Knowledge engineering applications in automated structural design

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Knowledge engineering applications in automated structural design
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Shankar, Nagarajan
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Structural design -- Automation   ( lcsh )
Expert systems (Computer science)   ( lcsh )
Aerospace Engineering, Mechanics and Engineering Science thesis Ph. D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaves 118-121)
Statement of Responsibility:
by Nagarajan Shankar.
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Typescript.
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Vita.

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University of Florida
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Full Text










KNOWLEDGE ENGINEERING APPLICATIONS IN AUTOMATED
STRUCTURAL DESIGN
















By

NAGARAJAN SHANKAR


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

and

Ravi & Hari














ACKNOWLEDGEMENTS


I would like to express my sincere gratitude to Dr. Prabhat Hajela for his

encouragement and support throughout the research. I would also like to thank Drs.

Nevill, Navathe, Sun, and Zimmerman for consenting to serve on the thesis

committee.

I am extremely thankful to my parents for their blessing and enduring support

during my graduate studies. I also thank all my friends who made my stay in

Gainesville enjoyable.


iii














TABLE OF CONTENTS



ACKNOWLEDGEMENTS ..................................... iii

TABLE OF CONTENTS ........................................ iv

LIST OF TABLES ............................................. vi

LIST OF FIGURES ........................................... vii

ABSTRA CT ................................................ ix

CHAPTER 1 INTRODUCTION .................................. 1

Background ............................................. 1
Relevant Literature ....................................... 3
Prototype Design System ................................... 8
Expert System s .......................................... 10
O verview ............................................... 12

CHAPTER 2 DEVELOPMENT OF DESIGN METHODOLOGY ......... 14

Generic Problem Solving Systems ............................. 14
Knowledge Level .................................... 15
Function Level ..................................... 18
Program Level ...................................... 19
Problem Definition and Primitives ............................ 20
Optimal Structural Synthesis ................................ 23
Data M management ........................................ 26

CHAPTER 3 IMPLEMENTATION OF DESIGN METHODOLOGY ...... 29

Knowledge Representation in CLIPS .......................... 29
Knowledge Base for Topology Generation ...................... 31
CLIPS Integration ........................................ 34
Adhoc Decomposition ..................................... 35
Topology Analysis ........................................ 37








Heuristic Optimization ..............................
Illustrative Examples ...............................
Topology Dependence on Decomposition ................


CHAPTER 4 GLOBAL SENSITIVITY BASED DECOMPOSITION ....... 50

Decomposition Based Design Methodology ..................... 51
Clustering .............................................. 55
Partial Structure Generation ................................ 58
Inter-Cluster Connectivity .................................. 61
Global Sensitivity Analysis .................................. 61
Design Constraints .................................. 67
Heuristics for Assembly of Structures ..................... 69
Implementation ......................................... 70

CHAPTER 5 STAGEWISE PROBLEM DECOMPOSITION ............. 73

Dynamic Programming ..................................... 74
DP Adaptation in Stagewise Decomposition ..................... 77
Implem entation .......................................... 79
G enetic Search .......................................... 83
Basic Operations .................................... 84
Optim ization ....................................... 85

CHAPTER 6 GENETIC ALGORITHMS IN TOPOLOGY GENERATION 93

Adaptation in Structural Synthesis ............................ 93
Integration with SG Approach .......................... 94
Fitness Function Formulation .......................... 95
Implementation and Examples ............................... 98


CHAPTER 7 CLOSING REMARKS .....................

Conclusions ...................................
Recommendations for Further Research .............

APPENDIX A ORGANIZATION OF THE DESIGN SYSTEM

APPENDIX B TYPICAL RULE IN THE KNOWLEDGE BASE


......... 106

......... 106
......... 110

......... 112

........ 113


APPENDIX C EXECUTION OF DESIGN SYSTEM ................


REFERENCES .....................................

BIOGRAPHICAL SKETCH ...........................


......... 118

........ 122


. 115














LIST OF TABLES


Table page

2-1 Table of primitives ......................................... 24

3-1 Optimized areas for topology in figure 3-3 ....................... 46

3-2 Optimized areas for topology in figure 3-4 ....................... 46

5-1 Optimized areas for topology in figure 5-4 ....................... 91

6-1 Optimized areas for example in figure 6-5 ...................... 105














LIST OF FIGURES


Figure page

1-1 Conventional design methodology ............................... 9

1-2 Schematic of a typical expert system ........................... 11

2-1 A general framework for design ....... ........................ 16

2-2 Organization of the three levels ............................... 21

2-3 Typical design problem ................................... 22

2-4 Data management facility .................................... 27

3-1 Topology refinement procedure ............................... 38

3-2 Backtracking strategy ....................................... 40

3-3 An example domain and proposed topology ................... .. 44

3-4 Problem domain with distributed loads .......................... 44

3-5 Incremental design synthesis .................................. 48

4-1 Different modules in the implementation ............. ... ........ 53

4-2 Example problem domain ................................... 54

4-3 Clustering for the example problem ............................ 58

4-4 Partial structures ........................................ 60

4-5 Allowable connectivities between clusters ........................ 62

4-6 Possible elements of connectivity .............................. 63


vii








4-7 An intrinsically coupled system ................................ 65

4-8 Complete structure ......................................... 71

5-1 M ultistage serial system ..................................... 75

5-2 Information flow between stages ............................... 80

5-3 Example design problem .................................... 82

5-4 Final topology for the example problem ......................... 90

5-5 Variation of maximum fitness during evolution .................... 92

6-1 Example design domain ..................................... 99

6-2 Topologies in generation #1 ................................ 100

6-3 Topologies in generation #10 ................................ 101

6-4 Topologies in generation #20 ................................ 102

6-5 Final topology ........................................... 104


viii













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

KNOWLEDGE ENGINEERING APPLICATIONS IN AUTOMATED
STRUCTURAL DESIGN

By

Nagarajan Shankar

May 1991


Chairperson: Dr. Prabhat Hajela
Major Department: Aerospace Engineering, Mechanics and Engineering Science

The present work focuses on the development of AI based design systems.

The work proposes a general methodology for integrating AI methods with

algorithmic procedures for the development of preliminary configurations of load

bearing truss and frame structures. The proposed framework is an adaptation of a

recently proposed model for generic problem solving systems. The methodology is

characterized by the use of algorithmic processing to enumerate the design space,

and utilizing this information in a design process dominated by heuristics.

The design process deals with the synthesis of preliminary structural

configurations for a given problem specification. The approach is based upon the

systematic decomposition of the design space, and repeated application of design

heuristics to incrementally generate an optimal structural topology to account for the








applied loads. The principal shortcomings of a heuristic decomposition approach

are underscored, and more rational quasi-procedural methods are proposed.

The research sheds light on two distinct methods of decomposition for large

structural design problems. The first method which has its basis in global sensitivity

analysis, proposes an approach to decompose the design space into subproblems for

which solutions can be readily obtained, and to integrate these partial solutions into

the system solution by consideration of first order sensitivities of subproblem

interactions. The second approach employs the use of dynamic programming,

wherein the size of the solution space is managed by a decomposition of the design

problem into many stages.The use of this strategy generates a number of possible

configurations, which are then subjected to a genetic algorithm based stochastic

search to yield a near-optimal structure. Several design examples are presented to

demonstrate the validity of the design methodology and the decomposition methods.













CHAPTER 1
INTRODUCTION



Background



Optimal structural design for minimum weight is a multistage process, initiated

with the definition of a structural topology to transmit applied loads to specified

support points. Topology generation forms the initial and perhaps a crucial step in

the design process. In most problems, there is no unique topology for a given load-

support specification. Once an acceptable topology has been generated, the members

of the structure are resized to satisfy structural and dynamic response requirements

using formal optimization techniques. The final outcome of this procedure is a

minimal weight structure, the characteristics of which are strongly linked to the

original topology definition.

Generic CAD systems available today are generally limited to problem solving

tasks in the algorithmic or deterministic domain. However, creative aspects in design

are often ill structured, and require significant intervention on the part of the

designer. Therefore, attempts have been made in recent years to incorporate

decision making capabilities in automated design procedures; the topology definition

problem described above is naturally amenable to this approach. Such decision








2

making characteristics warrant the use of artificial intelligence methods with unique

abstraction capabilities. Artificial intelligence methods, such as search techniques

and expert systems, allow us to manage problem decomposition and knowledge

representation in a very efficient manner. Additionally, expert systems help in the

dissemination of design procedures to a large user community. Use is made of the

knowledge and experience of the human designer in this process. However, there

exist two major drawbacks of such decision support systems as far as optimal

structural design is concerned. The first pertains to the rather limited computational

power of such systems which renders them inefficient for a meaningful structural

design problem. The low computational efficiency is usually attributed to the

language selected for the development of such systems. The second problem involves

its limited interfacing capabilities with algorithmic computing. Therefore, most of

the expert systems developed for structural design do not extend into the analysis and

optimization domains. Hence, the development of systems, which realize the

advantages of integrating heuristic and algorithmic computations, has to be explored.

The common feature to all the knowledge based systems was interaction with

the user. The user responds to a series of queries posed by the system, and the

system uses this information in conjunction with the knowledge built into the system

to arrive at pertinent conclusions. Therefore, such structural design systems do not

comply with the accepted definition of automation. Hence, attempts have to be

made to deemphasize interaction, to achieve a larger degree of automation.

Problem decomposition plays a major role in the design process. The basic








3

theme of decomposition is to divide and conquer. The design process involves

successive refinement in which the problem is decomposed into simpler, preferably

single goal problems. Ultimately, the solutions to all the single goal problems are

integrated in a rational manner. Decomposition methods have been employed in the

design of complex structures, but the dependence of design on such methods has not

been completely studied. A study of problem dependence is of paramount

importance in establishing the right approach for problem decomposition.

