NASA 

CR 

3226 

c. 1 

NASA Contractor Report 3226 

Dual Methods and Approximation 

Concepts in Structural Synthesis 

Claude Fleury and Lucien A. Schmit, Jr. 

GRANT NSG- 1490 

DECEMBER 1980 
TECH LIBRARY KAFB, NM 

llnlllllilnmIIlllllll

00b2028 

NASA Contractor Report 3226 

Dual Methods and Approximation 

Concepts in Structural Synthesis 

Claude Fleury and Lucien A. Schmit, Jr. 

University of California, Los Angeles 

Los Angeles, California 

Prepared for 

Langley Research Center 

under Grant NSG-1490 

MSA 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1980 

Preface 

This report presents some major findings of a continuing research 

program entitled "Fundamental Studies of Methods for Structural Synthesis," 

sponsored by NASA Research Grant No. NSG-1490. The research effort reported 

herein was carried out in the Department of Mechanics and Structures at UCLA 

during the period from February 1978 to August 1979. 

The ACCESS 3 computer program, which implements the new methods set forth 

in this report, is a research type computer program that was written by adding the dual formulation as well as the DUAL 1 and DUAL 2 maximization algorithms into the previously developed ACCESS 2 program. The ACCESS 3 computer 

program was delivered to the NASA Langley Research Center in June 1979. 

Dr. Claude Fleury carried primary responsibility for the development of ACCESS 3. 

Professor Lucien A. Schmit serves as principal investigator and Dr. J. Sobieski, 

of the NASA Langley Research Center, is the cognizant NASA Technical Officer 

for this research program. 

The authors want to take this opportunity to express their gratitude to 

Dr. G.N. Vanderplaats of the NASA Ames Research Center for his cooperation 

and help in preparing the ACCESS 3 program for delivery to the NASA Langley 

Research Center. We also want to thank Deborah Haines of the Mechanics and 

Structures Department for her careful attention to detail in typing this report. 

iii 

CONTENTS 

LISTOF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 

LISTOFFIGURES ........................... 

SUMMARY ............................... 

1. INTRODUCTION. .......................... 

2. PRIMAL AND DUAL APPROACHES TO STRUCTURAL SYNTHESIS. ....... 

2.1 

2.2 

Formulation of the Structural Synthesis Problem. . . . . . . 

The Constrained Minimization Techniques. . . . . . . . . . . 

2.2.1 The Primal Methods (Direct Approach). . . . . . . . . 

2.2.2 The Penalization Methods (Transformation Approach). . 

2.2.3 The Linearization Methods (Indirect Approach) . . . . 

2.2.4 The Multiplier Method . . . . . . . . . . . . . . . . 

2.2.5 The Dual Methods in Convex Programming. . . . . . . . 

The Approximation Concepts Approach. . . . . . . . . . . . . 

2.3.1 Reduction of the Problem Dimensionality . . . . . . . 

10 

14 

15 

16 

17 

18 

19 

2.3 19 

20 

2.3.2 Linearization Process . . . . . . . . . . . . , . . . 21 

2.3.3 Primal Solution Scheme. . . . . . . . . . . , . . . . 23 

2.4 Joining Approximation Concepts and Dual Formulation. . . . . 

2.4.1 Primal and Dual Solution Schemes. . . . . . . . . . . 

26 

26 

2.4.2 The Dual Method Formulation . . . . . . . . . . . . . 29 

2.5 Relations with the Optimality Criteria Approaches. . . . . . 34 

2.5.1 Conventional and Generalized Optimality Criteria. . . 35 

2.5.2 The Constraint Gradients: Pseudo-Loads Versus 

Virtual Load Techniques . . . . . . . . . . . . . . . 39 

2.5.3 The Stress Constraints: Zero Versus First Order 

Approximations. . . . . . . . . . . . . . . . . . . . 42 

xii 

1 

5 

9 

‘-A 
2.5.4 The Scaling of the Design Variables. ......... 45 

3. DUAL METHODS FOR CONTINUOUS DESIGN VARIABLES ........... 47 

3.1 The Second Order Discontinuity Planes ............ 47 

3.2 Characteristics of the Dual Function-Continuous Case. .... 50 

3.3 Dual 2 -NewtonTypeMaximizer ................ 51 

4. DUAL METHODS FOR DISCRETE DESIGN VARIABLES ............ 57 

4.1 The First Order Discontinuity Planes. ............ 58 

4.2 Characteristics of the Dual Function-Mixed Case ....... 60 

4.3 The Pure Discrete Case. ................... 62 

4.4 Construction of a Unique Ascent Direction .......... 64 

4.5 Dual 1 - Gradient Projection Type Maximizer ......... 68 

4.5.1 Direction Finding Process. ........... ; .. 70 

4.5.2 Restart of the Algorithm ............... 74 

4.5.3 Retrieval of the Primal Variables. .......... 76 

4.5.4 One Dimensional Maximization ............. 80 

5. THE ACCESS 3 COMPUTER PROGRAM. .................. 87 

5.1 Scope of the ACCESS 3 Code. ................. 88 

5.2 Program Organization .................... 93 

6. NUMERICALEXAMPLES ........................ 97 

6.1 lo-Bar Truss (Problem 1). .................. 97 

6.1.1 Case A: Equality Contraints on Displacements ..... 98 

6.1.2 Case B: Pure Continuous Problem. ........... 88 

6.1.3 Case C through Case E: Assessment of DUAL 1. ..... 100 

6.1.4 Case F: Pure Discrete Problem. ............ 180 

6.2 25-Bar Truss (Problem 2). .................. 101 

6.2.1 Case A: Pure Continuous Problem ........... 102 

vi 
6.2.2 Case B through Case D: Pure Discrete Problems ..... 103 

6.3 72-Bar Truss (Problem 3) ................... 104 

6.4 63-Bar Truss (Problem 4) ................... 105 

6.5 Swept Wing Model (Problem 5) ................. 109 

6.5.1 Case A: Pure Continuous Problem ............ 110 

6.5.2 Case B: Puxe Discrete Problem ............. 112 

6.6 Delta Wing Model (Problem 6) ................. 112 

6.6.1 Case A: Pure Continuous Problem ............ 115 

6.6.2 Case B: Mixed Continuous-Discrete Problem ....... 117 

7. CONCLUSIONS ............................ 121 

REFERENCES .......................... ..12 5 

TABLES ............................ ..13 1 

FICUE=..............................18 5 

vii 
List of Tables 

Table 1 

Table 2 

Table 3 

Table 4A 

Table 4B 

Table 5A 

Table 5B 

Table 6A 

Table 6B 

Table 7A 

Table 7B 

Table 8 

Table 9A 

Table 9B 

Table 10A 

Table 10B 

Alternate Paths After Solving ODM (Dual 1) . . . . . . . . . . 131 

Available Options for Natural Frequency Constraints . . . . . 132 

Algorithm Options for Various Kinds of Problems . . . . . . . 132 

Definition of Problem 1: Planar lo-Bar Cantilever Truss . . . 133 

(SI Units) 

Definition of Problem 1: Planar lo-Bar Cantilever Truss . . . 134 

(U.S. Customary Units) 

Iteration History Data for Problem 1 (Case A) . . . . . . . . 135 

Planar lo-Bar Cantilever Truss 

(SI Units) 

Iteration History Data for Problem 1 (Case A) . . . . . . . . 136 

Planar lo-Bar Cantilever Truss 

(U.S. Customary Units) 

Final Designs for Problem 1 . . . . . . . . . . . . . . . . . 137 

Planar lo-Bar Cantilever Truss 

(SI Units) 

Final Designs for Problem 1 . . . . . . . . . . . . . . . . . 138 

Planar lo-Bar Cantilever Truss 

(U.S. Customary Units) 

Iteration History Data for Problem 1 (Cases B-F) . . . . . . . 139 

Planar lo-Bar Cantilever Truss 

(SI Units) 

Iteration History Data for Problem 1 (Cases B-F) . . . . . . . 140 

Planar lo-Bar Cantilever Truss 

(U.S. Customary Units) 

Available Discrete Values for all Example Problems . . . . . . 141 

Definition of Problem 2: 25-Bar Space Truss . . . . . . . . . 142 

(SI Units) 

Definition of Problem 2: 25-Bar Space Truss . . . . . . . . . 143 

(U.S. Customary Units) 

Iteration History Data for Problem 2 . . . . . . . . . . . . . 144 

25-Bar Space Truss 

(SI Units) 

Iteration History Data for Problem 2 . . . . . . . . . . . . . 145 

25-Bar Space Truss 

(U.S. Customary Units) 

viii 
List of Tables, continued 

Table 11A 

Table 11B 

Table 12A 

Table 12B 

Table 13A 

Table 13B 

Table 14A 

Table 14B 

Table 15A 

Table 15B 

Table 16A 

Table 16B 

Table 17A 

Final Designs for Problem 2 . . . . . . . . . . . . . . . . 146 

25-Bar Space Truss 

(SI Units) 

Final Designs for Problem 2 . . . . . . . . . . . . . . . . 147 

25-Bar Space Truss 

(U.S. Customary Units) 

Definition of Problem 3: 72-Bar Space Truss . . . . . . . . 148 

(SI Units) 

Definition of Problem 3: 72-Bar Space Truss . . . . . . . . 149 

(U.S. Customary Units) 

Iteration History Data for Problem 3 . . . . . . . . . . . 150 

72-Bar Space Truss 

(SI Units) 

Iteration History Data for Problem 3 . . . . . . . . . . . 151 

72-Bar Space Truss 

(U.S. Customary Units) 

Final Designs for Problem 3 . . . . . . . . . . . . . . . . 152 

72-Bar Space Truss 

(SI Units) 

Final Designs for Problem 3 . . . . . . . . . . . . . . . . 153 

72-E&r Space Truss 

(U.S. Customary Units) 

Definition of Problem 4: 63-Bar Space Truss . . . . . . . . 154 

(SI Units) 

Definition of Problem 4: 63-Bar Space Truss . . . . . . . . 155 

(U.S. Customary Units) 

Iteration History Data for Problem 4 . . . . . . . . . . . 156 

63-Bar Space Truss 

(SI Units) 

Iteration History Data for Problem 4 . . . . . . . . . . . 157 

63-Bar Space Truss 

(U.S. Customary Units) 

Final Designs for Problem 4 . . . . . . . . . . . . . . . . 158 

63-Bar Space Truss 

(SI Units) 

ix 
List of Tables, continued 

Table 17B 

Table 18A 

Table 18B 

Table 19A 

Table 19B 

Table 20A 

Table 20B 

Table 21A 

Table 21B 

Table 22A 

Table 22B 

Table 23A 

Table 23B 

Table 24A 

Final Designs for Problem 4 . . . . . . . . . . . . . . . . . 160 

63-Bar Space Truss 

(U.S. Customary Units) 

Definition of Problem 5: Swept Wing Model . . . . . . . . . . 162 

(SI Units) 

Definition of Problem 5: Swept Wing Model . . . . . . . . . . 163 

(U.S. Customary Units) 

Nodal Coordinates for Swept Wing Model (Problem 5) . . . . . 164 

(SI Units) 

Nodal Coordinates for Swept Wing Model (Problem 5) . . . . . 165 

(U.S. Customary Units) 

Applied Nodal Loading for Swept Wing Model (Problem 5) . . . 166 

(SI Units) 

Applied Nodal Loading for Swept Wing Model (Problem 5) . . . 167 

(U.S. Customary Units) 

Iteration History Data for Problem 5 . . . . . . . . . . . . . 168 

Swept Wing Model 

(SI Units) 

Iteration History Data for Problem 5 . . . . . . Y . . . . . 169 

Swept Wing Model 

(U.S. Customary Units) 

Final Designs for Problem 5 . . . . . . . . . . . . . . . . . 170 

Swept Wing Model 

(SI Units) 

Final Designs for Problem 5 . . . . . . . . . . . . . . . . . 171 

Swept Wing Model 

(U.S. Customary Units) 

Definition of Problem 6: Delta Wing Model . . . . . . . . . . 172 

(SI Units) 

Definition of Problem 6: Delta Wing Model . . . . . . . . . . 173 

(U.S. Customary Units) 

Nodal Coordinates for Delta Wing Model (Problem 6) . . . . . 174 

(SI Units) 

X 
List of Tables, continued 

Table 24B 

Table 25A 

Table 25B 

Table 26A 

Table 26B 

Table 27A 

Table 27B 

Table 28A 

Table 28B 

Table 29 

Nodal Coordinates for Delta Wing Model (Problem 6) . . . 

(U.S. Customary Units) 

Fuel Mass Distribution for Delta Wing Model (Problem 6) 

(SI Units) 

Fuel Weight Distribution for Delta Wing Model (Problem 6) 

(U.S. Customary Units) 

Iteration History Data for Problem 6 . . . . . . . . . . 

Delta Wing Model 

(SI Units) 

. 

. . . 178 

Iteration History Data for Problem 6 . . . . . . . . . . . . . 179 

. . 175 

. . 176 

. . 177 

Delta Wing Model 

(U.S. Customary Units) 

Initial and Final Designs for Problem 6 . . . . . . . . . . . 180 

Delta Wing Model 

(SI Units) 

Initial and Final Designs for Problem 6 . . . . . . . . . . . 181 

Delta Wing Model 

(U.S. Customary Units) 

Final Webs Thicknesses for Problem 6 . . . . . . . . . . . . . 182 

Delta Wing Model 

(SI Units) 

Final Webs Thicknesses for Problem 6 . . . . . . . . . . . . . 183 

Delta Wing Model 

(U.S. Customary Units) 

Detailed Iteration History Data for Problem 6 . . . . . . . . 184 

Delta Wing Model-Mixed Case (DUAL 1) 

UNITS 

All data and results presented in this report are given in SI or 

SI and U.S. Customary units (see A and B Tables,respectively). Unless 

otherwise noted, all inputs to and outputs from ACCESS 3 were in U.S. 

Customary units and the computations were executed on the IBM 360/91 

at CCN, UCLA using a single precision version of the program. 

xi 
List of Figures 

Fig. 1 Key to a Tractable Formulation - . . . . . . . - - . . . . . 185 

Fig. 2 Design Space for 3-Bar Truss - - . . - - . . . . . . . - - - 186 

Fig. 3 3-Bar Truss - Trajectories in Reciprocal Space . . . . . . . 187 

Fig. 4 3-Bar Truss - Convergence of Weight. . . . . . . . . . . . . 188 

Fig. 5 Zero and First Order Approximations. . . . . . . . . . . . . 189 

Fig. 6 Dual 2 Algorithm - Block Diagram ...... : ....... 190 

Fig. 7 Seek Max of a(x) in Subspace M ............... 191 

Fig. 8 Simple 2D Example - Pure Discrete Problem. . . . . . . . . . 192 

Fig. 9 

Fig. 10 

Fig. 11 

Fig. 12 

Fig. 13 

Projecting Multiple Gradients into Discontinuity Plane . . . 193 

Dual 1 Algorithm - Block Diagram . . . . . . . . . . . . . . 194 

One Dimensional Maximization Scheme (Dual 1) . . . . . . . . 195 

Basic Organization of ACCESS 3 ............... 196 

Planar Ten-Bar Cantilever Truss (Problem 1). ........ 197 

Fig. 14 Iteration History for Problem 1 (Case B) . . . . . . . . . . 198 

Ten-Bar Cantilever Truss 

Fig. 15 

Fig. 16 

25-Bar Space Truss (Problem 2) . . . . . . . . . . . . . . . 199 

Iteration History for Problem 2 (Case A) . . . . . . . . . . 200 

25-Bar Space Truss 

Fig. 17 72-Bar Space Truss (Problem 3) . . . . . . . . . . . . . . . 201 

Fig. 18 Iteration History for Problem 3. . . - . . . . . . . . . . . 202 

72-Bar Space Truss 

Fig. 19 63-Bar Space Truss (Problem 4) . . . . . . . . . . . . . . . 203 

Fig. 20 Iteration History for Problem 4. . - . - - - - . . . . . . . 204 

63-Bar Space Truss 

Fig. 21 Swept Wing Analysis Model (Problem 5). . . - - . - . . . . . 205 

Fig. 22 Swept Wing Design Model (Problem 5). . - - - - . . . . . . . 206 

xii 
Fig. 23 Iteration History for Problem 5 . . . . . . . . . . . . . . 207 

Swept Wing Model 

Fig. 24 Delta Wing Analysis Model (Problem 6) , . , . . , . . . . . 208 

Fig. 25 Delta Wing Design Model (Problem 6) . . . . . . . . . . . . 209 

Fig. 26 Iteration History for Problem 6 . . . . . . . . . . . . . . 210 

Delta Wing Model 

xiii 
SUMMARY 

Approximation concepts and dual method algorithms are combined to 

create a new method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of 

the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit 

dual functions, which are maximized subject to nonnegativity constraints 

on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality 

criteria approach. The dual method is successfully extended to deal with 

pure discrete and mixed continuous-discrete design variable problems. The 

power of the method presented is illustrated with numerical results for 

example problems, including a metallic swept wing and a thin delta wing with 

fiber composite skins. 

In Chapter 1, a brief literature review is presented, with an outline 

of the historical background of the present work, namely, the rigor of the 

mathematical programming approach and the efficiency of the optimality 

criteria approach. 

In Chapter 2, the structural synthesis problem considered is stated 

as a nonlinear programming problem. The constrained minimization methods 

available to solve this problem are briefly described. It is concluded 

that the well established approximation concepts approach can be interpreted 

as a mixed primal-linearization mathematical programming method. Indeed, 

the initial problem is transformed into a sequence of linearized problems, 

however each subproblem is solved partially using a primal method that 
insures generation of a sequence of steadily improved feasible designs. In 

view of the high quality of the approximate problem statement, it is suggested that each explicit problem can be solved exactly, rather than partially. For that purpose, dual algorithms of convex programming are seen 

to be especially appropriate, because the number of dual variables, which 

are associated with the linearized behavior constraints, is generally much 

smaller than the number of design variables. This dual solution scheme, 

which no longer produces a sequence of always feasible designs, is then 

related to the optimality criteria techniques, in which basically the same 

explicit approximate form of the constraints is achieved by neglecting internal force redistribution. 

It is shown in Chapter 3 that, when all the design variables are 

assumed to vary continuously, there are hyperplanes in the dual space where 

the second partial derivatives of the dual function exhibit discontinuity. 

Nevertheless, a 2nd order Newton type of maximization algorithm (called 

- 

DUAL 2) can be devised that is especially well suited to the solution of the 

dual problem in the pure continuous case. 

In Chapter 4, the dual method approach is extended to deal with discrete design variables, e.g., available cross-sectional areas of bars, available gauge sizes of sheet metal,the number of plies in a laminated composite 

skin, etc . ..When the primal structural synthesis problem involves discrete 

design variables, there are hyperplanes in the dual space where the first 

partial derivatives of the dual function exhibit discontinuity. Therefore 

a @t order gradient projection type of maximization algorithm (called DUAL 1) 

is devised, that can accommodate the local discontinuities in gradient of 

the dual function. The DUAL 1 algorithm can handle problems involving a 

2 
mix of discrete and.continuous design variables, as well as the two limiting special cases, namely the pure discrete case and the pure continuous 

case. 

Chapter 5 is devoted to description of the ACCESS 3 computer program, 

where approximation concepts and dual methods are effectively combined to 

produce an efficient minimum weight structural design capability. The scope 

and organization of the ACCESS 3 code are successively described. 

Finally detailed numerical results for various structural optimization 

problems are presented in Chapter 6. For pure continuous variable problems, 

the numerical results obtained with the DUAL 2 optimizer indicate that the 

improved analysis/synthesis capability developed by combining approximation 

concepts and dual methods is remarkably efficient. Computational effort 

expanded in the optimization portion of the program is reduced dramatically 

in representative examples and the number of reanalyses required to converge 

the overall optimization process is reduced significantly. Results for problems involving discrete design variables show that the DUAL 1 optimizer 

appears to have promise as a practical design tool. The collection of examples offered is made up of several well known truss test problems, a metallic 

swept wing and a thin delta wing with fiber composite skins. 

3 

1. INTRODUCTION 

While the coupling together of finite element methods of structural 

analysis and mathematical programming techniques was first suggested in 

1960 [see Ref. 11, computationally efficient practical capabilities such as 

those represented by the WIDOWAC [e.g., Refs. 2, 3,and 41 and the ACCESS 

[e.g., Refs. 5, 6 and 71 codes did not begin to emerge until the 1970's. 

During the late 1960's and early 1970's many investigators focused their 

efforts on constructing automated structural design procedures based on 

fully stressed design and discretized optimality criteria concepts [e.g. 

Refs. 8 through 171. These efforts to create practical automated design 

for large scale structural systems culminated in the development of the 

ASOP and FASTOP computer programs [Refs. 12 and 18 through 241. 

The main obstacles to the implementation of efficient mathematical 

programming based structural synthesis methods prior to 1970 were associated 

with the fact that the general formulation of the basic structural design 

problem involves: (1) large numbers of design variables; (2) large numbers 

of inequality constraints; (3) many inequality constraints that are computationally burdensome implicit functions of the design variables. The 

introduction of approximation concepts [Ref. 251 leading to a sequence of 

tractable approximate problems via the use of design variable linking 

(and/or basis reduction), temporary constraint deletion (regionalization 

and truncation), and the construction of high quality explicit approximations for retained constraints (intermediate variables and Taylor series 

expansion), has led to the emergence of mathematical programming based 

structural synthesis methods that are computationally efficient [e.g., 

Refs. 3 through 71. 

5 
Ihe development of discretized optimality criteria methods usually 

involves: (1) derivation of a set of necessary conditions that must be 

satisfied at the optimum design; and (2) construction of an iterative 

redesign procedure that drives the initial trial design toward a design 

which satisfies the established necessary conditions. Design procedure 

based on optimality criteria generally entail two distinct types of 

approximations: (1) those associated with identifying how many and which 

constraints will be critical at the optimum design; and (2) those associated 

with development of the iterative redesign rule. As first noted in 

Ref. 1261, the essential difficulties involved in applying optimality 

criteria methods to the general structural synthesis problem are those 

related to identifying the correct critical constraint set and the proper 

corresponding set of passive members [see also Refs. 15 and 171. These 

difficulties were recognized and addressed with varying degrees of success 

in studies such as those reported in Refs. 127, 28 and 291. However it was 

only with the advent of the dual formulation set forth in Refs. [30 and 311 

that these obstacles were conclusively overcome. Introduction of the dual 

formulation resolves the essential difficulties inherent to the optimality 

criteria method because determining the critical constraint set and keeping 

track of the status of each design variable (active or passive) becomes an 

intrinsic part of the algorithm used to find the maximum of the dual function 

subject to nonnegativity constraints. In Ref. [32], the dual formulation 

is interpreted as a generalized optimality criteria method and it is shown 

to be well suited to the efficient solution of structural design optimization problems with relatively few critical constraints. In Refs. 133 and 

341, the dual method is presented as a basis for the coalescing of the 

6 
mathematical progranrming and optimality criteria approaches to structural 

synthesis. 

In this work, the approximation concepts approach to structural 

synthesis is combined with the dual method formulation to create a powerful new method for minimum weight design of structural systems. The dual 

method is successfully extended to deal with pure discrete and mixed 

continuous-discrete design variable problems. Approximation concepts are 

used to convert the general structural synthesis problem into a sequence 

of explicit primal problems of separable algebraic form. The dual method 

formulation, which exploits the separable form of each approximate problem, 

is used to construct a sequence of explicit dual functions. These dual 

functions are maximized subject to nonnegativity constraints on the dual 

variables. The efficiency of the method is due to the fact that the 

dimensionality of the dual space, where most of the optimization effort is 

expended, is relatively lo w+ for many structural optimization problems of 

practical interest. Furthermore, in the dual formulation the only inequality constraints are simple nonnegativity requirements on the dual 

variables. 

In contrast to the interior point penalty function methods used in 

Refs. 12-4 and 5-71, the dual methods employed in this work capitalize 

upon the separable form of the approximate problem at each stage and instead 

of seeking a partial solution to each approximate problem, they seek a 

complete solution for each approximate problem. Therefore, at the end 

f The dimensionality of the dual maximization problem is primarily dependent on the number of critical behavior constraints. 

7 
of any stage, the design may notbe strictly feasible, in which case scale 

up is needed to obtain a feasible design. The explicit dual methods presented in this work efficiently find the "exact" solution to each of the 

separable approximate problems generated in sequence. For the class of 

problems considered herein, the approximation concepts approach generates 

explicit constraint functions, that are identical to those employed in conventional optimality criteria techniques [see Refs. 33 and 341. Thus, in 

a sense, the joining together of approximation concepts.and dual methods 

has led to the envelopment of the optimality criteria method within the 

general framework of the mathematical programming approach to structural 

optimization. 

Use of trade names or names of manufacturers in this report does not 

constitute an official endorsement of such products or manufacturers, either 

expressed or implied, by the National Aeronautics and Space Administration. 
2. PRIMAL AND DUAL APPROACHES TO STRUCTURAL SYNTHESIS 

The structural synthesis problem considered in this work can be briefly 

stated as follows: minimize the weight of a finite element model of fixed 

geometry with limitations on the structural response (behavioral constraints) 

and on the design variables (side constraints). 

The most natural and rigorous way of attacking this problem is to 

make use of mathematical programming methods. This approach will be reviewed 

in this chapter, with emphasis on the practically important property of 

preserving the feasibility of the design. It will be shown why strict application of the available mathematical programming techniques to the structural 

synthesis problem has invariably failed to produce fully satisfactory results 

and how this led to the emergence of a powerful and now well established 

design procedure based on approximation concepts. 

The approximation concepts approach, as applied in this work, proceeds 

as follows: 

(1) construct an approximate problem by linearizing the behavioral 

constraints with respect to the reciprocal design variables; 

(2) partially solve the current explicit problem using a primal 

mathematical programming algorithm; 

(3) reanalyze the structure and update the approximate problem 

statement. 

This process facilitates generation of a sequence of steadily improved 

feasible designs. 

Pursuing further the approximation concepts idea, it can be argued 

that the approximate problem statement is of such high quality that it can 

be solved exactly, rather than partially, at each redesign stage. Adopting 

9 
this alternative viewpoint leads naturally to consideration of dual mathematical programming algorithms for solving the explicit problem. In&&, the 

number of dual variables associated with the linearized behavioral constraints 

is generally very small when compared to the number of design variables. 

This dual solution scheme, which no longer produces a sequence of 

always feasible designs, will be related to the well known optimality 

criteria techniques, in which basically the same explicit approximate problem is constructed by neglecting the internal force redistribution. The 

dual method approach, which can be viewed as a generalized optimality 

criteria approach, can handle large numbers of inequality constraints and 

it intrinsically contains a rational scheme for identifying the strictly 

critical constraints. Finally, the virtual load technique, the stress ratio 

algorithm and the scaling concept, widely employed in conventional optimality criteria techniques, will be investigated for possible use in conjunction with the dual method approach. 

2.1 Formulation of the Structural Synthesis Problem 

The structural synthesis problem considered in this work is restricted 

to the weight minimization of a finite element model with fixed geometry 

and material properties. The transverse sizes of the structural members 

(e.g. bar areas, shear panel and membrane thicknesses, etc...) are the design 

variables D.. I. They are subjected to the side constraints 

(2.1) 

where D!L) 1 and Di") are lower and upper limits that reflect fabrication and 

analysis validity considerations. For the moment, all the design variables 

are assumed to be continuous, but later in this work, treatment of discrete 

10 
design variables will be included in the structural synthesis problem (see 

Chapter 4). The behavioral constraints impose limitations on quantities 

describing the structural response, for example, the stresses and the displacements under multiple static loading cases, the natural frequencies, 

etc... They can be written as nonlinear inequality constraints: 

The number of inequality constraints Q is large since usually one behavioral 

constraint is associated with each failure mode (e.g. upper limit on deflection) in each load condition. The objective function to be minimized is 

the structural weight. It is a linear function of the design variables: 

gq6 2 0 ; q = 1,2,...,Q (2.2) 

M(s) = f miDi 

i 

(2.3) 

.th where mi denotes the weight of the i member when D i = 1 (i.e., specific 

weight times length of a bar truss member; specific weight times area of a 

membrane element). 

In equations (2.1) through (2.31, it has been assumed that the vector 

of design variables 5 contains one scalar component for each finite element 

in an idealized structural representation involving I finite elements. 

However it is neither necessary nor desirable for each finite element in 

the structural analysis model to have its own independent design variable. 

Design variable linking can be used to reduce the number of variables. As 

implemented in the ACCESS programs (see Refs. 5, 6 and 7), design variable 

linking simply fixes the relative size of some preselected group of finite 

elements, so that one independent design variable controls the size of all 

finite elements in that linking group. Hence the element sizes Di (e.g., 

bar areas and sheet thicknesses) are linked to the independent reciprocal 

11 
variables ab by the relation: 

1 D.=T - 1 ib(i) a 

; i k 1,2,...,1 

b(i) 

(2.4) 

where T ib(i) is the linking table constant and b(i) denotes an integer 

element of a pointer vector b'which, given the integer i, identifies the 

variable b to which the size of the element i is linked. 

Reciprocal variables {ab; b = 1,2,..., B) are used as the independent 

variables after linking, because the behavior constraints are much more 

shallow in the space of the reciprocal variables. Indeed it is well known 

that the stresses and the displacements are strictly linear functions of the 

reciprocal design variables for a statically determinate structure. Therefore it is reasonable to expect that they remain nearly linear in case of 

redundancy. Linear approximation in terms of the ab is the key idea of both 

the approximation concepts method and the optimality criteria techniques 

(see Sections 2.3 and 2.5). 

Design variable linking reduces the number of design variables while 

facilitating the imposition of constraints that make the final design more 

realistic. Linking makes it possible to introduce constraints based on 

symmetry, prior design experience, fabrication and cost considerations 

associated with the number of parts to be assembled. Taking account of the 

linking relations given by Eq. (2.41, the weight objective function defined 

in Fq. (2.3) is written as follows in terms of the independent reciprocal 

variables a : b 

w= F m.D. 

i=l 1 1 

(2.5) 

where the constant weight coefficients w b are given by 

12 
Wb = i~b c miTib(i) (2.6) 

Keeping in mind the linking relations, the structural synthesis 

problem, originally defined by Eqs. (2.1 through 2.31, can be concisely 

stated as a nonlinear mathematical programming problem of the following 

form: 

Find the vector of independent reciprocal variables z such that 

+ Min 

subject to behavioral constraints 

hq& L 0 q = 1,2,...,Q 

and side constraints 

(L) g, _ca b s a:) ; b = 1,2,...,B 

(2.7) 

(2.8) 

(2.9) 

Standard minimization techniques have been applied with varying degrees 

of success to the nonlinear programming problem embodied in Eqs. (2.7 

through 2.9). However this problem exhibits some characteristics that make 

it complicated when practical structural design applications are considered. 

