operation could be a remedy to this problem. Both options in vectorevaluated approach performed better than the sharing approach, with the VE,Â„^ performing slightly better than VE,,^. Furthermore, the only degradation encountered with mcreasing problem dimensionality is the increase in probability of schema disruption. Definition of shorter defining length strings is expected to reduce such degradation. 52.2 Dvnamicallv Loaded Ten Bar Truss This problem represents a variant of the statically loaded ten bar truss, where a harmonically varying load f = 20,000Sin45.2t N was applied in the d, direction at node 5 as shown in Figure 5.22. A truss bar length L = 100 in. was
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146 W1 W2 W3 Figure 5.21 Best fitness values obtained using the sharing function approach in the 3criterion problem.
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147 20000 sin45 . 2 lb Figure 5.22 Dynamically loaded ten bar truss structure. Element numbers are indicated by 0.
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148 chosen. A minimum weight design with minimum steady state vertical displacement d^ was sought, with constraints on dynamic displacements and member stresses. The design space for this problem is nonconvex [42]; however, it offers no significant difficulty to the GA approach. The overall fitness function for the problem was as follows: F ^ F weight ,^_^^p 0.493 ^' ^ 98.97 (5.13) As before, problems with three different dimensionalities were constructed (Table 5.7). However, only results for the sharing function approach and for cases 13 are presented. Crossover and mutation probabilities of 0.8 and 0.01, respectively, were used in this exercise. W, and W, both range between 0.0 through 1.0, and the condition Wi + W, = l was enforced. The population size at each generation was 100, and the genetic search allowed to proceed through 20 generations. The iteration histories of the best fitness value for different weighting coefficient combinations in cases 13 are shown in Figures 5.23ac, and the final designs of the truss for ten weighting combinations for each case are shown in Table 5.8ac. 5.2.3 WingBox Structure A wingbox structure comprised the final example with 16 design variables and 38 constraints [96]. This problem, as shown in Figure 5.24, has a combination of stress and displacement constraints; the objective of the optimization problem
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149 5.7 Description of the dynamically loaded ten bar truss problem. TEN BAR TRUSS (DYNAMIC LOAD) CASE 1 Design Variable Element Type Constraint XI X2 X3 1,5 6,7,8,9 2,3,4,10 Continuous Discrete Discrete d, < 0.5 in. dg < 0.5 in. CASE 2 Design Variable Element Type Constraint XI X2 X3 X4 X5 1,3 2,4 5,6 7,8 9,10 Continuous Discrete Discrete Discrete Discrete d, < 0.5 in. dj < 0.5 in. d^ < 0.3 in. CASE 3 Design Variable Element Type Constraint XI X2 X3 X4X10 1 2 3 410 Continuous Discrete Continuous Discrete d^ < 0.5 in. ds < 0.5 in. dj < 0.3 in. Axial stresses (elements 110) < 45,000 psi Elements of discrete set 0.130, 0.165, 0.175, 0.190, 0.220, 0.245, 0.260, 0.270, 0.280, 0 300 0 315 0 325 0 340 0 3Sn 0.610, 0.640, 0.670, 0.700, 0.725, 0.765, 0.800, 0.840, 0.880, 0.920, 0.960 1 000 "''"'^j^;"'
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150 (w^1vy,) ^ure 5.23a Optimal designs for the dynamically loaded ten bar truss (case 1), using the sharing function approach.
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151 w, (V\^= 1W^) Figure 5.23b Optimal designs for the dynamically loaded ten bar truss (case 2), using the sharing function approach.
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152 (V\^= 1W^) Figure 5.23c Optimal designs for the dynamically loaded ten bar truss (case 3), using the sharing function approach.
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153 Table 5.8a Optimal designs for the dynamically loaded ten bar truss (case 1), using the sharing function approach. DYNAMICCASE 1 [sharing] Weighting factors Design variables Objective Weight (lb) Displacement (in) Wl W2 XI X2 X3 0.0 ].0 0.370 0.165 0.840 0.441 142.4 0.217 0.1 0.9 0.300 0.175 0.765 0.440 131.6 0.168 0.2 0.8 0.510 0.175 0.800 0.666 147.4 0.227 0.3 0.7 0.309 0.150 0.400 0.690 86.8 0.301 0.4 0.6 0.210 0.165 0.840 0.772 133.4 0.191 0.5 0.5 0.397 0.165 0.460 0.804 101.0 0.290 0.6 0.4 0.206 0.150 0.390 0.784 79.8 0.370 0.7 0.3 0.102 0.165 0.300 0.766 66.1 0.490 0.8 0.2 0.141 0.165 0.300 0.748 68.4 0.481 0.9 0.1 0.154 0.150 0.340 0.735 71.2 0.430 1.0 0.0 0.102 0.165 0.300 0.668 66.1 0.490
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154 Table 5.8b Optimal designs for the dynamically loaded ten bar truss (case 2), using the sharing function approach. DYNAMICCASE 2 [sharing] Wei fac ghling tors Design variables Objective Weight (lb) Displacement (in) Wl W2 XI X2 X3 X4 X5 0.0 1.0 0.556 0.840 0.670 0.540 0.150 0.003 172.2 0.002 0.1 0.9 0.268 0.340 0.920 0.500 0.340 0.168 153.7 0.007 0.2 0.8 0.204 0.270 0.725 0.315 0.340 0.273 120.3 0.019 0.3 0.7 0.1 SO 0.840 0.175 0.245 0.315 0.398 112.5 0.040 0.4 0.6 0.179 0.765 0.375 0.190 0.150 0.448 101.9 0.030 0.5 0.5 0.141 0.150 0.460 0.435 0.315 0.643 102.5 0.123 0.6 0.4 0.141 0.165 0.920 0.520 0.150 0.763 123.0 0.021 0.7 0.3 0.151 0.350 0.190 0.260 0.500 0.775 99.9 0.113 0.8 0.2 0.159 0.350 0.190 0.150 0.500 0.815 91.6 0.183 0.9 0.1 0.567 0.390 0.340 0.220 0.165 1.028 104.2 0.394 1.0 0.0 0.310 0.375 0.375 0.270 0.165 0.958 94.8 0.263
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155 Table 5.8c Optimal designs for the dynamically loaded ten bar truss (case 3), using the sharing function approach. DYNAMICCASE 3 [sharing] fac iors Design variables Obj. Weight (lb) Disp. (in) Wl W7 XI X2 X3 .\4 X5 X6 X"? X8 X9 XIO 0.0 1.0 0.129 0.175 1.000 0.450 1.087 0.15O 0.190 0320 0.960 0.640 0.499 171.4 0.246 0.1 0.9 0J20 0.8Â«0 0.610 03Z5 0.974 0.175 0.175 0.800 0.165 0320 0.668 161.0 0.277 0.2 0.8 0.753 0315 0.435 0.245 1.070 0.150 0315 0.670 0380 0350 0.809 1583 0301 03 0.7 0.749 0.670 0.270 0.700 0J57 0.150 0325 0.435 0360 0.400 0.905 1533 0..309 0.4 0.6 1.020 0390 0.400 0.460 0305 0.150 0350 0.450 0300 0.190 0.907 1283 0319 O'i 0S 0.522 0.420 0.S40 0380 0392 0315 0.270 0.190 0.175 0315 1.079 124.9 0.442 0.6 0.4 0.439 0.485 0340 0.270 0J29 0.150 0340 0315 0.640 0.15O 1.034 1203 0375 0.7 03 0341 0.400 0300 0.245 0.289 0.175 0375 0.245 0.485 0.165 0.975 1003 0.434 O.S 0.2 0.424 0375 0^80 0.460 0313 0.175 0315 0.340 0.420 0350 1.180 1263 0387 0.9 0.1 0.936 0.150 0350 03O0 0391 0.150 0.450 0340 0.670 0.150 1.29.1 KVl3 0.353 1.0 0.0 0374 0315 0.220 0.2*0 1.106 0.190 0300 0.450 0340 0.640 1.438 1423 0.435
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156 100001b z t 50001b t Figure 5.24 Statically loaded wingbox structure.