The aim of the present work was to study various facets of heuristic methods

in automated structural design. First, an attempt was made to provide a formal

architecture for the proposed design system. Second, this architecture, which

provided for automation and involved both heuristics and algorithmic computing, was

implemented to design an optimal structure for given load-support specifications.

Third, decomposition methods with a basis in dynamic programming and sensitivity

analysis were studied, with the goal of enhancing the strategies of problem

decomposition. Also included in the decomposition studies was an examination of

genetic algorithms as a viable optimization approach for large scale structural system

synthesis. Emphasis was placed on heuristic reasoning which was accomplished using

the CLIPS rule based expert system environment.



Relevant Literature


Numerous knowledge based systems have been developed in the area of








4

design of engineering systems. A general observation about most such systems is that

there is limited emphasis on attempting to optimize the final design.

Ohsuga [1] discussed the conceptual design of a CAD system, which

emphasized the importance of representing and processing knowledge to enhance

modeling and manipulation. The design system does not address the optimization

of the final design. Airchinnigh [2] addressed the problem of interfacing knowledge

systems and conventional CAD software. He discussed the need for development of

a universal programming language for both CAD software and knowledge

representation software.

Rehak et al. [3] discuss a possible architecture for structural building design,

based on a distributed network of knowledge based processing components.

However, the architecture does not address the automation of the iterative process.

Shah and Pandit [4] discuss the synthesis of forms for machine structures. A

taxonomy of primitives serves as the component database, and an expert system

synthesizes forms using these primitives. This is one of the first reported works in

the area of preliminary design synthesis. Nevill et al. [5] reported the importance of

problem abstraction in preliminary synthesis. This work involved the implementation

of a design methodology for 2-d mechanical and structural systems. However, there

is no reference to algorithmic computing in the reported approach. Brown [6]

studied the treatment of subproblems in preliminary design. This work discussed the

dependence of preliminary synthesis on design abstraction. Brown and

Chandrashekaran [7] report the development of an expert system for the design of








5

mechanical components. The problem of design refinement is treated using

heuristics and a hierarchical representation; however, optimization of the final design

abstraction is not addressed. The paper sheds light on the advantages of hierarchical

representation of the design process. Davies and Darbyshire [8] present an expert

system for process planning to decide metal removal in a manufacturing system. In

case of process planning, where algorithmic methods of solution tend to be

inadequate, expert systems are proposed as the viable alternative. Rooney and Smith

[9] present an investigation of a design methodology with an emphasis on the usage

of AI methods. This is one of the earliest attempts at AI based design methods but

does not include the concept of optimization as an end module in the procedure.

The key feature is the design knowledge being reintroduced as an integral part of a

data management facility. Dixon and Simmons [10] reported one of the first

attempts to introduce an expert system as a viable tool in the design of mechanical

systems. The work also emphasized the role of conceptual design, validation, and

evaluation of the derived design, to be represented as information modules for the

expert system. Liu [11] suggested the importance of integrating the decision making

capabilities of knowledge based systems into mainstream design methods. The

advent of supercomputers in the design workplace can be viewed as complimenting

the strength of expert systems, and both systems in tandem can be effectively

employed in the engineering decision process. The approach suggested in the paper

is robust and can be easily implemented. The importance of supercomputers in

uncertain domains like the design synthesis problem is underscored. Arora and








6
Baenzinger [12] report the implementation of AI methods in tandem with

optimization methods. Heuristics are used to analyze the results after each iteration

and optimization process is enhanced. Bennett and Englemore [13] report an

attempt in the development of expert system for structural analysis, SACON. This

was the first reported work in the area of structural analysis, and can be categorized

as a consultative system. Rogers and Barthelemy [14] discuss an expert system

implemented to aid users of the Automated Design Synthesis (ADS) optimization

program. Sriram et al. [15] describe an implementation of design methodology

integrating expert systems in structural design for civil engineering. The analysis

system does not emphasize the use of optimization to derive the final design.

Expert systems have also been successfully applied in manufacturing. Most

of these applications have been in the form of diagnostic tools and performance

monitors for machinery. Lusth et al. [16] present a knowledge based system for arc

welding. This work reported the experiences in development of a system to aid in

the welding of thin metal plates. Liu [17] reports a knowledge based system for finite

element modeling for strip drawing applications. The paper highlights the use of

knowledge based systems in other areas of mechanical engineering. Shodhan and

Talavage [18] present an AI based approach to manufacturing system design. The

simulation of a manufacturing process including selection of design, manufacturing

process, and inspection method is aided by the incorporation of a knowledge based

system. Phillips et al. [19] report research efforts in integrating AI methods with

existing CAD and CAM technologies. A knowledge based system is used to convert








7
the design for manufacturing. Murthy et al. [20] discuss a methodology for the

organization of design data and relevant information for use with knowledge based

design methods. A graph based data representation scheme has been reported for

use with primitives based design. The scheme can be translated into a relational

database system for use with quasi-procedural design methods. In general, most of

the aforementioned knowledge based systems are of a consultative type, which

impedes the total automation of the design task.

A careful analysis of the relevant literature in knowledge based design

identifies two important drawbacks. The first is a lack of formalism in the

development of a design architecture for structural design, although most of the

knowledge based design systems contain similar computational modules. Tong [21]

identified the need for formalism in knowledge based design and proposed a general

framework for knowledge based problem solving systems, recognizing the type of

information involved. The second is that a formal attempt to derive an automated

design procedure, including all formal stages in the design process and a conceptual

design module in the form of a knowledge based system, has not been successfully

accomplished. One possible explanation for the latter could be that the system tends

to get very complex, considering that a tedious finite element procedure has to be

incorporated for design analysis. Issues such as problem decomposition and

alternative optimization methods have to be addressed in the context of available

efficient AI methods. Such efforts would probably assist in the complex process of

structural design.








8

Prototype Design System



Figure 1-1 indicates a conventional design methodology for the design of

structural systems. The different stages indicated in the design constitute the

standard methodology adopted in a typical design procedure. The first phase relates

to the generation of an initial design, which is typically either an existing design or

one which evolves through heuristics derived from prior experience. The design,

which is primitive at this stage, is modified in a detailed design process to meet all

the requirements.

The next phase actions relate to the analysis and subsequent changes in the

design from the previous step. The analysis of the design is carried out to evaluate

the design constraints and check for their satisfaction. If all the constraints are

satisfied, then the design is considered feasible and can be promoted to the

optimization phase. If the constraints are not satisfied, then the design is

iteratively modified through the use of heuristics or the application of formal

optimization methods. In case a design is modified using heuristics to achieve a

satisfactory design, a procedural optimization derives the final design. This is

iterative also, and the final outcome is an optimal design. The methods adopted are

usually gradient based, although the role of new algorithms based on evolution theory

are presently under examination.





































Figure 1-1. Conventional design methodology.








10

Expert Systems



Expert systems are computer programs which simulate the actions and

decisions of a proven expert in a given domain. Architecture of a typical expert

system and the interactions between the different modules are shown in figure 1-2.

The main ingredients are a data storage capability, a knowledge representation

scheme, a mechanism to operate on the knowledge and arrive at decisions, and last

but not the least, a user-friendly interface.

The knowledge representation module in figure 1-2 is a representation scheme

for all the domain specific information in an usable form, depending on the

application involved. There are many ways of representation, with the frequently

used method being in the form of IF... THEN type rules. All the domain relevant

heuristics are stored as rules and are referred to as the 'knowledge base' of the

expert system.

The data storage module for the expert system contains all the relevant

information pertaining to the existing conditions. This is also referred to as the

database. Information retrieval and storage is performed from this database.

External databases can be utilized for large scale expert systems.

Inference module is an integral part of the expert system. This is a

mechanism which operates on the knowledge base, utilizing information relating to

the existing conditions in the problem domain, to provide decisions that may effect

those conditions. There are two recognized methods in inferencing, namely forward















INTERFACE


I Ill I


v
INFERENCE
FACILITY



Sno KNOWLEDGE
EDITOR


Figure 1-2. Schematic of a typical expert system.


KNOWLEDGE BASE

DOMAIN
INFORMATION

DATABASE


I I


I-








12
and backward chaining. The forward inferencing method is best suited for problems

such as design, where there are numerous outcomes for a given set of initial

conditions. Backward inferencing is preferred when a well defined set of conditions

is known, and events that led to those conditions are known to occur.

The user interface is an added component to the expert system, to make it

amenable to the user. Extensive graphics are usually incorporated in support of such

a user interface.



Overview



A modification of the conventional structural design approach, to incorporate

the advantages of established AI tools, is the principal focus in chapter 2. An AI

based design framework is outlined which recognizes the various levels at which

information must be organized to facilitate an effective integration of heuristics and

procedural methods. All the tools and procedures are classified depending on the

type of information being processed, and their interactions are defined. Chapter 3

details the implementation of the proposed AI framework. Preliminary examples

which illustrate the effectiveness of such an implementation are presented. A

decomposition method based on the recently developed global sensitivity analysis is

the focus in chapter 4. Details of the development and implementation of the

method, including illustrative design problems are provided in this chapter. A

stagewise (SG) decomposition approach which is thematically similar to dynamic








13

programming (DP) is presented in chapter 5. Also described in this chapter is a

genetic search (GA) approach to optimization, utilized for member resizing.

An altogether different scheme for structural topology synthesis, which is a

consequence of the SG/GA approach of chapter 5, is the focus in chapter 6. In this

chapter topology generation and optimization are considered from the GA point of

view, with support stemming from heuristics directed SG. The chapter describes the

principal issues in such an approach. Finally, concluding remarks and some

directions for further research are presented in chapter 7.













CHAPTER 2
DEVELOPMENT OF DESIGN METHODOLOGY



The development of a framework for a design system, which could act as a

template for future implementation of knowledge based structural design systems,

was the first objective in the present work. This framework was an extension of the

general approach provided by Tong [21] to encompass the structural design domain.