The main difficulty arises from the fact that the hq(z) appearing in Eq. (2.8) 

are in general implicit functions of the design variables and their precise 

numerical evaluation for a particular design z requires a complete finite 

element analysis. Since the solution scheme is essentially iterative, it 

involves a large number of structural reanalyses. Therefore the computational cost often becomes prohibitive when large structural systems are 

dealt with. 

. 

13 
2.2 The Constrained Minimization Techniques 

The structural synthesis problem stated in Section 2;l is a nonlinear 

mathematical programming problem for which a wide variety of solution methods 

are available. Before describing briefly these various constrained minimization techniques, it is worthwhile mentioning that all of them seek a local 

optimum, which must necessarily satisfy the following first order KUHNTUCKER conditions [see Ref. 351: 

aw 

a% 

- E Eq 

xq a% - pb + vb = 

0 

q=l 

khq = O 

ub(ab - ab (L)) = 0 

v,(aF) - %) = 0 

x 

q 

10 

!Jb L 0 

vb 2 0 

b = l,B (2.10) 

q = 1,Q (2.11) 

b = 1,B (2.12) 

b = 1,B (2.13) 

The quantities IX 

9 

; q=l,Q), associated with the behavioral constraints 

(Eq. 2.81, and hb, vb; b=l,B), associated with the side constraints 

(Es. 2.91, are called dual variables. They have the meaning of Lagrangian 

multipliers conjugated to the constraints. Depending upon whether a given 

constraint becomes an equality or not at the optimum (i.e.,is active or 

inactive), the corresponding dual variable is positive or equal to zero. 

The KUHN-TUCKER relations embodied in Eqs. (2.10'; 2.13) are in general 

necessary conditions for local optimality. In the special case of a convex 

problem, they become sufficient conditions for global optimality. They can 

then be used to relate the primal variables - i.e., design variables - to the 

dual variables - i.e., Lagrangian multipliers -. 

The classification of the constrained minimization techniques given 

14 
in the sequel is of course not the only one possible. However it is convenient for organizing the discussion of the solution algorithms that have 

been applied to the structural optimization problem stated in Eqs. (2.7 - 

2.9). This classification also shows clearly why the strategy recommended 

in the present work - combination of approximation concepts and dual methods - 

emerges as one of the best approaches available at this time. 

2.2.1 The Primal Methods (Direct Approach) 

The well known and widely used direct constrained minimization techniques employ a sequence of search directions in the space of the primal 

variables, such that the constraints remain satisfied and that the objective 

function is minimized along each search direction. They are thus very 

similar to the unconstrained minimization techniques such as steepest descent, con3ugate ' gradient etc..., where a sequence of one dimensional minimizations are carried out. Essentially two kinds of algorithms belong in 

this category: the feasible direction methods and the projection methods. 

They have been very popular in the structural synthesis field [see Refs. 36- 

391 , mainly because they generate a sequence of feasible designs with decreasing structural weight. Even when the optimization process is terminated before convergence has been achieved, a practical and meaningful design, 

better than the initial one, is generally obtained. 

Since the direct constrained minimization techniques start from a 

feasible design and gradually improve it by working on the primal variables, 

they are often referred to as "primal" methods [see Ref. 401. Although this 

appelation could sometimes be ambiguous, it is very convenient, and throughout this work, a primal solution scheme will denote one in which the design 

is continuously improved while remaining feasible. It will be seen subse15 
quently that not only the direct constrained minimization methods - feasible 

direction and projection algorithms -.enjoy this important "primal" property. 

2.2.2 The Penalty Function Methods (Transformation Approach) 

The main drawback of the primal methods arises from the special 

treatment of the constraints they require. Except in the simple case where 

the constraints are linear, keeping them satisfied is an arduous task which 

always demands a sophisticated algorithm. 

In an attempt to circumvent these difficulties, penalty function 

methods have been introduced that transform the original problem in a sequence 

of unconstrained problems, by adding to the objective function a penalty 

term reflecting the degree of non-satisfaction of the constraints. The 

exterior point penalty function formulation leads to generation of a sequence 

of infeasible designs and therefore it has received relatively little attention in structural synthesis applications [see Ref. 411. The interior 

point penalty function methods - or barrier methods - are especially attractive since they yield a sequence of feasible points corresponding to decreasing values of the objective function. Such a formulation clearly 

adheres to a primal philosophy. The only difference is that strict primal 

methods - projection and feasible direction algorithms - produce boundary 

points (critical designs), while barrier methods generate interior points 

(noncritical designs). In the context of structural synthesis, this kind 

of method was used in Refs. [42-441. 

It is worth mentioning that the primal and penalty methods have 

exhibited rather poor convergence properties when applied to structural 

optimization problems. They require a large number of iterations, each 

involving at least one reanalysis of the structure. Moreover the number 

16 
of iterations grows with the number of design variables. That troublesome 

trend led many investigators to believe that the mathematical programming 

approach to structural synthesis would not work for large practical systems 

[see Ref. 451. This viewpoint fails to recognize that the primal and penalty methods are only a subset of the mathematical programming techniques 

available. 

2.2.3 The Linearization Methods (Indirect Approach) 

Probably the simplest approach to a nonlinear programming problem 

is to transform it into a sequence of linear programming problems. Each 

iteration consists of linearizing the objective function and the constraints 

at the current design point and solving the resulting linear problem. Applied 

as such this technique usually fails because it tends to converge to a vertex in the design space or indefinitely oscillate between two vertices 

[see Refs. 46 and 471. By introducing move limits, which restrain the range 

of the design variables to the neighborhood around the point where the 

linearization is made, the method of approximation programming is able to 

overcome these drawbacks and, though very simple, constitutes one of the 

most powerful and versatile optimization techniques currently available 

[see Refs. 48 and 491. 

In contrast with the primal and barrier methods, the linearization 

methods do not maintain the feasibility of the design point at each iteration+. On the other hand, their convergence properties are not related to 

the number of design variables, but to the degree of nonlinearity of the 

problem. This is a much more attractive dependence for structural synthesis 

f An exception is the method of inscribed hyperspheres [see Refs. 251 

This special linearization technique usually generates a sequence of 

feasible designs. 

17 
applications. I 

2.2.4 The Multiplier Method 

The multiplier method, which has enjoyed considerable popularity 

in recent years, has not yet been extensively applied to structural synthesis 

and it is mentioned in this classification only for the sake of completeness 

[see Ref. 50 for more details]. The multiplier method is a general purpose 

mathematical programming method whose algorithmic philosophy is similar 

to the usual exterior quadratic penalty function formulation, in that a 

constrained nonlinear programming problem is transformed into a sequence of 

unconstrained minimization problems. The penalty term is added to the 

Lagrangian function, rather than simply to the objective function, so that 

the multiplier method is sometimes referred to as the "augmented Lagrangian 

function method." The updated Lagrangian multiplier estimates at each stage 

are used to accelerate the overall optimization process. An attractive 

feature of the multiplier method is that each unconstrained minimization 

problem tends to be well behaved, which is a significant improvement over conventional penalty function methods. When the Lagrangian multipliers are 

regarded as the dual variables, the method can be viewed as seeking a saddle 

point by working alternatively in the primal and dual spaces. Therefore 

the multiplier method is also called a "primal-dual method." In its usual 

implementation the algorithm tends to generate a sequence of infeasible 

designs, like the regular exterior penalty function method. 

The method was applied in Ref. [50] to optimum design of truss 

structures considering both configuration and sizing type design variables. 

18 
2.2.5 The Dual Methods in Convex Progrannninq 

All the previously mentioned methods are quite general and they can 

be applied to obtain a local optimum for any nonlinear programming problem. 

In the special but important case of a convex problem, it is well known that 

every local optimum is also global. Furthermore the Lagrangian multipliers 

associated with the constraints have the meaning of dual variables in terms 

of which an auxiliary and equivalent problem can be stated. Under some 

unrestrictive conditions, this dual problem can be reduced to the maximization of the Dagrangian functional with simple nonnegativity requirements 

on the dual variables. If, in addition, the problem is separable, the dual 

formulation leads to a very efficient solution scheme since each primal 

variable can be independently expressed in terms of the dual variables. 

As the present work seeks to point out, dual methods should play an 

important role in the structural synthesis field. Used in conjunction with 

a special linearization technique - the approximation concepts approach 

reviewed in the next section - they facilitate creation of a 

tural synthesis method. This method is, in its own right, a 

programming approach, as usually defined, but it can also be 

generalized optimality criteria approach. 

2.3 The Approximation Concepts Approach 

powerful strucmathematical 

viewed as a 

As described in the previous section the use of primal and barrier 

methods had only a limited success in structural synthesis due to their 

prohibitive cost when large numbers of design variables were considered 

[see for example Pefs. 36 and 371. On the other hand, recourse to pure 

linearization methods, with or without move limits, failed to be efficient 

19 
because the behavioral constraints, expressed in terms of direct sizing 

variables, exhibit a rather high degree of nonlinearity. 

It is then not surprising that the combined use of a primal philosophy and of linearization techniques (using reciprocal variables) has 

finally led to a very efficient method, known as the "approximation 

concepts approach" (see Refs. 5,6,7 and 25). Briefly stated this approach 

replaces the initial problem with a sequence of approximate - but explicit and tractable - problems while retaining the important features 

of the primary problem. This is achieved through the coordinated use 

of various approximation concepts: 

(1) design variable linking; 

(2) temporary deletion of unimportant constraints; 

(3) generation of high quality explicit approximations 

for the surviving behavioral constraints. 

2.3.1 Reduction of the Problem Dimensionality 

Design variable linking, previously described in Section 2.1, leads 

to a significant reduction in the number of independent variables, which 

helps make the initial structural synthesis problem described by Eqs. 

(2.1 through 2.3) more tractable. Similarly, constraint deletion 

techniques are used to decrease the large number of behavioral constraints usually embodied in Eqs. (2.8) (see Ref. 5, Sections 2.4.1, 

2.4.2, and 2.4.3.). These constraint deletion techniques are nothing 

more than the computer implementation of traditional design practice. 

At the beginning of each stage in the iterative design procedure a 

complete finite element structural analysis is executed and all of the 

constraints (see Fq. 2.8) are evaluated. 

20 
Constraint deletion techniques are then used to temporarily ignore redundant 

and unimportant constraints. Let the relatively small set of surviving constraints for the p 

th stage be denoted by Q, (P) eQ- The constraints retained 

during the p 

th stage of the design procedure, as a function of the independent reciprocal design variables after linking (a,), are represented by 

hq(& t 0 ; qeQRtP)e Q 

As a result of constraint deletion only the critical and potential critical 

constraints (design drivers) are considered during the p 

th stage of the 

iterative design process. It is important to understand that at the beginning of each stage in the design process, the status of all of the constraints 

in the set +Q is assessed and the subset of constraints to be retained is 

re-established. Thus constraints that are ignored during an early stage 

may appear during a later stage if they become design drivers. 

It is worth noticing that, while design variable linking leads to 

reducing the number of primal variables in the structural synthesis problem, 

constraint deletion techniques result in a decrease in the number of dual 

variables. The net result is to reduce the dimensionality of the problem 

in both its primal and dual forms. 

2.3.2 Linearization Process 

The most important feature of the approximation concepts approach 

lies in the construction of simple explicit expressions for the set of 

constraints retained during each stage. This is achieved by using linearization of these constraints with respect to the linked reciprocal design 

variables a b' At each stage p, the following explicit approximate problem - 

referred to as the "linearized problem" - is thus generated: 

f The notations Q (or QR) are used to represent either the number of behavioral 

constraints (retained) or the set of indexes q corresponding to these constraints. 

21 
Find z such that 

B w 

W(Z)= 1 b+Min 

b=l *b 

subject to 

(L) 

abp -<a 

(U) 

b 5 abp ; b = l,B 

(2.15) 

(2.16) 

(2.17) 

where W(z) is the weight objective function, -(PI -f h q (a) represents the 

th linearized form of the q constraint function constructed at the beginning 

of the p 

th (PI stage, Q, denotes the reduced set of constraints to be retained 

during the p 

th stage, a (L) 

bp 

and a:(), respectively, represent the lower and 

upper move limits for the p 

th stage. 

The objective function (Eq. 2.15) does not need to be linearized, 

since it is an exact explicit function of the ab. The linearized behavior 

constraints (Fq. 2.16) are obtained using a first order Taylor series 

expansion in terms of the reciprocal variables f : 

hq 6 (2, = hq(zp) + 

5 (ab-odp)) 2 (b) ; seQ2) 

, 

b=l 

(2.18) 

where z 

P 

and a:' denote the design at the beginning of the p 

th stage in 

vector and scalar form respectively. The side constraints defined by 

Eqs. (2.17) arise from the original side constraints expressed in 

fNote that the finite element analysis must include auxiliary sensitivity 

analyses, which evaluate first partial derivatives of approximate response 

quantities. 

22 
Es. (2.91, but they can be modified at each stage p to include move limits 

which restrict the design modifications, during the p 

th stage, to a region 

in the z space over which the linearized expressions of the constraint 

functions in Eq. (2.18) are accurate enough to guide the design improvement process. 

In summary, then, design variable linking, constraint deletion 

techniques and linearization of the behavior constraints retained are used 

to generate a sequence of relatively small explicit mathematical programming 

problems which retain the essential features of the primary structural 

synthesis problem stated in equations (2.11, (2.2) and (2.3). This use of 

approximation concepts as the key to generating tractable approximate 

problems is summarized schematically in Fig. 1. In the p 

th stage, the 

original problem, expressed in terms of the linked reciprocal design variables (see Eqs. 2.7, 2.8 and 2.91, is replaced with its linearized form 

at the current design point p (WS- 2.15 through 2.18). Except for the 

fact that the explicit objective function is not linearized, the approximation concepts approach proceeds therefore as a classical linearization 

method in mathematical programming (see Section 2.2.3). It should be 

recognized that while recourse to the reciprocal variables Ctb is initially 

motivated by the observation that the linearized forms of static stress 

and displacement constraints are exact for a statically determinate 

structures, a more analytic justification is also available (see Section 

2.5.4 and Fig. 5). 

2.3.3 Primal Solution Scheme 

The linearized problem stated in Eqs. (2.15 through 2.18) is still 

a nonlinear mathematical programming problem, because of the nonlinear 

23 
objective function, but it is now explicit and easily treated by standard 

minimization techniques. In order to maintain a primal philosophy 

(sequence of steadily improved feasible designs), the approximation concepts 

approach, as initially proposed in Ref. [51, employed either a feasible 

direction method or an interior penalty function method to solve the 

linearized problem. In this way, it was possible to solve it only partially and to preserve, at each stage of the process, the feasibility of 

the design point with respect to the primary problem (Eqs. 2.7-2.9). In 

addition, the minimization algorithms were designed to permit introduction, 

in the approximate problem statement, of more sophisticated explicit constraints than the simple linear constraints of Eq. (2.181, such as spherical 

displacement constraints, second order Taylor series expansions, etc. 

In the ACCESS-l computer program [Ref. 51, two distinct optimizer 

options were available: (1) CONMIN - a general purpose optimizer based on 

a modified feasible direction method [see Ref. 511 and (2) NEWSUMT 1 - a 

sequence of unconstrained minimization techniques based on the linear 

extended interior penalty function formulation of Ref. [43] and a modified 

Newton method minimizer introduced in Ref. [21. Subsequently the ACCESS-2 

program [see Ref. 71 employed an improved optimization scheme called 

NEWSUMT 2, based on the quadratic extended penalty function set forth in 

Ref. 131. NEWSUMT 2 uses a rational method for determining a suitable 

transition parameter [see Ref. 521. This new optimizer is capable, for 

moderately infeasible designs, of guiding the design back to the feasible 

region. It is worth noticing that, when starting from a feasible interior 

point, the NEWSUMT optimizers tend to generate a sequence of designs that 

"funnel down the middle" of the feasible region. This represents an 

24 
attractive feature in the context of approximation concepts and from an 

engineering point of view. 

On the other hand, starting from an optimality criteria approach, 

a method similar to the approximation concepts approach was independently 

initiated in Ref. 1531. Using virtual load considerations, a first order 

approximate problem is generated, which is identical to the linearized 

problem posed by Eqs. (2.15 - 2.18). This problem is also solved partially 

using a primal solution scheme, with the aim of preserving the design 

feasibility, as in the approximation concepts approach. However, the 

method is less general since it relies on first or second order projection 

algorithms restricted to the case of linear constraints. The first order 

algorithm is very similar to the well known gradient projection method. 

The second order algorithm uses a weighed projection operator to generate 

a sequence of Newton's search directions that are constrained to reside in 

the subspace defined by the set of active constraint hyperplanes. A partial 

solution of the linearized problem is obtained by prescribing an upper 

limit on the number of one dimensional minimizations performed before updating the explicit problem statement [see Refs. 54 and 553. 

In summary, the approximation concepts approach can be classified 

as a mixed primal-linearization method. The initial problem is transformed into a sequence of linearized problems, which is classical in the 

mathematical programming linearization methods. However each subproblem 

is solved using a primal solution scheme that insures feasibility of the 

intermediate designs at each stage. 

25 
2.4 Joining Approximation Concepts and Dual Formulation 

2.4.1 Primal and Dual Solution Schemes 

A partial solution of the current explicit problem (Eqs. 2.15 - 

2.18) reduces the weight while maintaining feasibility with respect 

to the constraints. An exact solution of the current explicit problem finds the minimum weight, subject to the constraints, recognizing 

that one or more of the constraints will be critical at the solution. 

So far a primal philosophy has been adopted that leads only to 

partial solution of the linearized problem (Eqs. 2.15 through 2.18), 

using for example an interior point penalty function formulation with 

only a small number of response surfaces (typically 1 or 2) and a 

rather high response factor decrease ratio (typically 0.5). A structural reanalysis is then performed, the linearized problem is reformed 

and again solved partially. This primal solution scheme produces a 

sequence of feasible designs with decreasing values of the structural 

weight, an attractive feature of practical interest to the designer. 

An alternative viewpoint is to recognize that the approximation 

made by linearizing the constraints with respect to the reciprocal design variables is of such high quality that the current explicit problem can be solved exactly, and not partially, after each structural 

reanalysis. This idea leads to abandoning the primal philosophy in 

favor of a pure linearization approach. f 

In order to illustrate this concept consider the classical 3 bar 

truss shown in Fig. 2. By symmetry only 2 design variables define the 

f It should be noted that in conventional linearization methods the 

objective function is also linearized. This is not the case in the 

present work. 

26 
problem which therefore admits the simple geometrical representation shown 

in Fig. 2 in the space of the direct design variables. The behavioral 

constraints consist of tensile and compressive stress limits and the side 

constraints reduce to non-negativity of the bar cross-sectional areas. 

At the optimum only one constraint is active (tensile stress in member 1); 

the associated constraint surface is tangent to a constant weight plane 

(W = 0.074 kg). This problem has been solved using the approximation 

concepts approach with a penalty function formulation using the ACCESS 3 

program. Three different couples of values have been successively adopted 

for the response factor decrease ratio and the number of response surfaces: 

(0.5 x 11, (0.3 x 2) and (0.1 x 3). Thus increasingly exact solutions are 

generated for each linearized problem and the approximation concepts approacl 

gradually changes from a pure primal method, with partial solution of the 

explicit problem, to a pure linearization technique, with complete solution 

of the explicit problem. The trajectory of the design point toward the 

optimum is shown for each case in Fig. 3, the space of the reciprocal 

variables, where the constraints are linearized. The approximation concepts 

approach, as initially formulated, leads to a sequence of interior points; 

the trajectory "funnels down the middle" of the feasible region. OTI the 

other hand, forgetting the primal philosophy by solving almost exactly 

each explicit approximate problem produces a trajectory very close to the 

boundary of the feasible region (see Fig. 3). The convergence curves of 

the weight with respect to the number of structural reanalyses are represented on Fig. 4 for the three previously mentioned cases. The benefit 

gained from a complete solution of each linearized problem is clearly 

illustrated. 

27 
Once a primal philosophy is abandoned in favor of a pure linearization approach, any minimization algorithm can be chosen to solve the explicit approximate problem posed by Eqs. (2.15-2.18) since only its final 

exact solution needs to be known at each redesign stage. In order to 

improve the computational efficiency it is advisable to select a specialized 

nonlinear programming algorithm, well suited to the particular mathematical 

structure of the explicit problem. The objective function is strictly 

convex and all the constraints are linear, so that the problem is a convex 

programming problem. Moreover all the functions involved in this problem 

are explicit and separable. In such a case the dual method formulation is 

attractive, because the dual problem presents a much simpler form than 

the primal problem (see Section 2.2.5). 

Numerical experiments and engineering practice indicate that the 

number of strictly critical behavioral constraints is most often small 

when compared to the number of independent design variables. That is the 

reason why the convex, separable problem stated in Sqs. (2.15-2.18) can 

be very efficiently handled with the dual methods of convex programming, 

in which the variables become the Lagrangian multipliers (or dual variables) 

associated with the linearized constraints (Eq. 2.16). Therefore the 

dimensionality of the dual problem is much lower than that of the original - 

or primal - problem. The dual methods are thus likely to provide the 

most efficient solution scheme to the linearized problem, provided the 

original behavioral constraints are not too nonlinear in the reciprocal 

variables. This is actually true for most problems involving stress, displacement, frequency and buckling constraints [see Ref. 561. The extension to more sophisticated constraints - such as flutter and time parametric 

28 
dynamic responses - remains to be proven feasible. 

Another important advantage of the dual methods is that they allow, 

without weakening the efficiency of the optimization process, the introduction of discrete design variables, e.g., available cross-sectional areas 

of bars, available gage sizes of sheet metal, the number of plies in a 

laminated composite skin, etc.. (see Chapter 4). Finally a philosophically 

important feature of the dual formulation lies in its interpretation as a 

generalized optimality criteria approach (see Section 2.5). 

2.4.2 The Dual Method Formulation 

For the purpose of forming the explicit dual function it will be 

convenient to restate the primal problem at the p 

th stage as follows 

(see Eqs. 2.15-2.18): 

Find z such that 

Bw 

w(z) = 1 2 + Min 

b=l % 

subject to linear constraints 

where 

; WQR 

(2.19) 

(2.20) 

(2.21) 

and the side constraints are written separately: 

(L) 

% _<a 

NJ) 

b -< ab ; b=l,B (2.22) 

The wb in Eq. (2.19) are positive fixed constants (see Eq. 2.6) corresponding to the weight of the set of elements in the b th linking group when 

*b = 

1. Equations (2.20) represent the current linearized approximations 

29 
of the retained behavior constraints, in which the C 

w 

are constant. The 

(L) and (U) 

% % respectively denote lower and upper limits on the independent 

reciprocal design variables. Q R is the set of retained behavioral constraints 

for the current stage. For convenience, the index p denoting the stage in 

the iterative design process has been dropped in Eqs. (2.20-2.22). However 

it should be kept in mind that Eqs. (2.19-2.22) represent only the approximatef primal problem for the p 

th stage of the overall iterative design 

process. 

Let a Lagrangian function corresponding to the foregoing primal 

problem be defined as follows: 

L&I) = 

b!, : - qJQ 

xq (; 

R 

q - bil % ab) 

with the understanding that the nonnegativity conditions 

x LO; 

q seQ, 

(2.23) 

(2.24) 

must be satisfied. In view of the separability of each function involved 

in the primal problem, the Lagrangian function L(g,x) is also separable. 

By regrouping terms, L&,x) can be put in the following form: 

bqJQ 's'bs 

R 1 -,E, "q'q 

R 

(2.25) 

Let I\ denote the set of all dual points satisfying the nonnegativity conditions expressed by Eq. (2.24) and let A define the set of all primal points 

satisfying the side constraints embodied in Eq. (2.22). Now (z*,x*) is 

f Note that for statically determinate structures subject to static stress 

and displacement constraints, the primal formulation given by Eqs. (2.20 

and 2.21) is exact. 

30 
said to be a saddle point of L(z,x) if 

L(g*,T*) ,< L(;t,I*) for all 2 e A 

and 

L(Z*,X*) 1 L(Z*,T) for all T e A 

It is known that if (:*,I*) is a saddle point of L(z,x), then z* is a 

solution of the primal problem [see pages 83-91 of Ref. 571.‘ Furthermore 

the existence of a unique saddle point of L(z,I) can be proven because the 

approximate primal problem posed by Eqs. (2.19-2.22) is demonstrably convex (since the w b are positive and all the constraints are linear). 

The saddle point of L(z,T) can be obtained by a two phase procedure 

as follows: 

+Max Min 

i'.fA &A 

L&T) 

or, alternatively, 

Max 

len 

II (3 

where 

k?(x) = +Min L(z,x) 

a 6A 

(2.26) 

(2.27) 

(2.28) 

is defined as the dual function. Substituting Eq. (2.25) into Eq. (2.28) 

leads to the following expression of the dual function: 

iI(X) = +Min 

a,ZA 

Since the last term in this equation is a constant and the set A is separable, 

the minimum value of the sum of B single variable functions is equal to the 

sum of the minimum values of each single variable function. Therefore 

31 
Eq. (2.29) can be written in the alternative form: 

"b 4, + *b q$QR ‘q‘bs - q JQR ‘q % (2.30) 

Focusing attention on the single variable minimization problems 

hq %q 

I 

; 

b = 1,2,...,13 

R 

let 

f(ab) = 

"b 

<+abqJQ 'qCbq 

R 

Taking the first derivative and setting it equal to zero yields 

df "b 

=b=-?+ 

c x c =o 

"b qeQ, ' w 

Solving Eq. (2.33) for ab locates the extremum point Bb 

-2 "b 

"b = 

1 +lcbs seQ, 

(2.31) 

(2.32) 

(2.33) 

(2.34) 

which is the minimum point of f(ab), since, for db > 0, 

d2f 2w b -z-,0 

da: 3 

*b 

(2.35) 

because wb is known to be positive. Since ab is subjected to side constraints, the minimum of f(ab) is given by ab = Bb in Eq. (2.34) provided 

it resides in the acceptable interval a 

(L) (U) 

b <B ia b b' If db A aLL), 

then ab = a:) or if db L f), then ab = a:). Note also that in view of 

Eq. (2.35), f(ab) has positive curvature for any ab > 0 and is consequently 

32 
unimodal. 

From the foregoing development, it can be concluded that the dual 

problem has the following explicit form: 

Find 1 such that the explicit dual function 

Bw 

26) = 1 2 

b=l. ab 

subject to nonnegativity 

+qJQ 

Aq tuq& -“91 + Max 

R 

constraints 

(2.36) 

x 20; 9 seQ, 

where 

uq(Z = jl ‘bs ‘b 

(2.37) 

(2.38) 

and the primal variables a b 

are given explicitly in terms of the dual 

variables Aq by: 

g, = [sQw:qcbq / 1’2 if [$L)12 < [2Qw:qcw ] < [ar)12 (2.39) 

R R 

43 = a:) if Lb1 

qeQ AS% 

_< [aLL)12 

R 

L [aF)12 

(2.40) 

(2.41) 

The key to being able to construct this explicit dual problem resides in 

the convexity and separability of the approximate primal problem (i.e., 

Eqs. 2.19 - 2.22) and the simplicity of the single variable minimization 

problems embodied in Rq. (2.31). 

33 
An attractive feature of the dual problem is that it is a quasiunconstrained problem, because taking care of the nonnegativity constraints 

(Rq. 2.37) is straightforward. Two maximization methods will be subsequently described in this work: a second order Newton type algorithm 

(DUAL2; see Chapter 3) and a first order conjugate gradient type algorithm 

(DUALl; see Chapter 4). In addition the dual method formulation will be 

extended to deal with pure discrete and mixed continuous-discrete problems, 

and a specially devised gradient projection type of algorithm will be 

developed (see Chapter 4). 

2.5 Relations with the Optimality Criteria Approaches 

Most of the earlier optimality criteria techniques are based on 

the consideration of a statically determinate truss subject to stress and 

displacement constraints. In such a case, the behavior constraints take 

on explicit forms which can be expressed using virtual load techniques 

and/or stress ratio formulas (see Sections 2.5.2 and 2.5.3 respectively). 

As a result, the minimum weight design can be defined analytically, provided an appropriate algorithm is available for selecting the critical 

constraints. In the case of a statically indeterminate structure, the 

explicit redesign relations must be employed recursively, by constructing 

new explicit forms of the behavior constraints after each structural 

reanalysis. Therefore, the basic assumption is that the amount of force 

redistribution induced when the design variables are modified will 

generally be moderate enough to insure the convergence of the redesign 

process. This is the central idea of the optimality criteria approach 

and, not too surprisingly, it is also the main reason for the success of 

34 
the mathematical programming approach using approximation concepts. 

In fact, as shown in Ref. 1321, the whole process of combining the 

linearization of the behavioral constraints with respect to the reciprocal 

design variables and a dual solution scheme can be viewed as a generalization of the optimality criteria approach. In other words, a generalized 

optimality criteria approach can be defined as a special form of the 

linearization methods in mathematical programming. It amounts to replacing the original problem with a sequence of explicit approximate problems 

where the behavior constraints are linearized with respect to the reciprocal design variables. 

Conversely the joining together of approximation concepts and dual 

methods (see Section 2.4) can be interpreted as a powerful mathematical 

programming approach that contains and generalizes the conventional optimality criteria techniques. 

2.5.1 Conventional and Generalized Optimality Criteria --__ - IIThe generalized optimality criteria approach set forth in Xef. [32] 

consists in solving exactly, after each structural reanalysis, the linearized problem stated in Eqs. (2.15 - 2.18), which can be recast as 

follows in terms of the direct design variables Di (assuming no linking 

nor constraint deletion and dropping the stage index p, for sake of 

simplicity): 

I 

minimize W = 2: miDi 

i 

.I. c. 

subject to ; - 

q c 

-+ 0 seQ 

ii 

(2.42) 

(2.43) 

D% D (2.44) i i 2 DlL) 

35 
Instead of employing primal or dual mathematical programming methods, 

an alternative approach, which is typical of the optimality criteria 

philosophy, is to use the explicit character of the approximate problem 

embodied in Eqs. (2.42 - 2.44) in order to express analytically the optimal 

design variables. This can be achieved through the use of the KUHN-TUCKER 

conditions (see Eqs. 2.10 - 2.13) which, in view of the convexity of the 

linearized problem, are sufficient for global optimality. These conditions 

lead to a generalized optimality criterion yielding explicitly the design 

variables: 

active design variables: 

if [DjL) 12mi < 1 Ciq Aq < [DfU) 1 21’ni + Di = IL 1 ‘iq ~~1% 

seQ mi si3Q 

(2,451 

passive design variables: 

if 

if 

C ‘iq Aq ’ 

seQ 

[DjL)12rni + Di = DIL) 

C ‘iq ‘q -> 

qeQ 

[D:")]2mi -f Di = Dj') 

(2.46) 

(2.47) 

In these expressions, the Lagrangian multipliers h are associated with 

q 

the linearized behavior constraints (Rq. 2.43). They must satisfy the 

complementary conditions given in Eqs. (2.111, namely: 

critical constraint: 

I c. 

x 20 if 

9 

Ix=; 

i=l Di 9 

(2.48) 

36 
non critical constraint ~- 

k = O 

I c. 

if 1 --=<; D i=l i q (2.49) 

The Eqs. (2.45 - 2.47) relating the design variables Di to the 

Iagrangian multipliers Xq provide a basis for separating the design variables 

in two groups. The passive variables are those that are fixed to a lower 

or an upper limit (see Fqs. 2.46 and 2.47) while the active variables are 

explicitly given in terms of the Lagrangian multipliers using Eq. (2.45). 