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157 was to achieve the minimum weight of the wingbox structure and a maximum value of the sum of the first two natural frequencies. Problem parameters used in the numerical study are described in Table 5.9. The overall fitness function was defined as follows: weight ^ 500 0.5xJ2_.0.5x.l50 freq. (5.14) The population size was maintained at 100 and the search was carried out to 30 generations. Sample results for both the sharing function and vectorevaluated approaches are presented here. The evolution of the best fitness for different weighting combinations are shown in Figures 5.25ac. These trends are similar to those obser\'ed in the smaller dimensionality problems. A comparison with results obtained using nonlinear programming methods show that a larger number of evolution cycles are required for convergence in both the sharing function and the VEg^.^ approaches. The results are very close to converged values when using the ^Einc approach. In general, the current implementation of the sharing function approach was consistently outperformed by the vectorevaluated approach. Additionally, there is a trend that the VE^^ tends to outperform the VE^^, approach, particularly with increasing problem dimensionality. It is difficult to compare the computational efficiencies of the gradientbased nonlinear programming method and the genetic search method. A nonlinear programming algorithm will iterate until a relative minimum or
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158 Table 5.9 Description of the statically loaded wingbox structure. WINGBOX Design V ul IclUlC Element Element Type Node Connectivity Constraint VI X2 1,2 3,4 1 E TE 13, 810 35, 1012 U, < 2.0 in. for all unconstrained nodes X3 5,6 TE 24, 911 X4 X5 7,8 9,10 TE TE 46, 1113 67, 1314 Axial or equivalent stresses < 10,000 psi for elements (128) X6 11,12,13,14 TM 124, 431, 8911, 11108 X7 15,16,17,18 TM 346, 653, 101113, 131210 X8 19,20 TM 567, 121314 X9 21 QM 18103 XIO 22 QM 310125 Xll 23 QM 29114 TE: Truss element X12 24 QM 411136 TM: Triangular membrane X13 25 QM 613147 QM: Quadrilateral membrane X14 26 QM 310114 X15 27 QM 512136 0.01 < X,., < 3.50 in' X16 28 QM 512147 0.01 < X^,^ < 3.50 in
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159
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160 Wingbox l8t oeneration + 10th generation o 20th oÂ«r>eration A 30th generation V Nonlrear pr oyaming Figure 5.25b Best fitness values obtained using the VEj^, approach.
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161 Fftness 3 Wingbox 1ct generation + 10th flenoration * 20th floneration ^ 30th generation V Nonlhear pro^aming W (W^. 1Figure 5.25c Best fitness values obtained using the VE^^^ approach.
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162 maximum is reached, regardless of how many function evaluations are necessary. Genetic search algorithms either finish when convergence to a best solution is reached, or when the number of fitness function evaluations reaches a prescribed maximum cap. In genetic search, the optimum is usually obtained before the entire population converges to the optimal design. It is, therefore, practical to perform genetic searches by assigning a maximum value to the number of fitness function evaluations which are computationally affordable. An interesting, though indirect comparison can be made between the efficiencies of these two approaches, which have totally different convergence criteria. For the fivevariable case of the statically loaded tenbar truss problem, it took an initial total of 2,748 function evaluations to perform 11 optimizations, based on different weighting combinations. More function evaluations will be required when the optimization problem was rerun with different starting design estimates. This procedure is recommended in working with gradient based methods, as the existence of relative minima in the design space is not generally known apriori. Eleven 'globally optimal' solutions obtained are shown in Figures 5.19a5.19c. As a comparison, 4,000 fitness function evaluations were necessary in the genetic search, and the results obtained by the VE,Â„, and VE,,, approaches were observed to be as good as these 'globally optimal' designs. It is noted that only four discrete design variables are involved in this optimization problem. When a large number of discrete/integer design variables are presem, the branchandbound approach used in nonlinear programming techniques to accoum for the
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163 noncontinuous design variables requires many more function evaluations. Genetic search algorithms have no additional computational requirements due to an increased number of discrete/integer design variables in the optimization problem. 52.4 Viscoelastic allv Damped Laminated Composite Re,?^m An integrated approach to the multicriterion optimal design of a viscoelastically damped aluminum cantilever beam was investigated in Chapter 3. To investigate the discontinuous application of damping material on the beam, a genetic search approach was used. The results described below were obtained by considering a population size of 100. A combination of the VE^^, and VEÂ„,^ strategies was used to account for the multicriterion nature of the problem and values of P, = 0.8 and PÂ„, = 0.01 were used. In this problem, the base structure was a graphiteepoxy composite laminate with a layup sequence given as [0, 45, 90, 45, 0]y^. The material properties, as shown in Table 5.10, were used in this simulation. The beam was divided into 20 equal segments along its length, and a 0/1 type of variable was used to denote the absence/presence of damping material on a given segment. This resulted in a 20digit string of O's and I's defining the placement of the damping material. The structural system of the composite sandwich beam is shown in Figure 5.26. The design variable set included variables indicating the presence of damping material on a beam segment X^^^ [Xj, X,, X20], the thickness of the viscoelastic layer X^^, the
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164 Table 5.10 Material properties of the structural system. Constraining layer Viscoelastic layer Graphiteepox}' lamina (GN/m) 69.0 0.0021 127.9 (GN/m) 10.27 0.3 0.499 0.24 0.20 Damping factor 0.0 0.10 0.005 Density (kg/m^) 2700.0 970.0 1580.0
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165 CONSTRAINING LAYER VISCOELASTIC LAYER GRAPHITEEPOXY LAMINATE 1' Xcon Xvis 20*Xply 0.4 m Figure 5.26 Viscoelastically damped laminated composite beam structure.