The underlying approach is quasi-procedural, wherein heuristics suggest a design

which is then verified by an algorithmic process.



Generic Problem Solving Systems



The process of automated conceptual design proceeds with a synergistic

application of design heuristics and domain analysis programs, resulting in a tightly

coupled quasi-procedural approach. The organization of a problem solving system

to achieve this task determined the overall efficiency of the process. Separation of

various components of such a system was important, as it retained a level of

modularity in the system which allowed for component updates. At the very outset,

it was important to recognize three distinct levels at which the automated design

process was organized--the knowledge, function, and program levels. A typical








15
arrangement of the three levels in relation to each other is shown in figure 2-1.

The figure also portrays typical component modules for the different levels.



Knowledge Level. A detailed description of the design domain is relegated

to this level. This included all pertinent specifications which define the acceptability

of a solution. All necessary and applicable requirements for analysis are also made

available at this level. Furthermore, the types of design applications envisaged for

such systems would very seldom generate designs that bear no resemblance to their

predecessors. A possibility exists, therefore, to develop at this level a general

taxonomy of problem types, of the most typically encountered design constraints, and

of possible solution strategies. The domains that can be considered in this exercise

include geometric modeling, structural analysis, and optimization methodology.

In creating a taxonomy of designs based on problem specifications, one is

essentially identifying abstract features that result from an imposition of such

specifications. Structural design requirements may be classified on the basis of

strength, stiffness, elastic stability, degree of redundancy, types of support conditions,

dynamic behavior, and a requirement of least weight or least complexity in

manufacturability. Clearly, each of these requirements has an influence on the

design that distinguishes it from designs dominated by other requirements. As an

example, a structure that is governed by structural stability requirements will be

dominated by elements that can withstand compressive loads, and further, such

elements will typically have appropriate aspect ratio and stiffness properties.









16
























I


PROGRAM LEVEL

INFERENCEE FACILITY
DATA MANAGEMENT


Figure 2-1. A general framework for design.


KNOWLEDGE LEVEL

TOOLS
* TOPOLOGY GENERATION
* OPTIMIZATION
DOMAIN KNOWLEDGE
* DESIGN SPECIFICATIONS
* LIBRARY OF PREVIOUS DESIGNS
* DATABASE


FUNCTION LEVEL

CONTROLLER
PROBLEM DECOMPOSITION
ANALYSIS
TESTER








17

However, in as far as possible, it is advantageous to relate one abstract feature with

one problem requirement. Failing this, the classification must clearly indicate the

relative contribution of a requirement to a salient characteristic.

Clearly, the information available at the knowledge level determines the class

of problems for which solutions can be obtained. The coupled algorithmic and

heuristic process can be computationally demanding in some situations. It is in such

cases that a taxonomy of designs based on problem requirements is particularly

useful. It is conceivable to abstract partial designs in a primitive form, where such

primitives adequately model the displacement field of more refined models, but may

have considerably fewer structural elements. These primitive forms may be retained

in the analysis to a point where it becomes necessary to introduce greater refinement

in order to meet a specific class of design requirements. One can think of a

macroelement that has specific stiffness characteristics to satisfy displacement

requirements. Such a macroelement can then be defined in terms of component

elements with the same aggregate stiffness, and which account for strength

requirements.

All the heuristics that aid in the synthesis of topology are maintained at this

level. This includes the heuristics employed for design decomposition and

subproblem integration as well as for optimization purposes. Constraint

requirements that are to be satisfied, if any, by any chosen topology, are retained

here. A global database, which stores domain information, taxonomy of topologies,

allowable primitives and design requirements, is also provided at this level.








18

Function Level. Perhaps the most significant feature of this level of

organization is a controller which directs the flow of the conceptual design process.

The controller is essentially an embodiment of the problem solving strategies that

may be invoked for the problem at hand. The design specifications handed down

from the knowledge level are attempted to be satisfied at the function level.

Although designs can be generated by considering all requirements simultaneously,

this methodology is not considered appropriate for the task at hand. Instead, a more

natural process of refinement in steps is adopted, wherein the problem is

decomposed into smaller, more tractable, and preferably, single goal problems. The

underlying principle in such a refinement is that the solution space is more likely to

be unique in the presence of a higher degree of specification detail.

In a problem where the design philosophy is one of sequential refinement to

satisfy an ordered set of goals, there is a need for evaluating the proposed partial

concepts for acceptance. Such testing operators are made available at the function

level and simply interact with the available information base at the knowledge level

to recover the pertinent information. The failure of a proposed concept to meet the

acceptability test is generally followed by alternative design moves. Such moves are

facilitated by the initial decomposition of the problem by design specifications; this

allows for construction of tree-like deduction paths.

Finally, the controller must have the option of modifying the design rules,

particularly if it assists in realizing the design specifications. The acceptability tests

can themselves be relaxed to admit marginally infeasible designs. Yet another option








19

available at this stage is to extend the design by adding features which enable it to

pass the acceptability requirements. It is important to understand that for any given

task, there may be several transfers of control to the program and knowledge level

as deemed appropriate by the controller at the function level. This transfer of

control was repetitively invoked for each of the component tasks.

Pertinent tools available at this level included domain descriptors, analysis

modules, dynamic programming and genetic algorithms for design decomposition and

optimization, respectively, and other modules for nonlinear optimization.

It is important to emphasize the interactions that exist between this level and

the other two levels. Since, the function level constitutes all the required evaluation

and problem description tools, it has an implicit dependence on the program level,

which in turn couples to the knowledge level.



Program Level. At this level, no problem solving knowledge is explicitly

available. However, the implementation of all design steps, on the basis of

information received from other levels, is carried out at this level. In addition, tasks

of data management and programming procedures are assigned to this level. The

database management capabilities of such systems are of particular importance:

significant amounts of data are generated and must be managed for a design system

to work efficiently. This is even more crucial as large amounts of algorithmically

generated numerical data must be stored and post-processed to be used effectively

in the iterative process.








20

The inference facility is perhaps the most significant feature at this level. The

inference approach in the current implementation was forward chaining, which is a

logical choice considering the fact that the design process has numerous solutions.

In structural design problems that constitute the focus of the present work, the

inference mechanism available in the rule-based C Language Integrated Production

System (CUPS) [22] is used. This utility can be invoked from within a Fortran

program, making available a convenient link between algorithmic and heuristic

processing of information.

Since the design process is inherently complex and tedious, a good I/O

capability is an essential ingredient. In the present development, emphasis is placed

on design automation and hence a very minimal I/O interaction is permitted. The

key I/O capability is a graphical display of the generated topology, and a topology

plot file was created during the design process. Once the design procedure is

completed, the plot file can be displayed to view the topology.

In keeping with the proposed framework of developing generic problem

solving systems, the various tools necessary for the task were assembled into each of

the three levels of organization as shown in figure 2-2. Appendix A presents a

detailed representation of each module within a particular level of organization.



Problem Definition and Primitives


Figure 2-3 shows a representative design problem considered for structural




















































Ijn UCUJLAL.J ALi
optimization

KNOWLEDGE
LEVEL


Figure 2-2. Organization of the three levels.


FUNCTION
LEVEL


PROGRAM
LEVEL




















(0, 45) (60, 45)


500 lb/in
I I I I I I I I


(5,30)


5,5)
(5,5)


(25, 30)






(20,5)


500. lb
[0., 1.1

[.,] Direction
(45, 30) Cosines

(.,) Nodal
Coodlnates


(45,5)


(0, 0)


(50, 0)


Figure 2-3. Typical design problem.


(0, 45)


(60, 45)








23

synthesis. The domain consists of a set of concentrated and distributed loads,

moments, possible support points, and regions where structural members are not

permitted. The task involved the determination of a least weight structural topology

to transfer the given loads to the prescribed support points, and avoiding the

forbidden zones to whatever extent possible. The design was performed in

conjunction with analysis methods to evaluate the relative merits of each topology.

The topologies were designed for constraints dictated by displacement or stiffness

requirements, strength requirements, and for dynamic considerations as represented

in frequency constraints.

Such a problem usually incorporates a set of prescribed primitives, from which

the structure is evolved. Table 2-1 lists the various primitives that were considered

for loads, supports, materials, etc. A database of all the primitives with

corresponding properties was maintained, and from which relevant information was

sought during the design process.



Optimal Structural Synthesis



The framework described in the preceding sections was implemented in an

automated design system for the generation of near-optimal, two dimensional

structural configurations. The problem required the development of both the

structural form and member cross-section dimensioning, to carry a prescribed set of

concentrated and distributed loads and moments. The least weight structural weight



















Table 2-1. Table of primitives.


Primitives Attributes

Element Beam, Truss

Load Point Loads, Moments,
Distributed Loads
Support Hinge, Clamp
Material Steel, Aluminum, Aluminum Alloy
Cross Sections I-section, C-section, Tube, Rod








25

was sought with constraints on allowable displacements, fundamental frequency, and

structural strength. The normalized constraints for the displacements, natural

frequency and stress response can be expressed as follows:


i -1<0 (2-1)
dall





1- i <0 (2-2)





___-1<0 (2-3)
all

i -150 (2-3)




where the subscript 'all' denotes allowable value of the response quantity. In the

present work, these limits were set to the following values:

dan = 104 in

oal = 12.5 Hz

oai = 250. psi

In addition, side constraints were established for all design variables (x,) as follows:


0.01" < xi 5 5.0" (2-4)




At the function level, the topology analysis module was responsible for

developing a table of pertinent features for each of the loads and supports. These








26

features include load-load distances, load-support distances, their angular

orientations, nature of the loads, type of supports, and information regarding location

of forbidden zones. Two other modules at the function level were programs for

finite element analysis, and a nonlinear programming optimization program. The

latter is available to perform procedural/ heuristic optimization on a given geometry.