This subdivision of the design variables into active and passive groups is 

classical in the optimality criteria approaches [see Refs. 13-17 and 26-291. 

When the Lagrangian multipliers satisfying Fqs. (2.48 and 2.49) are known, 

the optimal design variables can be easily computed using the explicit optimality criterion stated in Eqs. (2.45 - 2.47). Therefore the problem has 

been replaced with a new one, which is defined in terms of the Lagrangian 

multipliers only. To solve this new problem, the conventional optimality 

criteria techniques usually make the assumption that the behavior constraints critical at the optimum are known a priori, avoiding thus the 

inequality constraints on the Lagrangian multipliers appearing in Eqs. (2.48, 

2.49). An update procedure for the retained Lagrangian multipliers is then 

employed, so that the optimal design variables can be sought in an iterative 

fashion by coupling the update procedure and the explicit optimality criterion defined by Eqs. (2.45 - 2.47). 

As first noted in Ref. 1261, the essential difficulties involved in 

applying these optimality criteria methods to the general structural synthesis problem are those associated with identifying the correct critical 

constraint set and the proper corresponding set of passive members [see 

37 

I. - 
also Ref. 171. These difficulties were recognized and addressed with 

varying degrees of success in studies such as those reported in Refs. [27-291. 

However, it was only with the advent of the dual formulation set forth in 

Refs. [30 and 311 that these obstacles were conclusively overcome. 

The dual method approach inherently contains a mechanism for iteratively seeking the optimal Lagrangian multipliers satisfying the generalized 

optimality criterion embodied in Eqs. (2.45 - 2.49). In fact, the equivalence between this generalized optimality criterion and the Eqs. (2.39 - 

2.41) derived in the dual method formulation is straightforward (the only 

difference is the change from direct to reciprocal variables). Therefore 

it is apparent that the dual method formulation, which consists in maximizing the Lagrangian function subject to nonnegativity constraints on the 

Lagrangian multipliers, can be viewed as an update procedure for the 

Lagrangian multipliers. After the update procedure is completed, the primal 

design variables can be evaluated using the optimality criteria equations 

(2.45 - 2.47). 

The main difference between the conventional and the generalized 

optimality criteria approaches can now be identified as lying in the 

iterative process used to seek the dual variables (or Lagrangian multipliers). The conventional optimality criteria techniques replace the 

inequality relations (2.43) with equalities, postponing the selection of 

the active constraints to a subsequent part of the iterative process (or 

simply assuming that the active constraint set is known a priori). Consequently, simple recursive relations can be derived. The low computational cost of these recursive relations is the attractive feature of the 

conventional optimality criteria approaches. On the other hand, the dual 

38 
method formulation employed in the generalized optimality criterion method 

demands, at least formally, solution of an auxiliary mathematical programming problem (see Eqs. 2.36 - 2.41). However this maximization problem is 

remarkably simple and its exact solution can be generated at a low computational cost, which is comparable to that required by the recursive 

techniques of conventional optimality criteria. The dual algorithms can 

handle a large number of inequality constraints. They intrinsically contain a rational scheme for identifying the critical constraints through 

the nonnegativity constraints on the dual variables. They also automatically sort out the active and passive design variable groups using the 

explicit relations between primal and dual variables. 

In conclusion, while the coupling together of approximation concepts 

and the dual method formulation represents a pure mathematical programming 

approach, it can also be viewed as a generalized optimality criteria 

approach. 

2.5.2 The Constraint Gradients: Pseudo-loads Versus Virtual Load Techniques ~- 

So far, no attention has been given, in this work, to the way the 

constraint gradients are evaluated. In the approximation concepts method, 

which has its genesis in the mathematical programming approach to structural 

synthesis, the pseudo-loads technique is used to compute the gradients of 

the nodal displacements under a given set of load conditions [see Ref. 58, 

page 2421. The stress and displacement constraint gradients are then 

readily evaluated. This proce.dure requires that a certain number of additional loading cases be treated in the structural analysis phase. Introducing the pseudo-load vectors 

39 
Gbk = - aK 4, [ a% I (2.50) 

the gradients of the nodal displacements are computed by solving the systems 

of linear equations 

I b = l,B 

k = 1,K (2.51) 

where z k is the displacement vector for the kth load condition and [K] is 

the system stiffness matrix [see Ref. 5 page 831. The number of pseudoload vectors is directly related to the number of load conditions and the 

number of independent design variables after linking and it is independent 

of the number of behavior constraints. 

On the other hand, the generalized optimality criterion reported in 

Ref. [301 uses, as do most of the conventional optimality criteria approaches, 

the virtual load technique to generate first order explicit approximations 

of the stress and displacement constraints: 

I c.. 

?k = 

1 *-<; D i=l i j 

(2.52) 

with 

c = ijk (gjT IKil+. 1 (2.53) 

In these expressions, zk denotes the displacement vector for the k th load 

condition, [Ki] represents contribution to system stiffness matrix of the 

i th element and z 

j 

is the displacement vector due to a virtual loading case 

conjugated to the j th behavior constraint. As shown in Ref. [30], the 

coefficients C ijk' which have the meaning of energy densities in an optimality criteria context, are also the gradients of the constraints with 

respect to the reciprocal design variables 6 i = l/Di: 

40 
au 

C ijk 

=jk=$;T 

acii bi j Wil ;I (2.54) 

Consequently, the explicit expressions defined in Eq. (2.52) are first 

order approximations of the behavior constraints. Recast in terms of the 

linked reciprocal variables cb, they turn out to be identical to the first 

order Taylor series expansions used in the approximation concepts approach. 

The virtual load technique is widely used in conventional optimality 

criteria approaches [see Refs. 13-171. It employs a few additional unit 

loads to generate first order explicit approximations for preselected displacement constraints. In Ref. [30], this technique has been extended to 

stress constraints, for which the virtual loading cases are no longer 

represented by unit loads. Introducing virtual load vectors v' 

j 

conjugated 

to the behavior constraints, the corresponding virtual displacement vectors 

are evaluated by solving the systems of linear equations 

[K]zj = Gj j = 1,Q R (2.55) 

The coefficients C ijk are then computed using Eq. (2.53), and the explicit 

forms of the behavior constraints defined by Eq. (2.52) are available. 

This alternative approach to the evaluation of the constraint gradients 

requires as many additional virtual loading cases as the number of stress 

and displacement constraints retained, regardless of the number of design 

variables and of the number of real loading conditions. 

The decision as to which procedure should be selected to compute the 

constraint gradients can be based on a comparison of the total number of 

additional loading cases introduced into the structural reanalysis at each 

41 
given stage: 

(1) if the pseudo-loads technique is used, the number of additional 

loading cases is equal to the number of independent design 

variables after linking times the number of applied loading 

conditions; 

(2) if the virtual load technique is adopted, the number of additional loading cases is equal to the number of potentially 

active stress and displacement constraints retained for the 

current stage (provided each stress constraint involves only 

one stress component; see Ref. 1301). 

It is worthwhile noticing that a primal versus dual opposition appears in 

the number of additional loading cases, which, on one hand, ("optimality 

criteriaU), is equal to the number Q, of dual variables, while, on the 

other hand ("mathematical programming"), it is proportional to the number 

B of primal variables. 

2.5.3 The Stress Constraints: Zero Versus First Order Approximations 

In the approximation concepts approach that is adopted in this work, 

as well as in the generalized optimality criteria approach proposed in 

Ref. 1301, all the behavior constraints are replaced by first order explicit 

approximations. In many conventional optimality criteria techniques, such 

as those reported in Refs. 113-171, only the displacement constraints are 

approximated by first order expansions, while the stress constraints are 

treated using the classical "Fully Stressed Design" (FSD) concept. In this 

approach, the implicit nonlinear stress constraints 

i = 1,2 ,-a-, I 

k = 1,2,...,K (2.56) 

42 
(where cik denotes a suitable reference stress in the i th element for the 

k th (U) loading condition and ai is the corresponding allowable stress limit) 

are transformed, at each stage p, into simple side constraints: 

"b 

s g(p) (2.57) 

by using the well known stress ratio formula: 

As shown in Ref. [33], this FSD procedure can be interpreted as using 

zero order approximation of the stress constraints, because it relies on 

explicit expressions that preserve only the value of the stress constraints, 

and not of their derivatives. 

The zero order approximation of stress constraints offers two important advantages. First when the virtual load technique is used to compute 

the constraint gradients, the number of additional loading cases is significantly reduced because no virtual load cases have to be associated with the 

stress constraints. Secondly, the number of behavior constraints retained 

in each explicit approximate problem (see Eq. 2.20) is also substantially 

reduced, since all the stress constraints are now transformed into side 

constraints. This feature is especially beneficial when dual methods are 

employed to solve the explicit problem, because the dimensionality of the 

dual problem corresponds to the number Q, of first order approximated constraints embodied in Eq. (2.20). 

Cm the other hand, it is well known that the FSD procedure, because 

it employs a zero order approximation of the stress constraints, does not 

43 
always converge to the true optimum and sometimes is the source of instability or divergence of the optimization process. In practical structures, 

it is observed that many of the stress constraints can be approximated with 

sufficient accuracy by the FSD procedure, while others require a more 

sophisticated approximation using, for example, first order Taylor series 

expansion with respect to the reciprocal design variables. 

The selection of constraints requiring first order approximation can 

be made automatically on the basis of the following criterion [see Ref. 321. 

A retained potentially critical stress constraint must be linearized with 

respect to the reciprocal variables if, 

+ 

aub 

- (g,<< 

Ub (apI 

a% % 

(2.59) 

where u b denotes the appropriate reference stress in an element whose size 

is controlled by the b th independent design variable. That condition arises 

from the fact that, in a statically determinate structure,'zero and first 

order approximations of the stress constraints coalesce, since then: 

sub 

-= 0 

aaa 

for a # b (2.60) 

It should be clearly recognized that the selection criterion stated in 

Eq. (2.59) must be repeated at each design stage of the overall optimization 

process, exactly like the well known truncation procedure for deleting temporarily redundant and unimportant constraints (see Section 2.3.1). 

Mixing the FSD criterion and the virtual load procedure for generating accurate representation of the stress constraints has been presented 

in Ref. 1321 as a hybrid optimality criterion. It can be interpreted in 

44 
the present work as replacing some of the high quality, first order 

approximations of the constraints with computationally inexpensive, 

zero order approximations. 

2.5.4 Scaling of the Design Variables 

To close this section, it is worthwhile giving a geometrical 

interpretation of the approximation concepts approach. This interpretation is based upon the concept of scaling, which is classically 

used in optimality criteria approaches. Scaling simple sizing type 

design variables (e.g.,bar areas and sheet thickness) does not lead 

to any force distribution. That is, when all the member sizes are 

multiplied by the same factor, the stresses and the displacements are 

simply divided by the scaling factor (assuming the applied loads do 

not depend on the design variables). Therefore scaling is a convenient procedure for bringing the design point back to the boundary of 

the feasible region (see Refs. 54 and 55). 

In the design space, scaling corresponds to a move along a straight 

line joining the origin to the point where the structural analysis is 

made. Along a scaling line, the gradients of the stress and displacement constraints with respect to the reciprocal variables remain constant (see Ref. 33). Therefore the linearized forms of the constraints 

embodied in Eq. (2.18) furnish the exact values of the constraints and 

of their gradients all along the scaling line passing through the design point gp where the linearization is accomplished. Consequently, 

in the space of the reciprocal variables, the approximation concepts 

approach can be interpreted as replacing each real constraint surface 

by its tangent plane at its point of intersection with the scaling 

line (see Fig. 5). 

45 
When zero order approximation is used, the stress constraints are 

transformed into the simple side constraints embodied in Eq. (2.57). It 

can be shown [see Ref. 341 that each approximate constraint surface 

43 = 62' is again represented by a plane passing through the point of 

intersection of the corresponding real constraint surface with the scaling 

line. However it is no longer the tangent plane, but the plane perpendicular 

to the b th axis of the design space (see Fig. 5). 

Finally, the criterion for automatic selection of zero or first order 

approximation can be geometrically interpreted as follows: the condition 

posed by Eq. (2.59) is satisfied when the relevant stress constraint for the 

b th independent design variable is represented in the design space by a 

surface that is roughly parallel to the b th base plane. 

46 
3. DUAL METRODS FOR CONTIWOUS DESIGX VARIABLES 

In this chapter, solution methods for the dual problem formulated 

in Section 2.4.2 are examined. All the design variables are assumed to 

vary continuously and the dual problem posed by Eqs. (2.36 - 2.41) corresponds to the primal problem stated in Eqs. (2.19 - 2.22). It will be shown 

that, although there are hyperplanes in the dual space where the second 

partial derivatives of the dual function exhibit discontinuity, a second 

order Newton type of maximization algorithm can be devised that is especially 

well suited to the solution of the dual problem in the pure continuous case. 

3.1 The Second Order Discontinuity Planes 

An attractive feature of the dual method formulation is that the first 

derivatives of the dual function are readily available, because they are 

given by the primal constraints (Eq. 2.20): 

This is a well known theorem in convex programming [see, for instance, 

Ref. 35, 40 and 571 which, for the explicit dual problem considered here, 

can be easily demonstrated. Taking the first derivatives of the dual function embodied in Eq. (2.36) yields: 

(3.1) 

ai B wbacb -= - 

ax 

9 

.I, 25 + k;QR 'k 2 + uq - 'q (3.2) 

From Eqs. (2.39 - 2.41), it follows that: 

47 
$% 

acb 

-= 

/ 

- 

b 

if [aF)12 <wb 

c 

qeQ 'qcbq 

< [$u)12 

ah R 

q 

0 otherwise 

Substituting Eq. (3.3) into the first term of Rq. (3.2) gives 

-j, :>=t,i, 'bq4, 

bq 

(3.3) 

(3.4) 

where the summation on the index b is over the set of free primal variables f 

i = {bla?) < Qb < c’} (3.5) 

Cn the other hand, using the explicit definition of uq(G) (Eq. 2.38) 

yields 

auk 

ah= 

q 

jl 'bk 2 

9 

(3.6) 

so that, taking successively account of Eq. (3.3) and Eqs. (2.39 - 2.41), 

and rearranging the terms under summation, the second term in Eq. (3.2) 

becomes: 

Finally, comparing Rq. (3.4) and Eq. (3.7), it is seen that the first and 

second terms in Eq. (3.2) cancel and the first derivatives of the dual 

function are given by Eq. (3.1). The simplicity of Eq. (3.1) is a comf A primal variable is said to be "free" if it has not taken on its upper 

or lower bound value (cb(U) or cb(L)), that iS if it iS given by 

Rq. (2.39) rather than Eq. (2.40) or Eq. (2.41). 

48 
putationally important property of the dual method formulation. When a 

numerical maximization scheme is employed to solve the dual problem, the 

evaluation of the dual function (Eiq. 2.36) requires the determination of 

the primal constraint values (u - iq), so that the first derivatives 

P 

given by EQ. (3.1) are available without additional computation. 

In the DUAL 2 algorithm described subsequently, the Newton 

method is used to maximize the dual function and therefore the second 

partial derivatives of %(I) must be evaluated. Let the elements of the 

be represented by the notation F 

qk' 

Hessian matrix associated with L(x) 

then, from Eq. (3.1): 

F 

sk = & 6) 

Interchanging the indices 

B 

P 

au 

= 9 (1) 

k 

(3.8) 

k and q in Eq. (3.6), it follows that 

a aI3 F 

sk 

=j,c - 

b=l bq q 

(3.9) 

Changing the index q to k in EQ. (3.3) and substituting Eq. (3.3) into 

Eq. (3.9) gives the explicit form of the second derivatives: 

F c-1 1 CbqCbk 3 

sk 2b6g "b eb (3.10) 

where the summation on the index b is over the set of free primal variables 

(see Eq. 3.5). 

From Eq. (3.10), it can be concluded that the second derivatives 

of the dual function are discontinuous, because the F 

sk 

elements jump to 

other values each time the set % of free primal variables is modified. 

Now, the explicit relationships between primal and dual variables (see 

Eqs. 2.39 - 2.41) indicate that changes in the status of primal variables 

(from free to bound), which signal discontinuities in the second deriva49 
tives,occur on hyperplanes in the dual space given by 

and 

1 xc = 

"b 

seQ, q W 

c xc = 

"b 

seQ, ' bs 

(3.11) 

(3.12) 

The hyperplane defined by Eq. (3.11) subdivides the dual space into a 

half-space where ctb = ab (L) (b ounded primal variable) and another half-space 

where a b 

> asL’ (free primal variable). Clearly, the same argument holds 

(L) for Eq. (3.12), with ab replaced by a:'. Consequently the dual space 

is partitioned into several domains separated by the second order discontinuity planes embodied in Eqs. (3.11 and 3.12). In each domain, the set 

B of free primal variables remains constant. However when passing from one 

domain to another, across a second order discontinuity plane, the set B is 

modified and the second derivatives of the dual function change abruptly 

(see Eq. 3.10). 

3.2 Characteristics of the Dual Function - Continuous Case 

The explicit dual function for the pure continuous variable case, 

defined by Eqs. (2.36 - 2.411, has several interesting and computationally 

important properties, which are summarized as follows: 

(1) it is a concave function and the search region in dual space 

is a convex set defined by Eq. (2.37); 

(2) it is continuous and it has continuous first derivatives 

with respect to Aq over the entire region defined by Ekq. (2.37); 

(3) the first derivatives of k(l) are easily available because 

50 
they are given by the primal constraints, that is: 

B 

q = Ll 'bq ab - "q (3.13) 

(4) the second derivatives of k(x) are given explicitly by: 

(3.14) 

where B denotes the set of free primal variables (see Eq. 3.5); 

(5) discontinuities of the second derivatives exist on hyperplanes 

in the dual space defined by Bqs. (3.11 and 3.12), which locate points where there is a change in status of the b th 

design variable from "free" to "bound". 

3.3 DUAL 2 - Newton Type Maximizer 

In this section, a second order Newton type algorithm for finding 

the maximum of the dual function (see Eqs. 2.36, 2.38, 2.39, 2.40 and 

2.41), subject to nonnegativity constraints (see Bq. 2.37), is described. 

The method has been found to be very efficient in practice, even though 

there are hyperplanes in the dual space where the second partial derivatives are not unique (see Eqs. 3.11 and 3.12). The algorithm involves 

iterative modification of the dual variable vector as follows: 

It+1 = 

-xt + dt zt (3.15) 

where s t denotes the modification direction in dual space and dt represents the distance of travel along that direction. Alternatively, in 

scalar form, the modification is given by 

x q,t+l = h qt +d S 

t St i seQ, (3.16) 

51 
In the DUAL 2 algorithm, the Newton method is used to seek the maximum 

of the dual function in various dual subspaces 

M = {qjhgt ' 0 i seQ,) (3.17) 

which exclude those X components that are not currently positive. The 

q 

move direction in such a dual subspace is given by 

2 = -t - tF dt) I-’ Vdt) (3.18) 

where F(xt) denotes the Hessian matrix of the dual function evaluated at 

It (see Eq. 3.10) and the subscript _ indicates that the collapsed vector 

(matrix) includes only those components (elements) corresponding to strictly 

positive values of the dual variables at xt (i.e., entries for A 

St 

> 0 only). 

If the initial starting point in dual space is such that the Hessian 

matrix (see Eq. 3.18) is non-singular, and additional non-zero components 

A > 0 are added one at a time, 

9 

each subsequent Hessian [F(it)] will be 

non-singular [see Ref. 30, page 501. In the first stage (p=l), it is convenient to select the starting point so that the only non-zero dual variable 

corresponds to the most critical constraint (based on the structural 

analysis of the primal design used to generate the current approximate 

primal problem). For subsequent stages (p>l), the starting point is given 

by the dual variable values at the end of the dual function maximization 

in the previous stage. This procedure is employed in DUAL 2 and therefore 

the dimensionality of the maximization problem generally does not exceed 

the number of strictly critical constraints excluding side constraints 

(see Eq. 2.22). 

The DUAL 2 algorithm is outlined in the block diagrams shown in 

52 
Figs. 6 and 7. Given a set of values for the dual variables A qt' seQ, 

(see block 1) attention is directed to identifying the set of non-zero 

dual variables M (block 2). The integers in the set M define a dual subspace and in that subspace the maximum of the dual function P,(x) is sought 

subject to nonnegativity constraints (see block 3; Fig. 6 and Fig. 7). Let 

the maximum of L(x) in the subspace defined by the set M be denoted as xM. 

At TM evaluate the first partial derivatives of R(x) with respect to those 

Xq not included in the subspace defined by the set M (block 4, Fig. 6). 

Test to see if the maximum of E(x) in the dual space (qEQR) has been 

obtained (block 5), if so store the primal variables corresponding to the 

current dual variables r M' end the stage and go to the overall design process convergence test. If any of the first partial derivatives 

+f (lM);q6QR are positive find the largest one, denote the corresponding 

q 

index as q+ (see block 6, Fig. 6), add this component to the set M (increasing the dimensionality of the dual subspace), and continue to seek the 

maximum of the dual function E(x) associated with the current stage. 

The procedure followed in order to find the maximum of P.(i) in a 

dual subspace M (see block 3, Fig. 6) is elaborated on in Fig. 7. The 

START block in Fig. 7 is entered from block 3 of Fig. 6. Given the initial 

values of the non-zero dual variables X 

St 

>O; qeM (block 1, Fig. 7), evaluate 

the partial derivatives g (It); qEM (block 2, Fig. 7), and then test to 

9 

see if the maximum point in the subspace defined by qi%M has been found 

(block 3, Fig. 7). If the absolute value of the gradient IV%(q) 1 is equal 

to or less than s,the maximum of J?(X) in the subspace defined by qf?M has 

been found. Let It replace xM (block 4, Fig. 7) and go to point G on Fig. 6. 

53 
Cn the other hand, if /vE(T) 1 is greater than E, the maximum of g(x) in the 

subspace defined by qeM has not been found and the search for the maximum 

is continued by using Eq. (3.18) to generate a new search direction St 

(see block 5, Fig. 7). 

The next step is to determine the maximum step length (dmax) along 

the direction $ such that none of the X 

q,t+l 

become negative. Setting the 

x 

q,t+l 

to zero in Eq. (3.16) and focusing attention on only the negative 

components (S 

St 

CO) it follows that the maximum step length is given by 

d = Min LL 

max =ie'M 

S 

St 

<o I S Ii qt 

(3.19) 

Determine d and let the value of q which gives d 

IlElX 

max be denoted by the 

symbol q- (see block 6 of Fig. 7). Test d to see if it is less than 

max 

unity, if so then the move distance d t is replaced by dmax (block 7a, 

Fig. 7), otherwise the move distance is set equal to unity (block 7b, 

Fig. 7). The dual variables are now updated using the move direction generated in block 5 and the move distance dt generated in either block 7a or 

block 7b. Also the primal variables (cb t+l; b = 1,2,...B) (corresponding 

, 

to the X ti qeM) are evaluated using Eqs. (2.39, 2.40 and 2.41) (see q,t+l' 

block 8, Fig. 7). The next step is to determine whether or not the move 

from It to Xt+l has involved passing through any discontinuity planes 

(see Eqs. 3.11 and 3.12). This is accomplished by comparing the set of 

free primal variables at design point gt with those at 2 t+1- If there is 

no change in the set of free primal variables, then it follows that none 

of the hyperplanes defined by Eqs. (3.11 and 3.12) have been traversed in 

moving from Xt to Zt+l in dual space. Now if It and $+l are in the same 

domain (i.e.,the move from xt to lt+l has not involved passing through any 

54 
discontinuity planes) and dt # d,,, (see block 10 -False; Fig. 7), then the 

scheme behaves like a regular Newton method taking a unit step in the g -t 

direction and going to block 2 to continue the iteration. When xt and 

%+1 are in the same domain and d t = d max 

(see block 10 -True; Fig. 7), 

evaluate the directional derivative at 1 t+l (block 11, Fig. 7), using the 

following well known relation 

P(d,) = g V&+l) = (3.20) 

Note that the partial derivatives g (xt+l) are easily evaluated using 

q 

Eq. (3.13) since the primal variables gt+l were previously computed and 

stored. When xt and xt+l are not in the same domain (block 9 -False; Fig. 7) 

the directional derivative at t+1 should also be evaluated (block 11, 

Fig. 7). If in block 12, l/'(dt) is positive and dt does not equal d,,, 

(block 13 -False), move the distance d t along the direction 2 -t and go to 

block 2 to continue the iteration. If in block 12, i'(d,) is positive and 

dt = dmax then go to H (i.e.,return to H in Fig. 6) and delete the component 

q- tagged in block 6 of Fig. 7 when dmax was evaluated. Finally, if Il' (dt) is 

not positive then cut dt in half (block 14, Fig. 7) and go to block 8. 

The scheme for determining the step length along a direction 2 

-t 

described by blocks 6 through 14 of Fig. 7 does not seek the maximum of the 

dual function along the direction !t, rather it is designed to assure that 

either: (a) a regular Newton unit step is taken without any change in the 

set of free primal variables; or (b) the move distance is selected so that 

the value of the dual function increases. Note that in contrast to the 

DUAL 1 algorithm, which will be described subsequently, the move distance 

selection scheme employed in the DUAL 2 algorithm does not calculate dis55 
- _- ._ - .- .-.-._ .__ 

tames along St locating the intercepts with the 2nd order discontinuity 

planes defined by ?Zqs. (3.11 and 3.12). 

56 
4. DUAL XETHODS FOR DISCRETE DESIGN VARIABLES 

Attention is now directed toward extending the explicit dual formulation to problems involving discrete design variables. There are many 

occasions in structural optimization where the design variables describing 

the member sizes must be selected from a list of discrete values. For 

example, conventional metal alloy sheets are commercially available in 

standard gauge sizes and cross-sectional areas for truss members may, in 

practice, have to be chosen from a list of commercially available member 

sizes. Furthermore the growing use of fiber composite materials in aerospace structures also underscores the importance of being able to treat 

structural synthesis problems where some or even all of the design variables 

are discrete. 

In the structural optimization literature, relatively little attention 

has been given to dealing with discrete variables. Those efforts that have 

been reported [see Ref. 59 for a review of this literature] generally attack 

the discrete design variable optimization problem by employing integer 

programming algorithms to treat the problem directly in the primal variable 

space. In this chapter it will be shown that the combined use of approximation concepts and dual methods, set forth in chapters 2 and 3 for continuous sizing type design variables, can be extended to structural synthesis 

problems involving a mix of discrete and continuous sizing type design 

variables. The mixed case formulation and the implementing algorithm DUAL 1, 

described in the sequel, can also handle the two limiting special cases, 

namely, the pure discrete and the pure continuous variable cases. 

It is worthwhile noticing that when discrete design variable% are 

- 57 
introduced, the approximate primal problem is no longer convex and therefore, 

the dual formulation presented in this chapter does not necessarily yield 

the exact solution of the approximate primal problem (duality gap). However the computational experience reported in Chapter 6 shows that useful 

and plausible discrete designs are readily generated using the DUAL 1 

algorithm. These numerical results confirm the observation made in Refs. 

[57 and 601 to the effect that although the extension of the dual formulation to discrete variables lacks rigor, it frequently gives good results. 

4.1 The First Order Discontinuity Planes 

The explicit dual method previously described can be extended to 

mixed continuous-discrete variable primal problems of the form given by 

Eq.5. (2.19 - 2.21), with the side constraints of Eq. (2.22) replaced by 

(L) 

cb ,<a b for continuous cb 

and 

abeAp for discrete cb (4.21 

where 

(D) 

% k = 1,2,...nb} 

(4.1) 

(4.3) 

represents the set of available discrete values for the design variable CL b' 

listed in ascending order. For convenience the index p denoting the stage 

in the iterative design process has been dropped from Fqs. (4.1 - 4.3) as 

well as from Eqs. (2.19 - 2.21). However, it should be kept in mind that in 

general Eqs. (2.19 - 2.21) and Eqs. (4.1 - 4.3) represent only the approximate primal problem for the p 

th stage of the overall iterative design process. 

58 
The primal variables in terms of the dual variables are given implicitly by (see Eqs. 2.31): 

Min %+a 

0.J ab 

b q%Q 'q 'bs 

"b R 

(4.4) 

and explicitly by Eqs. (2.39 - 2.41) for continuous ab. In an analogous 

manner, for discrete ab it is assumed that 

b q;Q 'q 'bs 

R t 

(4.5) 

relates the continuous dual variables to the discrete primal variables a b' 

The dual function !L(x) is still given by Eq. (2.36) and the first 

derivatives g (x) are still given by Eq. (3.13). It is apparent from 

Eq. (3.13) thaz discrete values for some of the primal variables g, will 

cause discontinuities in the first derivatives of the dual function to arise. 

When the solution of Eq. (k) (4.5) shifts from one value of ab to the next 

(k+l) 

"b the following identity maintains continuity of the dual function 

Wb 

- +a (k) 

(k) b 

% 

+ atk+l) 1 (k+l) b A c 

seQ, q bq 

Equation (4.6) can be reduced to the following form 

Wb 

(k) (k+l) 

cb "b 

(4.6) 

(4.7A) 

59 
which defines hyperplanes in the dual space where the dual function 11(x) 

exhibits first order discontinuities. The hyperplane defined by Eq. (4.7) 

subdivides the dual space into a half-space where a = a (k) 

b b and another 

(k+l) half-space where ab = ab . Similarly the hyperplane defined by 

c xc = 

"b 

seQ, sbq (k-1) (k) 

ab 'b 

(4.7B) 

(k-1) is associated with a shift in the solution of Eq. (4.5).from ab to a:’ 

(k-1) and it subdivides the dual space into a region where ab = ab and another 

region where % = a;). 