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166 thickness of the constraining layer X^^n' the ply thickness of the base structure Xpiy. Here, X^^^, X^^ and Xpy were discrete variables, selected from lists of allowable values shown in Table 5.11. The finite element analysis described in Chapter 3 was used to evaluate the structural damping and natural frequencies, as well as the static displacements that resulted when different types of damping treatments were applied. The vector of criterion contained four elements; the first three corresponded to the damping ratio in the first three modes and the last was the total weight of the beam. The constraint set comprised lower and upper bounds on the first three natural frequencies and a maximum allowable displacement under a static load of 1N applied at the tip of the beam. These design constraints are summarized in Table 5.11. The mathematical statement of the optimum design problem can be written as follows: Minimize damp^ damp^_ ^ damp^ ^weight"'' (5.15) Subject to gj < 0, y=l,2,...,7 where damp^ represent the i'^mode damping, and weight represents the total weight of the structural system. The superscript av represents the populational average value of the system variable in a given generation. A slowed searching rate in the genetic algorithm may result from the presence of significantly low
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167 Table 5.11 Design constraints and allowable discrete values of design variables ^ly Xcon, and X^,. .000150 .000175 .000200 .000225 (m) V con .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008 .0009 .0010 .0011 .0012 .0013 .0014 .0015 .0016 .0017 .0018 .0019 .0020 .0021 .0022 (m) .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008 .0009 .0010 (m) Design Constraints 20.0 < freqi < 50.0 (Hz) 130.0 < freq, < 200.0 (Hz) 400.0 < freq3 < 700.0 (Hz) tip vertical displacement < 0.002 m
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168 damping values in certain designs. These low values would most probably be present only in early generations. In order to avoid such situations, the average value of a population, instead of a constant value, was used to scale the magnitudes of these criteria. Wj, 1 = 1,4, represent weighting parameters and each weight was used to emphasize the relative importance of the i"* criterion. To account for the problem constraints, a penalty term was introduced as an exterior penalty function. The scaled, overall fitness function for this probl em was defined as follows: F = F max damp^ damp 2 ^ damp^ weight'^' * (5.16) For five different weighting combinations of these candidate criteria, the optimal designs obtained from the genetic search procedure are presented in Table 5.12. It is noted replacing the aluminum beam with a graphiteepoxy laminated beam gave a 50% increase in damping and curtailed the weight by 30% m this application. According to the results, a continuous application of damping tapes, which spans from the root of the beam to a point close to the end, is a reliable selection for obtaining high damping values in all three modes. For maximum damping in the 2"'^ and 3^'^ modes, the most effective region of damping tape application was near the middle area. Final designs with discrete damping treatments were obtained for the weighting combination in which the firstmode damping was weighted with 0.5, and for the weighting combination wherein the
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169 Table 5.12 Multicriterion optimal designs. Weighting combination uam \ 1 /u/ L oJ Loss Factor W T W, Jply Weight V con Mode 1 (N) W2 Mode 2 W3 (m) Mode 3 W 0.5 11111111111101100000 0.0230 U.U .000200 0.0155 1.2415 0.0 .0022 0.0149 0 5 .UUUl U.U 11011111111111111000 0.0135 U.J .000200 0.0266 1.3125 0.0 .0022 0.0197 0.5 .0001 0.0 00111111111111111110 0.0136 0.0 .000200 0.0250 1.3481 0.5 .0022 0.0233 0.5 .0001 0.2 11111111111111111100 0.0221 0.2 .000200 0.0252 1.3836 0.2 .0022 0.0230 0.4 .0001 0.1 11111111111111111000 0.0196 0.1 .000200 0.0216 1.1860 0.1 .0016 0.0185 0.7 .0001
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170 secondmode was weighted with 0.5. The noncontinuous damping treatment configurations used in these two designs would not be feasible for optimization using the two continuoustype models presented in Chapter 3. Gradientbased nonlinear programming methods are essentially inept in solving the problems presented in this section due to the presence of the large numbers of discrete design variables.
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CHAPTER 6 APPROXIMATION CONCEPTS IN GENETIC ALGORITHM BASED OPTIMIZATION 6.1 Introduction A definitive gap can be identified between a generally efficient, gradientbased mathematical programming algorithm, and a computationally intensive direct enumerative scheme. As discussed earlier, the former approach is the method of choice in problems where the design space is continuous and convex. It IS, however, prone to converging to a relative optimum when the design space is nonconvex. While not adversely affected by nonconvexities, enumerative schemes place a severe requirement on the computational resources. Genetic algorithms represent a relatively new approach for bridging this gap. They inherently possess a better chance of locating the global optimum than nonlinear programming algorithms, and though computationally intensive, require less effort than enumerative search techniques. New genetic search strategies having significantly improved efficiency may be available in the future; however, present methods usually require thousands of fitness function evaluations to achieve an optimum solution for a structural optimization problem having only moderate dimensionality. More computational effort is needed when the optimizati tion 171
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172 problem involves a large and complicated design space. Computationally expensive finite element analysis methods are often used to evaluate structural response during an optimal design cycle. Since genetic algorithms require that this analysis be performed several times, it is often prudent to replace the computationally intensive analysis by a less demanding approximation. Hence, approximation concepts in genetic algorithms are a subject worthy of investigation. A computationally efficient alternative to the finite element analysis method, which also yields an acceptable level of accuracy, is desirable for use in genetic algorithms. The applicability of using an approximation, based on a trained neural network, is proposed as an alternative to the finite element analysis method. The problem under consideration is one pertaining to the design of a viscoelastically damped composite beam structure. 6.2 Neural Networks The neural computing concept has its philosophical basis in the biological cerebral process. Neural networks are parallel computing models which use many small elements, referred to as neurons or nodes, that are densely interconnected. A set of inputs to the network relays through the neurons in a pattern that can alter network characteristics in a learning phase, or be subsequently recognized in a memory operation. It has been shown [97] that a neural network can model the behavior of any linear or nonlinear continuous system. The use of neural networks in the analysis of structural systems holds promise as a new tool for efficiently
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173 solving problems that may be computationally intensive or hold solutions that are not readily apparent by traditional methods. Recent studies in this area have been performed by Hajela [98], where a network was trained to perform automated structural design. A similar approach was used by Rehak [99], who trained a network to represent the stiffness matrix for a two degreeoffreedom spring system. Neural networks have been trained to model the forcedisplacement relationship in a number of simple structural systems. The applicability of using neural networks in genetic search optimization problems to model complex relationships between input and output of the structural system, is examined in this work. The configuration of a neural network is defined in terms of individual neurons, the network connectivity, the weights associated with interconnections between neurons, and the activation function for each neuron. The network maps a set of input vectors to a set of output vectors. With appropriate training, the neural network can "learn" to perform reasonably accurate mapping. The functional operation of the neuron is first described. Consider the single neuron shown in Figure 6.1. This neuron, referred to as neuron q, receives a set of m inputs, x^, j = l,m and produces an output N^. There is a weight associated with each input. The inputoutput relation is obtained as N = F{A) = F f m (6.1)
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174 INPUT m Figure 6.1 A single neuron.
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175 where A is the weighted sum of all inputs comiected to the neuron q, and the function, F, is the activation function which transforms the weighted sum on inputs to an output response. This activation function can be the source of nonlinearity in the neuron, and in the present work, a sigmoid function was used for this purpose. The sigmoid function is defined as, = (6.2) Â• , r where B is the bias parameter used to modulate the output of the neuron. The principal advantage of using the sigmoid function is its ability to handle both large and small numbers. Referring to Figure 6.2, it can be seen that the outputs for very large and very small numbers approach 1 and 0, respectively, while numbers in between have smooth variations. The output of the neuron, obtained from the sigmoid function is then treated as an input for neurons in next layer. The neural network of this study used an input layer consisting of a number of neurons that receive design variables values of the structural systems, and an output layer consisting of a number of neurons that produce values representing the resultant behavioral response of the structural system. If no hidden layers exist between the input and output layers, the network is referred to as a flat network. Simple flat networks show limited applicability in the representation of nonlinear relationships. A neural network with a number of hidden layers is referred to as hiddenlayer network. Each hidden layer may have
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176 STiall (AB) 0 large (AB) Figure 6.2 A typical sigmoid function.
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177 an arbitrary number of neurons. Sketches of a flat network and two different hiddenlayer networks are shown in Figure 6.3. A flat network which has a 3neuron input layer and a 2neuron output layer is referred to as having a (32) architecture. The same network with a 2neuron hidden layer is described as having (322) architecture; likewise, the presence of two 2neuron hidden layers would be denoted by a (3222) architecture. After a neural network is built, it must be trained to produce an acceptable output vector for a given set of input vectors. One supervised learning technique used for this purpose is the backpropagation training algorithm [100]. This is especially well suited for neural networks with multilayer architectures. 6.3 BackPropagation Training Algorithm Prior to training a neural network, the input and output vectors representing the structural system parameter values must be determined, either analytically or by numerical methods. Randomly generated small values are initially assigned to all network weights and bias parameters. The first output vector of the network is computed from the input vector using these initial weights and the bias parameters. The error between an element of the output vector and the expected output at that element is computed as = (6.3) where is the network output of the neuron i, as defined in (6.1), and T, is the
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178 INPUT LAYER OUTPUT LAYER (32) OUTPUT LAYER INPUT LAYER (322) OUTPUT LAYER (3222) re 6.3 Representative neural network architectures.