Also present at this level was a controller which directed the flow of the synthesis

process. Two distinct procedures were adopted in the present work. In the first

phase, an arbitrary decomposition of the problem was prescribed to the controller,

wherein the structural design proceeded in ordered steps of satisfying the

displacement, frequency, and stress constraints. More rational decomposition

strategies based on dynamic programming and global sensitivity methods constituted

the second phase of the work and are described in chapters 4, 5 and 6.



Data Management



The process of structural design described in the preceding section cannot be

successfully implemented without reference to an extensive and detailed description

of the features of the design space. Such large quantities of information required an

efficient data handling capability. In the present framework, this data management

facility was provided at the knowledge level. A global database is central to the

knowledge level organization. Figure 2-4 represents the data organization in the

present framework. Both domain and design specifications were resident in this































/ I


Information Information
for load #1 .."" for load #k .


LOCAL DATABASES


Figure 2-4. Data management facility.


nfor m action
or load #n








28
database, as were the results obtained from any enumeration of the design space

during the process of sequential refinement. Local databases were created from this

global database for use in both algorithmic and heuristic processing in the knowledge

and program levels. The creation of these local databases was closely linked to

passing of control from one level to another. A typical local database consists of

information pertaining to a particular load, such as load-support distance, type of

support, forbidden zone information, etc. The advantage of creating a global

database is that the information can be utilized any time later to either alter the

existing design or synthesize a new design.

Two levels of data management were implemented in the current system. At

the core level was a global database that records information for long term usage.

In the current implementation, this was achieved by simply writing all the information

pertaining to a load such as the distances, orientations, supports, etc., for all the loads

into a data file. Pertinent information was retrieved from this data file when

necessary. Problem and subproblem related databases were extracted from the main

data file and were local to the knowledge level. Such an approach provided a

convenient blackboard for constraint posting and propagation as the design was taken

through a process of incremental refinement.













CHAPTER 3
IMPLEMENTATION OF DESIGN METHODOLOGY


This chapter describes the implementation of a design system conforming to

the general framework presented in the previous chapter. All the relevant

implementation details are discussed with reference to illustrative examples. The

system was implemented on a VAX 3100 platform, utilizing the VMS Command

Language feature wherever possible for automation. Issues ranging from

representation of heuristics, to development and integration of procedural and

knowledge base [CLIPS] modules, are described in the following sections.



Knowledge Representation in CLIPS



All the design heuristics were represented in the form of rules, written in a

format required by the inference environment of CLIPS. A general rule syntax in

CLIPS is given as follows:

(defrule ["comment"]

(< first pattern >)


[( ;LHS
;LHS








30
(< nth pattern >)]

=>

(< first action >)

[( .. ) ;RHS

( .. )

()])

The LHS consists of one or more patterns which form the conditions) to be

satisfied, before the rule can be fired. An implicit 'and' is present in the LHS if

more than one pattern is present. The RHS of the rule is a list of one or more

actions) to be performed as a result of the firing of the rule. The rule base also

included a few meta-rules; these were also treated as rules in the inference

environment, each of which is a superset of two or more rules. The advantage of

using meta-rules is the efficiency which results from concatenating task specific rules.

Furthermore, a better control over the evaluation of rules is possible.

Both an interactive mode of execution and one in which the expert system is

embedded in a set of algorithmic and procedural programs, are available in the

present implementation. Initially, all illustrative examples were run interactively.

This provides the opportunity to view the execution process, editing the knowledge

base, and the ability to maintain a record of the status of the system. A typical usage

involves creating a rules and facts file, which are used during CLIPS execution.

The other mode of execution is the use of CLIPS as an embedded expert

system. In this mode, CLIPS is executed from a FORTRAN program, preceded by








31
the loading of the rules file. All later implementations involved embedded CLIPS

execution. A basic feature in the CLIPS environment is that rules and data are two

separate entities. The inference engine uses a forward chaining algorithm to apply

the knowledge (rules) to the data.

In a basic execution cycle, the knowledge base is examined to see if conditions

of any rule are satisfied. This is done by simple pattern matching of the data with

the conditions. If the fields of the data which are asserted as variables, match the

conditions of a rule, then that rule is activated for firing. In case of more than one

matched rule, all such rules are activated and pushed onto a stack. The most

recently activated rule has the lowest priority and is put at the bottom of the stack.

The top rule is selected and executed. As a result, new rules may be activated and

some previously activated rules may be deactivated. This cycle is recursively applied

until no further pattern matches are obtained.



Knowledge Base for Topology Generation



The definition of the design domain was placed into the knowledge level. In

particular, this included specification of the magnitude, type, and orientation of the

loads with respect to the supports in the chosen reference axes system. The type and

location of support points were also part of such a database. This extended database

was used in conjunction with the stored knowledge base to invoke the necessary steps

in generation of the structural topology.








32

The underlying philosophy in the present sequential procedure was to stabilize

one load at a time. The pertinent load information was retrieved from the database.

If the applied load happened to be a moment, one design decision was immediately

obvious. The element that transferred this load to a support must allow for

rotational degrees of freedom at its end nodes. Hence, a beam element was selected

to transfer this load. To stabilize this load and to maintain a minimum weight, the

nearest support was examined. If this support was of clamp type, then a beam

element of nominal cross sectional dimensions was chosen. In case the support was

not a clamp, then the next nearest support was considered. If this support was a

clamp, then a beam element was chosen between this support and the load. If not,

the load was connected to both the supports using two beam elements. A further

level of refinement is possible by checking to see if the third nearest support is a

clamp, and if so assess the weight penalty of connecting to this support against the

use of two beam elements to the two nearest supports (a frame configuration).

This strategy was also used in the event that at a given point, both applied

forces and moments were present. For the situation where the loads are the only

applied forces with nonzero x and/or y components, the orientations of the loads

stored in the database were used. The angle between the force resultant and the

hypothetical element connecting the point of load application and support, was

computed. If this angle was less than 20 deg, then the load was considered to have

a large axial component, and an axial rod element was preferred.

The other possible case was one in which a large transverse component of the








33

load existed in addition to an axial load. The orientation of the load resultant with

respect to the next nearest support was considered. In the event that the load could

be considered to be largely axial for such an element, an axial rod element was

introduced between the load and this support. In the event that this condition was

also violated, the first two supports were considered. If the nearest support was a

clamp, then a beam element was chosen between the load and support. If not, then

a beam element was chosen between the load and the second support, provided it

was a clamp. Otherwise, two beam elements were chosen between the load and the

two supports. Multiple loads acting at a point were simply resolved into the x and

y components, and handled as above. Likewise, distributed loads were replaced by

equivalent end loads and moments, which were computed by assuming the distributed

load to act on a beam segment, and these loads and moments were connected to

supports as explained in the foregoing discussion.

Once a load was stabilized by the process described above, the point of

application of the load became available as a support point. This is physically an

elastic support but may be modeled as a simple support point, for subsequent

generation of structural topology. In case of truss structures, stability was verified

based on the relationship between the number of members, nodes and degrees of

freedom. The controller used the above heuristics until all the loads in the input had

been accounted for and were stabilized.








34

CLIPS Integration



As described in an earlier section, the input to the design process included a

specification of loads, type and location of supports, and specification by coordinates

of any forbidden zones in the design domain. Using this input, a FORTRAN

program was used to extract all relevant data about the design domain such as

distances between loads, between loads and supports, any possible non-permissible

members, and the orientation of loads with respect to available supports. One of the

major tasks for this module included the creation of a facts file. This file contained

the description about each load, as shown below.

(load#, load location, x component, y component, moment, support#, type of

support, support location, distance, angle, (next nearest support information). .

)

A CLIPS batch file was created which uses the facts and the rules to generate

a topology. The topology generated was stored in a data file in a two dimensional

array as shown below.

member load# support/load# type

1 1 3 b

2 1 2



The array shown above contains the connectivity information of the structure. The

symbol in the column 'type' denotes the type of member between the specified load#








35

and the support/load#. The allowable symbols are 't' for truss, and 'b' for beam

elements. The numbers in the array indicate the specified member#, load#, or

support#.

Once the topology was generated so as to stabilize all loads, the completed

structure was available for finite element analysis to check satisfaction of the design

constraints and possible modification of topology using heuristics, as described in an

earlier section. In each of these incremental modifications to obtain constraint

satisfaction, the controller used the output of the CLIPS based topology generation

routine, and created an input runstream for the EAL (Engineering Analysis

Language) [23] finite element analysis program. Upon execution of this program, the

necessary constraint and objective function information was read and returned in the

proper format for it to be read as a CLIPS input file. It is evident from the

foregoing discussion that there is a significant transfer of information between various

levels of organization in such an integrated design system.



Adhoc Decomposition



The generation of structural topology was subject to constraints on the

displacements at the nodes, natural frequencies of the structure, and the load

carrying capacity of the individual members. Since the approach adopted in this

work inherently examines several topologies, it was essential to use the constraint

satisfaction as a factor to make the choice of a candidate topology from the available








36
set of topologies. To achieve this objective, an 'adhoc' decomposition procedure was

used to satisfy the design requirements in a predetermined order. Alternate

topologies were generated and analyzed when the initial topology failed to satisfy any

of the structure behavioral constraints; a specific order was assumed for satisfaction

of these constraints. The displacement constraints were considered first, followed by

frequency, and finally the stress constraints. The implementation proceeded as a

depth first search to identify the least weight topology that satisfied all the

constraints, with set of topologies represented in a tree-like structure; the first level

contains all topologies satisfying the displacement requirement, the second and third

level with topologies satisfying dynamic and strength requirements, respectively. The

satisfaction of the constraints need not necessarily follow the order suggested here.

The problem of ordering the constraint satisfaction schedule is not as formidable as

it may first appear, as a set of topologies were made available which helps in

choosing a feasible topology, as detailed in the next few sections.

The topology synthesis proceeded until a satisfactory structure was synthesized.