It is apparent from the foregoing interpretation of Eq. (4.7), that 

the discrete primal variables ab are explicitly related to the continuous 

dual variables A9 as follows: 

(k) ii “b 

*b = % 

-- < 

(k) (k-1) c AC < 

Wb 

SW (k) (k+l) (4.8) 

%I cb qcQ, ab *b 

In summary, the dual problem corresponding to the mixed continuousdiscrete primal problem posed by Eqs. (2.19 - 2.21) and Eqs. (4.1 - 4.3) is 

taken to have the form: find x such that L(x) -+ Max (see Eq. 2.36), subject 

to the nonnegativitl\ constraints embodied in Eq. (2.37), where the continuous ab are given in terms of the dual variables X 9 

by Eqs. (2.39 - 2.41) 

and the discrete ab are given explicitly by Eq. (4.8). 

4.2 Characteristics of the Dual Function-Mixed *se 

The explicit dual function for the mixed continuous-discrete variable 

case, defined by Eqs. (2.36) through (2.41) and Eq. (4.8) has the following 

60 
interesting and computationally important properties: 

(1) it is a concave function and the search region in dual space 

is a convex set defined by Eq. (2.37); 

(2) it is a continuous function and it has continuous first 

derivatives with respect to h 

9 

over the region defined by 

Eq. (2.37) except for points located in hyperplanes defined 

by Eq. (4.7) - these first order discontinuities are associated 

with shifts in the discrete variable solution of the one dimensional minimization problem represented by Eq. (4.5); 

(3) the first derivatives of k(l) are easily available because they 

are given by the primal constraints 

(4.9) 

and on the first order discontinuity planes two distinct values 

of the first derivative arise, because at such a point there is 

a shift in the discrete value of a particular primal variable, 

say 

(k) (k+l) g,fromg toab which gives 

aktk) (k) - 

ah 9 jzb js a*+CWab I -% 

and 

(4.10) 

(4.11) 

2 

(4) disccntinuities of the second derivatives $& (X) exist 

q k 

61 
on hyperplanes in the dual space defined by Fqs. (3.11 and 3.12) 

for continuous ab variables - these second order discontinuity 

planes locate points in the dual space where there is a change 

in status of the b th continuous primal variable from "free" to 

bound. 

4.3 The Pure Discrete Case 

In the pure discrete variable case, the explicit dual function is 

piecewise linear, that is, its contours are sections of intersecting hyperplanes. The dual space is partitioned into several domains, each of which 

corresponds to a distinct combination of available discrete values of the 

primal variables. The following simple two dimensional example may help to 

clarify the foregoing points. 

The example illustrated in Fig. 8 concerns a 2-bar truss subjected 

to a single horizontal load Isee Ref. 30, page 591. The vertical and 

horizontal displacements are limited and the problem takes the explicit 

form: 

find al, a2 such that 

[weight] (4.12) 

and 

3 a1 + a2 5 z [horiz. displ.] (4.13) 

1 al - a2 ..s y [vert. displ.] (4.14) 

In the space of the reciprocal variables (a,, a2), the continuous optimum 

occurs at the point ($ , $1 * Only one constraint is active (horizontal 

displacement; see Fig. 8A). A pure discrete problem has been constructed 

62 
by restricting the cross-sectional areas of both bars to the discrete values 

1, 1.5 and 2 corresponding to cl, a2 e{$ , $ , 1). 

The formulation of the dual problem involves 2 dual variables associated with the two displacement constraints (4.13) and (4.14). The first 

order discontinuity planes are given explicitly by the equations 

1 

Al + A2 = $ 

1 

hl 

-AZ=+ 

Al + A2 = 3 . A1 -ha=3 

(4.15) 

They subdivide the dual space in 9 regions each corresponding to a different 

primal point (see Fig. 8-B). The dual objective function is written 

R(Al,A2) = 1 +1 +h(a +a al a2 1 1 2 - $1 + A2 (a1 - a2 - +). (4.16) 

where the primal variables a 1 and a 2 are given in terms of the dual variables 

A 1 ma A 2 according to the explicit inequalities (see Eq. 4.8): 

cl =l if Al + A2 c $ 

2. a1 = J If 3 5 < x1 + x2 < 3 

1. a1 = 2 If 3 <A 12 +x 

and 

a2 =l if A1 -A2<+ 

2. 3 a2 = 7 rf y < Xl - h2 < 3 

1 . a2 = y If 3 < x1 - A2 

(4.17.A) 

(4.17-B) 

63 
The contours of the dual function are represented in Fig. 8.C. The maximum of the dual problem lies at the dual point ($ , 0) where the dual 

function value is 2.75. The optimal subdomain is cross hatched in Fig. 8.B. 

It corresponds to the primal point ($ , $1 I with the weight equal to 3. 

4.4 Construction of a Unique Ascent Direction 

The main difficulty associated with the explicit dual formulation of 

the mixed continuous-discrete variable case is linked to the existence of 

hyperplanes in the dual space, where the gradient of the dual function 

Vi(x) is not uniquely defined, because of the previously described first 

order discontinuities (see Eqs. 4.10 and 4.11). The existence of these 

first order discontinuity hyperplanes in dual space complicates the task of 

devising a computational algorithm for finding the maximum of the explicit 

dual function. Fortunately, it turns out that at points in the dual space 

where the gradient VL(x) is multivalued, the orthogonal projection of each 

distinct gradient into the subspace defined by the set of pertinent discontinuity hyperplanes, yields a single move direction 2 and furthermore 

the directional derivative ($1 of the dual function along the move direction 5 is unique and positive. 

An intuitive understanding of the basic scheme used to cope with the 

existence of first order discontinuity planes can be gained by examining 

a simple example, with a single discontinuity hyperplane, such as that 

depicted schematically in Fig. 9. Let the equation of the first order 

discontinuity plane (line a-a in Fig. 9) be represented by (see Eq. 4.7A 

with k = 1) 

Wb f,(T) = F zb - (1) t2) = 0 

'b ab 

(4.18) 

64 
then the normal to the discontinuity plane is 

Vfb = Zb (4.19) 

Let ;;1 and G2 denote the two distinct values of the gradient at point t on 

the first order discontinuity plane (see Fig. 9). Components of the 

vectors Gl and ti2 are given by Eqs. (4.10) and (4.11) with k = 1, that is 

(1) ap +’ 

gq = c (AtI = hq + sq ail) 

9 

and 

g(2) (2) 

q 

= E (I,) =hq + 

9 

%q d2) 

where 

hq = j)b 'jb aj - 'q 

Rewriting Eqs. (4.20) and (4.21) in vector form gives 

(4.20) 

(4.21) 

(4.22) 

(4.23) 

(4.24) 

The projections of Gl 

+ 

and g2-into the discontinuity plane are given by 

and 

z=; -- 

2 

'b '2 e 

+T+ b 

'b cb 

(4.26) 

65 
To confirm that the move direction given by Eqs. (4.25 and 4.26) is unique, 

simply substitute Fq. (4.23) into (4.25) or Eq. (4.24) into (4.26) to find 

in either case. 

To show that the directional derivative along 2 is unique and positive 

'b 'b + -gqh (4.27) 

use Eq. (4.27) and (4.23) to show that 

and use Eqs. (4.27) and (4.24) to show that 

dR +T+ 

-= z g2 dz 

Furthermore, since it follows from Eq. (4.27) that 

-a?-+ 

'b 'b 

(4.28) 

(4.29) 

(4.30) 

also, it is apparent that 

+I?+ 

z g1 

= pG2 = 

+I?-+ ZZ>O ifZ#Z; (4.31) 

and therefore, provided 2 # 6, the directional derivative along z iS 

unique and positive. 

The foregoing development can be generalized to the case where the 

current point in the dual space 1 t resides in the subspace defined by P 

first order discontinuity planes (see Eq. 4.7). For convenience assume 

that the primal variables are numbered so that the first P variables are 

66 
those associated with the discontinuity planes pertinent to the current 

point in dual space xt. The equations of these P first order discontinuity 

planes are 

b = 1,2,...P 

At such a point in dual space there are 2 P different gradients (denote 

-t(L) them as g ; R = 1,2 ,...2') corresponding to the 2' possible 

of the values a:) or CX~" for b = 1,2,...P and they can be 

as follows 

+(2) 4 = rt + i ap Zb ; R = 1,2,...2p 

b=l 

where the components of h' are given by 

hq = 

(4.32) 

combinations 

represented 

(4.33) 

(4.34) 

Now the orthogonal projection operator, which will yield the projection of 

any vector into the subspace defined by the set of discontinuity planes, 

is given by [see p. 177,Ref. 58 1 

[PI = 11 - N (NTN) -lNT] (4.35) 

where I is a Q, x Q R identity matrix and N denotes a Q, x P matrix with 

columns corresponding to the vectors ..Zp appearing in Eq. (4.321, 

that is 

INI = I$,, z2, . . .Zb.. .spl (4.36) 

Q,xP 

To show that the projection of the 2 P distinct gradients (see Rq. 4.331, 

into the subspace defined by the P discontinuity hyperplanes (see Eq. 4.32), 

67 
yields a unique direction of travel 2, write 

-+(a) z = [Pig (4.37) 

-+(a) and substitute in for g from Eq. (4.331, then 

(4.38) 

since [P]Zb = 6 ; b = 1,2,...P (4.39) 

Furthermore it can also be shown that the directional derivative along 

the move direction z given by Eq. (4.38) is unique and positive, that is 

dll +T -+(a) -=z g dz = F it = iTyP]ii = ([P]iT)TIPliT = ?i?i > 0 if;+0 

(4.40) 

In the DUAL 1 algorithm described in the next section, direction vectors 

are generated using the projection matrix, whenever the current point 1, 

resides in one or more first order discontinuity planes. However, for 

computational efficiency the [PI matrix is actually generated by employing 

update formulas rather than by using Fq. (4.35). 

4.5 DUAL 1 - Gradient Projection Type Maximizer 

In this section a first order gradient projection type algorithm for 

finding the maximum of the explicit dual function k(l) (defined by 

Eqs. 2.36 through 2.41 and Eq. 4.8) for mixed continuous-discrete variable 

problems, subject to the nonnegativity constraints of Eq. (2.37), is described. The existence of hyperplanes in the dual space (see Eq. 4.7) where 

the dual function l,(l) exhibits first-order discontinuities, for pure discrete and mixed continuous-discrete variable problems, requires the use of 

a specially devised first order algorithm akin to the well known gradient 

68 
projection method. For each stage P of the overall design process, the 

DU-AL 1 algorithm seeks 1 such that !L(T) + Max subject to h 2 0; qcQR(P). 

9 

The dual variable vector is modified iteratively as follows 

%+1 = 

$ + dt gt (4.41) 

The maximization algorithm consists of a sequence of gne dimensional 

maximizations (ODM'S) executed along ascent directions zt obtained by pro- - 

jecting the dual function gradient into an appropriate subspace. 

To help fix ideas consider the pure discrete variable case where 

each ODM necessarily terminates on either a discontinuity plane or a boundary plane where some Xq becomes zero. In either case it is possible to 

construct a new projection matrix by updating the old one, avoiding the 

costly matrix inversion which would be required if the projection matrix 

was obtained from‘ Eq. (4.35). The authors are not aware of a comparably 

efficient scheme for directly updating the projection matrix itself when 

a first order discontinuity plane must be dropped from the current set, 

or when a zero dual variable must re-enter the set of nonzero dual variables. 

Since the foregoing scheme does not provide for the selection and release 

of discontinuity or base plane equality constraints, the maximization process can terminate at a "vertex" of the dual space [number of discontinuity 

planes equal to the number of nonzero (Aq>O) dual variables] that is not 

necessarily the optimum. The DUAL 1 algorithm copes with this difficulty 

by restarting the maximization procedure releasing all or all but one of 

the previously accumulated equality constraints # . 

# If the last ODM executed prior to the restart test terminated on either a 

discontinuity plane or a base plane, that corresponding single equality 

constraint is retained. 

69 
Now if the previous maximum of the dual function was really the true dual 

maximum, then the new updating sequence will generate the same projection 

matrix and the dual function maximum point will be located in the same 

subspace as before. On the other hand, if the previous maximum of the 

dual functional (just prior to the restart test) was not the true dual 

maximum, then the algorithm will sequentially accumulate a new set of 

discontinuity and boundary planes and terminate at a different vertex with 

a higher dual function value. 

4.5.1 Direction Finding Process 

Turning attention to the general mixed continuous-discrete variable 

case, the DUAL 1 algorithm is described using the schematic block diagram 

shown in Fig. 10. At each step the direction 2 t is taken as the gradient 

V9.(lt) or a projection of the gradient into an appropriate s&space. The 

scheme for generating the next search direction depends upon the nature of 

the previous ODM's termination point. 

Initially [block 11, or when the dual point xt does not reside in any 

of the discontinuity planes, the move direction z t is taken as the gradient 

at -kt modified so as to avoid violation of the nonnegativity constraints 

X$0; seQR, that is [block 21 

S 

St 

=0 if X 

qt 

= 0 and++ (xt) = u q (Z t ) - ii 

q 

q-< 0 

otherwise 

S 

qt 

=+“,, =u qt (;:, -; q 

9 

(4.42.A) 

(4.42.B) 

The foregoing procedure for generating the move vector is equivalent to 

70 
projecting the gradient vector into the subspace represented by the set 

of base planes {h = 0; qEN] where 

q 

N = (q(Aqt = 0; $+ (It, 50; seQ, 1 

q 

(4.43) 

Typically, at this point, the convergence test Izt/ < E [block 31 will 

not be satisfied, therefore go to block 4 and determine whether or not the 

conjugate direction modification is appropriate. 

Whenever it makes sense successive move directions are conjugated to 

each other using the well known Fletcher-Reeves formula [see Ref. 58 p. 871 

(4.44) 

(4.45) 

The second and subsequent move directions within a subspace are generated 

using Eqs. (4.44 and 4.45) Iblock 51. In the DUAL 1 algorithm several ODM's 

can take place without a change in subspace, provided these ODM's do not 

terminate on either a new first order discontinuity plane or a new base 

plane. In any event the conjugacy modification is reinitialized (8, = 0) 

if the number of ODM's executed within a single subspace becomes equal to 

the dimensionality of the subspace. The dimensionality of a subspace is 

equal to Q,, less the number of zero dual variables N, less the number of 

first-order discontinuity planes encountered so far. 

With the move direction zt established in block 4 or 5 go to block 6 

and solve the one dimensional maximization (ODM) problem. The scheme - - 

employed to solve the ODM problem in DUAL1 will be described subsequently, 

71 
however it should be clearly recognized that in contrast to the DUAL 2 

line search scheme, which simply assures an increase in e(x), the DUAL 1 

ODM accurately locates the maximum of !L(R] along the direction zt. After 

solving the current ODM [block 61 there are six possible paths leading to 

the calculation of a new move direction zt in either block 15 or block 2 

of Fig. 10. The six paths are summarized in Table 1 and each of them will 

be briefly described in the sequel: 

Path 1: The updated zt emerging from block 6 does not reside in either 

a new discontinuity plane [block 7 + F] nor a new base plane 

Iblock 8 -f F], but at least one first order discontinuity 

plane has been previously encountered [block 9 + F] # leading 

via point B to block 15, where the move direction is calculated 

without updating the projection matrix according to the relation 

QZ= [Plr: (4.46) 

which is based on Fq. (4.38). 

Path 2: The updated It emerging from block 6 does not reside in either 

a new discontinuity plane [block 7 + F] nor a new base plane 

[block 8 -+ F] and it is true that no discontinuity planes have 

been previously encountered [block 9 + T] leading to block 2, 

where the next move direction is calculated using Eqs. (4.42.A 

and 4.42.B). 

Path 3: The updated At emerging from block 6 does not lie on a new 

discontinuity plane [block 7 + F], but it does reside on a new 

# FOD denotes a Boolean variable which is zero when no first order 

discontinuity planes have been encountered. 

72 
base plane [block 8 + T] and no discontinuity planes have been 

previously encountered [block 10 + T], leading to block 2 where 

the next move direction is calculated using Eqs. (4.42-A and B). 

Path 4: The updated It emerging from block 6 does not reside on a new 

discontinuity plane Iblock 7 -f F], but it does reside on a new 

base plane [block 8 + T] and one 

have been previously encountered 

block 14 where the IP] matrix is 

where z is a unit vector normal 

q 

or more discontinuity planes 

[block 10 + F], leading 

updated by letting 

to 

(4.47) 

to the newly encountered base 

pl=-=, and then modifying the projection matrix as follows 

(4.48) 

The next move direction is calculated in block 15 using the 

updated [R matrix from Eg. (4.48). 

Path 5: The updated It emerging from block 6 resides on a new discontinuity plane [block 7 + T] and it is the first discontinuity 

plane encountered lblock 11 + T] leading to block 12 where the 

projection matrix is initialized according to the following 

procedure. bet zb denote the gradient to the first discontinuity 

plane encountered. Construct a trial projection matrix as 

follows 

z* 

[PI = III - L-Z!? +T+ (4.49) 

'b 'b 

73 
and project either value of the gradient at x into the subspace t 

defined by the first discontinuity plane using Rg. (4.271, that 

is 

(4.50) 

If z 

St 

< 0 for h 

St 

= 0 set the corresponding elements of the 

vector Zb to zero (C 

w 

= 0) and recalculate [PI and zt. When 

Z 

qt 

L 0 for'all X 

qt 

= 0 the initial projection matrix has been 

obtained. The end result of this iteration is to generate a 

[PI matrix that projects any vector into the subspace defined 

by the first discontinuity plane and the appropriate current set 

of Aq = 0 base planes. The next move direction is calculated 

in block 15. 

Path 6: The updated Tt emerging from block 6 lies on a new discontinuity 

plane [block 7 -+ T] but it is not the first discontinuity plane 

encountered [block 11 + F] leading to block 13. The projection 

matrix is updated as follows: 

y’ -+ IPIZb (4.51) 

where $ b is understood to denote the gradient to the new discontinuity plane and 

w [PI+-[PI - yyjq 

YY 

(4.52) 

The next move direction is calculated in block 15. 

4.5.2 Restart of the Algorithm 

At the end of each of the six paths that may be followed after solving the ODM and updating It [block 61 the result is a new move direction zt 

74 
If the new move direction has an absolute value equal to or greater than 

E [block 3 + F] the search for the maximum of the dual function in the 

current subspace continues (i.e.,go to block 4). On the other hand if 

IS.: 1 < E or if the subspace defined by the set of base planes h = 0; 

q 

qeN and the set of P first order discontinuity planes has collapsed to a 

single point (i.e.,QR = N + P) go to block 16. If no first order discontinuity planes have been encountered (i.e. FOD = 0 -+ T] then the maximum 

of the dual function, subject to the nonnegativity constraints 1 

q 

L 0; 

seQ,r has been obtained, the stage is complete, and the values of the 

primal variables are stored. 

Gn the other hand, if one or more first order discontinuity planes 

have been encountered [block 16 + PI, go to block 17 and make the following restart tests: 

(1) if the current value of the dual function L(5t) is equal to 

or greater than the upper bound weight z associated with the 

current 1 t restart and go to block 18; 

(2) if the current value of the dual function !L($) is less than 

the upper bound weight f associated with the current 1 t' compare 

!L(xt) with its value when the restart block 17 was previously 

entered, and if the difference is small go to block 21, otherwise go to block 18. 

It should be noted that unless the stage ends without encountering 

# upper bound weight is given by selecting the smaller of the two candidate 

values (er), dk+l) ) for each discrete primal variable (associated with a 

first order discontinuity plane) in calculating the weight (n.b. the primal 

variables CI b are reciprocal variables). 

75 
any discontinuity planes [block 16 -f T] there will always be at least one 

restart. Once it is determined that the maximum of &(I) subject to 

Xq 2 0; qeQ, may not have been obtained yet, the algorithm is restarted 

releasing all of the previously accumulated equality constraints. However 

.if the last ODM prior to restart terminated on either a discontinuity plane 

[block 18 + T] or a new base plane [block 19 -f T] this single constraint is 

retained while all the others are dropped. This scheme guards against the 

possibility of two successive restarts leading to traversal of exactly the 

same sequence of subspaces. If the last ODM prior to restart is on a discontinuity plane, retention of that constraint is handled by going to 

block 12 and initializing the projection matrix. If the last ODM prior to 

restart is on a new base plane, retention of that constraint is handled by 

modifying Eq. (4.42.A) to read [see block 201 

S 

qt 

=0 if h 

qt 

=0 (4.42.A') 

where q is associated with the base plane encountered by the last ODfl prior 

to restart. Finally, if the last ODM prior to restart does not reside on 

either a discontinuity plane or a new base plane, go to block 1 and restart 

dropping all the previously accumulated base and discontinuity plane constraints. 

4.5.3 Retrieval of the Primal Variables 

For mixed continuous-discrete problems, a stage usually ends by 

exiting block 17 + F and entering block 21 with a dual point It that resides 

on one or more first order discontinuity planes (see Eq. 4.7). For each 

of these P discontinuity planes the corresponding primal variable CI b has 

76 
two candidate values denoted a (k) 

b and a:"). The upper bound solution is 

obtained by selecting the smaller discrete value for each such discrete 

variable. If the upper bound design is feasible f 

, the lowest weight 

feasible design is selected from the set of 2' possibilities that exist. 

On the other hand, if the upper bound solution is not feasible, then a 

feasible design, or by default the design which is most nearly feasible, 

is selected from the set of 2' possibilities. This is done by finding the 

design for which the most seriously violated constraint exhibits the smallest 

infeasibility. Computational experience indicates that when the upper 

bound design is infeasible, none of the other (2' -1) designs are feasible. 

The foregoing discrete search through 2' possible designs is organized 

in such a way that, when passing from one primal candidate to the next, only 

one design variable changes. As a consequence, the new weight and the 

associated constraint values can be computed very efficiently as follows. 

When the b th (k) design variable changes from a discrete value ab to the next 

available discrete value a:+') [with cc:) < abkfl)], the weight becomes 

W(k+l) = W(k) + w 

b i 

1 1 --~ 

(k+l) (k) 

cb cb 1 

and the corresponding constraint values are: 

U 

(k+l) = JW Ia (k+l) 

q + 'bq b - a 

9 

;k)l: qeQ, 

(4.53) 

(4.54) 

The second term on the right hand side of both equations (4.53) and (4.54) 

can be computed and stored once and for all prior to starting the search 

f With respect to the approximate constraints for the p 

th stage. 

77 

IL - 
,.,..,.,,., .,.. .,----.---.-.--.. . . -. . . . --... . . . .._ .. -....-._-....-.- 

through the 2' possible combinations of discrete primal points. Hence subsequent calculations involve only (Q, + 1) additions [or subtractions if the 

(k+l) design variable cb changes from ctb to ctLk) 1 for each new candidate 

optimum. It is worth mentioning that, as soon as it is known that a feasible 

primal solution exists, all the design points for which the weight is lower 

than the maximum dual function value can be disregarded, because the final 

optimal weight must satisfy the following relation: 

w* = g* - 1 

+Q, 

hc; (u (;-iq) 1R” 

since 

(4.55) 

(4.56) 

[Note that in Eqs. (4.55, 4.56), the upperscript * is associated with optimal quantities]. As a result, if the weight computed using Eq. (4.53) turns 

out to be lower than L*, it is superfluous to evaluate the constraint values 

(Eq. 4.54) to check design feasibility and the next discrete point can be 

treated. I 

In the DUAL 1 algorithm, the discrete search through 2' possible 

primal points is based on a sequence of integer numbers, each of which 

corresponds to a distinct discrete solution. To help understand the procedure followed, it is useful to represent each possible discrete solution 

as a point in a P-dimensional integer space where only two different values 

exist along each axis. Such a point can be described using binary coding, 

0 corresponding to one discrete value and 1, to the other. This binary 

representation facilitates creation, in the P-dimensional integer space, 

of a sequence of discrete points such that, when passing from a point to 

78 
the next, only one component (0 or 1) changes (to 1 or 0). This can be 

achieved as follows, depending upon the value of P: 

binary coding decimal equivalent 

P=l 0 0 

1 1 

P=2 

P=3 

010 0 

I 

Ol l 1 

--- 

11 1 3 (= 1+2) 

I 110 2 (= 0+2) 

010 0 0 

0 I 0 1 

0 I 

1 

1 1 

1 

3 

011 0 2 

--lT-O-- I 6 (= 2+4) 

111 l 7 (= 3+4) 

110 1 5 (= 1+4) 

1'0 I 0 4 (= 0+4) 

The foregoing developments permit construction of a computational procedure for automatically defining the sequence of discrete points to be 

successively processed. For example, it is easily verified that, for 

P = 4, the following sequence of integer numbers is generated: 

P=4 0 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 

The difference between two sucessive integers in the foregoing sequence 

gives all information about the next discrete primal point to be processed. 

79 
The absolute value of this difference indicates the discrete number to be 

changed, according to the following prescription: 

difference + discrete variable number 

21 1 

+2 2 

f4 3 

f8 4 

f. . . . . . 

+2 P-l 

If the difference is positive, then the corresponding discrete variable 

shifts to a smaller value. If the difference is negative,then the discrete 

P 

variable changes to a larger value. With this process, all the 2 possible 

primal points are taken into account without repetition and in such a way 

that only one design variable is modified each time. 

It is important to recall that the number P of first order discontinuity planes cannot be larger than the number (Q, - N) of nonzero dual 

variables. In other words, the number of ambiguous design variables, for 

which one out of two candidate values must be selected, never exceeds the 

number of strictly critical behavioral constraints, which is precisely the 

effective dimensionality of the dual space. This number is usually small 

for practical problems and therefore dual methods retain their attractiveness when discrete variables are introduced. 

4.5.4 One Dimensional Maximization 

In the DUAL 1 algorithm,once a move direction is established (see 

blocks 2, 15, 4 and 5; Fig. lO),it is necessary to find the maximum of the 

80 
explicit dual function R(x) along the direction z t emanating from the 

point It, using Eq. (4.41). It follows that %(I) becomes a function of the 

scalar move distance d along the direction 2 t' The one dimensional maximi- - - - 

zation (ODM) of this concave function Ii(d) is equivalent to seeking the 

vanishing point of the directional derivative (see Eq. 3.20): 

e'(d) = q V& = ; Z 

b=l bt b 

ci (d) - Xt 

where 

'bt = c s 

seQ, qt 'bq 

and 

x = t 1 s ii 

V=Q, qt 9 

(4.57) 

(4.58) 

(4.59) 

The primal variables ccb(d) are known explicit functions of d along the 

direction zt in dual space (by using Eqs. 2.39, 2.40 and 2.41 for the 

continuous variables and Eq. 4.8 for the discrete variables), except for 

the indices b related to the current P discontinuity planes (see Eq. 4.7) 

in which the gradients are projected. For these indices the discrete 

primal variables crb(d) can take on either of two distinct values. Fortunately the corresponding values of Zbt happen to vanish, since the direction 

+ 

St lies in the discontinuity planes (see Eq. 4.32), so that: 

'bt tb= 

c-p; O;beP (4.60) 

With this remark the ODM problem is uniquely defined and consists 

of seeking the vanishing point of a'(d) over the interval 0 < d < d 

max. 

81 
The maximum allowable step length dm, is selected so that none of the 

dual variables can become negative. The computation of d 

max 

is accomplished 

employing the same scheme as in Dual 2 (see Eq. 3.19), namely 

d = Min S 

I I 

29s 

max S 

qt 

<o qt 

(4.61) 

It should be recognized that the dual space is subdivided into several 

domains separated by first and second order discontinuity planes defined 

in Eq. (4.7) and Eqs. (3.11, 3.12), respectively. Typically the move vector zt will intersect the discontinuity planes at several values of d which 

fall in the interval 0 < d < d,,, and therefore the ODM search is subdivided 

into several distinct intervals within which the function k'.'(d) and a"(d) 

remain continuous. 

Substituting Eq. (4.41) into Eqs. (3.11 and 3.121, it is easily 

shown that the vector z 

t intersects the second order discontinuity planes 

at values of d given by 

d(L) "b =- 

b 'bt ,,&2 - 'bt 

I ; 

b = 1,2,...B 

and 

,(‘I - “b 

b Z bt 

where 

Y bt = c A 

SCQ, qt 'bq 

(4.62) 

(4.63) 

(4.64) 

and Z bt is defined by Eq. (4.58). The function a'(d) exhibits slope discontinuities for values of d equal to the intercept distances (L) 

53 and 

82 
(VI 

4, (i.e., R"(d) is discontinuous). In a similar way, the intercept 

distances to the first order discontinuity planes are obtained by substituting Eq. (4.41) into Eq. (4.7): 

(4.65) 

The function a'(d) is discontinuous for values of d equal to the intercept distances (k) 

53 - 

The key idea of the ODM procedure employed in DUAL 1 is that the 

intercept distances to first and second order discontinuity planes (see 

Eys. 4.62, 4.63 and 4.65) are evaluated and used to locate either the one 

dimensional maximum or an appropriate uncertainty interval within which 

a'(d) vanishes and both L'(d) and L"(d) are continuous. The procedure will 

be described qualitatively using Fig. 11 to help clarify the basic approach 

followed. It will be convenient to consider the pure continuous, pure discrete and mixed continuous-discrete cases separately. 

(1) Pure Continuous Variable Case. A hypothetical plot of the dual 

function versus d along the direction zt is shown in Fig. ll.A. 

With d max already known from Eq. (4.61), the intercept distances 

CL) 

4, and dr) to second order discontinuity planes (see Eqs. 4.62 

and 4.63) that intersect zt between 0 and d,, are computed and 

stored in ascending order (i.e.,0 < d. < d,,, ; j = 1,2,...). The 

3 

slope of dual function (see Eq. 4.57) along the direction zt is 

then evaluated at each intercept location until the sign of L'(d) 

changes from plus to minus [aa > 0 and !L'(d. 

I+1 1 < 01. Once it 

is known that the maximum of L(d) resides in the interval 

83 
d.<d<d 

I j+l' in which a"(d) is continuous, the Newton Raphson 

method is used to refine the location of the maximum point: 

dbJ+l) = ,w !L' (d") 

-- 

R"(d') 

(4.66) 

Referring to Eq. (4.57), it is easily verified that 

2 3 

'bt cb c"(d) = - + 1 - 

b& Wb 

(4.67) 

where the set B is that associated with the free primal variables 

(see Eq. 3.5). The free primal variables are known explicit 

functions of d: 1 

- 2 

cb = 

; be< (4.68) 

The Newton-Raphson iteration described in Eq. (4.66) furnishes 

the point d* where L'(d*) = 0, which corresponds to the maximum 

point of !L(d) and this distance is selected as the solution of the 

ODM (i.e. dt + d*). 