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179 expected output of the same neuron. This error can be reduced by changing the values of weights connecting all neurons in the previous layer to the neuron that is under examination. Weight adjustments are computed from the following equation: , ^^PÂ„ = y^.,N^j ' ' ' (64) where Npj denotes the output of neuron, p, in the hidden layer, j, which directly precedes the output layer, k; AWp,^ is the change in value of the weight between the neuron, p, in the previous hidden layer and the neuron, i, in the output layer k; Y represents the learning rate coefficient. The learning rate coefficient, typically selected between 0.2 and 0.9, functions as a step size parameter in the backpropagation process, and determines the speed of the network training; 6,^ is the product of the error signal defined in (6.3) and the derivative of the output N of the sigmoid function F(A) of the present neuron, and is obtained as follows: ''^Â• = ar^'>=^'^(^ ^^^^ where the subscripts i and k denote the i"^ neuron in k"^ layer (k"' layer is the output layer of the reural network here). A modification of (6.4) is proposed by Rumelhart et al [101], who added a momentum term into (6.4) to stabilize the training and incorporate the memory into the adjustment of the weights. This ' , " modifies (6.4) as follows.
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180 where superscript t represents the cycle of the iteration, and a represents the momentum coefficient. The error signal, EÂ„ is obtainable in the form defined in (6.3) only for neurons in the output layer because expected output T, is unknown for the previous layers. For neurons in the hidden layer adjacent to the output layer, the following equation is used to replace (6.5). E ^M(6.7) i where 6^^ is the <5 corresponding to the p"^ neuron in the hidden layer j. After 8 is obtained for neurons in this hidden layer, the adjustment of weights between neurons in the next hidden layer and the neuron under consideration is obtained using (6.4) or (6.6) with appropriate changes in indices. This process is repeated for all hidden layers. The backpropagation training process repeatedly considers all training patterns umil the error is reduced to less than a prescribed value. Another approach involves the consideration of the accumulated squared error sum from all the training inputoutput vectors. The training process is then itself an error minimization problem.
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181 64 Miilticriteri on Optima] Design of ThreeDimensional Viscoelastically Damped Laminated Composite Beam A genetic search approach was implemented for the optimal design of a threedimensional, viscoelastically damped, laminated composite, cantilevered tapered beam. The design required a simultaneous minimization of the structural weight and maximization of structural damping. A diagram detailing the structure of the beam is shown in Figure 6.4. The structural system comprised of a 20ply, unidirectional graphiteepoxy laminated composite base structure on which viscoelastic damping treatment was applied. Each ply had a uniform thickness that was determined by a discrete design variable Xjg. This design variable assumes a value selected from the set of [0.000100, 0.000125, 0.000160, 0.000225] (m). The composite laminate was symmetric and balanced; therefore, the stacking sequence could be represented by the five design variables, [x^., x,^, ^i5Â» Xjo, ^vh/sThese five variables were defined as discrete variables that represented the fiber orientations of the top five plies of the laminate. Each of the five design variables was selected from the following set [90, 85, 80, 80, ' 85] (degrees). Design variables which defined the composite base beam and the stacking sequence of 20 graphiteepoxy laminae are shown in Figure 6.5. The top surface of the composite beam was partitioned into twelve sections. Six sections of the same length were distributed on each side of the cantilevered tapered beam. The damping treatment of the beam was specified by twelve design variables, x^ through x,^, of 0/1 type. The damping treatment method is illustrated in Figure 6.4. Each of the 12 sections could have the
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182 X 0.03 m 1 / / /// /_ A CONSTRAINING LAYER VISCOELASTIC LAYER LAMINATED COMPOSITE TAPERED BEAM 1 m t F X19 .X20 20*X18 X Figure 6.4 Viscoelastically damped laminated composite beam.
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Figure 6.5 Ply sequence arrangement of laminated beam.
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184 damping tape independently applied. When two consecutive sections were both subjected to damping treatment, a onepiece damping tape was applied. The ply thickness of each graphiteepoxy laminae, Xjg, the thickness of the constraining layer, Xjg, and the thickness of the viscoelastic layer, Xjq, were all defined as discrete quantities with admissible values selected from the set shown in Table 6.1(a). As stated earlier, the dual goal of this optimization problem was to both maximize the damping of the first three vibrational modes and minimize the structural weight. The design constraint set comprised of the lower bounds on the first three natural frequencies, and the maximum displacement when a static load of 1N was applied at the tip of the beam. The maximum work theory [102], referred to as the TsaiHill theory, was used as a failure criterion on the graphiteepoxy composite laminate. The external loading condition used in the failure criterion was the combination of N,=250,000 N/m and M, = 150 Nm/m. Material properties of the structural system and design constraint specifications are shown in Table 6.1(a) and (b). The scaled, overall fitness function was expressed as F = F max damp^ damp, damp^ ^,p;Â„Uf [ damp^ damp, damp^ ' weight ' (6.cS) where wÂ„ i1,4, represents the weighting parameters used to emphasize the relative importance of the i"' criterion. For the calculation of the penalty term, PÂ„ the following scaled quantities need to be defined.
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185 Table 6.1(a) Constants and discrete sets of design variables Xjg, Xjg, and X. ^18 .000100 .000125 .000160 .000225 (m) Xjg .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008 .0009 .0010 (m) .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008 .0009 .0010 (m) Design constraints 1) 4.0 < freqi (Hz) 2) 20.0 < freq, (Hz) 3) 50.0 < freqj (Hz) 4) tip vertical displacement < 0.05 m 5) failure criterion with 1 = 250kN/m, and = 150Nm /m  Table 6.1(b) Material properties of the structural system constraining layer Viscoelastic layer Graphiteepox7 lamina El (GN/m) 69.0 0.0021 127.9 E, (GN/m) 10.27 0.3 0.499 0.24 ^2 0.20 Damping factor 0.0 0.10 0.005 Density (kg/m^) 2800.0 970.0 1580.0
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186 5.0 freq^ 20.0 , freq2 s.^l (6.9) ^ 005 ' ^5 = ( Â— ) ( Â— )( Â— ) {^) (^) 1 where g^, i = 1,5, represents the scaled value of the i"" constraint. A design is feasible when these quantities are all negative. These constraint values were used directly in the inputoutput mapping available in the neural network approach. A total of twenty design variables and eight output quantities were involved in the optimization problem. These eight output quantities consisted of damping in the first three modes, and five previously defined scaled constraint quantities, g,. An alternative to this arrangement would be to use actual values of natural frequencies and displacement as elements of the output vector of the network. In order to explore the effectiveness of different architectures in this problem, a number of neural network architectures were implemented. An initial model consisted of an input layer of 20 neurons, two hidden layers, each consisting of 30 neurons, and an output layer with 8 neurons. The twenty neurons in the input layer represented the twenty design variables of the structural system.
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187 The eight neurons in the output layer represent the eight outputs of the structural system discussed earlier. This neural model is referred to as (2030308), and was used to represent the inputoutput relationship of the structure. A number of smaller models were also used as an alternative to the (2030308) architecture for the present problem. These refined models included a (20303) architecture for predicting damping in the first three modes, a (20303) model for predicting the first three natural frequencies, a (20201) model for predicting the static displacement, and a (6151) model for predicting the failure criterion. Only six input neurons were used in the input layer of the last model, as only 6 design variables, x^g through x^g, representing the stacking sequence of the laminate and its ply thickness, were required to determine the failure criterion of the laminate. Another numerical experiment was conducted with the following architectures, (2030203), (2030303), (2020201), and (620201) where one additional hidden layer was added in the previous four models. These four models determined 3 dampings, 3 frequencies, one displacement and one failure criterion, respectively. A further refinement of the neural network models which compute damping and frequencies were also built. These entailed six (2030201) models, each of which predicted either one damping or one frequency in a specific vibrational mode. A subscript was placed after the parenthesis representing the specific neural network, such as (2030308)ioo, where the subscript, 100, indicates the total number of inputoutput training sets used to train the model.