The initial task was to identify the least weight topology based on heuristics described

in the previous section. Also, all other topologies in the order of increasing weight

were generated in the event that a lower weight topology did not meet any of the

design requirements. This leads to the generation of many possible topologies,

including some which may be infeasible. Such a taxonomy gives the power to choose

a suitable topology depending on the current problem requirements. This is essential

as either a change in the order of decomposition (satisfaction of constraints), or a








37
change in any of the constraints during synthesis, may result in constraint violation

in the chosen topology. However, the set of topologies generated for a given

problem specification remains unchanged as these topologies are generated on the

basis of parametric information for the particular design domain. In such situations,

alternate topologies provide a greater scope for feasible topology identification.



Topology Analysis



The analysis of the structure was performed using a general purpose finite

element program EAL. Displacement constraints were first checked for feasibility,

the failure of which invokes a set of heuristics designed to obtain constraint

satisfaction. These heuristics were based on incremental changes in the structure

designed to incur the least weight penalty. An alternate structure is generally

proposed by addition of a member at the node location for which the displacement

constraint was violated. The member is connected to nearest support point to which

it was not previously connected. Figure 3-1 depicts the topology refinement

procedure that was employed. This process was repeated for a satisfaction of the

frequency and stress constraints by invoking a similar set of heuristics.

The concept of backtracking included in the problem to identify the least

weight topology, helps in managing different topologies that are a consequence of the

adopted procedure. Simply stated, the backtracking strategy helps in choosing a least

weight candidate topology, which satisfies all the design requirements (chapter 2).











































Figure 3-1. Topology refinement procedure.








39
In the event that after all possible connections the current constraint was still

violated, the controller used a backtracking strategy (figure 3-2), to return to the

previous constraint and look for the next best configuration acceptable for that

constraint. This design was then passed to the immediate lower level to check for

the satisfaction of the failed constraint.

The chosen topology was first checked for displacement constraint violation.

If a violation was established, then the next heavier topology was chosen and

displacement constraint verified. In the event that the topology passed the

displacement criterion, then it was verified for the dynamic constraint. Successful

validation of the dynamic constraint was followed by a verification of the stress

constraint. In case, the dynamic constraint was violated, then backtracking procedure

of figure 3-2 returned control to the previous level at which the displacement

constraints were satisfied. Such a step by step analysis and backtracking was

performed at three levels, with the next higher weight topology chosen if constraints

were violated at any stage. Upon satisfactory completion of constraint satisfaction,

the backtracking procedure was terminated and the candidate topology was passed

on for subsequent optimization. A feasible structure may or may not result from the

exercise. This would depend upon the extent of redundancy available in the problem

domain. In the event of an infeasible design, one may either discontinue the exercise

or attempt constraint satisfaction by varying member cross sectional properties.




















level 1


analysis


struct 3


level 2
frequency analysis


level 3


stress analysis


satisfactory design


Figure 3-2. Backtracking strategy.


struct








41

Heuristic Optimization



Once an initial topology has been generated, the next task in the optimal

design process is the resizing of the cross sections of the structural members. In the

present work, an attempt was made to reduce the extensive numerical evaluations

required in an iterative procedural process, by incorporating some heuristics in the

optimization procedure. The objective of this work was to simulate the search

process of a constrained minimization scheme, but to reduce the computational effort

by making gross heuristic assumptions regarding the suitability of a chosen search

direction.

The sensitivities of the objective and constraint functions with respect to the

design variables were first evaluated. Likewise, the initial values of the normalized

constraints and the objective function were obtained as an assessment of the starting

design. These were evaluated on the basis of a finite element analysis. The section

areas were the design variables for the problem.

The design variables were ordered on the basis of the objective function

sensitivities, with the first corresponding to the one that gave the best improvement

in the objective function. The controller first attempted to satisfy any violated

constraints, by changing the design variable that resulted in the least weight penalty.

The actual step length was determined on the basis of a piecewise linearization of

the violated constraint about the initial point.

Once an initially feasible design was established, the objective function and








42

constraint sensitivities were recomputed and ordered as before. While some of the

constraints may be critical, others could be satisfied by large buffer margins, holding

out hope for further decrease in the objective function. The design variable with the

maximum possible improvement in the objective function was then chosen.

Constraint gradients were checked to see if a change in this design variable could be

accommodated without violating a constraint. If the piecewise linearization indicates

the possibility of an infinite move in this direction, then a 25% change in the design

variable was allowed. Otherwise, the constraint closest to critical for a move in this

direction was checked to see if at least a 10% improvement was possible before the

constraint was violated. If such an improvement was confirmed, then the allowable

change was affected. Failure of this check resulted in examination of the next best

available move direction. Note that if an unbounded move is suggested for a design

variable in more than one iteration, it indicates a member that can possibly be

removed from the structure, provided it does not render the structure unstable. It

is worthwhile to note that such deletion of members is a difficult step in strictly

procedural optimization.



Illustrative Examples



The specific organization of design tools in the context of the three level

approach described in the previous chapter, is outlined in this section. At the

function level, the topology analysis module was responsible for developing a table








43
of pertinent features for the loads and supports. Two other modules at the function

level were programs for finite element analysis, and a nonlinear programming-based

optimization program. The latter is available to perform procedural optimization on

a given geometry in lieu of the heuristic optimization. Also present at this level was

a controller which directed the flow of the synthesis process. In this implementation,

the controller assumed an adhoc decomposition of the problem, and addressed the

various design requirements in that order. A more rational approach of using global

sensitivity analysis and dynamic programming concepts for this purpose is reported

in later chapters. As stated in the previous chapter, no problem solving knowledge

was introduced at the program level. However, the inference facility which uses

information from the knowledge and function level to suggest new design moves, was

available at this stage.

The approach presented in the preceding sections was applied to the

generation of optimal topologies for 2-dimensional structural systems. Two problems,

as described in figures 3-3 & 3-4, illustrate the proposed approach. The design

domain contains both concentrated loads and moments, as well as forbidden zones

in which no element may be introduced. The example of figure 3-3 used standard

nonlinear programming algorithm [CONMIN] for member sizing optimization, and

that of figure 3-4 was optimized on the basis of embedded heuristics for optimization.

Tables 3-1 and 3-2 show the optimized areas for the two example problems, based

on procedural approach and heuristics directed optimization. The difference in the

weight of the optimized structure of figure 3-4 was 18% between procedural and






















(60,45)


Figure 3-3. An example domain and proposed topology.






























500. lb
50. lb/ln 4530)
I I I i i I i Y I I
(5,30) (25,30)

(.,.) nodal
coord.

[.,.] direction
cosines



(5,5) (25,5) -- (45,5)
) X (50,0)



50.0 lb/in. 500. lb.

B D

A C E



^711 .17 0^


Figure 3-4. Problem domain with distributed loads.


(50,55)


(0,55)













Table 3-1. Optimized areas for topology in figure 3-3.


Member Area (sq. in.)
1 2.52
2 3.80
3 3.80
4 4.53
5 4.53



Table 3-2. Optimized areas for topology in figure 3-4.


Element By heuristics NLP approach
Area (sq. in.) Area (sq. in.)
A 5.6 4.53
B 4.5 3.9
C 3.1 2.97
D 5.8 5.1
E 3.3 2.65








47

heuristics based optimization, with the heuristics based optimization yielding the

heavier structure. Figure 3-5 portrays the incremental refinement of the structural

topology in example 1.



Topology Dependence on Order of Decomposition



For the problem shown in figure 3-3, the order of constraint satisfaction was

varied to determine its influence on the final structural topology. Frequency

constraint, stress requirements, and displacements were satisfied in that order, as

opposed to previous order of decomposition. The topology obtained in this approach

was the same as before, because the candidate topology corresponds to the least

weight topology thus far synthesized, that satisfies all the three design requirements.

The design domain under consideration was simple enough so that the

generated topologies were unaffected by the order in which the constraint satisfaction

was introduced. The more general case would be one in which the topology would

indeed be influenced by the order of constraint satisfaction. As an example, if there

are two equivalent solutions to satisfying a particular constraint, choosing one over

the other may effectively eliminate the possibilities of some topologies as other

constraints are satisfied.

Similarly, the dependence of final topology on design requirements is also

paramount. Any change in the design requirements alters the design space under

consideration. If such changes are critical, different requirements may dominate the






































Figure 3-5. Incremental design synthesis.








49

development of topology. This is dependent on the amount of change in the design

requirements, whether it causes a relaxation or tightening of the constraints.














CHAPTER 4
GLOBAL SENSITIVITY BASED DECOMPOSITION


Synthesis and design of complex structural systems is an involved process

which must simultaneously consider a large number of possibly conflicting design

requirements. The problem is difficult to handle by traditional mathematical tools

which have been shown to perform very poorly in the presence of large

dimensionality problems. Furthermore, in problems of topology generation, the

design space is disjoint and is not amenable to standard solution techniques. Quasi -

procedural methods present an alternative solution strategy for this class of problems.

A look into the characteristics of the problem usually sheds light on some form of

intrinsic coupling of characteristics of the problem. Such couplings, if identified, can

be used as breakaway points, to alleviate the design process. Decomposition

methods have been proposed to take advantage of such couplings. The aim of

decomposition being to reduce the complex design problem into many simpler,

preferably single goal problems.

Two methods of decomposition have been used in prior work. The first

approach is the one presented by the 'adhoc' decomposition procedure described in

chapter 2. That decomposition stipulated that the structure be designed satisfying

the design requirements in some specified order. In such an approach, the design

is dependent on the prescribed order in which the constraints were satisfied. A

50








51

second and more recent approach proposes the decomposition of complex systems

into smaller subsystems to simplify the computation of coupled system sensitivity [24].