(2) Pure Discrete Variable Case. A representative one-dimensional 

plot of the dual function versus d along the direction zt is 

shown in Fig. ll.B. This plot consists of a sequence of linear 

segments with discontinuous first derivatives at distances d (k) 

b 

locating the points where the vector zt intersects first-order 

discontinuity planes (see Eq. 4.65). With d 

max 

already known 

(k) from Eq. (4.61), the intercept distances db between 0 and dmax 

are computed and stored in ascending order (i.e., 0 < d. < d 

7 maxi 

j = 1,2,...). The slopes of the dual function along the direction 

84 
-+ St are evaluated at each intercept location until the sign of 

Ill(d) changes from plus to minus [L'(di) > 0 and e'(df) < 01, indicating that the maximum is at dj(i.e. dt + dj). 

(3) Mixed Continuous-Discrete Variable Case. A representative plot of 

the dual function versus d is shown in Fig. 11-C. With d 

max 

already known from Eq. (4.61), the intercept distances to both 

first (Eq. 4.65) and second order (Eqs. 4.62 and 4.63) discontinuity 

planes that reside between 0 and dmax are computed and stored in 

ascending order (i.e. 0 < d. c amax; j = 1,2,...). 

3 

The slope(s) 

of the dual function along 2 t is then evaluated at each intercept 

location until a sign change signals the location (see Fig. 11.U) 

or trapping (see Fig. 11.C2) of the maximum. In the first case, 

the maximum has been located at d 

j 

since !?'(a;) > 0 and &'(df) < 0 

(Fig. U-Cl). In the second case, the maximum has been trapped in 

the interval d 

j 

<a cd. J+l' since iI > 0 and Il'(d: 

I+1 

) < 0. The 

Newton-Raphson method (see Eq. 4.66) is then used to locate the 

point a* where !L'(d*) = 0 and this distance is taken as the solution 

of the ODM (i.e. d t f a*). 

It is important to notice that, with this ODM procedure, the discrete 

primal variables do not need to be explicitly computed from Eq. (4.8), 

where a lot of tests have to be completed before finding the right discrete 

values. They are directly deduced from "status" vectors identifying the 

design variable (index b) and the discrete value (index k) to which each 

intercept distance d (k) 

b corresponds (see Eq. 4.65). The status vectors 

are constructed, stored and reordered when the intercept distances 

85 
(0 c d. c d 

3 max 

1 are evaluated and put in ascending order. As a result, 

it is only in block 1 in Fig. 10 that the discrete primal variables are 

evaluated using Eq. (4.8) (i.e.,in order to start or restart the maximization procedure). Subsequently, the primal variables are always determined 

in the ODM part of the algorithm using the intercept values and corresponding "status" vectors. 

86 
5. THE ACCESS 3 COMPUTER PROGRAM 

The ACCESS+ 

. 

computer programs have been developed to demonstrate the 

effectiveness of an automated structural synthesis capability formed by 

combining finite element analysis techniques and mathematical programming 

algorithms. The ACCESS 1 program demonstrated the efficiency of the coordinated use of approximation concepts on problems of relatively small scale, 

subject to simple static constraints [see Refs. 5 and 61. Subsequently the 

ACCESS 2 program was developed to permit consideration of more complicated 

constraints than those treated in ACCESS 1 and to build a body of experience 

that can be used to set sound guidelines for future developments of large 

scale industrial application problems [see Refs. 7 and 611. 

The basic ideas set forth in this work, which combine approximation 

concepts and the dual method formulation, have been implemented in a further 

improved computer program called ACCESS 3. In contrast to its predecessors 

ACCESS 1 and ACCESS 2, which employed feasible direction and/or interior 

penalty function algorithms without exploiting the special algebraic form 

of the explicit approximate problem (see Eqs. 2.19 - 2.22), the new ACCESS 3 

program uses the dual formulation as the basis for adding two powerful 

optimization algorithms into the ACCESS framework (namely, DUAL 1 and DUAL 2). 

ACCESS 3 retains all of the ACCESS 2 capabilities as a subset and the data 

preparation formats are fully compatible [see Ref. 621. 

f sproximation Concepts code for Efficient Structural 3nthesis 

- 

87 
5.1 Scope of the ACCESS 3 Code 

The ACCESS 3 program assumes that the structural topology, configuration and material are preassigned parameters given by the user. The topology 

is specified via element-node connectivity data, the configuration is established by giving nodal positions (for the undeformed system) relative to a 

fixed reference coordinate system, and the given material is represented by 

its specific weight, stiffness, strength and thermal expansion properties. 

The program treats sizing quantities (i.e., truss cross-sectional areas and 

thicknesses of shear panel or membrane elements) as design variables. The 

design variables can be continuous or discrete variables. In the case of 

discrete variables, the user supplies the set of available values in ascending order. The ACCESS 3 code accepts user supplied side constraints on 

continuous design variables and a rather general capability for design 

variable linking is also built into the program. Move limits can also be 

specified restricting the percentage change in the design variables within a 

given stage of the overall iterative design process. 

Four distinct optimization algorithms are available in the ACCESS 3 

program. The user can select a specific optimizer, depending upon the 

nature of the constraints, the expected number of strictly critical first 

order approximated constraints, the number of design variables, and their 

continuous or discrete character. The four optimization algorithms are as 

follows: 

(1) the NEIWSUMT optimizer implements a sequence of unconstrained 

minimizations technique using a quadratic extended penalty 

function feature [see Ref. 73; 

88 
(2) the PRIMAL 2 optimizer uses a second order projection algorithm 

to generate a sequence of feasible search directions [see 

Ref. 301; 

(3) the DUAL 2 optimizer employs a second order Newton type of 

algorithm to find the maximum of the aual function when all 

the design variables are continuous (see Chapter 3); since it 

has been found to be very efficient in practice, it is the 

recommended option for pure continuous variable problems (see 

Chapter 6); 

(4) the DUAL 1 optimizer employs a gradient projection type of 

algorithm to maximize the dual function when the design variables 

are all discrete or mixed continuous-discrete (see Chapter 4); 

when all the design variables are continuous, the DUAL 1 algorithm reduces to a special form of the conjugate gradient method; 

however it is generally less efficient than the DUAL 2 optimizer 

for pure continuous variable problems. 

The two primal optimizers (i.e., NEWSUMT and PRIMAL 2) tend to generate a 

sequence of steadily improved feasible designs, because they are employed to 

solve only partially each explicit approximate problem, This feature can be 

used to control the convergence of the overall optimization process when the 

constraints of the primary problem are highly nonlinear. On the other hand, 

the two dual optimizers (i.e., DUAL 1 and DUAL 2) produce a sequence of not 

necessarily feasible designs, because they find the "exact" solution to each 

of the separable approximate problems generated in sequence. However it has . 

been observed that the design infeasibility, if any, is usually small and it 

decreases stage by stage. 

89 
The program includes provision for guarding against a variety of 

failure modes including strength, deflection, slope (relative deflections) 

and natural frequency limits. For truss members independent tension and 

compression allowables can be specified. In shear panels and isotropic 

membrane elements, where multiaxial stress states exist, strength constraints 

are introduced by limiting the value of an equivalent stress based on the 

distortion energy criterion. In the orthotropic membrane elements used to 

model fiber composite lamina at a preassigned orientation, three separate 

strength failure criteria options are available: the maximum strain criteria, stress interaction formulas or the Tsai-Azzi criterion [see Refs. 7 

and 613. These strength failure criteria for fiber composite lamina take 

into account differences in the longitudinal, transverse and shear allowables 

as well as differences in the tension and compression allowables. When the 

explicit problem is formed at each stage, all the stress and strain constraints can be replaced with either first order approximations or with zero 

order ones. The zero order explicit approximations are obt,ained using 

classical stress ratio formulas (see Eq. 2.58). They can be expressed as 

simple side constraints, which is especially beneficial when dual methods 

are employed. A selection criterion permits automatic subdivision of the 

stress and strain constraints in two categories: those requiring first 

order approximation (full linear Taylor series expansion) and those for 

which zero order approximation (side constraint) is sufficiently accurate 

(see Section 2.5.3). 

The program also contains provisions for placing lower and upper 

limits on the first several natural frequencies. In addition to the structural mass, which varies as the sizing design variables change, fixed nodal 

90 
masses can also be prescribed. For example, these fixed nodal masses can 

be used to simulate fuel inertia or engine masses in wing problems. There 

are three distinct approximation options available for frequency constraints 

in ACCESS 3, namely, the X = w2 are approximated by: (1) a first order 

Taylor series in terms of the independent reciprocal sizing variables after 

linking; (2) a first order Taylor series in terms of the direct independent 

sizing variables after linking; (3) a full second order Taylor series expansion in terms of direct independent sizing variables after linking [see 

Ref. 631. It should be noted that only option 1 above can be used with any 

one of the four optimization algorithms options (NFWSUMT, PRIMAL 2, DUAL 2, 

DUAL 1) available in ACCESS 3. Cm the other hand the NEWSUMT optimization 

algorithm can be used with any one of the three A = WI. approximation options. 

The available combinations of X = w2 approximation and optimization algorithm 

are shown in Table 2. 

It is also important to recognize that while ACCESS 3 can handle three 

distinct kinds of sizing type structural optimization problems t(1) pure 

continuous variable problem, (2) pure discrete variable problems, and (3) 

mixed continuous-discrete variable problems], only the DUAL 1 optimization 

algorithm is applicable to pure discrete and mixed continuous-discrete problems. For the case of pure continuous design variable problems all four 

optimization algorithm options (NEWSUMT, PRIMAL 2, DUAL 2 and DUAL 1) are 

applicable and DUAL 2 is the preferred choice f ,because it will generally 

be the most efficient. The algorithm options available for various kinds 

f Unless the approximation selected for the frequency constraints requires 

the use of NFWSUMT (See Table 2). 

91 
of problems are summarized in Table 3. 

The set of finite element types available in ACCESS 3 is the same as 

that in its precursor program ACCESS 2. They include uniform bar (TRUSS), 

isotropic constant strain triangle (CSTIS), orthotropic constant strain 

triangle (CSTOR), isotropic symmetric shear panel (SSP), pure shear panel 

(PSP) and thermal shear panel (TSP) element types. The program data structure can accommodate four additional finite element types. A detailed 

description of the basic characteristics of the six element types currently 

included will be found in Appendix A of Refs 161 and 621. All finite element 

types include provisions for representing thermal and body force loads. For 

each of several distinct loading conditions temperature changes and gravity 

field loads may be specified. These design variable dependent loads are 

included in addition to specified external applied loading conditions. The 

external applied loads may take the form of specified pressure loadings 

and/or given nodal forces for each loading condition. The objective function 

in ACCESS 3 is taken to be the total weight of the idealized finite element 

representation of the structural system. 

The ACCESS 3 computer program is a research type program, however, 

it is capable of treating example problems that are large enough to clearly 

demonstrate the generality and efficiency that can be achieved by combining 

approximation concepts and dual methods. Such research programs provide a 

knowledge and experience base on which to build full scale analysis/synthesis 

capabilities for widespread use by industry. The current version of ACCESS 3 

has a data structure which permits it to handle problems with up to 1000 

finite elements, 600 displacement degrees-of-freedom, 200 independent design 

variables and 20 distinct load conditions. The current problem size limits 

92 
are due primarily to the restriction that the compact vector form of the 

system stiffness and mass matrices must fit in core simultaneously. A 

further discussion of restrictions and limitations applicable to bothf 

ACCESS 2 and 3 will be found in Section 2.3 of Ref. [621. 

In summary, the main feature of ACCESS 3 lies in the joining together of approximation concepts and dual methods. This solution scheme 

can be interpreted as a generalized optimality criteria method. Another 

new capability is the zero order approximation of the stress constraints 

based on the conventional "Fully Stressed Design" optimality criterion. 

Therefore the ACCESS 3 program can be regarded as an advanced research tool 

where mathematical programming and optimality criteria approaches coalesce 

to provide an efficient and reliable structural weight minimization method. 

5.2 Program Organization 

The organization of the ACCESS 3 computer program is, in principle, 

similar to that of its precursor ACCESS 2. The function of the "preprocessor" 

(see Fig. 12) is to compute and store all necessary information that is 

independent of the design variables after linking. A typical stage in the 

overall iterative design process begins with the control block supplying a 

"primal trial design" to the "approximate problem generator" block (see 

Fig. 12). This primal trial design is subjected to a detailed finite element 

structural analysis and the results are used to evaluate all of the constraints. Deletion techniques are employed to temporarily drop unimportant 

constraints. This is followed by calculation of partial derivatives 

f The only additional limitations on problem size for ACCESS 3 arise from 

storage requirements for discrete variable data sets. 

93 
(sensitivity analysis) and construction of first or zero order explicit 

approximations for the constraints that survived the deletion process. 

The approximate primal problem for the p 

th stage is passed back 

through the design process control block and this primal problem plus a 

set of initial trial values for the dual variables (if the DUAL 1 or DUAL 2 

option has been selected) or the primal variables (if the NEWSUMT or PRIMAL 2 

option has been selected) are handed off to the optimization algorithm block. 

In the case of dual methods, which are of primary interest herein, the 

explicit dual function is formed and its maximum is then sought (subject to 

nonnegativity constraints on the dual variables). If all the design variables 

in the primal problem are continuous, either algorithm (DUAL 1 or DUAL 2) 

may be used, but the second order Newton type maximizer of DUAL 2 is recommended because it is usually more efficient. For problems where the design 

variables are all discrete or mixed continuous-discrete, the DUAL 1 algorithm 

must be used, because it can accommodate the local discontinuities in gradient of the dual function which arise in such problems. It should be emphasized that when dual methods are used, a precise solution to the approximate problem posed at each stage p is sought while when interior point 

penalty function methods (i.e., NEWSUMT optimizer) or projection methods 

(i.e., PRIMAL 2 optimizer) are used in primal space, the goal of each stage 

is to produce an improved noncritical design (as in ACCESS 1 and 2). 

Once the set of dual variable values corresponding to the maximum 

th 

of the dual function for the p stage has been found, the corresponding 

set of primal variables is stored. This improved set of primal variables 

is then subject to an overall design process convergence test and if con94 
vergence has not been achieved, the improved set of primal variables (as 

well as the associated dual variables f ) are passed back to the design process 

control block and another stage begins. It should be clearly recognized 

that only one detailed finite element structural analysis is executed per 

stage and none of the constraints included in the original problem statement 

are permanently deleted (unless they are strictly redundant). 

In summary, one stage of iteration includes one finite element structural analysis, one constraint deletion process, one sensitivity evaluation 

for retained constraints, and one optimization of an approximate problem 

using either primal or dual algorithms. Since the final design is subjected 

to a detailed finite element analysis, the total number of structural analyses 

equal the number of iteration stages plus one. The iterative design process 

is terminated when one of the specified convergence criteria is satisfied, 

which will be typically after about 10 redesign stages. 

f The dual variables generated at the end of a stage are used as a starting 

point for the next maximization problem. 

95 

6. NUMERICAL EXAMPLES 

In this.chapter, detailed results for various structural optimization 

problems are presented. For pure continuous variable problems, attention 

is focused on results obtained with the DUAL 2 optimizer and efficiency is 

assessed using comparable results obtained with the previously available 

[Refs. 5-71 NEWSUMT algorithm. The numerical results reported here indicate 

that the improved analysis/synthesis capability developed by combining dual 

methods and approximation concepts is remarkably efficient. Computational 

effort expended in the optimization portion of the program is reduced dramatically in representative examples (by at least a factor of ten) and the total 

computer time required to converge the overall optimization process is also 

reduced significantly. Results for pure discrete and mixed continuous-discrete 

variable problems show that although the extension of dual methods represented 

by the DUAL 1 optimizer is lacking in mathematical rigor, it appears to have 

promise as a practical design tool (see Chapter 4). Unless otherwise specified, all problems have been run using a single precision version of ACCESS 3 

on the IBM 360/91 at CCN, UCLA. 

6.1 lo-Bar Truss (Problem 1) 

In this section, consideration is given to the planar lo-bar cantilever 

truss shown on Fig. 13. The structure is subject to a single load condition 

consisting of 444.8 kN (100 kip) downward loads applied at nodes 2 and 4 (see 

Fig. 13). The truss element material properties, as well as initial crosssectional area, minimum member size and displacement limits, are summarized 

in Table-4. Detailed tabular input data can be found in Ref. [5], where this 

97 
example was designated as problem 3. Note that only lower limit side constraints are imposed on the member sizes and uniform stress limits are prescribed. No design variable linking is specified and therefore this problem 

has ten independent design variables. Several cases will be considered; 

they include pure continuous, pure discrete and mixed continuous-discrete 

variable cases. 

6.1.1 Case A: Equality Constraints on Displacements 

An interesting feature of the dual algorithms implemented in ACCESS 3 

is that they permit treatment of equality constraints (simply by assigning 

the same value to the lower and upper limits). As an example, the previously 

described ten bar truss problem is considered with equality constraints on 

the vertical displacements at nodes 1 [-5.08 cm (-2.0 in.)] and 3 I-2.54 cm 

(-1.0 in.)], in addition to the usual stress limitations and side constraints 

(see Table 4; Case A). The iteration history presented in Table 5 shows that 

the DUAL 2 algorithm generates the optimal design after only 9 reanalyses, 

despite the difficulty of the problem. Table 5 also gives for each stage, 

the values of the constrained displacements and of the stress in member 6, 

which reaches the allowable limit [172,375 kN/m2 (25,000 psi)] at the optimal 

design. The final design presented in Table 6 is identical to that given in 

Ref. [30], where this problem was first solved. 

6.1.2 Case B: Pure Continuous Problem 

The ten-bar truss problem will now be discussed in its conventional 

form, namely with inequality constraints imposed on the displacements in 

the Y direction for all nodes [f 5.08 cm (2 2.0 in)]. Stress limitations 

[? 172,375 kN/m2 (+ 25,000 psi)] and minimum area [0.6452 cm2 (0.1 in')] constraints are also taken into account. The final designs presented in Table 6 

98 
and the iteration history data given in Table 7 and plotted in Fig. 14 

permit comparison of the results generated by ACCESS 3 using the NEWSUMT 

and DUAL 2 optimizers for the pure continuous variable case. 1tcanbe 

seen that when compared to NEWSUMT, DUAL 2 leads to a significant improvement 

in the rate of convergence, a lower final mass and a dramatic reduction in 

the amount of computer time required by the optimizer (see Table 7). For 

comparison with previously published results for this now classical problem, 

Rafs. [5 and 301 can be consulted, where various mathematical programming 

and optimality criteria techniques are discussed. 

It is well known that the lo-bar truss problem presents at least two 

distinct local optima [2302.78 kg (5076.67 lhn) and 2295.60 kg (5060.85 lbm); 

see Refs. 30, 34 and 641. The present approach, combining approximation concepts and dual methods, leads to the lowest mass design [2296 kg (5061 lbm)] 

in 13 reanalyses. The optimal design with mass 2296 kg (5061 lbm) exhibits 

the interesting property that member 5 is simultaneously constrained by 

stress and minimum size limitations. Only one displacement constraint is 

strictly critical (node Z), while the displacement constraint at node 1 is 

almost critical C5.057 cm < 5.080 cm (1.991 in < 2.0 in.)]. With regard to 

the final design generated by NEWSUMT 12309 kg (5090 lbm)], the downward 

vertical deflections at nodes 1 and 2 both attain the limiting value and no 

stress constraint is critical. 

It should be noted that the iteration history data given in Table 7 

for this example (Case B; pure continuous case) contains both the unscaled 

and scaled mass (feasible, strictly critical) for each iteration using 

DUAL 2. The iteration history for NEWSUMT contains the unscaled mass only 

since NBWSUMT generates a sequence of feasible, noncritical designs. 

99 
6.1.3 Case C through Case E: Assessment of DUAL 1 

In order to validate the DUAL 1 optimizer, several simple discrete 

problems have been derived from the foregoing 10 bar truss example. The 

data are the same as in the previous case, except that for all or some members, 

the cross-sectional areas can only take on the available discrete values given 

in Table 8. These values have been selected as follows: they are the numbers 

c6.452 cm2 , 12,904 cm', 19.356 cm'.. . . ...25.08 cm'}, that is the integer 

sequence Cl in', 2 in2 , 3 in',...... -40 in'} in which the optimal values of 

the design variables obtained in the pure continuous case have been inserted. 

In each of the test cases to be discussed subsequently (pure discrete and 

mixed continuous-discrete), the DUAL 1 optimizer should be capable of retrieving the previously generated continuous variable optimum design with a 

mass of 2296 kg (5061 lbm). The results are summarized in Table 7 under the 

headings Case B through Case E. Case B is the pure continuous problem, previously described. Case C is the pure discrete problem, where all the crosssectional areas can only take on the available discrete values given in 

Table 8. Case D is a mixed discrete-continuous problem, where only the 

design variables 1, 3, 6, 8 and 10 are discrete. Case E is also a mixed 

discrete-continuous problem, where now the only discrete variables are those 

numbered 2, 4, 5, 7 nad 9. The iteration histories given in Table 7 illustrate the efficiency of DUAL 1. In all cases, the expected optimal solution 

is retrieved 12296 kg (5061 lbm)], within 13 to 15.iterations. 

6.1.4 Case F: Pure Discrete Problem 

Finally, Case F is another pure discrete variable problem where the 

set of available discrete values is IO.6452 cm', 3.226 cm2 , 6.452 cm', 

9.678 cm', 12.904 cm', 16.130 cm2 . . . . . . ..I. that is the sequence IO.1 in', 

0.5 in', 1.0 in', 1.5 in', 2.0 in', 2.5 in2 . . . . . . . ..I. Unlike Cases C 

100 
through E, this set no longer contains the optimal continuous solution 

values of the design variables. The iteration history is given in Table 

7 and it is not very different from the iteration history for the pure 

continuous variable case (i.e.,case B). The final design given in Table 

6 is slightly infeasible (0.04%), which explains why the mass for this pure 

discrete case 12295 kg (5060 lbm)] is lower than the mass obtained in the 

pure continuous case [2296 kg (5061 lbm)]. It should be noted that the 

design generated at iteration 11 12303 kg (5078 lbm)] is feasible and it 

could be chosen as the final design (see Table 7, case F and Table 6). 

It is worth mentioning that all the masses given in Table 7 do not 

correspond to a feasible design. This is because a dual solution scheme 

is used, yielding the exact solution to each explicit approximate problem 

(see Eqs. 2.19 - 2.22). Furthermore, when discrete variables are involved 

scaling cannot be used to produce a feasible critical design after each 

full structural analysis. 

6.2 25-Bar Truss (Problem 2) 

Attention is now focused on the 25-bar space truss represented in 

Fig. 15. The structure is assumed to be symmetric with respect to both the 

X-Z and Y-Z planes and therefore the problem involves eight independent 

design variables after linking in order to impose symmetry (see Table 11 

for the linking scheme). Material properties and other data sufficient to 

fully describe the problem are given in Table 9. Constraints are placed on 

member sizes [0.06452 cm2 (0.01 in') minimum area], displacements in X, y 

and Z directions for the two upper nodes [? 0.889 cm (f 0.35 in)], tensile 

stresses [275,800 kN/m2 (40,000 psi)] and compressive stresses (reduced 

101 
stress limits based on the Euler buckling formula; see Ref. 5, Problem 5). 

Two distinct loading conditions are applied to the structure. Several cases 

will be considered, including pure continuous and pure discrete variable 

problems. 

6.2.1 Case A: Pure Continuous Problem 

The NEWSUMT and DUAL 2 options of the ACCESS 3 program have been 

employed to solve the 25-bar truss problem with all continuous design 

variables. Detailed numerical results are given in Tables 10 and 11, while 

Fig. 16 presents the iteration histories graphically. Again the use of the 

DUAL 2 algorithm dramatically reduces the computer time expended in the 

optimizer portion of the program and improves the convergence properties of 

the overall optimization process. Only 3 reanalyses are sufficient to produce a nearly optimal design (within 0.4% of the final mass) and convergence 

is achieved after 6 reanalyses (see Table 10). For this problem, the truncation procedure used in ACCESS 3 [see Refs. 5-71 does not significantly 

reduce the number of potentially active constraints retained at each stage. 

In the last iteration, 13 constraints are still retained, which is larger 

than the number of independent design variables (eight). However, the 

efficiency of the DUAL 2 optimizer is not adversely affected, because the 

effective dimensionality of the dual problem does not exceed the number of 

strictly active, linearly independent constraints (three). At the optimum 

design, the critical constraints are the Y components of displacements at 

nodes 1 and 2 under both load conditions as well as the compressive stress 

in member 19 and 20 (both in linking group 7) under load condition 2. 

102 
In this particular example the dual method approach -generates a 

sequence of feasible, noncritical designs despite the fact that the explicit 

approximate problem is solved completely, rather than partially (NEWSUMT), 

during each redesign stage. As a result, scaling down the design variables 

to generate a critical design leads to further reduction in the structural 

mass (see Table 10; DUAL 2 scaled and unscaled mass). It can also be observed that the iteration history produced by the DUAL 2 algorithm is the 

same as the one generated by the generalized optimality criterion [Ref. 301 

and it is also very similar to the iteration history of the conventional 

optimality criteria technique reported in Ref. [13]. These results offer 

numerical confirmation of the fact that the general (mathematical programming based) capability represented by ACCESS 3 generates, with comparable 

efficiency, iteration histories and final designs that are very similar to 

those produced by conventional optimality criteria techniques. This parallel 

performance is observed and can be expected for those problems where conventional optimality criteria techniques are found to be adequate. 

6.2.2 Case B through Case D: Pure Discrete Problems 

The 25-bar truss is one of the few discrete problems for which a reference solution is available in the literature [see Ref. 651. Therefore, it 

offers an opportunity to relate the present work to past experience on discrete variable problems. The data are the same as for the pure continuous 

case (Case A). The available discrete values for the cross-sectional areas 

are given in Table 8 for the three cases under consideration. The differences 

between two successive values are IO.6452 cm2 (0.1 in') Case B], 12.5808 cm2 

(0.4 in') Case C] and 15.1616 cm2 (9.8 in') Case D]. Case A is the pure 

103 
continuous case. The iteration histories generated by DUAL 1 are given in 

Table 10 while the final designs are compared in Table 11 with solutions 

reported in Ref. [65] for Cases C and D. In all cases, ACCESS 3 produces 

very good results. It leads to plausible designs (when compared with the 

continuous solution), which are superior to those given in Ref. [65]. It 

is interesting to note that all designs presented in Table 11 are feasible. 

Comparison of the minimum mass achieved in Cases B, C and D gives a quantitative measure of the increasing mass penalty (1.4%, 5.5'10, 14.6%) associated 

with larger increments between available discrete member sizes. 

6.3 72-Bar Truss (Problem 3) 

In this section, attention is directed to the 72-bar four level 

skeletal tower depicted in Fig. 17. Definition data for this widely studied 

example are summarized in Table 12. In addition to stress [* 172,375 kN/m2 

(* 25,000 psi)] and minimum area LO.6452 cm2 (0.1 in']] constraints, displacement limits [? 0.635 cm (k 0.25 in)] are imposed on the four uppermost 

nodes in the X and Y directions. Two distinct loading conditions are applied 

(see Table 12). By symmetry the problem involves 16 independent design variables after linking (see Table 14 for the linking scheme). In this example, 

the capability available in ACCESS 3 of treating the stress constraints by 

using zero order explicit approximations is exploited (see Section 2.5.3). 

The program automatically finds out that none of the stress constraints has 

to be first order approximated, so that, at each stage, all stress constraints 

are replaced with simple side constraints. 

The numerical experiment conducted in Section 2.4.1 with a three bar 

truss example was reproduced with the 72 bar truss problem. Namely, the problem was solved using the NEWSUMT option of ACCESS 3, with different values 

104 
for the control parameters, in such a way that increasingly exact solutions 

are generated for each explicit approximate problem. The iteration history 

data given in Table 13 and plotted in Fig. 18 clearly show that the more 

precise solutions of the explicit problems lead to faster convergence of 

the mass, with respect to the number of structural reanalyses. In the limiting case where the explicit problems are solved exactly at each stage, the 

NEWSUMT optimizer would of course generate the same sequence of design points 

as the DUAL 2 optimizer. In addition to the iteration history data corresponding to NEWSUMT and DUAL 2, Table 13 and Fig. 18 also contain results 

produced by conventional optimality criteria techniques. The close similarity between the results from ACCESS 3 [NEWSUMT (0.1x3) or DUAL 2 options] 

and those of Refs. [15, 17 and 301 numerically confirms the interpretation 

of the dual method approach as a generalized optimality criteria method 

(see Section 2.5). 

For the final design obtained with ACCESS 3 [as well as with the 

methods of Refs. 15, 17 and 301, the critical constraints are the compressive 

stress in members 1 through 4 (linking group 1) under load condition 2, the 

X and Y displacements of node 1 under load condition 1 and the minimum member 

size requirements for the members of linking groups 7, 8, 11, 12, 15 and 16. 

The member sizes corresponding to this design are given in Table 14. 

6.4 63-Bar Truss (Problem 4) 

The next example involves a 63 bar-truss idealization of the wing carry 

through box for a large swing wing aircraft subject to two distinct loading 

conditions (see Fig. 19). Detailed data defining this problem are given in 

Table 15 (see Table 17 for element-node connectivity data). Minimum mass 

105 
design is sought considering stress and minimum size constraints [? 689,500 

kN/m2 (100,000 psi) and 0,06452 cm2 (0.01 in2) limits, respectively], as well 

as a torsional rotation limit. The torsional rotation constraint is introduced 

by imposing an upper bound [2.54 cm (1.0 in)] on a relative displacement of 

nodes 1 and 2 in the x direction. Since design variable linking is not used, 

the problem involves 63 independent design variables. The problem was first 

proposed in Ref. [17] and it has been studied further in Refs. 15, 6, 7, 30, & 321. 

In Table 16, the iteration history data reported in these references 

are compared with the results generated by ACCESS 3. The NEWSUMT option 

leads to a sequence of noncritical feasible designs with monotonically decreasing mass, which corresponds well to the primal philosophy of this 

solution scheme. Once again, when the explicit approximate problem is solved 

with more accuracy at each stage, the convergence of the mass becomes faster, 

but the computational cost increases substantially [when passing from NEWSUMT 

(0.5x1) to NSWSUMT (0.5x2) in Table 161. Solving each explicit approximate 

problem exactly using the DUAL 2 optimizer yields a sequence of infeasible 

designs (unscaled masses in Table 16). Consequently scaling produces feasible mass from one iteration to the next (scaled masses in Table 16). A 

graphical comparison of NEWSUMT and DUAL 2 performance is shown in the convergence curves of Fig. 20. The net result is that DUAL 2 furnishes an 

optimal design after a smaller number of structural reanalyses than NEWSUMT, 

at a much lower computational cost (60 set for DUAL 2 and 163 set for NEWSUMT). 