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188 The finite element analysis technique [33] was used to compute the structural characteristics of the threedimensional viscoelastically damped beam. The beam was analyzed as having plate characteristics. Plates treated with viscoelastic layers can be modeled as shown in Figure 6.6, wherein a three layer sandwich is modeled by two layers of nodes. Both the upper and lower layers can be arbitrarily laminated. Two hundred and twenty randomly generated input vectors were created, and the finite element analysis approach was used to generate the output vectors corresponding to these inputs. One hundred sets of inputoutput data were initially used to train selected neural models. An additional set of one hundred training patterns were then combined with the original set to make a new training set of two hundred. The two hundredset data was used in training the same neural models. The last twenty inputoutput mapping data sets were used for comparison with the outputs of all trained models to determine the accuracy of results predicted by the neural networks. All neural network training was done on a Micro Vax 3300 computer. The CPU time (minutes) required to train these models and the accuracies of the mapping by these models are shown in Table 6.2. The CPU time obviously increased when training was performed using all 200 inputoutput patterns. It was most apparent when using the largest model, (2030308), where fourteen times more CPU time was used to train the model with 200 training pairs than with 100 pairs. Smaller models were observed to require less CPU time in completing the training, and provided more accurate
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189 Figure 6.6 Damped plate finite element model.
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190 a o Â«3 Ic c E E D U
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191 mapping. As a result of the high degree of nonlinearity in the inputoutput relationships, and the large dimensionality of the structural system, higher accuracies were difficult to obtain without a larger number of training pairs. Better results of training of these models will be obtained by choosing narrower ranges of input design variable values. Although the average error of these models, when calculating the values of frequencies and failure status, was large, the percentage of accurate prediction of constraint violations was quite acceptable. Accurate values of these quantities were not as important when designs were feasible. Best models for predicting damping in the first three modes were the 3 models identified as (2030201)200, i" which each determined the damping of a single mode. Similarly, for predicting frequencies the models which had the least average errors were from the three (2030201)200 models. Best predictions of scaled displacement and failure status were models (2020201)200, and (620201)200' respectively. The six trained neural networks mentioned above were subsequently used in a genetic search to determine approximations of damping, frequencies, displacement and failure status for the structural system. The genetic search was completed after 50 generations with a population size of 120. The crossover probability P, was 0.8, and the mutation probability PÂ„ was 0.01. The VE^^^ approach described in Chapter 4 was used to simultaneously generate optimal solutions according to different weighting combinations of design criteria. The
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192 final designs corresponding to the four combinations of weighting parameters are shown in Table 6.3, and the configurations of the damping treatments of these final designs are shown in Figure 6.7.
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193 Table 6.3 Final designs of genetic search. damping. freq, [Hz] X,, through Xn fdeel w. damping. freq, [Hz] X,o [m] W3 damping3 freq3 [Hz] Xi9 [m] W4 weight [N] disp [m] X20 [m] failure 0.5 0.1529(0.1311)' 4.0(4.3) 20, 30, 30, 75, 55 0.0 0.0834(0.0687) 23.3(21.0) 0.000160 0.0 0.0341(0.0241) 63.9(50.1) 0.0006 0.5 2.1857 0.034(0.040) 0.0005 0.23(0.34) 0.0 0.1387(0.1471) 4.6(5.4) 20, 15, 85, 65, 10 0.5 0.1738(0.1393) 30.4(26.7) 0.000160 0.0 0.0398(0.0257) 69.9(60.1) 0.0009 0.5 2.5685 0.029(0.023) 0.0005 0.12(0.78) 0.0 0.1406(0.1307) 4.2(5.3) 5, 0, 80, 90, 10 0.0 0.0917(0.0602) 23.0(26.1) 0.000160 0.5 0.5 0.0517(0.0317) 76.1(64.5) 0.0009 2.3378 0.034(0.026) 0.0005 0.42(0.31) 0.2 0.1506(0.1573) 4.1(4.6) 5, 85, 85, 85, 20 0.2 0.1590(0.1172) 29.3(21.5) 0.000160 0.2 0.0414(0.0262) 61.7(51.3) 0.0007 0.4 2.4154 0.039(0.033) 0.0005 0.32(0.15)
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194 to. 20. 20. 20. 4] Figure 6.7 Damping treatment patterns for final designs.
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CHAPTER 7 CONCLUDING REMARKS 7.1 Conclusions The present study has examined the feasibility of using genetic search methods in optimization problems characterized by a mix of continuous, discrete, and integer design variables. Design variable representation schemes required to facilitate this application are proposed and evaluated for their performance in representative design problems. Genetic search methods require less computational effort than the traditional branchandbound schemes in problems where the number of integer and discrete design variables is large. Additionally, the methods have an inherently better probability of locating the global optimum than the gradientbased mathematical programming methods. This study also examines the issue of transforming constrained optimization problems into unconstrained function maximization, as required in the genetic search approach. A penalty function approach was used, and preliminary findings underscore the need for a very careful evaluation of the objective function magnitude in relation to that of the penalty term. Further numerical experimentation on large scale problems is necessary to build upon the general observations available from this study. It was found that caution must be 195
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196 exercised when using genetic search methods to solve constrained optimization problems. This study also examines the use of genetic algorithms in the simultaneous generation of a family of Pareto optimal designs. Three distinct strategies for implementing such a procedure were proposed and examined in the context of structural optimization problems. The approach was shown to be a viable option for multicriterion design problems. It is particularly powerful for problems with known nonconvexities in the design space. The extensive numerical experience derived in this implementation was considered extremely useful for the further refinement of this class of search techniques. The genetic search strategies presented in this study, were implemented on a multicriterion optimization problem involving the optimal design of a twodimensional, viscoelastically damped, laminated composite, cantilever beam. In this approach, the structural damping was enhanced by a combination of redesigning the base structure and by application of a viscoelastic damping tape on the surface of the base structure. Results of the optimization problem showed that discrete types of surface treatment of viscoelastic damping tapes were favored over a continuous type of treatment when the damping criteria in higher modes were heavily weighted. Discontinuous damping tape on the beam led to a noticeable loss in transverse stiffness for a thin beam. Such discrete treatment is not preferred when a tight static tip displacement constraint is in effect.