Although this decoupling may lead to many subsystems, the analysis in such systems

may be conducted simultaneously. The final result of the analysis of all subsystems

is a first order sensitivity of the behavior response, information that can be used

during the optimization of the design. A third method for problem decomposition

has its basis in dynamic programming methods. This research effort is discussed in

chapter 5.

The present effort may be considered as a precursor to a design system for

large structural systems. Here, a number of factors which influence the synthesis of

a structural system are used to decompose the design domain into many relatively

small design domains. The basic idea is to synthesize the partial structures in the

subdomains, and to integrate these partial designs on the basis of a global sensitivity

analysis. Once the partial structures have been integrated into a single structural

system, members of the structure are optimized. Towards this end, both a heuristics

based strategy and a procedural approach were implemented. Subsequent sections

describe the design system implementation.



Decomposition Based Design Methodology



The design process is essentially quasi-procedural, wherein a design

synthesized based on heuristics is validated using standard algorithmic procedures.








52

Figure 4-1 is a representation of the interaction of the various modules in the

method. There are three main modules to be considered. These are synthesis,

sensitivity analysis, and optimization modules, each one of which will be discussed in

subsequent sections. The problem is described in terms of a design domain with

concentrated loads, supports and forbidden zones where no structural members may

exist. The aim of the design process is to evolve a minimum weight structure by

decomposition of the problem domain, evolution of partial structures, and integration

of the partial structures. Interaction of partial designs were taken into consideration

through the use of global sensitivity analysis equations.

Figure 4-1 features the various processors embedded in the different modules

of the design system. Procedural subroutines are invoked to evaluate the design

domain, to determine the sensitivity analysis information and to perform gradient

based optimization. The most important feature is the determination of the linking

members required in the integration of partial structures. This is accomplished using

heuristics, which in turn use procedurally developed data such as the sensitivity

information for the structures under consideration.

Figure 4-2 depicts a typical design problem. In the presence of large number

of such problem attributes, a topology assignment based on design heuristics cannot

proceed in a rational manner. Hence, the first step in the proposed methodology was

to decompose the problem domain into smaller subsystems, and to create partial

designs for each of these sub-domains. The analysis and grouping of the parameters

(loads, supports, forbidden zones etc.) was carried out to clearly identify the



























STRUCTURAL SYNTHESIS


SENSITIVITY ANALYSIS


OPTIMIZATION


Figure 4-1. Different modules in the implementation.


Procedural









































Figure 4-2. Example problem domain.








55
clusters(subsystems). Structures were then synthesized for each of the clusters. The

next task involved the sensitivity analysis for structures generated for each of the

clusters. In the present approach, these partial designs were also required to satisfy

a subset of the design constraints for the problem. Most typically, these constraints

were those that were influenced by the local variables. Global type constraints such

as natural frequency constraints, were accounted for in the assembled structure.

Assembly of the partial structures into a single structure is critical and requires

careful consideration of the effect of coupling on both the local and global behavioral

response of the system. A more detailed description of the process and its

implementation are discussed in subsequent sections of this chapter. In all cases, the

framework developed in chapter 2 was adopted to whatever extent possible.



Clustering



The parameters in a given design domain (figure 4-2) such as the load points,

supports, types of loads and supports, etc. are characterized into a number of groups

or clusters. Each group is considered as a design synthesis problem and a suitable

structural topology is generated. This is the essence of the clustering process

employed in the present research effort.

The division of the given domain into many subdomains is governed by

geometry considerations; human designers approach such complex problems in much

the same way, and use proximity of loads and supports to generate partial designs.








56

The method of clustering used in the current implementation is based on geometry

and parametric information. The given design domain including any forbidden zones,

was divided into quadrants, each containing a maximum of five loads and supports.

The mean distances are evaluated between loads and supports for the domain.

The distances are measured along the line of sight, such that no member between

any two loads or between a support and a load intersects a forbidden zone. If such

an intersection occurs, a penalty is assigned by increasing the computed length

substantially, thereby rendering its selection unattractive from the standpoint of

increased weight. A vector VL with components consisting of mean distances

between loads, mean distances between loads and supports, number of loads, and

number of supports, is generated for each load point in the domain.


E dL
nL


VL dLs (4-1)
nLS
nL
ns



Here dL is the load-load distance, dLs is the load-support distance, nL is the number

of loads considered, and ns is the number of supports considered.

Clusters are formed on the basis of the vector of characteristics described in

(4-1), using the following heuristics. Any cluster may not contain more than a total

of five parameters. In addition, it was necessary to include at least one support and








57
one load point in each cluster. Within a cluster, the shortest load-to-load and load-

to-support distances are preferred, as they are most likely to yield a lower weight

configuration. Finally, and to whatever extent possible, an average number of

parameters is maintained for all clusters. Given that the maximum number of

parameters must not exceed five, and the total number of parameters known, a

decision can be made on the number of clusters and on the number of parameters

that each must contain. An arbitrary decision was made to begin generating clusters

from the load located closest to the origin of the coordinate system for the design

domain. The load closest to the first load, but not included in the first cluster was

the starting point for the second cluster. This pattern was repeated until all loads

and supports were assigned to some cluster. The only binding factor in limiting the

number of parameters in any cluster is the guarantee of a simple structural topology

within the cluster. Once a cluster was established, a cluster center was identified.

The center for each cluster was determined by taking the average of the position

coordinates of all parameters in the cluster. The mean distances of all loads and

supports from the center of the cluster, were evaluated for each cluster. Figure 4-3

depicts the clusters for the example problem shown in figure 4-2.

Also important to each cluster are the I/O points at which all relevant load

transfers are to be monitored. It is at these points that the influence of an adjacent

cluster will be transferred to the cluster under consideration. The choice of any node

as an I/O point is based on its proximity to the adjoining cluster boundary. Nodes













































Figure 4-3. Clustering for the example problem.








59
which lie in close proximity to adjoining cluster boundary are all considered I/O

points. These points are very important since the information available at these

points such as loads and displacements for each partial design, have a direct bearing

on the introduction of the connectivity elements between clusters.



Partial Structure Generation



The successful clustering of all input parameters leads to the next important

stage in which a topology is actually generated for each cluster. All the parameters

(both loads and supports) that fall within a cluster are recognized and accounted for.

Such parameters along with their pertinent information are then passed into a

subroutine which prepares an output file, to be used in conjunction with previously

developed knowledge based structural synthesis programs detailed in chapter 2. This

synthesis is sequential, with each load stabilized one at a time. Once a load is

stabilized, the point of application of this load serves as additional support point.

Additionally, nodes are introduced when the shortest distance between some load

and available connection points becomes unacceptably large, along the line of sight.

An example of this could be a situation where a long member were to become

critically unstable under a compressive load. A suitable structure is generated based

on the parameter information within each cluster, and this process is repeated for all

clusters. The partial designs generated for the problem are depicted in figure 4-4.




































































Figure 4-4. Partial structures.


~








61

Inter-Cluster Connectivity



Once all the partial designs are available, the next objective is to identify the

possible I/O points and the design of corresponding connecting elements. The

approach adopted was to consider only adjacent clusters for connectivity as shown

in figure 4-5. Nodes located close to the cluster boundary as measured from the

cluster center, and also in close proximity to the adjacent cluster were tagged as

possible I/O nodes for inter-cluster connections. Connection members of nominal

cross section were introduced from I/O points in the first cluster to all the I/O points

in the adjacent clusters. The I/O points in adjacent clusters were treated as simple

supports, and the reactions at these supports were determined for the applied loads

in the first cluster. Figure 4-6 shows the connectivity elements for the problem under

consideration.

The integration of partial structures generated for all the clusters, to form a

single structure, constitutes the next step in the design procedure. At this stage,

global sensitivity analysis is performed to determine the influence of the design

variables in any partial structure with regards to the load effects introduced during

integration with other partial structures.



Global Sensitivity Analysis.


A description of the method of global sensitivity analysis can be found in
















































Figure 4-5. Allowable connectivities between clusters.




















































A, B, C, D Connectivity members


Figure 4-6. Possible elements of connectivity.


1








64
Sobieski [24]. However, a summary of the approach is presented here for

completeness.

A generic coupled system is represented schematically in figure 4-7. The

system consists of two participating subsystems A and B. Complete coupling is

present as the output of one subsystem influences the output of the other. The

functional analysis equations for these two disciplines can be represented as


((XA YB) YA) = 0 (4-2)




((XB YA) YB) = 0 (4-3)



Here X's are the local or intrinsic variables, and Y's (also referred as extrinsic

variables) are their respective outputs. Some of the Y's comprise the interlinking

parameters between the subproblems as shown in figure 4-7. The total derivative of

the output of each subsystem with respect to their intrinsic variables are obtained on

the basis of first order Taylor series expression as follows:

dYA dY YA dY
= + (4-4)
dXA aXA 9YB dXA



dYB 3YB BYB dYA
= + (4-5)
dXB XB, aYA dXB

The cross sensitivities were derived using chain rule.

































X,



Y,


B


Figure 4-7. An intrinsically coupled system.


XA


7 Y,











dYA aYA dYB
dXY 9YB dXB




dYB aYB dY
dXA YA dXA


(4-6)





(4-7)


These equations can be written in a compact matrix notation as follows:


I

-YB
aY


I

YB,
A


aY,
aYB

I






aYA
9Y,
B

I


dYA
dXA
dYB
dXA


dYA
dX
B
dYB
dX


aYA
axA


0


0


aYB
lx,
Be


(4-8)


(4-9)


It is important to note that the total (coupled) derivatives of the output of each

subsystem with respect to the intrinsic variables of all subsystems depends on the

partial derivatives which may be obtained within each subsystem, and for some

nominal value of the outputs of other subsystems. These sensitivities can be used

later in the optimization process.