It is worthwhile noticing that the computer time expended in the optimizer 

portion of the program remains small when DUAL 2 is employed, despite the 

relatively large dimensionality of the dual problem at each stage (25 strictly 

106 
active behavior constraints at the optimum design). 

The final design generated by ACCESS 3 (DUAL 2 option) is compared 

in Table 17 to those reported in Refs. 15, 17, and 301. Except for the 

design of Ref. [17], all the other designs are very close to each other and 

they exhibit essentially the same set of critical constraints, namely: torsional rotation limit under load condition 2; minimum member size for elements 19, 20, 23, 24, 25, 29, 58, 59, 60, 61, 62,and 63; tension stress under 

load condition 1 for members 2, 4, 6, 8, 10, 12, 14, 16, 17, 21, 28,and 29; 

and compression stress under load condition 1 for members 1, 3, 5, 7, 9, 11, 

13, 15, 18, 22 f , 50,and 51. 

Looking at the results produced by the optimality criteria technique 

of Ref. [17] (see Tables 16 and 17), it appears that these results, without 

being as good as those generated by DUAL 2, are nevertheless acceptable for 

practical purposes. Since the approach of Ref. [17] employs the computationally inexpensive fully stressed design (FSD) concept to treat the stress constraints, it can be expected that the zero order stress approximation feature 

of ACCESS 3 should be efficient for solving the 63-bar truss problem. Using 

this capability, each retained potentially critical stress constraint is 

replaced with its first order approximation only if the test stated in 

equation (2.59) is satisfied within a given tolerance, which must be supplied 

by the user [see Ref. 621. The parameters permitting control over stress 

constraint approximations and deletion were chosen as foilows: 

EPS - initial = 0.4 TRJ? - initial = 0.01 

EPS - min = 0.1 TRF - max = 0.8 

f Compression stress in member 22 is not critical in the design of Ref. [5]. 

107 

I- 
EPS - multiplier = 0.6 C- cutoff = 1.0 

TRF - multiplier = 3.0 

[see Ref. 62; Section Q-XVIII for EPS and Section 4.X1X for TRF]. The 

iteration history and runtime data obtained with these control parameter 

values are presented in Table 16 under the heading "DUAL 2 (with FSD)", 

as opposed to the results obtained with DUAL 2 when first order approximation is used for all the stress constraints ["DUAL 2 (without FSD)"]. It 

can be seen that the convergence characteristics of the overall optimization 

process remain attractive when zero order approximation is employed for 

representing part of the set of critical stress constraints. Also, the 

computational cost is reduced further. It is emphasized that only 13 out 

of 30 retain potentially active stress constraints are selected as requiring first order approximation. All the other stress constraints are replaced with side constraints, using a stress ratio formula (see Section 

2.5.3). As a result, the dimensionality of the dual space, which is equal 

to the number of linearized behavior constraints, decreases and the cost 

related to the DUAL 2 optimizer is reduced substantially. Among the 14 

linearized constraints (13 stress constraints and 1 slope constraint), 9 

are found to be critical by DUAL 2, so the effective dimensionality of the 

dual problem never exceeds 10 during the optimization process (see Section 

3.3). 

6.5 Swept Wing Model (Problem 5) 

The example problem treated in this section was set forth in Ref. 

[51. The system considered represents an idealized swept wing structure shown 

in Fig. 21. The structure is taken to be symmetric with respect to the 

X-Y plane which corresponds to the wing middle surface. The upper half of 

the swept wing is modeled using sixty constant strain triangular (CST) 

108 
elements to represent the skin and seventy symmetric shear panel (SSP) elements for the vertical webs. 

, 

Extensive but plausible design variable linking is employed and the total number of independent design variables after 

linking is eighteen, 7 for the skin thickness (see Fig. 22.A) and 11 for 

the vertical webs (see Fig. 22.B). The wing is subject to two distinct 

loading conditions and the material properties are representative of a 

typical aluminum alloy. Detailed input data for this problem including 

material properties, initial design, nodal coordinates, applied nodal 

loadings and constraint specification will be found in Tables 18 - 20. 

Element-node connectivity data and the linking scheme are depicted schematically in Fig. 21 and 22, respectively,and they can also be found in tabular form in Ref. [5], where this problem was designated as Problem 9. 

The minimum mass optimum design of this idealized swept wing structure is sought, subject to the following constraints: (1) tip deflection 

is not to exceed 152.4 cm (60 in) at nodes 41 and 44 in Fig. 21; (2) Von 

Mises equivalent stress is not to exceed 172,375 kN/mL (25,000 psi) in any 

finite element; (3) minimum gage of skin and web material is not to be 

less than 0.0508 cm (0.020 in.). Two cases will be considered, corresponding 

to pure continuous and pure discrete design variable problems. It should be 

noted that, unlike the other examples presented in this report, the swept 

wing problem was run using a double precision version of ACCESS 3 on the IBM 

360/91 at CCN, UCLA. 

109 
6.5.1 Case A: Pure Continuous Problem 

The pure continuous design variable case was run using both the 

NEWSUMT and the DUAL 2 optimizer options available in the ACCESS 3 program. 

Iteration history and runtime data are presented in Table 21. Iteration 

histories are also plotted in Fig. 23. Detailed material distribution 

data for the final design obtained are given in Table 22. Previously 

reported results from Refs. [5 and 281 are included in Tables 21 and 22, 

as well as in Fig. 23, to facilitate comparison. In Table 21, the unscaled 

DUAL 2 results correspond to a sequence of "exact" solutions obtained for 

each approximate primal problem and the mass at iteration 2 does not 

correspond to a feasible design. The scaled DUAL 2 results in Table 21 

are all feasible and critical. They are obtained by scaling the "exact" 

solutions for each approximate problem so that a feasible design with at 

least one strictly critical constraint is produced. In Fig. 23, the convergence curve corresponding to DUAL 2 is plotted using the feasible scaled 

mass values of Table 21. It is emphasized that this procedure was employed 

for all examples previously discussed in this report (i.e.,iteration history plots contain only feasible design points). 

Examining Tables 21 and 22, it is seen that the final mass values and 

material distributions obtained by using the NEWSUMT and DUAL 2 options of 

ACCESS 3 are for practical purposes essentially the same. Those results 

are also seen to be in excellent agreement with those previously reported 

in Refs. 15 and 281. Comparing the DUAL 2 results with the NEWSUMT results, 

both obtained with the ACCESS 3 program, it is seen that the advantages of 

using the dual approach are: 

110 
(1) 

(2) 

(3) 

(4) 

the number of structural analyses required for convergence 

drops from 10 (NEWSUMT) to 5 (DUAL 2); 

the final mass obtained with DUAL 2 after 5 analyses is 

0.5% lower than the final mass generated by the NEWSUMT 

option after 10 stages; 

the total CPU time is reduced from 37.0 seconds for NEWSUMT 

to 19.4 seconds for DUAL 2; 

the computer times expended in the optimizer part of the ACCESS 

3 program are 4.5 seconds and 0.5 seconds for NEWSUMT and 

DUAL 2 respectively. 

Note that the ACCESS 3, DUAL 2 total CPU time (Table 21) for the swept wing 

problem (19.4 seconds) is lower than the ACCESS 1 CPU time (21.5 seconds) 

in spite of the fact that ACCESS 1 is an all core program limited to relatively small problems. It should be recognized that ACCESS 3, by virture 

of its greater generality and problem size capacity, carries a computational 

overhead burden (e.g., extensive use of auxiliary storage, etc.) when it is 

compared with programs like ACCESS 1 or that reported in Ref. 1281. Finally, 

it should be noted that the DUAL 2 final design has the following set of critical constraints: (1) minimum gage size for the skin elements 49-60 (see 

Fig. 21) in the outboard skin panel; (2) combined stress criteria in skin 

elements 8, 14 and 20 under load condition 1; combined stress criteria in 

web elements 20, 21, 30, 58 and 61 under load condition 1 as well as web 

elements 3, 5 and 42 under load condition 2. Several other stress constraints 

are nearly critical, but they are not identified as active constraints by the 

111 
DUAL 2 algorithm. This set of critical constraints at the DUAL 2 final 

design is essentially the same as that reported in Ref. [5] for the NENSUMT 

final design (see Fig. 25 of Ref. 5). 

6.5.2 Case B: Pure Discrete Problem 

A pure discrete design variable problem was derived from the previously described swept wing example, by assuming that the skin and web 

thicknesses can only take on the discrete values given in Table 8 for problem 5. These discrete values are representive of available gage sizes of 

aluminum sheet metal (2024 Aluminum Alloy). The other input data are the 

same as in the pure continuous case (see Tables 18 - 20). The iteration 

history and runtime data obtained with the DUAL 1 optimizer are presented 

in Table 21. Only 6 reanalyses are needed to obtain a discrete optimum 

design. It should be noted that this solution is generated by DUAL 1 in 

less computer time than that required by NEWSUMT to yield a continuous optimum design. 

The final discrete design produced by DUAL 1 is given in Table 22. 

For comparison, another discrete design is also presented, which is deduced 

from the continuous optimum design by rounding up all the thicknesses to 

the nearest available discrete value. It is seen that the DUAL 1 solution 

is 4% lighter than the intuitively derived design (both designs are 

feasible). 

6.6 Delta Wing (Problem 6) 

The last example treated here is a thin (3% thickness ratio) delta 

wing structure with graphite-epoxy skins and titanium webs. The problem 

112 
has been previously studied in Refs. [5, 6 and 71. The structure is 

symmetric with respect to its middle surface which corresponds to the 

X-Y plane in Fig. 24. The skins are assumed to be made up of O", 245" and 

90° high strength graphite-epoxy laminates. It is understood that orientation angles are given with respect to the X reference coordinate in 

Fig. 24, that is material oriented at O0 has fibers running spanwise 

while material at 90" has fibers running chordwise. The laminates are 

required to be balanced and symmetric and they are represented by stacking 

four constant strain triangular orthotripic (CSTOR) elements in each 

triangular region shown in Fig. 24. Therefore, the upper half of the delta 

wing is modeled using 4x63 = 252 CSTOR elements to represent the skin and 

70 symmetric shear panel (SSP) elements for the vertical webs. According 

to the linking scheme depicted in Fig. 25, it can be seen that the total 

number of independent design variables is equal to 60 made up as follows: 

16 for 0" material; 16 for f45" material; 16 for 90" material; and 12 for 

the web material. 

The graphite epoxy and titanium material properties used in the 

delta wing example are listed in Table 23. The nodal coordinates defining 

the layout of the idealized structure shown in Fig. 24 are specified in 

Table 24. The wing is subjected to a single static load condition that is 

roughly equivalent to a uniformly distributed loading of 6.89 kN/m2 (144 psf). 

The corresponding nodal force components are given in Table 23. It should 

be noted that, since some of the fiber composite allowable strains are 

different in tension and compression, the structural analysis of the symmetric 

delta wing must consider two loading conditions, the second load condition 

113 
being simply the negative of the first. Designing the upper half of the 

symmetric wing for both load conditions is then equivalent to designing 

the entire wing for one load condition while imposing midplane symmetry. 

Static deflection constraints of f 256.0 cm (2 100.8 in.) are imposed at the 

wing tip nodes (see Table 23). The strength requirements for the laminated 

skins are based on the maximum strain failure criterion [see Refs. 7, 61, 

and 621. In addition, the fundamental natural frequency is required to 

be larger than 2 I&z, while fixed masses simulate fuel in the wing. The 

fuel mass distribution employed is taken to be roughly proportional to 

the wing depth distribution (see Table 25). Minimum gage requirements are 

also specified [O.OSOS cm (0.02 in.) for the titanium webs and 0.0127 cm 

(0.005 in.) for the fiber composite lamina]. The thermal analysis capability 

of ACCESS 3 was also employed in this delta wing problem. It is assumed 

that the wing is subjected to the static loading conditions previously described while operating at a uniform soak temperature of -34.44OC (-30°F). 

The laminated skin and the webs are considered to be stress free at 76.7OC 

(170°F) and 21.1°C (70°F), respectively. Therefore, the mechanical load 

conditions are combined with the following temperature change inputs: 

(a) -129OC (-200°F) in the laminated fiber composite skins; and 

b) -73.3OC (-lOOOF) in the titanium webs. 

In this connection, it is important to point out that ACCESS 3 contains 

special features for handling midplane symmetric wing structures when temperature change effects are taken into account. The thermal analysis, with 

its midplane symmetric response, is treated separately from the midplane 

antisymmetric response due to the pressure loading and the results are then 

114 
superimposed [see Refs. 61 and 621. 

The problem studied here has its genesis in an interesting scenario 

presented in Ref. 7. Using an all titanium structure it was possible to 

obtain a satisfactory wing weight even when a 2 Hz lower limit was placed 

on the fundamental frequency. However, when fixed fuel mass was added to 

the wing, it was necessary to introduce fiber composite skins in order to 

avoid an unacceptable increase in the minimum mass (approximately a factor 

of 4). Initially a high modulus graphite epoxy fiber-composite was employed, 

however subsequent consideration of temperature induced stresses made it 

necessary to switch to a high strength graphite-epoxy material. In this 

reportthe final version of the delta wing problem (Case 3B of Ref. 7) will 

be reconsidered using the dual method approach. It should be recalled that 

this problem involves: 

(1) the use of a laminated high strength graphite-epoxy skin; 

(2) temperature change effects; 

(3) consideration of fixed fuel mass; 

(4) a 2 Hz lower limit on the fundamental natural frequency 

(which is a primary design driver). 

6.6-l Case A: Pure Continuous Problem 

Initially the foregoing delta wing example will be studied as a pure 

continuous problem, with exactly the same data as in Ref. [73. The aim is 

simply to compare the efficiency of the NEWSUMT and DUAL 2 optimizers of 

the ACCESS 3 program. Results for this case are presented in Table 26 

(iteration histories) and Table 27 (final designs). Since the fundamental 

natural frequency constraint is the main design driver in this example, its 

115 
value as well as the mass for each design in the sequence is given in 

Table 26. Note that designs 3, 4 and 6 in the DUAL 2 sequence are slightly 

infeasible with respect to the frequency constraint. Tables 26 and 27 

show that the advantages of the dual method approach are significant for 

the delta wing example: 

(1) the number of structural reanalyses required for convergence 

falls from 29 (NEWSUMT) to 15 (DUAL 2); 

(2) the final mass given by DUAL 2 after 15 analyses is 5% lower 

than the final mass generated by NEWSUMT in 29 analyses; 

(3) the total computer time is reduced from 719 set f for NEWSUMT 

to 261 set' for DUAL 2; 

(4) the computer times expanded in the optimizer part of the program are 145 secf and 2 set f for NEWSUMT and DUAL 2, respectively. 

Looking at the final designs generated by NEWSUMT and DUAL 2 (Table 

271, it can be seen that the two designs are similar to each other. The 

smaller mass given by DUAL 2 appears to be due, at least in part, to the 

larger number of design variables that reach minimum gauge IO.0127 cm 

(0.005 in)]. In both cases, most of the fiber composite material in the 

laminated skin is oriented spanwise, with relatively small amounts placed 

at f45O. Over most of the skin, the 90° or chordwise material is minimum 

thickness critical [i.e.,0.0127 cm (0.005 in.)]. The web material distribution is given in Table 28. For both the DUAL 2 and the NEWSUMT results, 

the contribution that the shear web structure makes to the total mass of 

f These times are for runs on the IBM 360/91 computer at CCN, UCLA. 

116 
the wing is small (11%). The final designs generated by both NEWSUMT and 

DUAL 2 are governed primarily by the critical frequency constraint. However, several skin strength constraints are critical in design variable 

regions 1 through 6 and 16 (see Table 27). These critical strength constraints are transverse tension strain limits in the bottom skin for 

material oriented at f45O or 90°. 

It should be emphasized that the DUAL 2 optimization algorithm effort 

accounts for less than 1% of the total computer time. This remarkably 

small computational cost suggest that algorithms like DUAL 2, which combine 

the generality of mathematical programming and the simplicity of optimality 

criteria, should find wide-spread acceptance in the next few years as a 

basis for major structural optimization codes. 

6.6.2 Case B: Mixed Continuous-Discrete Problem 

Attention is now directed towards the results obtained in the mixed 

continuous-discrete variable case, where the number of plies for the CSTOR 

elements are considered as discrete variables (more precisely, integer 

variables). The thicknesses of the shear panels representing the webs 

are still taken as continuous design variables. It should be recalled 

that the laminates are assumed to be balanced and symmetric. Therefore, 

the smallest change in lamina thickness is necessarily equal to two plies 

[or 0.0254 cm (0.010 in)]. Consequently, the set of available discrete 

values for the thicknesses of the skin lamina is given by IO.0254 cm, 

0.0508 cm, 0.0762 cm,.......]({O.Ol in., 0.02 in., 0.03 in.,......}) (see 

also Table 8). Results for the mixed continuous-discrete variable case 

117 
are presented in Table 26 (iteration history) and Table 27 (final design). 

Since discrete variables are involved, the DUAL 1 optimizer must be 

employed. In order to further illustrate how the DUAL 1 algorithm works 

Table 29 contains detailed iteration history data for each stage, namely: 

the number Q, of potentially active constraints retained; the number 

(QR - N) of non-zero dual variables (i.e., the number of strictly active 

behavior constraints found by DUAL 1 for the current approximate problem); 

the number P of discontinuity planes at the end of the stage; the number of 

restarts; the total number of ODM's required for convergence. For information 

the lower bound mass W, the optimal dual objective function value R*, the 

final mass W* and the upper bound mass w at the end of each stage are also 

given. As expected, the inequality W < J?,* 5 W* -< w is satisfied at each 

-- 

stage (see Section 4.5.3). The DUAL 1 optimizer run time (12 set) is higher 

than that for DUAL 2 (2 set) but significantly lower than the NEWSUMT run 

time (145 set) (see Table 26). The final mass generated by DUAL 1 is 

slightly heavier (4%) than that produced by DUAL 2, mainly because the 

minimum size for the CSTOR members has been increased from 0.0127 cm 

(0.005 in) to 0.0254 cm (0.010 in.). 

In Table 27, the final design for the mixed continuous-discrete 

variable case is given as follows: the first value represents the thickness and the integer number in parentheses is the number of plies. Again 

most of the fiber composite material is oriented in the O" direction (spanwise). The design is still governed primarily by the frequency constraint 

but some skin strength constraints are critical in design variable regions 

118 
1 through 6 and 15 (see Table 27). The web thicknesses are presented in 

Table 28. It can be seen that the web mass remains small compared to the 

skin mass. 

To conclude the description of the delta wing example attention is 

focused on a comparison of the NEWSUMT and DUAL 1 results given in Tables 

26 and 27 and illustrated in Fig. 26 (convergence curves): 

(1) 

(2) 

(3) 

(4) 

the pure continuous variable problem solved by NEWSUMT 

requires 29 structural reanalyses while the mixed continuous-discrete variable problem is solved by DUAL 1 in 13 

reanalyses; 

despite the more realistic formulation of the problem 

(discrete variables and balanced laminate requirements) 

DUAL 1 is capable of producing a lighter design than NEWSUMT 

(6026.5 kg (13,286 lbm) versus 6111.8 kg (13,474 lbm)]; 

the total computer time is reduced from 719 set f employing 

NEWSUMT to 253 set f using DUAL 1; 

the computer times associated with the optimization effort 

alone are respectively 145 set ' (NEWSUMT) and 12 set+ (DUAL 1). 

Note that the DUAL 1 iteration history presented in Fig. 26 does not correspond to a sequence of all feasible designs (see Table 26), since scaling 

cannot be employed in this example (because it contains discrete variables). 

f CPU time on the IBM 360/91 computer at CCN, UCLA. 

119 

7. CONCLUSIONS 

Considering first the case where all the design variables are continuous, the fundamental reasons underlying the efficiency achieved by 

combining approximation concepts and dual methods are seen to reside in the 

following points: 

(1) dual methods exploit the special algebraic structure of the 

approximate problem generated at each stage; 

(2) since the approximate primal problem at each stage is convex, 

separable and algebraically simple, it is possible to construct an explicit dual function; 

(3) most of the computational effort in the optimization part of 

the program is expended on finding the maximum of the dual 

function subject only to simple nonnegativity constraints on 

the dual variables; 

(4) the dimensionality of each dual space, namely the number of 

critical and potentially critical behavior constraints retained during that stage, is relatively small for many problems of practical interest; 

(5) The DUAL 2 optimizer has been especially devised so that it 

seeks the maximum of the dual function by operating in a 

sequence of dual subspaces with gradually increasing dimension, 

such that the dimensionality of the maximization problem never 

exceeds the number of strictly critical constraints by more 

than one; 

121 
(6) finally, by seeking the "exact" solution of each approximate 

problem using the DUAL 2 option, rather than a partial solution 

of each approximate problem using the NEWSUMT option, the number 

of stages needed to converge the overall iterative design process is usually reduced. 

The joining together of approximation concepts and dual methods provides 

further insight into the relationship between mathematical programming methods 

and optimality criteria techniques. It is well known that the essential 

difficulties involved in applying conventional optimality criteria methods 

are those associated with identifying the correct critical constraint set 

and the proper corresponding subdivision of passive and active design variables . Special purpose maximization algorithms such as DUAL 2, which also 

operate on the Lagrangian multipliers associated with the behavior constraints, 

intrinsically deal with and resolve these two crucial difficulties. The subdivision of passive and active-design variables is dealt with by the closed 

form relations expressing the primal design variables as functions of the 

Iagrangian multipliers (i.e., dual variables). Identification of the critical 

constraint set is automatically handled by taking the nonnegativity constraints 

on the dual variables into account when seeking the maximum of the dual 

function. Thus, the combining of approximation concepts and dual methods 

leads to a perspective where optimality criteria techniques are seen to reside within the general framework of a mathematical programming approach to 

structural optimization. 

Another important achievement reported in this work is the treatment 

of discrete problems using the dual method approach. The description of fiber 

composite laminates, which are fabricated from individual plies, naturally 

122 
involves discrete (integer) design variables. It is also well known 

that conventional metal alloy sheet material is frequently only available 

in standard gauge thicknesses, which again leads to discrete variables. 

It is therefore interesting and significant that the dual method has been 

extended to deal with structural synthesis problems involving either pure 

discrete or mixed continuous-discrete design variables. This extension of 

the dual methods provides a remarkably efficient minimum mass design optimization capability for structural sizing problems involving discrete 

variables. This efficiency is due primarily to the following characteristics: 

(1) the dual method implemented herein treats discrete or mixed 

design variable problems by operating on a continuous dual 

function; 

(2) as in the pure continuous case, the dimensionality of the 

dual problem is considerably lower than that of the primal 

problems and it is independent of the number of design variables; 

(3) the DUAL 1 algorithm incorporates special features for handling 

dual function gradient discontinuities that arise from the 

primal discrete variables; 

It should be recognized that when discrete variables are introduced, the 

approximate primal problem is no longer convex and, therefore, the dual formulation does not necessarily yield the true optimum design. Nevertheless, 

the computational experience reported in this work shows that, although the 

extension of dual methods to discrete variable problems lacks rigor, it frequently gives useful and plausible results [see Refs. 57 and 601. 

It is concluded, based on the results reported in this work, that 

combining approximation concepts with dual methods provides a firm foundation 

123 
for the development of rather general and highly efficient structural synthesis capabilities. Although ACCESS 3 is a research type program of limited 

scope, a substantial body of computational experience supports the conclusion 

that the dual method approach leads to a powerful capability for minimum 

mass optimum sizing of structural systems subject to stress, deflection, 

slope, minimum gauge and natural frequency constraints. using this apprOaCh, 

the computational effort expended in the optimization portion of the program 

has been reduced to a small fraction (e.g., less than 1% in the delta wing 

example with the DUAL 2 option) of the modest total run time required to 

obtain a minimum mass design. 

It is important to point out that the method presented is not restricted 

to the specific type of application that has been made in ACCESS 3 (i.e., 

sizing optimization with bar and membrane finite element models), but it 

could form the basis of a powerful optimizer embedded in a more general structural synthesis program, such as the PROSSS program of Ref. [66] or the PARS 

program of Ref. [67]. In this connection, it should be recognized that, when 

using dual optimizers such as DUAL 1 or DUAL 2, the only essential requirement 

is that all the functions describing the primal problem must be of separable 

form. 

124 
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130 
E 

Table 1. Alternate Paths After Solving ODM (Dual 1) 

Block Path 1 Path 2 Path 3 Path 4 Path 5 Path 6 

Solve ODM Update 161 X X X X X X 

On New Discontinuity Plane? 171 F F F F T T 

On New Base Plane? [8181 F F T T 

Nc Discontinuity Planes? .[9]+[10] F I']-- T T [lOI F 

First Discontinuity Plane? [ill T F 

Initialize [P]I Matrix WI X 

UpdaF5J] UpdaF 5$yl Matrix Eqs. (4.51, 1131 X 

Update [PI Matrix Eqs. (4.47, [141 X 

4.48) 

Calculate Et [151 X X X X 

Calculate gt El X x 
Table 2. Available Options for Frequency Constraints 

\ x=2 

1st Order 1st Order 2nd Order 

Algori&ii'-Approx. Reciprocal Direct Direct 

"\ DV's DV's DV's 

NEWSUMT * * * 

PRIMAL2 

DUAL1 

DUAL2 

* 

- - 

* 

- - 

* 

- - 

* available combination in ACCESS-3 Program 

Table 3. Algorithm Options for Various Kinds of Problems 

--.. Pure Pure MixedA1gori.h~~ Continuous Discrete Continuous 

Discrete 

NEWSUMT * - - 

PRIMAL2 

DUAL1 

* 

- - 

* * * 

DUAL2 * 

- - 

* available for application in ACCESS-3 Program 

132 
Table 4A. Definition of Problem 1 

Material 

Young's modulus : 

Specific weight : 

Allowable stress : 

Minimum area 

Uniform initial : 

area 

Planar lo-Bar Cantilever Truss 

(SI Units) 

Aluminum 

E = 68.95 x lo6 kN/m2 

p = 2768 kg/m3 

CT 

a 

= + 172,375 kN/m2 

D 0-J) = 0.6452 cm2 

Do3 = 129.0 cm2 

Nodal Loading (1 load case) 

Node 

2 

4 

Load components (N) 

X Y Z 

0 -444,800 0 

0 -444,800 0 

Displacement Constraints 

Problem I Node I Direction 

1 Displacemen: limits (cm) 

Name Lower Upper 

. 

Case A 1 Y -5.08 -5.08 

3 Y -2.54 -2.54 

-- 

Cases B-F l-4 Y -5.08 +5.08 

133 
Table 4B. Definition of Problem 1 

Planar lo-Bar Cantilever Truss 

(U.S. Customary Units) 

Material Aluminum 

Young's modulus : E = lo7 psi 

Specific weight : p = 0.1 lbm/in3 

Allowable stress : u = 95,000 psi 

Minimum area D(Ly = 0.1 in2 

Uniform initial : D(o) = 20.0 in2 

area 

Nodal Loading (1 load case) 

Node , 

Load components (lbf) 

X Y Z 

2 0 -100,000 0 

4 0 -100,000 0 

Displacement Constraints 

Problem 

Name 

Node Direction 

Displacement limits (in) 

Lower upper 

Case A 1 Y -2.0 -2.0 

3 Y -1.0 -1.0 

Cases B-F l-4 Y -2.0 +2.0 

134 
Table 5A. Iteration History Data for Problem 1 (Case A) 

Planar lo-Bar Ca&iliever Truss 

(SI Units) 

Member 6 

135 
Table 5B. Iteration History Data for Problem 1 (Case A) 

Planar lo-Bar Cantilever Truss 

(U.S. Customary Units) 

y-Displacements (in) Stress 

Analysis Mass (psi 1 

No. (lbm) Node 1 Node 3 Member 6 

1 8392.92 -1.8975 -0.83717 2006 

2 4738.37 -2.1384 -1.0616 15325 

3 4390.14 -2.0441 -1.1324 21182 

4 4224.58 -2.0397 -0.94574 21264 

5 4040.49 -2.0844 -0.97773 20989 

6 4045.01 -2.0054 -0.99987 24842 

7 4049.03 -1.9999 -1.0000 25002 

8 4048.81 -2.0001 -1.0000 25001 

9 4048.96 -2.0000 -1.0000 25000 

ZPU Total 4.46 

Cime Anal. 2.73 

[Set) Optim. 0.28 

136 
Table 6A. Final Designs for Problem 1 

Planar lo-Bar Cantiliever Truss 

(SI Units) 

This design is slightly infeasible. The feasilbe design at iteration 

11 with mass 2303 kg (See Table 7A) is the same except that the area 

of member 3 is 151.622 cm2. 

Except for the minimum size members 12, 5, lo] the cross sectional areas 

in Case F are integer multiples of 3.226 cm2 as noted in parentheses ( 1. 

137 
Table 6B. Final Designs for Problem 1 

Planar lo-Bar Cantilever Truss 

(U.S. Customary Units) 

Member 

No. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

4ass (lbm) 

Qo. of 

walyses 

Cross-sectional Area (in2) 

Case A Case B Case F 

DUAL2 NEWSUMT DUAL2 DUAL1 

22.66 30.95 30.52 30.5 

1.401 0.1 0.1 0.1 

21.58 26.08 23.20 23.0(*) 

8.434 15.04 15.22 15.5 

0.1 0.1 0.1 0.1 

0.1 0.1960 0.5510 0.5 

12.69 8.182 7.457 7.5 

14.54 20.22 21.04 21.0 

11.93 20.22 21.53 21.5 

1.982 0.1 0.1 0.1 

4048.96 5089.80 5060.85 5059.88 

9 13 13 13 

(*) This design is slightly infeasible. The feasible design 

at iteration 11 with mass 5078 lbm (see Table 7B) is the 

same except that the area of member 3 is 23.5 in2. 