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197 A neuralnetworkbased approximation was considered in this work to alleviate the computational requirements of the genetic search procedure. The applicability of using a multilayer perceptron model of an artificial neural network, to map the inputoutput relationships between the design variables and the output response of a threedimensional, viscoelastically damped, laminated composite tapered beam was examined. Two distinct types of network models were considered. The first category (lumped network) was one where the input patterns were mapped into all of the required output quantities. The second approach (distributed network) was based on creating several subnetworks, where each subnetwork mapped the input quantities into a smaller subset of the output quantities. The training of distributed models required less computational effort and resulted in higher mapping accuracies than the lumped network models. This was even more significant as the number of training patterns needed for the complex structural system were increased. Furthermore, a group of distributed models with a relatively small number of output elements is advantageous when the structural system is reoptimized with a different set of objective functions and constraints. In the evem that an optimal design is to be generated with additional constraints, a small model must be trained which contains this constraim as an output node. Additionally, if the definition of one constraim is changed, it only requires that the small network be retrained. In the case of generating a design with fewer constraints, the optimization can proceed by simply discarding the models which map the input vector into the deleted output elements. A lengthy
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198 training process of a new neural network modeling an entire inputoutput relationship of the structural system would otherwise be necessary. It is also worthwhile to indicate that the training times can be reduced further by training distributed models in parallel if such a computational architecture is available. 7.2 Recommendations for Future Work The exterior penalty function approach was exclusively used in the genetic search algorithms to transform constrained optimization problems into unconstrained optimization problems in this study. Different approaches should be examined with the objective of seeking a possibly better approach to deal with constrained optimization problems. These include different types of control schemes to limit the magnitude of the penalty term. In the present work, it was found that without such explicit limits, the convergence rate was significantly '\ lowered. " " ' In the context of multicriterion optimization, the use of a population average value of a specific criterion for a given generation may be replaced by other schemes. Caution is necessary in the sharing approach for multicriterion optimization, as it is analogous to the problem of biological inbreeding. Sources of diversity in the population must be introduced. Mating restrictions, other than those based entirely on the weighting coefficient combinations, will be important to include in future work. Since the population of designs in each specific weighting group is based on the penalized fitness of the entire population, a better
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199 average fitness for a specific weighting group will eventually result in that group attracting more members in successive generations. The fitness or the overall utility function has to be carefully scaled so as to distribute designs evenly over each weighting combination, and for each weighting group to possess roughly similar exploitative abilities in the genetic search. _ Due to the possible small size of each subpopulation in the VEj^c and ^Ee,.c approaches, genetic drift might result, with associated undesirable characteristics such as convergence to a suboptimal design. A possible approach to prevent such a premature convergence is to introduce a more moderate sized population for genetic evolution. The underlying assumption here is that a larger population is likely to be more diverse, and hence offers a higher possibility of locating the true optimum. This can be achieved in a strategy where the crossover transformation is a twostage process. In the first stage, a crossover is performed between a subset of members within the same subpopulation (akin to inbreeding). The remaining members from each subset form a larger gene pool where a second stage of crossover is conducted. In this second stage, a mating restriction must be applied on the variable denoting the weighting combination associated with a design. This would prevent crossovers between designs corresponding to radically different weight combinations. When a neural network is used to approximate the inputoutput relationship of a complex structural system, a relatively large number of training pairs may be necessary. In this situation, distributed neural network models with a
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200 smaller number of elements in the output layer were shown to be effective in saving computational effort. Additional experiments, where a group of distributed models is chosen to replace a large model are necessary to establish more fundamental understanding of the method. Another approach which employs clustering techniques [98] to narrow the range of input vectors may be used for complex structural systems in future work. A clustering approach could also be used, in future work, in conjunction with the refinement approach presented in this work. Further research is required to investigate approximation schemes other than the neural network based approximation used in this work. These methods may include linear approximation based on first order sensitivity, and the accumulated approximation techniques proposed by Rasmussen [103].
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REFERENCES 1. Maxwell, C, Scientific Papers, 2, 1869. 2. Michel], A.G.M., "The Limits of Economy of Material in Frame Structures," Philosophical Magazine, 6(8), 1904. 3. Berke, L., and Khot, N.S., "Use of Optimality Criteria Methods for Large Scale Systems," AGARD LS No. 70, Structural Optimization, pp. 129, October 1974. 4. Gellatly, R.A., Berke, L, and Gibson, W., "The Use of Optimality Criteria in Automated Structural Design," 3rd Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Ohio, October 1971. 5. Levy, R., and Parzynski, W., "Optimality Criteria Solution Strategies in Multiple Constraint Design Optimization," 22nd Structures, Structural Dynamics and Materials Conference, Atlanta, Georgia, April 1981. 6. Schmit, L.A., Jr., "Structural Design by Systematic Synthesis," Proceedings of 2nd Conference on Electronic Computation, ASCE, pp. 105122 New York, 1960. 7. Schmit, L.A., Jr., "Structural Synthesis 19591969: A Decade of Progress," Recent Advances in Matrix Methods of Structural Analysis and Design, ' Edited by R.H. Gallagher, Y. Yamada, and J.T. Oden, Tuscaloosa, The University of Alabama Press, 1971. I Ragsdell, K.M., "The Utility of Nonhnear Programming Methods for Engmeering Design," 11th ONR Naval Structural Mechanics Symposium, Optimum Structural Design, Tucson, Arizona, October 1981. >. Zangwill, W.L, Nonlinear Programming: A Unified Approach, Prentice Hall, Englewood Cliffs, NJ, 1969. 201
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202 10. Kunar, R.R., and Chan, A.S.L., "A Method for the Configurational Optimization of Structures," Computer Methods in Applied Mechanics and Engineering, 7, pp. 331350, 1976. 11. Vanderplaats, G.N., and Moses, F., "Automated Design of Trusses for Optimum Geometry," Journal of Structural Division, Proceedings ASCE 98, pp. 671690, 1972. 12. Vanderplaats, G.N., "Automated Optimization Techniques for Aircraft Synthesis," AIAA Aircraft Systems and Technology Meeting, Dallas, Texas, September 1976. ..y Â„ ^ 13. Fox, R.L., and Kapoor, M.P., "Structural Optimization in the Dynamic Response Regime," AIAA Journal, 8(10), pp. 17981804, 1970. 14. Pope, G.G., and Schmit, L.A., Jr., Structural Design Applications of Mathematical Programming Techniques, AGARDograph No. 149, 1971. 15. Kicher, T.P., "Structural Synthesis of Integrally Stiffened Cylinders," Journal of Spacecraft and Rockets, 5(1), pp. 6267, 1968. 16. Moses, F., "Mathematical Programming Methods for Structural Optimization," Structural Optimization Symposium, AMD7 dd 3S47 1974. 'FK, 17. Fox, R.L., Miura, H., and Rao, S.S., "Automated Design Optimization of Supersonic Airplane Wing Structures Under Dynamic Constraints," AIAA Paper No. 72333, 13th Structures, Structural Dynamics and Materials Conference, April 1971. Pareto, V., "Cours D'economie Politique," F.Rouge, Lausanne, 1896. Baier, H., "Uber Algorithmen zur Ermittlung und Charakterisierung Paretooptimaler Losungen bei Entwurfsaufgaben Elastischer Tragwerke " ZAMM, 57, pp. 318320, 1977. Baier, H., "Mathematische Programmierung zur Optimierung von Tragwerken Inbesondere bei Mehrfachen Zielen," Ph.D. Thesis, Darmstadt University, D17, 1978. Koski, J., "Truss Optimization with Vector Criterion," Tampare University of Technology, Publications 6, Tampare, 1979. 18. 19. 20. 21.