67
Design Constraints. In the design problem under consideration, system A can

be viewed as the partial structure in a cluster, and system B used to denote all

possible linking members between two adjacent clusters. Each cluster is considered

individually and similar subsystems are identified. In the equations derived before,

XA's are the cross sectional areas of the members in a structure for cluster #1; YA's

are the outputs for cluster #1, which are a cumulative measure of constraint

satisfaction/violation for that cluster, and the weight of the cluster structure; Y,'s are

the weights (WB) of the interlinking members, and the output reactions which act at

the I/O node in adjacent cluster; X,'s are the cross sectional areas of the interlinking

members. Similar inputs and outputs are identified for cluster #2.

The output from the analysis of any structure consists of the sensitivities of the

cumulative constraint, the weight of the structure, and reactive loads at I/O points

in the adjacent cluster, with respect to all design variables. The cumulative

constraint as used in the present work is taken from Hajela [25] and is written as

follows:


1 m
S= -E + ln( (e9')) (4-10)
P i=1


In the above equation gi's are the m stress and displacement constraints for each

cluster, E is of the order of 103, and p is a parameter taking values between 20 and

30. The effect of the n function is to create an envelope function representative of

all the m individual constraints.







68
A finite element analysis is performed on the partial structure, including a

structural member introduced between chosen I/O nodes on the partial structure and

the structure of an adjacent cluster. The I/O nodes in adjacent clusters are

considered to be simple supports and the reaction forces computed at these nodes.

All such combinations of I/O nodes are considered. The total sensitivities of the

structural weight, cumulative constraints and the reaction forces at I/O points of

adjacent cluster are determined as follows:


dY dQ dWA
dX ldXA' dXA




SdYA d dWA
XA, I X
dX, dX, dXB




dYT dW dFJ
dXA ldXA' dXAj




IdY T dW, dF
dXB dXB' dXB


Reaction forces, and sensitivities of the cumulative constraint, weight and reaction








69
forces, determine those pairs of I/O nodes between which connecting members must

be introduced. This is done on the basis of heuristics embedded in a knowledge

base, and is discussed in the following section.



Heuristics for Assembly of Structures. Pertinent data utilized in the

determination of candidate connectivity elements include all the global sensitivities,

the reaction forces, and displacements at various nodes in the structure. Connectivity

members are evaluated one at a time, until all the connectivity members are

accounted for. The objective is to choose that member which promotes constraint

satisfaction or introduces the least constraint violation, while still minimizing the

structural weight penalty associated with that member. This translates quantitatively

into choosing a member which has the least total sensitivity value of the cumulative

constraint with respect to the cross sectional area of the member under

consideration, provided it also carries the least weight penalty. Towards this end, the

sensitivity of the structural weight to such an addition is obtained. The other

measure of merit in choosing a member is the dependence of the reaction forces at

I/O points in the adjacent cluster, on the member cross section. Such sensitivities

are compared for each possible connectivity member, and the hierarchy of

comparison is the sensitivity of the cumulative constraint, weight, and reaction forces,

in that order. A member which provides a minimum for all the three sensitivities,

or most components of the sensitivity as per the specified hierarchy is selected as the

connectivity element.








70
Implementation



The decomposition method was implemented in the general framework of a

design system described in chapter 2. All algorithmic analyses was performed in

FORTRAN, the heuristics were implemented using the CLIPS syntax, and structural

analysis performed using EAL.

The approach was applied in the generation of topologies of representative

structural systems as illustrated by the example. Figure 4-8 portrays the complete

structure for the example problem.

The identification of the connectivity members leads to the assembly of all

partial designs. Before the integration of partial designs, the members of each

individual structure in each cluster, are optimized for minimum weight. The

elements obtained by this cycle were fixed, and the procedure repeated for all

clusters. The total sensitivities are utilized in the optimization of the structures. Two

different methods of optimization were performed to obtain optimal design. The first

method involved the use of heuristic optimization routines as reported in chapter 2,

while the second was based on traditional procedural methods for optimization. The

structures were optimized subject to the respective cumulative constraint. The

individual constraints for each structure were those introduced due to local nodal

displacement and element stress considerations. The stress constraint on the

connectivity member is considered as a local constraint. The connectivity element

is considered as an integral part of the structure in a cluster, and the I/O node


































































Figure 4-8. Complete structure.


I








72
treated as a simple support, as described in an earlier section. The integrated

structure continually evolves until all clusters have been taken care of. Note that in

each design cycle, the number of design variables are only governed by the number

of members for that cluster. Finally, the integrated structure can be optimized to

accommodate any constraint of a global nature.

Procedural optimization was performed using feasible-usable search algorithm

in ADS[26], and was executed in a batch mode by the design system.













CHAPTER 5
STAGEWISE PROBLEM DECOMPOSITION


In the previous chapter, the concept of decomposition as a solution to complex

design problems was approached from a global sensitivity standpoint. In this chapter,

decomposition is based on concepts of dynamic programming, which are typically

applicable to those design problems that may be considered as a series of

subproblems. The nature of the structural synthesis problem considered here may

be thought of as a serial process, wherein the output from a subproblem is

considered as an input to the succeeding subproblem. Therefore, it is a prime

candidate for adaptation of the dynamic programming approach. The nature of the

formulation transforms a problem into a different form, frequently more suitable for

solution. Reported efforts in the application of dynamic programming to structural

design are somewhat limited. Kirsch [27] provides insight into the application of

dynamic programming to affirm the topology of an existing structure. This

publication also cites other rudimentary applications, such as design of cantilever

beams.

A system with distinct subproblems which is organized such that a change in

the design of a given subproblem influences only downstream subproblems, can be

treated as a serial process. DP is based on a theory proposed in the original work

by Bellman [28] (pp. 12):








74
An optimal policy has the property that whatever the initial state and
decisions are, the remaining decisions must constitute an optimal policy
with regard to the state resulting from the first decision.


Dynamic programming is useful when the decision sequence is long, and the number

of decisions is considerably large. The basic feature is that at each given

subproblem, the decision is affected only by the variables and functions present in

that subproblem, and is insulated from the other subproblems.

Two different approaches were taken for structural synthesis and are reported

in the following sections. The class of structural design problems under consideration

were transformed into series of serial stages, so that principles of dynamic

programming could be applied. Once a minimum weight topology was identified, the

topology was optimized for minimum weight using genetic algorithms. A second

approach which is the central theme of the next chapter, involves a genetic algorithm

based methodology for topology generation, where all the candidates for the initial

population are derived from a dynamic programming approach. Finally, the

successful topology is optimized using genetic algorithms. The following sections

outline principles of dynamic programming, genetic algorithms, and their adaptation

in problem decomposition. Also reported is an illustrative example of the proposed

implementation.



Dynamic Programming(DP)


Figure 5-1 shows a serial multistage system, for which dynamic programming













































Zn Z2


Figure 5-1. Multistage serial system.


2Z Z,-,








76
can be applied. Since dynamic programming involves the execution of subsystems

in a specific order, it is convenient to number the subsystems. The following terms

and definitions are used in the formulation of a dynamic programming problem.

'Sn' state variables are those which carry information from stage to stage. The

variable 'n' is referred to as the stage number. The input to nth stage is also the

output from stage n-1. Therefore, we have


S,nl=S, n=1,2...N-1 (5-1)



The functional relationship refers to flow of data between the subproblems (stages).

'dn' decision variable at stage n, is an input variable which supplies input

information to the nth subproblem. All the input variables which are not stage

variables are generally referred to as decision variables.

'n' system stage is the subsystem where transformation of all stage variables

and decision variables take place in accordance with functional relationships between

them. The output state variable Sn is related to the input variable S through the

transformation function, and can be represented as


Sn =T,(S,,d,) (5-2)



'Z,' stage cost measures the objective function value at stage 'n'. The stage

cost is a function of the input state and decision variables, and is represented as











Z, = F,(S.,dn) (5-3)



The total objective is the summation of all individual objectives in all the stages


Z = F,(Sn,d,) (5-4)
n=l



The above definitions and relationships constitute a typical DP formulation.

An adaptation to our problem needs some modifications as far as the variables are

concerned, and this is described in the following section.



DP Adaptation in Stagewise Decomposition



The application of the DP like method to the problem of structural synthesis

is largely dependent on the definition of the problem. The DP like approach is not

particularly applicable to structural synthesis if it involves a routine optimization.

Since DP is primarily a method of serial optimization, its adaptation will not be

satisfactory in case of nonlinear design problems such as in structural member sizing.

Identification of a serial structure is the first phase. This involves the creation

of subproblems as required in establishing a serial architecture. In the case of

structural synthesis problems considered here, the adaptation was viewed from the

point of individual load stabilization. The basic approach was again based on

stabilizing each load, taken one at a time. The load stabilization task was considered








78
as 'n' distinct subproblems, with each subproblem having an objective to stabilize an

individual load in all possible configurations, by introducing nominal cross sectional

area members. The problem is propagated in this manner from one subproblem to

the other, until all the loads are stabilized, simultaneously carrying forward the

optimal weight (cost) thus far produced. The final topology at the end of the n-th

subproblem is the one which yields the lowest weight. This topology was then

optimized by resizing the member cross sections, subject to strength, displacement,

and frequency constraints.

The basic design problem considered here is the same as reported in the

previous chapters. The objective is to design a minimum weight topology, given a

set of loads, possible support points, and domains where no members are allowed.

Heuristics directed load stabilization approach (chapter 2) is adopted to transmit

each load to the support points.

In accordance with the definitions reported in the previous sections, variables

can be associated with the structural synthesis problem. Each load is considered as

a 'stage' in the design process. All possible supports to which each load can be

connected are considered as 'state' variables for that stage. In addition, all previously

stabilized loads in the upstream stages, are also considered as 'state' variables. The

'cost' function for each stage is the weight of the partial structure for each stage.

Details of the adaptation of DP to our problem, are described in the next section.