138 
Table 7A. Iteration History Data for Problem 1 (Cases B-F) 

Planar lo-Bar Cantilever Truss 

(SI Units) 

P 

w 

W 

Mass (kg) 

Analysis Case B (Pure Continuous) Case C Case D Case E Case F 

No. (Pure (Mixed (Mixed j (Pure 

'NEWSUMT DUAL 2 Discrete) Continuous Continuous Discrete) 

1 (0.3x2) ! Unscaled ' Scaled Discrete)+ Discrete)* 

1 3807 3807 3749 3807 3807 3807 3807 

2 3190 2720 2719 2751 2723 2749 ) 2727 

3 2687 2547 2628 2537 2555 2522 2659 

4 2627 2602 2578 2601 2597 2588 2585 

5 2561 2551 2525 2562 2541 2566 2557 

6 2499 2495 2467 2538 2488 2515 2501 

7 2442 2433 2401 2466 2417 2459 2445 

8 2394 2358 2359 2418 2336 2396 2370 

9 2345 2259 2319 2338 2277 2323 2272 

10 2324 2295 2302 2288 2295 2285 2312 

11 2309 2298 2297 2301 2281 2295 2303 

12 2309 2296 2296 2296 2292 2300 2295 

13 2309 2296 2296 2296 2296 2299 2295 

14 2296 

15 2296 

CPU Total 5.88 4.39 7.71 6.32 6.19 6.58 

Time Anal. 3.12 3.12 3.79 3.18 3.17 3.59 

(SecIOptim. 1.62 0.14 0.47 0.38 0.26 0.52 

+ design variables 1,3,6,8,10 are discrete 

* design variables 2,4,5,7,9 are discrete 
Table 7B. Iteration History Data for Problem 1 (Cases B-F) 

Planar lo-Bar Cantilever Truss 

(U.S. Customary Units) 

Lnalysis 

No. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

Mass (lbm) 

8393 8393 8266 

7032 5996 5994 

5924 5614 5793 

5792 5737 5683 

5647 5623 5566 

5510 5500 5438 

5383 5363 5294 

5278 5198 5200 

5169 4980 5112 

5124 5059 5075 

5090 5067 5065 

5090 5061 5061 

5090 5061 5061 ; 

8393 8393 

6064 6002 

5592 5632 

5735 5725 

5648 5602 

5595 5486 

5436 5328 

5330 5149 

5155 5019 

5044 5060 

5073 5029 

5061 5053 

5061 5061 

7.71 

3.79 

0.47 

Case D Case E 

(Mixed (Mixed 

Continuouf Continuo 

Discrete)-i Discrete 

6.32 6.19 

3.18 3.17 

0.38 0.26 

8393 

6060 

5560 

5706 

5658 

5544 

5422 

5282 

5121 

5038 

5059 

5071 

5069 

5061 

5061 

Case F 

(Pure 

Discrete) 

8393 

6013 

5861 

5699 

5637 

5514 

5391 

5225 

,5009 

5096 

5078 

5060 

5060 

~6.58 

3.59 

0.52 

+ design variables 1,3,6,8,10 are discrete 

* design variables 2,4,5,7,9 are discrete 
Table 8. Available Discrete Values for All Example Problems 

Problem Name Normalized discrete areas (A/Ar) or thicknesses (t/t,) 

Problem 1 lo-Bar Truss 

Cases C through E 

'A/Ar) 

0.1 , 0.551 , 1.0 , 2.0 , 3.0 , 4.0 , 5.0 , 6.0 , 

7.0 , 7.457 , 8.0 , 9.0 , 10.0 , 11.0 , 12.0 , 13.0 , 

14.0 , 15.0 , 15.22 , 16.0 , 17.0 , 18.0 , 19.0 , 20.0 , 

21.0 , 21.04 , 21.53 , 22.0 , 23.0 , 23.20 , 24.0 , 15.0 , 

26.0 , 27.0 , 28.0 , 29.0 , 30.0 , 30.52 , 31.0 , 32.0 , 

33.0 , 34.0 , 35.0 , 36.0 , 37.0 , 38.0 , 39.0 , 40.0 

Problem 1 lo-Bar Truss 

Case F 'A/AI: 1 

0.1 , 0.5 , 1.0 , 1.5 , 2.0 , 2.5 , 3.0 , 3.5 , 

4.0 , 4.5 . . . . . . , 38.0 ) 39.5 , 40.0 

Problem 2 25-Bar Truss 

Case B 'A/Ar) 

0.01,. 0.1 , 0.2 , 0.3 , 0.4 , 0.5 , 0.6 , 0.7 , 

0.8 , 0.9 , . . . . , 5.4 , 5.5 , 5.6 

Problem 2 25-Bar Truss 

Case C (A/A_) 

0.01, 0.4 , 0.8 , 1.2 , 1.6 , 2.0 , 2.4 , 2.8 , 

3.2 , 3.6 , 4.0 , 4.4 , 4.8 , 5.2 , 5.6 

Problem 2 25-Bar Truss Case D 'A/Ar) I 0.01, 0.8 , 1.6 , 2.4 , 3.2 , 4.0 , 4.8 , 5.6 

Problem 5 Swept Wing 

Case B (t/t, 1 

0.020, 0.025 , 0.032 , 0.040, 0.050, 0.063, 0.071, 0.080, 

0.090, 0.100 , 0.125 , 0.160, 0.190, 0.250, 0.313, 0.375, 

0.500, 0.625 , 0.750 , 1.000 

Problem 6 Delta-Wing I 0.01 , 0.02 , 0.03 .'. 0.04 , 0.05 , 0.06 , 0.07 , 0.08, 

Case B (t/t,) 0.09, 0.10, . . , 1.98 , 1.99 , 2.00 

Ar = 6.452 cm2 (1.00 in21 ; tr = 2.54 cm (1.00 in) 
Table 9A. Definition of Problem 2 

25-Bar Space Truss 

(SI Units) 

Material : 

Young's modulus 

Specific mass 

Minimum area 

Uniform initial 

area 

Allowable Stresses 

Aluminum 

E = 68.95 x lo6 kN/m2 

p = 2768 kg/m3 

D(L) 

= 0.06452 cm2 

D (0) 

= 19.356 cm2 

Members 

1 

2-5 

6-9 

10, 11 

Stress limits (kN/mL) Stress limits (kN/mL) 

Members . 

tension compression tension compression 

275,800 -241,959 12, 13 275,800 -241,959 

275,800 - 79,913 14 - 17 275,800 - 46,603 

275,800 -119,318 la - 21 275,800 - 47,982 

275,800 -241,959 22 - 25 275,800 - 76,410 

Nodal Loading (2 load cases) 

I Load I I Load components (N) I 

Case Node 

X Y z 

, 

1 1 4,448 44,480 -22,240 

2 0 44,480 -22,240 

3 2,224 0 0 

6 2,224 0 0 

2 5 0 88,960 -22,240 

6 0 -88,960 -22,240 

Displacement constraints 

I I Displacement limits (cm) 

Node 

X Y Z 

1 20.889 +o.aas +o.aa9 

2 f0.889 +o.aas 20.889 

142 
Table 9B. Definition of Problem 2 

25-Bar Space huss 

(U.S. Customary Units) 

Material Aluminum 

Young's modulus : E = lo7 psi 

Specific mass p = 0.1 lbm/in3 

Minimum area D UJ) = 0.01 in2 

Uniform initial : D(o) = 3.0 in2 

area 

Allowable Stresses 

Stress limits (psi) Stress limits (psi) 

Members Members' 

tension compression tension compression 

1 40,000 -35,092 12,13 40,000 -35,092 

2 -5 40,000 -11,590 14 - 17 40,000 - 6,759 

6-9 40,000 -17,305 la - 21 40,000 - 6,959 

10, 11 40,000 -35,092 22 - 25 40,000 -11,082 

Nodal Loading (2 load cases) 

Load I I Load compone !nts (lbf) 

Zase Node F 

X Y Z 

6 500 0 0 

2 5 0 20,000 -5,000 

6 0 -20,000 .-5,000 

Displacemetit cdnstraints 

143 
E 

Table 10A. Iteration History Data for Problem 2 

25-Bar Space Truss 

(SI Units) 

I Mass (kg) 

Pure Continuous Case (Case A) Pure Discrete Cases 

Analysis 

No. NEWSUMT DUAL 2 Ref. [30] Ref. [13] Case B Case C Case D 

(0.3~2) unscaled scaled Fleury- GellatlySander Berke ; =O.l A A =0.4 $ =o.a 

r r r 

1 450.1 450.1 333.1 333.1 333.1 360.0 360.0 360.0 

2 318.6 256.6 256.4 256.4 252.1 258.6 268.3 294.3 

3 265.8 248.0 248.0 248.0 249.1 250.8 263.0 283.5 

4 252.6 247.6 247.6 247.6 247.9 250.8 261.0 283.5 

5 248.9 247.4 247.4 247.4 247.6 261.0 

6 247.8 247.3 247.3 247.3 247.4 

7 247.4 247.4 -- 

CPU Total 5.20 2.75 2.85 2.85 2.21 

Time Analys. 3.35 1.93 / ,/ 1.85 1.92 1.56 

(set) Optim. 1.02 0.05 0.15 0.13 0.10 
Table 10B. Iteration History Data for Problem 2 

25-Bar Space Truss 

(U.S. Customary Units) 

Mass (lbm) 

Pure Continuous Case (Case A) Pure Discrete Cases 

Analysis I 

No. I 

i NEWSUMT DUAL2 Ref.[30]! Ref.[l3] Case B Case C Case D 

I (0.3x2) unscaled scaled Fleury- Gellatly- A ,=O.l i A 

Ar 

Sander Berke ,=0.4 $=,.a 

r r r 

1 992.2 

2 702.3 

3 585.9 

4 556.9 

5 548.7 

6 546.2 

7 545.5 

992.2 734.4 

565.6 565.3 

546.8 546.8 

545.8 545.8 

545.4 545.4 

545.2 545.2 

734.4 734.4 793.7 793.7 793.7 

565.3 555.7 570.1 591.4 648.7 

546.8 549.1 553.0 579.7 624.9 

545.8 546.5 553.0 575.4 624.9 

545.4 545.9 575.4 

545.2 545.5 

545.4 
._,.,..__ -,. . . . - ._....- -.- 

. 

Table 11A. Final Designs for Problem 2 

25-Bar Space Truss 

(SI Units) 

Design 

Variable 

Group No. 

(members) 

1 (1) 

2 (2-5) 

3 (6-9) 

4 (10,ll) 

5 (12,131 

6 (14-17) 

7 (18-21) 

8 (22-25) 

Mass (kg 1 

No. of 

Analyses 

Cross-Sectional Area (cm) 2 

Case A Case B Case C 

(continuous) A&=0.1 

A/Ar=0.4 Case D A/Ar=0.8 

DUAL 2 DUAL 1 DUAL 1 Ref. [65] DUAL 1 Ref. [65] 

0.0645 0.6452 2.581 12.962 5.162 10.065 

12.820 12.904 12.904 15.485 15.485 15.485 

19.298 19.356 20.646 15.485 20.646 15.485 

0.0645 0.0645 0.0645 0.071 0.0645 0.071 

0.0774 0.6452 0.0645 0.071 5.162 0.071 

4.407 4.516 5.162 5.162 5.162 5.162 

10.833 10.968 12.904 12.904 10.323 15.485 

17.188 17.420 15.485 18.066 20.646 20.646 

247.31 250.84 261.01 270.07 283.44 291.39 

6 4 5 I -- ._ 

146 
Table 11B. Final Designs for Problem 2 

25-Bar Space Truss 

(U.S. Customary Units) 

L 

Design Cross-sectional Area (in2) 

Variable Case A Group No. Case B Case C A/A,=O.4 Case D A/A,=O.S 

(members) (continuous) A,Qr=O.l 

DUAL2 DUAL1 DUAL1 Ref[65] DUAL1 Ref.[65] 

1 (1) 0.010 0.1 0.4 2.009 0.8 1.560 

2 (2-5) 1.987 2.0 2.0 2.400 2.4 2.400 

3 (6-9) 2.991 3.0 3.2 2.400 3.2 2.400 

4 (10,ll) 0.010 0.01 0.01 0.011 0.01 0.011 

5 (12,13) 0.012 0.1 0.01 0.011 0.8 0.011 

6 (14-17) 0.683 0.7 0.8 0.800 0.8 0.800 

7 (18-21) 1.679 1.7 2.0 2.000 1.6 2.400 

8 (22-25) 2.664 2.7 2.4 2.800 3.2 3.200 

Mass 

(lbm) 545.22 553.00 575.41 595.4 624.87 642.4 

No. of 

Analyses 6 4 5 4 

147 
Table 12A. Definition of Problem 3 

72-Bar Space Truss 

(SI Units) 

Material Aluminum 

Young's modulus E = 68.95 x lo6 kN/m2 

Specific mass p = 2768 kg/m3 

Allowable stress : cl, = 2172,375 kN/m2 

Minimum area : DIL) = 0.6452 cm2 

Uniform initial : D(o) = 6.452 cm2 

area 

Nodal loading (2 load cases) 

Load 

Case Node 

1 1 

2 '1 

2 

3 

4 

Displacement constraints 

Load components (N) 

Z 

-22,240 

-22,240 

-22,240 

-22,240 

Displacement limits (cm) 

X 

20.635 kO.635 

20.635 20.635 

20.635 20.635 

20.635 LO.635 

Y 

1 

148 
Table 12B. 

Material 

Young's modulus 

Specific mass 

Allowable stress 

Minimum area 

Uniform initial 

area 

Definition of Problem 3 

72-Bar Space Truss 

(U.S. Customary Units) 

: Aluminum 

: E = lo7 psi 

P = 0.1 lbm/in 3 

u 

. .tS 

= Y5,OOO psi 

= 0.1 in2 

: Db) = 1.0 in2 

Nodal loading (2 load cases) 

Dad Load components (lbf) 

:ase Node X Y Z 

1 1 5,000 5,000 -5,000 

2 1 0 0 -5,000 

2 0 0 -5,000 

3 0 0 -5,000 

4 0 0 -5,000 

Displacement constraints 

Node 1 Diylacement limits (i: 

1 -10.25 kO.25 

2 i0.25 kO.25 

3 f0.25 kO.25 

4 kO.25 I 20.25 

149 
Analysis 

No. 

1 

6 

10 

Table 13A. Iteration History Data for Problem 3 

72-Bar Space Truss 

(SI Units) 

Mass (kg) 

NEWSUMT 

(0.5x1) 

387.0 

258.9 

231.6 

214.9 

202.3 

193.1 

186.7 

182.4 

179.4 

177.3 

12.20 

9.38 

1.35 

NEWSUMT 

(0.3~2) 

387.0 

224.2 

185.7 

176.3 

173.4 

172.6 

172.3 

9.71 

7.10 

1.46 

NEWSUMT 

(0.1x3) 

387.0 

189.4 

172.7 

172.2 

172.2 

8.41 

5.78 

1.89 

DUAL 2 

unscaled 

387.0 

183.2 

172.3 

172.2 

172.2 

scaled 

297.9 

175.7 

172.3 

172.2 

172.2 

6.52 

5.54 

0.02 

Ref. [15] 

Taig-Kerr 

Ref. t3Ol 

FleurySander 

297.9 

175.7 

172.3 

172.2 

172.2 

175.5 

172.2 

172.3 
Table 13B. Iteration History Data for Problem 3 

72-Bar Space Truss 

(U.S. Customary Units) 

Mass (lbm) I, 

Analysis 

DUAL2 Ref.1151 Ref.[l7] 

No. NEWSUMT NEWSUMT NEWSUMT unscaled scaled Taig-Kerr Berke- 

(0.5x1) (0.3~2) (0.1x3) Ref. 1301 Khot 

FleuqSander 

1 853.1 853.1 853.1 853.1 656.8 656.8 656.8 

2 570.7 494.3 417.6 403.9 387.3 387.3 387.0 

3 510.6 409.3 380.'8 379.9 379.8 379.8 379.7 

4 473.8 388.6 379.7 379.7 379.7 379.7 379.9 

5 445.9 382.3 379.7 379.7 379.7 379.7 

-- 

6 425.7 380.5 

7 411.7 379.9 

8 402.1 

9 395.4 

10 390.8 

CPU Total 12.20 9.71 a.41 6.52 

Time Anal. 9.38 7.10 5.78 5.54 

(set) Optim. 1.35 1.46 1.89 0.02 
Table 14A. Final Designs for Problem 3 

72-Bar Space Truss 

tS1 Units) 

Cross-sectional area (cm2) 

Design Ref. [15] 

Variable Members ACCESS 3 Taig-Kerr Ref. [17] 

Group (DUAL 2) Ref. [301 Berke-Khot 

No. Flew-y-Sander 

1 l-4 1.014 1.014 1.014 

2 5-12 3.456 3.456 3.474 

3 13-16 2.648 2.645 2.681 

4 17,la 3.667 3.671 3.555 

5 19-22 3.270 3.269 3‘.279 

6 23-30 3.355 3.355 3.352 

7 31-34 0.6452 0.6452 0.6452 

8 35,36 0.6452 0.6452 0.6452 

9 37-40 a.259 a.259 8.252 

10 41-48 3.321 3.321 3.332 

11 49-52 0.6452 0.6452 0.6452 

12 53,54 0.6452 0.6452 0.6452 

13 55-58 12.239 12.239 12.214 

14 59-66 3.328 3.328 3.336 

15 67-70 0.6452 0.6452 0.6452 

16 71,72 0.6452 0.6452 0.6452 

Mass (kg) 172.22 172.21 172.22 

No. of Analyses 5 5 4 

152 
Table 14B. Final Designs for Problem 3 

72-Bar Space Truss 

(U.S. Customary Units) 

_- 

Cross-sectional area (in21 

Design Ref.[lS] 

Variable Taig-Kerr 

Group Members ACCESS3 Ref.1301 Ref.1171 

No. (DUAL2) Fleury-Sander Berke-Khot 

1 l-4 0.1572 0.1571 0.1571 

2 5-12 0.5356 0.5356 0.5385 

3 13-16 0.4104 0.4099 0.4156 

4 17,la 0.5683 0.5690 0.5510 

5 19-22 0.5068 0.5067 0.5082 

6 23-30 0.5200 0.5200 0.5196 

7 31-34 0.1 0.1 0.1 

a 35,36 0.1 0.1 0.1 

9 37-40 1.280 1.280 1.279 

10 41-48 0.5148 0.5148 0.5149 

11 49-52 0.1 0.1 0.1 

12 53,54 0.1 0.1 0.1 

13 55-58 1.897 1.897 1.893 

14 59-66 0.5~58 0.5158 0.5171 

15 67-70 0.1 0.1 0.1 

16 71,72 0.1 0.1 0.1 

Mass (lbm) 379.67 379.66 379.67 

---.. .-- -...- 

No. of Analyses 5 5 4 

153 
Table 15A. Definition of Problem 4 

63-Bar Space Truss 

(SI Units) 

Material : Titanium alloy 

Young's modulus E = 110.32 x lo2 kN/m2 

Specific mass : p = 4428.8 kg/m3 

Allowable stress : cl 

a 

= 689,500 kN/m2 

Minimum area DtL) = 0.06452 cm2 

Uniform initial D(O) = 774.24 cm2 

area 

Nodal Coordinates 

Node 

1 

2 

3 

t 

4 

5 

6 

7 

a 

9 

Coordinates (cm) 1 

Node 

X 

0.0 355.6 50.8 10 76.2 

0.0 355.6 0.0 11 -76.2 

-76.2 304.8 53.34 12 -76.2 

-76.2 304.8 -2.54 13 76.2 

76.2 304.8 53.34 14 76.2 

76.2 304.8 -2.54 15 -76.2 

-76.2 203.2 76.2 16 -76.2 

-76.2 203.2 -7.62 17 76.2 

76.2 203.2 76.2 18 76.2 

Y Z 

Nodal Loading (2 load cases) 

T Coordinates (cm) 

X 

I I Load components (MN) 

Load Case I 

Node I 

1 

I 

1 

I 

11.12 

22.24 1.112 

t 

-11.12 1.112 

11.12 1.112 

203.2 -7.62 

101.6 139.7 

101.6 -12.7 

101.6 139.7 

101.6 

-T- 

-12.7 

0.0 152.4 

0.0 -17.78 

0.0 152.4 

0.0 -17.78 

Displacement constraint: 22.54 cm limits on the relative displacement 

at nodes 1 and 2 in the x-direction. 

154 
Table 15B. 

Material : 

Young's modulus : 

Specific mass 

Allowable stress : 

Minimum area : 

Uniform initial area : 

Nodal coordinates 

Node 

1 

2 

3 

t 

4 

5 

6 

7 

a 

9 

Definition of Problem 4 

63-Bar Space Truss 

(U.S. Customary Units) 

Titanium alloy 

E = 1.6~10~ psi 

o = 0.16 lbm/in3 

u = 100,000 psi 

a 

D(L) = 0.01 in2 

D(O) = 120 in2 

X 

0.0 140.0 20.0 10 30.0 80.0 

0.0 140.0 0.0 11 -30.0 40.0 

-30.0 120.0 21.0 12 -30.0 40.0 

-30.0 120.0 -1.0 13 30.0 40.0 

30.0 120.0 21.0 14 30.0 40.0 

30.0 120.0 -1.0 15 -30.0 0.0 

-30.0 80.0 30.0 16 -30.0 0.0 

-30.0 80.0 -3.0 17 30.0 0.0 

30.0 80.0 30.0 ia 30.0 0.0 

Y X Y Z 

-3.0 

55.0 

-5.0 

n 

55.0 

-5.0 

60.0 

-7.0 

60.0 

-7.0 

Nodal Loading (2 load cases) 

I Load I I Load components (lbf) I 

Case Node 

X Y Z 

1 1 2.5x106 -5.0~10~ 2.5~10' 

2 -2.5~10~ 5.0x106 2.5~10' 

---.. 

2 1 5.0~10~ 

-5.0x106 

-2:5x106 

m 

2.5x10' 

2 2.5x106 2.5x105 

Displacement constraint: S-0 in limits on the relative displacement 

at nodes 1 and 2 in the x-direction. 
Table 16A. Iteration History Data for Problem 4 

63-Bar Space Truss 

(SI Units) 

Mass (kg) 

Analysis 

No. NEWSUMT NEWSUMT DUAL 2 (W/O FSD) DUAL 2 (w/FSD) Ref. 1301 Ref. 1171 Ref. [5] 

(0.5x1) (0.5~2) Fleury- Berke- schmitunscaled scaled unscaled scaled Sander Khot Miura 

1 30222 30222 30222 13705 30222 13705 13705 

2 

13705 30222 

7672 5690 3042 3435 3060 3459 3484 3437 5907 

3 5052 3931 2865 3969 2861 3192 2990 

4 

3123 4332 

4236 3308 2810 3054 2815 3121 2902 

5 

3143 3422 

3767 3038 2793 2854 2802 2953 2844 3085 3088 

6 3456 2904 2784 2832 2782 2898 2833 2998 2929 

7 3245 2839 2780 2813 2778 2851 2812 2936 2851 

8 3101 2807 2778 2795 2776 2820 2794 2898 2813 

9 3003 2791 2776 2781 2776 2801 2779 2873 

10 

2794 

2933 2784 2776 2777 2776 2789 2777 

11 

2855 2794 

2886 2780 2776 2776 2775 2784 2776 

12 

2841 2786 

2853 2777 2775 2776 2775 2781 2776 

13 

2831 2778 

2830 2776 2775 2776 2776 

14 

2826 2776 

2814 2776 2775 

15 

2820 

2802 2776 2775 2821 

. . . . . . . . . 

50 2794 

CPU Total 108 163 60.1 33.7 87.5 

Time Anal 44 46 41.2 27.9 19.8 

(set) Optim 59 113 13.9 2.8 66.7 
Table 16B. Iteration History Data for Problem 4 

63-Bar Space Truss 

(U.S. Customary Units) 

Mass (lbm) 

Analysis 

No. 1 NEWSUMT NEXGUMT DUAL2(w/o FSD) DUAL2 (w/FSD) Ref.1301 Ref.[17] Ref. [51 

(0.5x1) (0.5~2) --- Pleury- unscaled scaled unscaled scaled Berke- SchmitSander Khot 

c 

Miura 

1 66628 66628 66628 30214 66628 30214 30214 30214 166628 

2 16914 12543 6706 7573 6746 7625 7680 7577 '13023 

'3 11137 8667 6316 6546 6307 7037 6591 6884 9551 

,, 

4 9338 7293 6195 6733 6207 6880 6398 6928 7544 

5 8305 6697 6157 6292 6177 6510 ' 6270 6801 6807 

6 7620 6402 6138 6243 6134 6388 6246 6609 6457 

7 7154 6259 6129 6201 6125 6286 6199 6473 6285 

8 6836 6189 6124 6161 6121 6216 6159 6388 6202 

9 6620 6154 6121 6132 6120 6175 6126 6333 6160 

10 6467 6137 6120 6123 6119 6149 6123 6293 6160 

11 6362 6128 6119 6121 6118 6137 6121 6263 6141 

12 6289 6123 6118 6120 6118 6130 6120 624.X. 6124 

- - 

13 6238 6121 6118 6119 6119 62.31 - 6121 

14 6203 .6120 6118 6216 

15 6178 6119 6118 6220 

. . . . . . . . . 

50 6159 

CPU Total 108 163 60.1 33.7 87.5 

rime Anal. 44 46 41.2 27.9 19.8 

(Set) Gptim. 59 113 13.9 2.8 66.7 
Table 17A. Final Designs for Problem 4 

63-Bar Space Truss 

(SI Units) 

Member 

No. 

NC 

1 

S 

2 

DUAL 2 

(w/o FSD) 

Ref. [30] Ref. [17] Ref. [Sl 

Fleury- Berke- SchmitSander Khot Miura 

1 1 3 242.5 242.8 237.8 242.3 

2 2 4 235.0 234.8 238.1 235.4 

3 1 5 339.2 339.0 344.1 339.8 

4 2 6 346.9 347.1 344.0 346.9 

5 3 7 153.0 152.7 155.7 153.5 

6 4 8 186.7 186.9 179.5 186.8 

7 5 9 111.3 111.4 111.9 111.4 

8 6 10 137.7 137.3 141.9 138.1 

9 7 11 168.1 168.4 151.1 168.1 

10 8 12 161.9 162.1 167.4 10.2 * 3 

11 9 13 56.76 56.74 60.91 56.67 

12 10 14 58.04 58.11 63.36 57.85 

13 11 15 150.8 151.1 144.3 151.2 

14 12 16 126.1 126.4 119.9 126.3 

15 13 17 33.40 33.37 37.36 33.32 

16 14 18 19.02 19.02 28.84 19.07 

17 3 5 239.4 239.5 238.0 239.2 

18 4 6 240.7 241.0 242.1 240.7 

19 7 9 0.065 0.065 0.065 0.065 

20 8 10 0.065 0.065 0.065 0.065 

21 11 13 1.038 0.639 0.968 1.407 

22 12 14 0.895 0.394 0.065 1.097 

23 1 2 0.065 0.065 0,.065 0.065 

24 3 4 0.065 0.065 0.161 0.065 

25 5 6 0.065 0.065 0.065 0.065 

26 7 8 26.63 26.08 39.42 27.04 

27 9 10 6.568 5.878 0.065 6.355 

28 11 12 20.94 20.98 28.58 21.19 

29 13 14 0.065 0.065 7.420 0.065 

30 3 9 52.72 53.31 44.78 50.72 

31 4 10 51.54 52.03 62.97 50.32 

32 5 7 58.17 57.38 71.17 60.00 

33 6 8 58.51 57.95 52.20 59.55 

34 7 13 61.10 60.27 74.78 63.03 

35 8 14 58.51 57.96 52.20 59.55 

36 9 11 55.38 56.00 47.10 53.27 

37 10 12 51.55 52.04 63.04 50.33 

38 11 17 52.75 52.80 45.03 50.86 

Connecting 

Cross-sectional Area (cm&) 

158 
Table 17A., contd. 

Cross-sectional Area (cm21 

Connecting Ref. [301 Ref. [17] Ref. [51 

Member Nodes DUAL 2 Fleury- Berke- Schmit-. 

1 2 ' (w/o FSD) Sander mot Miura 

39 12 18 51.82 52.16 63.04 50.66 

40 13 15 57.55 57.64 70.46 59.26 

41 14 16 58.26 57.86 52.20 59.24 

42 1 6 163.1 164.4 158.8 162.8 

43 1 4 175.0 176.3 158.3 174.7 

44 2 5 124.2 122.9 127.6 124.8 

45 2 3 136.0 129.6 139.6 136.7 

46 5 10 108.0 107.1 110.9 108.5 

47 3 8 81.30 80.39 83.75 81.62 

48 6 9 119.9 120.8 116.7 119.7 

49 4 7 128.7 129.6 126.9 128.5 

50 9 14 42.97 43.01 43.42 42.91 

51 7 12 36.72 36.69 48.33 37.15 

52 10 13 51.83 52.27 61.29 52.44 

53 8 11 24.52 24.75 0.065 23.50 

54 13 18 38.20 38.36 43.81 38.62 

55 11 16 76.84 77.23 57.42 76.97 

56 14 17 37.87 37.92 24.52 37.73 

57 12 15 10.79 10.28 21.81 10.91 

58 4 5 0.065 0.065 0.065 0.065 

59 3 6 0.065 0.065 0.065 0.065 

60 8 9 0.065 0.065 0.065 0.065 

61 7 10 0.065 0.065 0.065 0.065 

62 12 13 0.065 0.065 0.065 0.065 

63 11 14 0.065 0.065 0.065 0.065 

Mass (kg) 2775.11 2774.90 2793.86 2776.44 

.- 

No. of Analyses 13 17 50 13 

159 
Table 17B. Final Designs for Problem 4 

63-Bar Space Truss 

(U.S. Customary Units) 

Member 

No. 