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203 22. Koski, J., "Multicriteria Optimization in Structural design," Proceedings of the International Symposium on Optimum Structural Design, 11th ONR Naval Structural Mechanics Symposium, Optimum Structural design, Tucson, Arizona, October 1981. 23. Osyczka, A., "An Approach to Multicriterion Optimization problems for Engineering Design," Computer Methods in Applied Mechanics and Engineering, 15, pp. 309333, 1978. 24. Osyczka, A., "An Approach to Multicriterion Optimization for Structural Design," Proceedings of the International Symposium on Optimum Structural Design, 11th ONR Naval Structural Mevhanics Symposium, Tucson, Arizona, October 1981. 25. Rao, S.S., and Hati, S.K., "Game Theory Approach in Multicriteria Optimization of Function Generating Mechanism," J. Mech. Des. Trans ASME, 101, pp. 398405, 1979. 26. Leitmann, G., "Some Problems of Scalar and VectorValued Optimization in Linear Viscoelasticity," Journal of Optimization Theory and Applications, 23, pp. 9399, 1977. 27. Shih, C.J., and Hajela, P., "Multicriterion Optimum Design of Belleville Spring Stacks with Discrete And Integer Decision Variables," Journal of Engineering Optimization, 15, pp. 4355, 1989. 28. Stadler, W., Personal Communication, San Francisco State University 1988. ^' 29. Sun, C.T., Sankar, B.V., and Rao, V.S., "Vibration Damping of , ^ Unidirectional Composite Laminates using Viscoelastic Damping Treatments," 59th Shock and Vibration Symposium, Vol III, Sandia Report SAND882473C UC13, October 1988. 30. 31. Rao, V.S., Sun, C.T., and Sankar, B.V., "Damping and Vibration Control of Some Laminated Composite Beams Using Addon Viscoelastic Materials " Proceedings of Damping '89 Conference, West Palm Beach, Florida February 1989. Sun, C.T., Sankar, B.V., and Rao, V.S., "Damping and Vibration Control of Unidirectional Composite Laminates using Addon Viscoelastic Materials," Journal of Sound and Vibration, 139(2), pp. 277287, 1990.
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204 32. Rao, V.S., Sankar, B.V., and Sun, C.T., "Constrained Layer Damping of Composite Beams Subjected to Initial Stresses," Proceedings of the 5th Technical Conference of the American Society for Composites, East Lansing, Michigan, June 1990. 33. Rao, V.S., Finite Element Analysis of Viscoelastically Damped Composite Structures, Ph.D. Thesis, University of Florida, 1991. 34. Holland, J.H., Adaptation in Natural and Artificial Systems, The University of Michigan Press, Ami Arbor, 1975. 35. Stadnyk, L, "Schema Recombination in Pattern Recognition Problems," Proceedings of the 2nd International Conference on Genetic Algorithms, Cambridge, MA, July 1987. 36. Sannier, A.V., II, and Goodman, E.D., "Genetic Learning Procedures in Distributed Environments," Proceedings of the 2nd International Conference on Genetic Algorithms, Cambridge, MA, July 1987. 37. Goldberg, D.E., and Smith, R.E., "AI Meets OR: Blind Inferential Search Using Genetic Algorithms," TCGA Report No. 86002, University of Alabama, Tuscaloosa, October 1986. 38. Glover, D.E., "Solving a Complex Keyboard Configuration Problem Through Generalized Adaptive Search," Genetic Algorithms and Simulated annealing, Lawrence Davis, Morgan Kauffman Publishers, Los Altos, 1987. 39. Davis, L., and Steenstrup, M., "Genetic Algorithms and Simulated Annealing: An Overview," Genetic Algorithms and Simulated Annealing, Lawrence Davis, Morgan Kauffman publisher, Los Altos, 1987. 40. Sinnott, E.W., Dunn, L.C., and Dobhansky, T., Principles of Genetics, McGrawHill Book Company, New York, 1950. 41. Hajela, P., "Genetic Algorithms in automated Structural Synthesis," Optimization and Decision Support Systems, ed. BHV Topping, Kluwer Academic Publishers, 1989. 42. 43. Hajela, P., "Genetic Search An Approach to the Nonconvex Optimization Problem," AIAA Journal, 26(7), pp. 12051210, July 1990. Nashif, A.D., Jones, D.I.G., and Henderson, J.P., Vibration damping, John Wiley, New York, 1985.
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205 44. Plunkett, R., "Friction Damping," Damping Applications for Vibration Control, Edited by P.J. Torvik, ASME Publication AMD38, pp. 6574, 1981. 45. Lazan, B.J., Damping of Materials and Members in Structural Mechanics, Pergamon, New York, 1968. 46. Plunkett, R., and Lee, C.T., "Length Optimization for Constrained Viscoelastic Layer Damping," Journal of Acoustical Society of America, 48, pp. 150161, 1970. 47. Sun., C.T., Gibson, R.F., and Chaturvedi, S.K., "Internal Material Damping of Polymer Matrix Composite Under OffAxis Loading," Journal of Materials Science, 20, pp. 25752585, 1985. 48. Gibson, R.F., Chaturvedi, S.K., and Sun, C.T., "Complex Moduli of Aligned Discontinuous Fiber Reinforced Polymer Composites," Journal of Materials Science, 17, pp. 34993509, 1982. 49. Gibson, R.F., and Yau, A., "Complex Moduli of Chopped Fiber and Continuous Fiber Composites: Comparison of Measurement with Estimated Bounds," Journal of Composite Materials, 14, pp. 155167, 1980. 50. Sun, C.T., and Wu, J.K., "Stress Distribution of Aligned ShortFiber Composites Under Axial Load," Journal of Reinforced Plastics and Composites, 3, pp. 130144, 1984. 51. Hajela, P., and Shih, C.J., "Use of an Advanced Shear Lag Model to Obtain the Optimum Internal Damping in ShortFiber Composites," Applied Mechanics Reviews, 42(11), November 1989. 52. Wu, J.K., Optimization of Material Damping and Stiffness of Laminated FiberReinforced Composite Structural Elements, Ph.D. Thesis, University of Florida, 1985. 53. Mead, D.J., "The Effect of Certain Damping Treatments on the Response of Idealized Aeroplane Structures Excited by Noise," Air Force Materials Lab., AFMLTR65284, WPAFB, August 1965. 54. Plunkett, R., "Damping Mechanism in Fiber Reinforced Laminates," Proceedings of lUTAM Conference on Advances in Composite Materials Pergamon, Oxford, 1983.
PAGE 213
206 Kimball, A.L., Vibration Prevention in Engineering, John Wiley, New York, 1932. Ungar, E.E., and Kerwin, E.M., Jr., "Loss Factors of Viscoelastic Systems in Terms of Energy Concepts," Journal of Acoustical Society of America, 24(3), pp. 954957, 1962. Ferry, J.D., Viscoelastic Properties of Polymers, 2nd Edition, John Wiley, New York, 1970. Ross, D., Ungar, E.E., and Kerwin, E.M., Jr., "Damping of Plate Flexural Vibrations by Means of Viscoelastic Laminate," Structural Damping, ASME, pp. 4958, New York, 1959. Kerwin, E.M., Jr., "Damping of Flexural Waves by a Constrained Viscoelastic Layer," Journal of Acoustical Society of America, 31(7), pp. 952962, 1959. Sun, C.T., Rao, V.S., and Sankar, B.V., "Passive Damping of Composite Structures," Proceedings of the Symposium on Applied Mechanics Celebrating the Seventieth Birthday Anniversary of TsuTao Loo, Shanghai, China, September 1990. Johnson, CD., Kienholz, D.A., Austin, E.M., and Schneider, N.E., "Finite Element Design of Viscoelastically Damped Structures," Proceedings of Vibration Damping Workshop, pp. HH1HH8, February 1984. Mantena, P.R., "Vibration Control of Composite Structural Elements with Constrained Layer damping Treatments," Ph.D. Thesis, University of Idaho 1989. Haftka, R.T., Giirdal, Z., and Kamat, M.P., Elements of Structural Optimization, Kluwer Academic Publisher, Norwell, MA, 1990. Vanderplaats, G.N., Numerical Optimization Techniques for Engineering Design, McGrawHill, New York, 1984. Glover, F., and Sommer, D., "Pitfalls of Rounding in Discrete Management Decision Variables," Decision Sciences, 22(4), pp. 445460, December 1975. Gomory, R.E., "An Algorithm for Integer Solutions to Linear Programs " PrincetonIBM Mathematics Research Report, Technical Report 1 November 1958.