79
Implementation



The automated design procedure is implemented under the guidance of the

architecture in chapter 2. Initially, as before, the design domain is enumerated and

data retrieved pertaining to load-support and load-load distances. If such a

connection intersects a prohibited zone, a weight penalty is associated with the

connection. A topology knowledge base (chapter 2) drives the decision making

process to stabilize each load individually. Nominal cross section elements are

introduced when a load is stabilized. In the process all possible configurations are

derived and their cost function in terms of weight evaluated. Such evaluations are

performed for all stages.

The process of information transfer from an upstream stage to a downstream

stage as defined by stage function, consists of the optimal cost functions for each

state in the immediate upstream stage. Figure 5-2 depicts the flow of information

from an upstream stage to a downstream stage. The '*' indicates the optimal value

as transmitted from the upstream stage. At each stage in the subsystem, the optimal

cost function is given by the equation


fN(SN) = min(FN(SN) + fN,(SN)) (5-5)



where 'f is the optimal cost, 'N' is the current stage, and 'F' is the cost (weight) for

the current state. Therefore, the optimal cost function is cumulatively updated

during the progress of the analysis of subsequent stages. Ultimately, at the end of



















ia+ la

.2a+ b *

*


b+2a+ Ib *


Stage 2


Stage 3


Optimum : 3b+2a lb

Figure 5-2. Information flow between stages.


Ia


Stage 1








81

the nth stage, after all the loads have been stabilized, the topology with the least

weight is identified. At this point, the analysis is backtracked to identify the states

corresponding to each of the upstream stages which contributed to this optimal

solution.

The most important advantage in the application of DP to design synthesis is

that all infeasible designs are pruned at each stage in a methodical way. Paths which

hold promise are only allowed to propagate to succeeding stages.

An example problem that was considered for design synthesis based on the DP

approach is shown in figure 5-3. Clearly, the number of stages can be identified as

4, which correspond to the total number of loads to be stabilized. The stabilization

proceeds from load nearest to the origin of the coordinate system and moving away

from it. All previously stabilized loads are also considered as possible supports.

Hence, the number of states keep increasing with stages in the downstream. For the

problem considered here, more than fifty topologies are possible, but the pruning of

infeasible paths after the analysis of each subsystem reduces the available paths to

a manageable number (24). Therefore, combinatorial explosions in the topology

generation process is checked due to the pruning that is introduced at each stage.

To further demonstrate the pruning of alternatives, let us again consider the example

in figure 5-2. The number of stages for the problem is three. The three states in

stage 2, combined with two possibilities from stage 1, results in six possible outputs.

But since only optimal states are considered for propagation, only three states are

identified, one each corresponding to each of the three states at stage 2. Similarly,































800. lb

(40 80) (110.80) L
(40. 0)
300. Ib

0.40) 00. lb 500. Ib 500. b
0 I b (70. 40) (
(95.40)
(50.40) 500. lb




(140.10)

(5.10) (50,10) (70,10) (95.10) (110.10)


Figure 5-3. Example design problem.








83

there are two states at stage 3. Correspondingly, for each of the states from stage

2, there are two possible outputs from stage three, with a total of six. However, the

actual number of outputs from stage 3 would have been twelve if all the possibilities

in stage 2 had been considered. The use of this process, therefore, alleviates the

combinatorial explosion in such problems.

The topology corresponding to the least weight structure for the example

problem, is the focus of optimization using a genetic algorithm [29, 30, 31] outlined

in the next section.



Genetic Search



Genetic search has its basis in Darwin's theory of evolution. It belongs to the

general category of stochastic search techniques. A population consisting of design

alternatives are allowed to follow the principles of evolution, such as to reproduce

and cross among themselves, with a bias towards more fit members of the population.

Combination of most fit members in the population results in a progeny population

that is generally more fit than the parents. A measure related to the relative merit

of members in the population serves as the objective function. There are three basic

operations which constitute a genetic search process, namely selection, crossover and

mutation, and these are explained in the next section. Representation of the

evolution process itself requires completion of three preliminary steps. The first is

a definition of a bit string representation scheme to portray possible candidate








84

solutions to the problem. Second, a suitable fitness function to provide for the

ranking of the members in the population based on their merit. The last and most

critical is the development of transformations to duplicate the evolution process

consisting of the above mentioned operations.

Each design variable is represented as a fixed length string of O's and l's,

typically the components of a binary number. A scaling scheme provides for the

conversion of the binary number to real number and vice-versa. The binary strings

for each design variable are placed end-to-end; this constitutes the string for one

design. Several such designs constitute a population, with each member associated

with a fitness measure. This is a measure of the relative goodness of the design, and

is a composite measure of the objective function and the constraint satisfaction.



Basic Operations. Once a population of designs becomes available, then

selection is the first concept to be introduced. The aim of this process is to select

only those candidate designs based on their relative witnesses, which are better than

some others. This is meant to provide bias in the population to contain more fit

members and to rid the population of less fit members.

The second component of the search is crossover, and pertains to the

exchange of characteristics between select members of the population. Crossover

entails selecting a start and end position on a pair of mating strings at random, and

exchanging the O's and 1's between these positions on one string with that from the

mating string. This is illustrated as follows:










parents = 111000100101

parent = 010111001001



child = 111011001101

child = 010100100001

Mutation is the third transformation operator in the genetic refinement

process, and safeguards the premature loss of valuable genetic material during

selection and crossover. In the implementation, this corresponds to selecting few

members of the population, determining at random a location on the string and

switching the 0 or 1 at this location.

The above steps are repeated for successive generations of the population

until either convergence is achieved on the fitness function, or a predetermined

number of evolution cycles have been completed. The approach has shown promise

in the current application because of its ability to search in a discontinuous design

space where structural members can be either removed or introduced during

optimization.



Optimization. The topology obtained from the DP method is optimized using

genetic search. In this implementation the constraints were derived from

displacements, natural frequency and stress requirements. The objective is to

minimize the weight of the structure subject to the above constraints. A cumulative

function with penalties assigned to the constraints was considered as the objective










function, and is given by


z = w + R < gi >2 (5-6)



where 'w' is the weight, 'R' is a penalty parameter such that the second term on rhs

in (5-6) is of the same order of magnitude as the weight, and represents the

normalized constraints and is defined as follows:


< gi > = [g,0] if [g > 0gi < 0] (5-7)



Genetic search maximizes a prescribed fitness function. Since our problem is one

of function minimization, a transformation of the objective is necessary. This is as

follows:


Z = Zmax (5-8)



where zmax is the maximum value of z in the given population. The value z* can

serve as the fitness function for the population.

Initial population consisted of twenty different topologies, with cross sections

of the members in the structure considered as design variables. All the designs in

the population were randomly generated. A nine bit string representation scheme

was chosen to represent each individual cross sectional area. A finite element

analysis of each of these designs yields the values for the displacements, frequency,

and member stresses. Constraints were evaluated and the fitness function








87

determined using the equations in the previous section. Since the population is

randomly generated and is limited in size, if raw values of the fitness function are

used in reproduction, the more fit designs dominate the population and the process

may converge prematurely. Hence, the raw fitness function values were scaled

appropriately to prevent premature convergence. Also, in the event that the average

fitness is close to the maximum fitness, the scaling magnifies the difference in relative

merits between members. The scaling used in the present implementation was

linear, and was defined as follows:


f = c,1f + C2 (5-9)



The scaled maximum fitness is given as k*fa,, where k is typically in the range of 1.0

to 2.0. For a chosen value of k = 1.5, the c's in the above equation are derived from


c1 = 0.f (5-10)
f -fr
fmax J avg




2 = f(fma-1.5*favg) (5-11)
fmax J avg


where fag is the average fitness for a given generation.

Scaled fitness values f were used to detect the most fit members of the

population for reproduction. Simulations of weighted roulette wheel were used for

such a selection, where each member of the population occupied a sector on the








88
wheel in proportion to its scaled fitness value. This scheme was then used to select

the mating members of the population. In any population, if one can identify any

particular pattern in a string of length (L), for example

H(1) = ***10**0****

where H is the pattern, and indicates that the position can be occupied by either

a 0 or a 1, then we can predict the number of occurrences of the same pattern in the

succeeding generation [32]. In any schema H, an important parameter is a defining

length (6), which is the distance between the first and last specific digit in a string;

for H(1), 6 is (8 4) = 4. Also important is the number of specific digits in any

schema H, and is referred to as the order O(H); the order of H(1) is 3. If m(H,t)

represents the number of occurrences of the pattern in generation 't', then the

number of strings with the same pattern in the next generation is given by


m (H,t1) = m(Ht)f() (5-12)
favg



where f(H) is the fitness for the pattern. It is easy to conclude from the above

expression that patterns with higher than average fitness tend to increase

exponentially over cycles of evolution.

Next, the crossover operation described in the previous section was performed

on the selected population with the defined probability of crossover p, (- 0.6 to 0.8),

to yield the new designs. Finally, a low mutation probability pm of .009 was used

during mutation transformation. In this process, a certain position is chosen along








89
the string, and based on pm, the value at that position was simply inverted. The

crossover and mutation operations can be disruptive in the evolution process. As

shown by Holland [32], in terms of the probabilities of p, and pm, the growth or

decay in a succeeding generation of a pattern H can be written as follows:


m(H,t+l) 2 m(H,t)* *(-p* O(H)*p) (5-13)
avg L-1



The above expression modifies the growth in population due to reproduction, by

considering the effects of crossover and mutation transformations. Equation (5-13)

shows that the defining length (6) plays a significant role in the occurrence of the

schema in a succeeding generation. If 6 is large for schema that has a relatively low

fitness, then its number in succeeding generation would decrease rapidly.

The topology at the end of twenty five generations of evolutions are shown in

figure 5-4, and the optimum value of the design variables are given in table 5-1.

Figure 5-5 gives the variation of the maximum fitness value in each generation of the

genetic search process. As can be seen, there is a general trend towards increasing

magnitude for the fitness function.













































Figure 5-4. Final topology for the example problem.