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207 67. Beightler, C.S., Phillips, D.T., and Wilde, D.J., Foundations of Optimization, Prentice Hall, Englewood Cliffs, NJ, 1979. 68. Hajela, P., and Shih, C.J., "Optimal Design of Laminated Composites Using a Modified Mixed Integer and Discrete Programming Algorithm," Computers and Structures, to appear. 69. Garfinkel, R.S., and Nemhauser, G.L., Integer Programming, John Wiley and Sons, New York, 1974. 70. Vanderplaats, G.N., "An Efficient Feasible Direction Algorithm for Design Synthesis," AIAA Journal, 22(11), pp. 16331640, 1984. 71. Hajela, P., and Shih, C.J., "Multiobjective Optimum Design in Mixed Integer and Discrete Design Variable Problems," AIAA Journal, 28(4), pp. 670675, 1990. 72. Hwang, C.L., Masud, A.S.M., Multiple Objective Decision Making: Methods and Applications; A StateofArt Survey, SpringerVerlag, Lecture Notes in Economics and Mathematical Systems, 164, New York, 1979. 73. Johnson, E.H., "Optimization of Structures Undergoing Harmonic or Stochastic Excitation", AIAA Journal, 14(2), 1976. 74. Rao, S.S., Optimization Theory and Applications, Wiley Eastern, New Delhi, 1979. 75. Baker, J., "Adapting Selection Methods for Genetic Algorithms," Proceedings of an International Conference on Genetic Algorithms and Their Applications, pp. 101111, 1985. 76. Whitley, D., "The Genitor Algorithm, and Selection Pressure: Why RankBased Allocation of Reproductive Trials is Best," Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 116121, George Mason University, 1989. 77. Eshelman, L.J., Caruana, R.A., and Schaffer, J.D., "Biases in the Crossover Landscape," Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 1019, George Mason University, 1989. 78. Fogarty, T.C., "Varying the Probability of Mutation in the Genetic Algorithm," Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 104109, George Mason University, 1989.
PAGE 215
208 79. DeJong, K.A., An Analysis of the Behavior of a Class of Genetic Adaptive Systems, Ph.D. Thesis, The University of Michigan, 1975. 80. Grefenstette, J.J., "Optimization of Control Parameters for Genetic Algorithms," IEEE Transactions on Systems, Man, and Cybernetics, SMC16(1), pp. 122128, January/February 1986. 81. Davis, L., "Adapting Operator Probabilities in Genetic Algorithms," Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 6169, George Mason University, 1989. 82. Goldberg, D.E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley, Reading, PA, 1989. 83. Goldberg, D.E., and Segrest, P., "Finite Markov Chain Analysis of Genetic Algorithms," Genetic Algorithms and Their Applications: Proceedings of the 2nd International Conference on Genetic Algorithms, pp. 18, Cambridge, MA, 1987. 84. Goldberg, D.E., "Optimal Initial Population Size for BinaryCoded Genetic Algorithms," TCGA Report No. 85001, Tuscaloosa, University of Alabama, 1985. 85. Goldberg, D.E., "Sizing Populations for Serial and Parallel Genetic Algorithms," Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 7079, George Mason University, 1989. 86. Antonisse, J., "A New Interpretation of Schema Notation that Overturns the Binary Encoding Constraint," Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 8691, George Mason University 1989. 87. Goldberg, D.E., and Richardson, J., "Genetic Algorithms with Sharing for Multimodal Function Optimization," Genetic Algorithms and Their Applications: Proceedings of the 2nd International Conference on Genetic Â• Â• Â• Algorithms, pp. 4149, Cambridge, MA, 1987. 88. Deb, K., and Goldberg, D.E., "An Investigation of Niche and Species Formation in Genetic Function Optimization," Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 4250, Georee Mason University, 1989.
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209 89 Schaffer, J.D., Some Experiments in Machine Learning Using Vector Evaluated Genetic Algorithms, Ph.D. Thesis, Vanderbilt University, Nashville, 1984. 90. Granet, I., Strength of Materials for Engineering Technology, 2nd Edition, Reston Publishing, Reston, 1980. 91. Dimarogonas, A., Computer Aided Machine Design, Pretice, New York, 1989. 92. Wahl, A.M., Mechanical Springs, Penton Publishing, Cleveland, 1949. 93. Chironis, N.P., Spring Design and Applications, McGrawHill, New York, 1961. 94. Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing, Lancaster, 1987. 95. Haftka, R.T., and Walsh, J.L., "StackingSequence Optimization for Buckling of Laminated Plates by Integer Programming," Proceedings of the 32th AIAA/ASME/ASCE/AHS/ASC SDM Conference, Baltimore, Maryland, April 1991. 96. Schmit, L.A., and Miura, H., "Approximation Concepts for Efficient Structural Synthesis," NASA CR2552, March 1976. 97. Lorentz, G.G., "The 13th Problem of Hilbert," Mathematical Development Arising from Hilbert Problems, Edited by F.E. Browder, American Mathematical Society, Providence, RI, 1976. 98. Hajela, P., and Berke, L, "Neurobiological Computational Models in Structural Analysis and Design," Proceedings of the AIAA/ASME/ASCE/AHS/ASC 31th Structures, Structural Dynamics and Materials Conference, Part 1, pp. 345355, Mobile, AL, 1990. 99. Rehak, D.R., Thewalt, C.R., and Doo, L.B.," Neural Network Approaches m Structural Mechanics Computations," Computer Utilization in Strctural Engineering, Edited by J.K. Nelson, Jr., ASCE proceedings from Structures Congress, 1989. 100. Rosenblatt, F., Principles of Neurodynamics, Spartan, New York, 1962. 101. Rumelhart, D.E., and NcClelland, J.L., Parallel Distributed Processing 1 & 2, the MIT Press, Cambridge, MA, 1988.
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210 Agarwal, B.D., and Broutman, L.J., Analysis and Performance of Fiber Composites, John Wiley and Sons, Inc., New York, 1980. Rasmussen, J., "Structural Optimization by Accumulated Function Approximation," Report No. 20, Institute of Mechanical Engineering, Aalborg University, Aalborg East, Denmark, June 1990.
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BIOGRAPHICAL SKETCH ChyiYeu Lin was born on November 15, 1957, in Taipei, Taiwan, Republic of China. He received his B.E. degree in July 1980, with a major in mechanical engineering from the ChungYuan Christian University in Taiwan. After finishing a twoyear mandatory military service, he enrolled in the Department of Mechanical Engineering at Tamkang University in Taiwan. He completed his M.S. degree with a major in mechanical engineering in July 1985. He then worked as a patent engineer for two years in TaiE International Patent and Trademark Office at Taipei. In August 1987, he was admitted to the graduate program at the University of Florida. He received his Ph.D. degree with a major in engineering mechanics from the Department of Aerospace Engineering, Mechanics and Engineering Science at the University of Florida in December 1991. v i , : 211
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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 the Doctor of Philosophy. Prabhat Haj^lja, Chairman Associate Professor of Aerospace Engineering, Mechanics and Engineering Science 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 the Doctor of Philosophy. Lawrence E. Malvern Professor of Aerospace Engineering, Mechanics and Engineering Science 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 the Doctor of Philosophy. ChangTsan Mm Professor of (Aerospace Engineering, Mechanics and Engineering Science
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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 quahty, as a dissertation for the degree of the Doctor of Philosophy. Bhavani V. Sankar Associate Professor of Aerospace Engineering, Mechanics and Engineering Science 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 the Doctor of Philosophy. Assistant Professor of Civil Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1991 ^Winfred M. Phillips Dean, College of Engineering Madelyn M. Lockhart Dean, Graduate School
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