Connectin< 

Nodes DUAL2 

(W/O FSD) 

Ref. 1171 Ref. 151 

Berke-K?ot S&nit-Pliur' 

1 

- 

Ref. [30] 

FleurySander 

1 1 3 37.58 37.63 36.86 37.55 

2 2 4 36.43 36.39 36.90 36.49 

3 1 5 52.58 52.54 53.33 52.66 

4 2 6 53.76 53.80 53.31 53.76 

5 3 7 23.71 23.67 24.13 23.79 

6 4 8 28.94 28.97 27.82 28.95 

7 5 9 17.25 17.26 17.35 17.26 

8 6 10 21.34 21.28 22.00 21.40 

9 7 11 26.06 26.10 23.42 26.06 

10 8 12 25.10 25.13 25.95 25.15 

11 9 13 8.798 8.794 9.44 8.784 

12 10 14 8.996 9.007 9.82 8.966 

13 11 15 23.38 23.42 22.37 23.43 

14 12 16 19.55 19.59 18.59 19.57 

15 13 17 5.176 5.172 5.79 5.165 

16 14 18 2.948 2.948 4.47 2.956 

17 3 5 37.10 37.12 36.89 37.07 

18 4 6 37.31 37.35 37.52 37.30 

19 7 9 o.ol- 0.01 0.01 0.01 

20 8 10 0.01 0.01 0.01 0.01 

21 11 13 0.1609 0.099 0.15 0.218 

22 12 14 0.1387 0.061 0.01 0.170 

23 1 2 0.01 0.01 0.01 0.01 

24 3 4 0.01 0.01 0.18 0.01 

25 5 6 0.01 0.01 0.01 0.01 

26 7 8 4.127 4.042 6.11 4.191 

27 9 10 1.018 0.911 0.01 0.985 

28 11 12 3.245 3.251 4.43 3.285 

29 13 14 0.01 0.01 1.15 0.01 

30 3 9 8.171 8.263 6.94 7.861 

31 4 10 7.988 8.064 9.76 7.799 

32 5 7 9.016 8.894 11.03 9.300 

33 6 8 9.068 8.982 8.09 9.229 

34 7 13 9.470 9.342 11.59 9.769 

35 8 14 9.069 8.983 8.09 9.230 

36 9 11 8.583 8.679 7.30 8.257 

37 10 12 7.990 8.066 9.77 7.801 

38 11 17 8.176 8.183 6.98 7.883 

Cross-sectional Area(in2) 

160 
Table 17B., contd. 

Kern&r 

NO. 

DUAL2 

(w/o FSD) 

Ref. [53 

Scl-xiG.t-Miur 

2 

Ref. 130 

FleurySander 

39 12 18 8.032 8.084 9.77 7.852 

40 13 15 8.919 8.934 10.92 9.184 

41 14 16 9.030 8.968 8.09 9.181 

42 1 6 25.28 25.48 24.61 25.23 

43 1 4 27.12 27.32 24.54 27.07 

44 2 5 19.25 19.05 19.78 19.35 

45 2 3 21.08 20.09 21.63 21.18 

46 5 10 16.74 16.60 17.19 16.81 

47 3 8 12.60 12.46 12.98 12.65 

48 6 9 18.59 18.73 18.09 18.55 

49 4 7 19.95 20.09 19.67 19.91 

50 9 14 6.660 6.666 6.73 6.650 

51 7 12 5.692 5.686 7.49 5.758 

52 10 13 8.033 8.102 9.50 8.128 

53 8 11 3.801 3.836 0.01 3.642 

54 13 18 5.921 5.946 6.79 5.986 

55 11 16 11.91 11.97 8.90 11.93 

56 14 17 5.870 5.877 3.80 5.848 

57 12 15 1.672 1.593 3.38 1.691 

58 4 5 0.01' 0.01 0.01 0.01 

59 3 6 0.01 0.01 0.01 0.01 

60 8 9 0.01 0.01 0.01 0.01 

61 7 10 0.01 0.01 0.01 0.01 

62 12 13 0.01 0.01 0.01 0.01 

63 11 14 0.01 0.01 0.01 0.01 

Mass (lbm) 6117.97 6117.5 6159.3 6120.9 

No. of Analyses 13 17 50 13 

Cross-st?ctionK! Art33 (ii12 

161 
Table 18A. Definition of Problem 5 

Swept Wing Model 

(SI Units) 

Material 

Young's modulus 

Poisson's ratio : 

Specific mass 

Allowable stress 

Minimum thickness 

Initial thickness 

skin (CST) 

webs (SSP) 

Displacement constraints 

Aluminum 

E = 73.09 x lo6 kN/m2 

v = 0.3 

p = 2657 kg/m3 

0 

a = 2172,375 kN/m2 

DIL) = 0.0508 cm 

D(O) = 0.762 cm 

D(O) = 0.381 cm 

Displacement limits (cm) 

Node 

X Y Z 

41 - 2152.4 

44 - 2152.4 

162 
Table 18B. 

Material 

Young's modulus 

Poisson's ratio 

Specific mass 

Allowable stress 

Definition of Problem 5 

Swept Wing Model 

(U.S. Customary Units) 

: Aluminum 

: E = 1.06x107 psi 

: v = 0.3 

: p = 0.096 lbm/in3 

: u = -125,000 psi 

a 

Minimum thickness :-J (L) = 0.02 in 

Initial thickness 

skin (CST) :D(O) = 0.30 in 

webs (SSP) :D(OL 0.15 in 

Displacement constraints 

Displacement limits (in) 

Node 

X Y Z 

41 - 260.0 

44 - 260.0 

163 
Table 19A. Nodal Coordinates for Swept Wing Model (Problem 5) 

(SI Units) 

Node No. Y 

(cm) 

1 0.0 762.0 25.4 

2 0.0 635.0 38.1 

3 0.0 469.9 33.02 

4 0.0 254.0 12.70 

5 254.0 656.1 21.80 

6 254.0 544.1 32.59 

7 254.0 399.3 28.14 

8 254.0 211.7 11.01 

9 482.6 560.8 18.56 

10 482.6 462.0 27.64 

11 482.6 335.5 23.77 

12 482.6 173.6 9.482 

13 660.4 486.9 16.045 

14 660.4 398.3 23.79 

15 660.4 286.0 20.36 

16 660.4 143.9 8.298 

17 825.5 418.1 13.706 

18 825.5 339.1 20.21 

19 825.5 240.1 17.20 

20 825.5 116.4 7.196 

21 977.9 354.6 11.547 

22 977.9 284.5 16.91 

23 977.9 197.7 14.28 

24 977.9 91.01 6.180 

25 1117.6 296.4 9.568 

26 1117.6 234.5 13.89 

27 1117.6 158.8 11.62 

28 1117.6 67.74 5.250 

29 1244.6 243.4 7.767 

30 1244.6 189.0 11.13 

31 1244.6 123.5 9.164 

32 1244.6 46.56 4.402 

33 1358.9 195.8 6.149 

34 1358.9 148.1 8.656 

35 1358.9 91.67 6.975 

36 1358.9 27.51 3.640 

37 1447.8 158.8 4.890 

38 1447.8 116.2 6.731 

39 1447.8 66.93 5.271 

40 1447.8 12.70 3.048 

41 1524.0 127.0 3.810 

42 1524.0 88.90 5.080 

43 1524.0 45.72 3.810 

44 1524.0 0.0 2.540 

164 
Table 19B. Nodal Coordinates for Swept Wing Model (Problem 5) 

(U.S. Customary Units) 

. 

X Y Z 

Node No. (in) (in) (in) 

1 0.0 300.0 10.00 

2 0.0 250.0 15.00 

3 0.0 185.0 13.00 

4 0.0 100.0 5.000 

5 100.0 258.3 8.583 

6 100.0 214.2 12.83 

7 100.0 157.2 11.08 

8 100.0 83.33 4.333 

9 190.0 220.8 7.308 

10 190.0 181.9 10.88 

11 190.0 132.1 9.358 

12 190.0 68.33 3.733 

13 260.0 191.7 6.317 

14 260.0 156.8 9.366 

15 260.0 112.6 8.017 

16 260.0 56.67 3.267 

17 325.0 164.6 5.396 

18 325.0 133.5 7.958 

19 325.0 94.54 6.771 

20 325.0 45.83 2.833 

21 385.0 139.6 4.546 

22 385.0 112.0 6.658 

23 385.0 77.84 5.621 

24 385.0 35.83 2.433 

25 440.0 116.7 3.767 

26 440.0 92.33 5.467 

27 440.0 62.53 4.567 

28 440.0 26.67 2.067 

29 490.0 95.83 3.058 

30 490.0 74.42 4.383 

31 490.0 48.62 3.608 

32 490.0 18.33 1.733 

33 535.0 77.08 2.421 

34 535.0 58.29 3.408 

35 535.0 36.09 2.746 

36 535.0 10.83 1.433 

37 570.0 62.50 1.925 

38 570.0 45.75 2.650 

39 570.0 26.35 2.075 

40 570.0 5.00 1.200 

41 600.0 50.00 1.500 

42 600.0 35.00 2.000 

43 600.0 18.00 1.500 

44 600.0 0.00 1.000 

165 
Table 20A. Applied Nodal Loading for Swept Wing Model (Problem 5) 

(SI Units) 

For all nodes, P = 0.0 and P = 0.0 

X Y 

Node 

pz 

Node P Node 

Z 

No. (N) No. (N) No. 

Load Condition 1 

pz 

(NJ 

5 5702.3 19 6462.9 

6 11480.3 20 4701.5 

7 15114.3 21 2041.6 

8 10586.2 22 4261.2 

9 4350.1 23 5564.4 

10 8953.8 24 3789.7 

11 11533.7 25 1610.2 

12 7846.3 26 3362.7 

13 3233.7 27 4385.7 

14 6164.9 28 2984.6 

15 8477.9 29 1254.3 

16 5769.1 30 2619.9 

17 2535.4 31 3416.1 

18 5293.1 32 2321.9 

5 10501.7 19 4559.2 

6 17240.4 20 1579.0 

7 10266.0 21 3749.7 

8 3527.3 22 6111.6 

9 7881.9 23 3669.6 

10 12877.0 24 1263.2 

11 7583.8 25 2957.9 

12 2588.7 26 4857.2 

13 5826.9 27 2895.6 

14 9496.5 28 996.4 

15 5595.6 29 2304.1 

16 1926.0 30 3785.2 

17 4657.1 31 2259.6 

18 7646.1- 32 778.4 

Load Condition 2 

33 916.3 

34 1917.1 

35 2504.2 

36 1703.6 

37 640.5 

38 1343.3 

39 1757.0 

40 1196.5 

41 275.8 

42 573.8 

43 751.7 

44 516.0 

33 1788.1 

34 2873.4 

35 177013 

36 685.0 

37 1383.3 

38 2143.9 

39 1361.1 

40 600.5 

41 591.6 

42 916.3 

43 582.7 

44 258.0 

166 
Table 20B. Applied Nodal Loading for Swept Wing Model (Problem 5) 

(U.S. Customary Units) 

For all nodes, P = 0.0 and 

X pY = O-O 

Node 

pZ 

Node P Node P 

No. (lbf) No. (1Ef) No. (1Ef) 

Load Condition 1 

. 

5 1282.0 19 1453.0 33 206.0 

6 2581.0 20 1057.0 34 431.0 

7 3398.0 21 459.0 35 563.0 

8 2380.0 22 958.0 36 383.0 

9 978.0 23 1251.0 37 144.0 

10 2013.0 24 852.0 38 302.0 

11 2593.0 25 362.0 39 395.0 

12 1764.0 26 756.0 40 269.0 

13 727.0 27 986.0 41 62.0 

14 1386.0 28 671.0 42 129.0 

15 1906.0 29 282.0 43 169.0 

16 1297.0 30 589.0 44 116.0 

17 570.0 31 768.0 

18 1190.0 32 522.0 

Load Condition 2 

5 2361.0 19 1025.0 33 402.0 

6 3876.0 20 355.0 34 646.0 

7 2308.0 21 843.0 35 398.0 

8 793.0 22 1374.0 36 154.0 

9 1772.0 23 825.0 37 311.0 

10 2895.0 24 284.0 38 482.0 

11 1705.0 25 665.0 39 306.0 

12 582.0 26 1092.0 40 135.0 

13 1310.0 27 651.0 41 133.0 

14 2135.0 28 224.0 42 206.0 

15 1258.0 29 518.0 43 131.0 

16 433.0 30 851.0 44 58.0 

17 1047.0 31 508.0 

18 1719.0 32 175.0 

167 
Table 21A. Iteration History Data for Problem 5 

Swept Wing Model, 

(SI Units) 

Mass (kg) 

Ref. [S] Ref. [281 DUAL 1 

Analysis NHNSUMT DUAL 2 Schmit- Rizzi Discrete 

(0.5~2) Scaled . No. Unscaled Miura Case 

1 2249 2249 1751 2249 2249 2249 

2 1609 1114 1313 1534 1303 1235 

3 1365 1121 1121 1226 1216 1245 

4 1235 1119 1119 1147 1163 1238 

5 1174 1118 1118 1125 1128 1230 

6 1146 1120 1116 1230 

7 1133 1117 1118 

8 1127 1117 1118 

9 1125 1118 

10 1123 1117 

. . . . . . 

17 1117 

CPU Total 37.0 19.4 21.5 25.6 

Time Anal. 30.8 17.7 17.0 21.1 

(Set) Opt. 4.5 0.5 3.1 0.44 3.0 

f f f f f f 

f IBM 360/91 

ff CDC 7600 (for comparison, time should be multiplied by 5) 

168 
Table 21B. Iteration History Data for Problem 5 

Swept Wing Model 

(U.S. Customary Units) 

Analysi 

No. 

Mass (lbm) 

NEWSUM’ 

(0.5x2 

DU; 

unscalec 

2 

scaled 

Ref.[5 

Schmit 

Miura 

Ref.[28 

Rizzi 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

f . . 

17 

4959 

3548 

3009 

2723 

2588 

2526 

2498 

2484 

2480 

2475 

30.8 

f IBM 360/91 

4959 3861 

2455 2894 

2471 2471 

2466 2466 

2464 2464 

19.4 21.5 

17.7 17.0 

0.5 3.1 

f Y 

4959 

3381 

2702 

2528 

2480 

2469 

2463 

2463 

4959 

2873 

2681 

2563 

2486 

2460 

2464 

2464 

2464 

2463 

. . . 

2462 

0.44 

f f 

DUALS 

discrete 

case 

4959 

2722 

2744 

2730 

2712 

2712 

25.6 

21.1 

3.0 

f 

ff CDC 7600 (for comparison, time should be multiplied by 5) 

169 
Linked 

Design 

Variable 

Region 

CST 1 0.5210 0.5187 0.5179 0.5166 0.6350 0.6350 

2 0.4539 0.4516 0.4514 0.4503 0.4826 0.4826 

3 0.4013 0.3988 0.3985 0.3970 0.4064 0.4064 

4 0.3327 0.3287 0.3292 0.3284 0.4064 0.4064 

5 0.2969 0.2944 0.2929 0.2819 0.3175 0.3175 

6 0.2629 0.2621 0.2609 0.2423 0.2286 0.3175 

7 0.0510 0.0508 0.0508 0.0508 0.0508 0.0508 

SSP 1 0.0758 0.0682 0.0745 0.0819 0.0635 0.0813 

2 0.0563 0.0562 0.0553 0.0508 0.0635 0.0635 

3 0.1097 0.1426 0.1128 0.0878 0.1600 0.1600 

4 0.0924 0.0907 0.0897 0.1159 0.0813 0.1016 

5 0.5288 0.5014 0.5306 0.5573 0.4826 0.6350 

6 0.0717 0.0773 0.0948 0.2421 0.0635 0.0813 

7 0.2389 0.2371 0.2296 0.2261 0.2540 0.2540 

8 0.2058 0.1999 0.2032 0.1514 0.1803 0.2032 

9 0.0834 0.0763 0.0827 0.0887 0.0635 0.0813 

10 0.1123 0.1282 0.1247 0.1533 0.1016 0.1600 

11 0.1470 0.1694 0.1634 0.2576 0.1600 0.1803 

Skin Mass 

(kg) 

Webs Mass 

(kg) 

Total Mass 

(kg) 

No. of 

Analyses 

Table 22A. Final Designs for Problem 5 

Swept Wing Model 

(SI Units) 

NEWSUMT 

(0.5~2) 

DUAL 2 

Ref. 151 

schnitMiura 

Ref. 1281 

Rizzi 

Discre 

DUAL 1 

2 case 

iounding 

UP 

995.67 989.75 988.22 1106.13 1128.79 

127.14 128.12 128.91 -- .24.03 151.48 

1122.81 1117.87 1117.14 1116.65 1230.23 1280.27 

10 5 8 17 6 -- 

Thickness (cm) 

170 
Table 22B. Final Designs for Problem 5 

Swept Wing Model 

(U.S. Customary Units) 

Linked 

Design 

Variable 

Resion 

CST 1 

2 

3 

4 

5 

6 

7 

SSP 1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Skin Mass 

(lbm) 

Webs Mass 

(lbm) 

Total Mass 

(lbm) 

No. of 

Analyses 

YEWSUMT 

(0.5~2) 

0.2051 

0.1787 

0.1580 

0.1310 

0.1169 

0.1035 

0.02008 

0.02983 

0.02216 

0.04320 

0.03636 

0.2082 

0.02821 

0.09405 

0.08104 

0.03283 

0.04422 

0.05788 

2195.04 

280.3C 

2475.34 

0.2042 0.2039 0.2034 

~ 0.1778 0.1777 0.1773 

I 0.1570 0.1569 0.1563 

0.1294 0.1296 0.1293 

0.1159 0.1153 0.1110 

0.1032 0.1027 0.09541 

0.02 I 0.02 0.02 

0.02686 0.02932 0.03223 

0.02213 0.02177 0.02 

0.05613 0.04439 0.03455 

0.03570 0.03531 0.04563 

0.1974 0.2089 0.2194 

0.03045 0.03732 0.09530 

0.09335 0.09038 0.08901 

0.07869 0.07999 0.05959 

0.03005 0.03255 0.03494 

0.05047 0.04911 0.06036 

0.06669 0.06435 0.1014 

Thickness (in) 

m 

DUAL2 Ref[5] Ref.[28] 

Schmit- Rizzi 

Miura 

2181.98 2178.62 - 

I 282.46 284.20 - 

2464.44 2462.82 2461.76 

5 8 17 

Discrete case 

DUAL1 Rounding 

. up 

0.250 0.250 

0.190 0.190 

0.160 0.160 

0.160 0.160 

0.125 0.125 

0.090 0.125 

0.020 0.020 

0.025 0.032 

0.025 0.025 

0.063 0.063 

0.032 0.040 

0.190 0.250 

0.025 0.032 

0.100 0.100 

0.071 0.080 

0.025 0.032 

0.040 0.063 

0.063 0.071 

2438.56 2488.52 

273.56 333.94 

2712.14 2822.46 

6 

171 
Material 

Young's moduli 

Shear modulus 

Poisson's ratio 

Specific mass 

Thermal expansion 

coefficients 

Allowable stress 

Allowable strain 

Table 23A. Definition of Problem 6 

Delta Wing Model 

(SI Units) 

Skin Webs 

graphite-epoxy Titanium 

: EL = 144.8 x 106 kN/m2 E = 113.1 x 106 kN/m2 

ET = 11.72 x lo6 kN/m2 

G LT = 4.482 x lo6 kN/m2 

'LT = 0.21 v = 0.3 

p = 1550 kg/m3 p = 4429 kg/m3 

Cl= -0.378 lJm/m"c a = 10.08 p/mot 

cx = 28.8 m/mot 

t 

EL = 0.008571 m/m 

ca 

= 861,875 kN/m2 

EC L =-O-O08571 m/m 

t 

ET = 0.004706 m/m 

EC T =-O-O17647 m/m 

'LT = 0.018462 

Minimum thickness DIL) = 0.0127 cm DcL) = 0.0508 cm 

Initial thickness See Table 27A. D(O) = 0.381 cm 

Nodal Loading (2 load cases) 

Displacement Constraints 

Z[ 

172 
Table 23B. Definition of Problem 6 

Delta Wing Model 

(U.S. Customary Units) 

Material : 

Young's moduli 

Shear modulus 

Poisson's ratio : 

Specific mass 

Thermal expansion 

coefficients 

Allowable stress : 

Allowable strain : 

Skin Webs 

graphite-epoxy titanium 

EL = 21x106 psi E = 1.64~10~ psi 

E T = 1.7x106 psi 

G LT = 0.65x106psi 

'LT = 0.21 v = 0.3 

p = 0.056 lbm/in 3 

p = 0.16 lbm/in 3 

aL= -0.21x10 -6 in/inOF c1 = 5.6x10e6 in/inOF 

= 16x10 -6 

aT in/inOF 

t 

EL = 0.008571 in/in. ca = 125,000 psi 

C 

EL =-O-O08571 in/in. 

t 

ET = 0.004706 in/in. 

C E 

T =-0.017647 in/in. 

'LT = 0.018462 

Minimum thickness : D (L) 

= 0.005 in. 

Initial thickness See Table 27.A 

Nodal Loading (2 load cases) 

Load Node 

Load components (lbfp 

Case X Y Z 

1 10-44 0 0 8075 

2 10-44 0 0 -8075 

Displacement constraints 

D(L) = 0.02 in, 

Db) = 0.15 in. 

Node Displacement limits (In) 

X Y 1 z 

43 1 flOO.8 

44 +10() 8 - . 

173 
Table 24A. Nodal Coordinates for Delta Wing Model (Problem 6) 

(SI Units) 

Node No. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

X Y z 

(cm) (cm) (cm) 

0.0 2438.4 16.43 

0.0 2133.6 29.13 

0.0 1828.8 38.13 

0.0 1524.0 43.38 

0.0 1219.2 44.93 

0.0 914.4 42.77 

0.0 609.6 36.88 

0.0 304.8 27.28 

0.0 0.0 13.95 

254.0 2133.6 16.22 

254.0 1828.8 28.30 

254.0 1524.0 36.22 

254.0 1219.2 40.03 

254.0 914.4 39.67 

254.0 609.6 35.20 

254.0 304.8 26.57 

254.0 0.0 13.80 

508.0 1828.8 15.95 

508.0 1524.0 27.23 

508.0 1219.2 33.86 

508.0 914.4 35.79 

508.0 609.6 33.07 

508.0 304.8 25.68 

508.0 0.0 13.62 

762.0 1524.0 15.61 

762.0 1219.2 25.88 

762.0 914.4 30.78 

762.0 609.6 30.33 

762.0 304.8 24.54 

762.0 0.0 13.38 

1016.0 1219.2 15.15 

1016.0 914.4 24.04 

1016.0 609.6 26.64 

1016.0 304.8 22.99 

1016.0 0.0 13.06 

1270.0 914.4 14.50 

1270.0 609.6 21.44 

1270.0 304.8 20.81 

1270.0 0.0 12.61 

1524.0 609.6 13.52 

1524.0 304.8 17.49 

1524.0 0.0 11.93 

1854.2 213.4 11.07 

1854.2 0.0 10.06 

174 
Table 24R Nodal Coordinates for Delta Wing Model (Problem 6) 

(U;S. Customary Units) 

I 

Node : No. X Y Z 

(in) (in) (in) 

1 0.0 960.0 6.468 

2 0.0 840.0 11.47 

3 0.0 720.0 15.01 

4 0.0 600.0 17.08 

5 0.0 480.0 17.69 

6 0.0 360.0 16.84 

7 0.0 240.0 14.52 

8 0.0 120.0 10.74 

9 0.0 0.0 5.492 

10 100.0 840.0 6.385 

11 100.0 720.0 11.14 

12 100.0 600.0 14.26 

13 100.0 480.0 15.76 

14 100.0 360.0 15.62 

15 100.0 240.0 13.86 

16 100.0 120.0 10.46 

17 100.0 0.0 5.434 

18 200.0 720.0 6.281 

19 200.0 600.0 10.72 

20 200.0 480.0 13.33 

21 200.0 360.0 14.09 

22 200.0 240.0 13.02 

23 200.0 120.0 10.11 

24 200.0 0.0 5.362 

25 300.0 600.0 6.146 

26 300.0 480.0 10.19 

27 300.0 360.0 12.12 

28 300.0 240.0 11.94 

29 300.0 120.0 9.660 

30 300.0 0.0 5.268 

31 400.0 480.0 5.966 

32 400.0 360.0 9.463 

33 400.0 240.0 10.49 

34 400.0 120.0 9.051 

35 400.0 0.0 5.143 

36 500.0 360.0 5.710 

37 500.0 240.0 8.441 

38 500.0 120.0 8.193 

39 500.0 0.0 4.966 

40 600.0 240.0 5.322 

41 600.0 120.0 6.887 

42 600.0 0.0 4.696 

43 730.0 84.0 4.360 

44 730.0 0.0 3.959 

175 
Table 25A. Fuel Mass Distribution for Delta Wing Model 

(Problem 6) 

(SI Units) 

Node Fuel Mass Node Fuel Mass 

No. (kg) No. kg) 

10 866.4 22 1765 

11 1510 23 1370 

12 1932 24 725.8 

13 2132 25 831.0 

14 2118 26 1377 

15 1878 27 1642 

16 1415 28 1619 

17 734.8 29 1293 

18 849.1 30 714.0 

19 1452 31 808.3 

20 1810 32 1279 

21 1910 33 1420 

Node Fuel Mass 

No. , (kg) 

34 1225 

35 694.9 

36 772.0 

37 1143 

38 1107 

39 671.3 

40 721.2 

41 934.4 

42 635.0 

43 589.7 

44 535.2 

Total fuel mass = 42,480 kg 

176 
Table 25B. Fuel Weight Distribution for Delta Wing Model 

(Problem 6) 

(U.S. Customary Units) 

Total fuel mass = 93,650 lbm. 

177 
Table 26A. Iteration History Data for Problem 6 

Delta Wing Model 

(SI Units) 

NEWSUMT(continuous) DUAL 2 (continuous) DUAL 1 (mixed) 

Analysis ' 

No. Mass Frequency Mass Frequency Mass Frequency 

(Mg) (Hz) t&r) (Hz) (Mg) (Hz) 

1 39.38 2.829 39.38 2.829 39.38 2.829 

2 31.87 2.650 9.703 2.016 9.448 2.009 

3 26.36 2.516 7.598 1.961 7.788 2.000 

4 22.30 2.396 6.505 1.937 6.985 1.974 

5 19.34 2.293 6.736 2.007 6.799 2.000 

6 17.25 2.209 6.341 1.987 6.532 1.994 

7 15.39 2.127 6.264 2.003 6.387 1.996 

8 13.54 2.042 6.178 2.007 6.314 2.003 

9 12.15 2.010 6.083 2.005 6.133 1.998 

10 11.14 2.009 6.006 2.004 6.083 2.003 

11 10.27 2.010 5.942 2.002 6.065 2.009 

12 9.503 2.010 5.892 2.001 6.028 2.003 

13 8.836 2.010 5.856 2.001 6.028 2.000 

14 8.260 2.010 5.829 2.000 

15 7.766 2.009 5.811 2.000 

. . . . . . . . . . 

20 6.378 2.003 

25 6.151 2.002 

29 6.110 2.002 

CPU Total 719 261 253 

Time Analy 564 252 234 

(sec)Opt 145 2 12 

178 
Table 26B. Iteration History Data for Problem 6 

Delta Wing Model 

(U.S. Customary Units) 

NEWSUMT(continuous) DUAL2 (continuous) DUAL1 (mixed) 

Analysis --- 

No. Mass Frequency Mass Frequency Mass Frequency 

(x1031h) (HZ) (x1031bm) (Hz) (x1031bm) (Hz) 

1 86.82 2.829 86.82 2.829 86.82 2.829 

2 70.26 2.650 21.39 2.016 20.83 2.009 

3 58.11 2.516 16.75 1.961 17.17 2.000 

4 49.16 2.396 14.34 1.937 15.40 1.974 

5 42.64 2.293 14.85 2.007 14.99 2.000 

6 38.04 2.209 13.98 1.987 14.40 1.994 

7 33.93 2.127 13.81 2.003 14.08 1.996 

8 29.86 2.042 13.62 2.007 13.92 2.003 

9 26.78 2.010 13.41 2.005 13.52 1.998 

10 24.56 2.009 13.24 2.004 13.41 2.003 

11 22.64 2.010 13.10 2.002 13.37 2.009 

12 20.95 2.010 12.99 2.001 13.29 2.003 

13 19.48 2.010 12.91 2.001 13.29 2.000 

14 18.21 2.010 12.85 2.000 

15 17.12 2.009 12.81 2.000 

. . . . . . . . 

20 14.06 2.003 

25 13.56 2.002 

29 13.47 2.002 

. 

3PU Total 719 261 253 

rime Analy 564 252 234 

(set : opt. 145 2 12 

179 

I 
Table 27A. Initial and Final Design for Problem 6 

Delta Wing Model 

(SI Units) 

Linked Design Fiber 

Variable Region Orientatio 

1 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

00 

f450 

900 

00 

2459 

900 

00 

f450 

900 

00 

2450 

900 

00 

?45” 

900 

00 

+450 

900 

00 

545” 

900 

00 

2450 

900 

00 

2450 

900 

00 

2450 

900 

00 

?45Q 

900 

00 

245” 

900 

00 

+45” 

900 

00 

+45” 

900 

00 

+450 

900 

00 

f450 

900 

Skin Mass (kg) 

Web Mass (kg) 

Total Structural Mass (kg) 

No. of AnalYSeS 

* Transverse tension Strain 

Initial NEWSJMT DUAL 2 DUAL 1 

Design continuous continuous mixed 

(Cm) case (cm) case (cm) case (cm) 

0.3810 0.0244 0.0198 0.0254 (2) 

0.3810 0.0193 0.0127* 0.0254 (2) 

0.3810 0.0213* 0.0127+ 0.0254*(2) 

1.5240 0.0714 0.1191 0.1016 (8) 

1.2700 0.0414f O-0127* 0.0254* (2) 

0.2540 0.0160* 0.0127* 0.0254’(2) 

3.8100 0.3787 0.2306 0.2540 (20) 

2.5400 0.0594* 0.0325* 0.0508*(4) 

0.7620 0.0127’ O-0127* 0.0254*(2) 

3.8100 2.3731 2.9464 2.7940 (22C 

2.5400 0.0899 l O-0607* 0.0762*(6) 

0.7620 0.0348 0.0203 0.0254 (2) 

1.5240 0.0343 0.0531 0.0508 (4) 

1.2700 0.0249 0.0127 0.0254 (2) 

0.2540 0.0127 0.0127 0.0254*(2) 

3.8100 0.2891 0.1412 0.1778 (14) 

2.5400 0.0803 0.0277 0.0254 (2) 

0.7620 0.0127 O-0127* 0.0254*(2) 

3.8100 1.9205 2.1064 2.1082 (166 

2.5400 0.0859 0.0328 0.0508 (4) 

0.7620 0.0404 0.0300 0.0254 (2) 

0.50RO 0.0175 0.0127 0.0254 (2) 

0.5080 0.0127 0.0127 0.0254 !2) 

0.2540 0.0127 0.0127 0.0254 (2) 

2.5400 0.2055 0.0775 0.1016 (8) 

1.2700 0.1031 0.0450 0.0508 (4) 

0.5080 0.0127 0.0127 0.0254 (2) 

2.5400 1.4732 1.5890 1.5494 (122 

1.2700 0.1029 0.0638 0.0762 (6) 

0.5080 0.0127 0.0127 0.0254 (2) 

2.5400 0.1044 0.0338 0.0508 (4) 

1.2700 0.1130 0.0569 0.0762 (6) 

0.5080 0.0127 0.0127 0.0254 (2) 

2.5400 0.9517 1.0132 0.9906 (78) 

1.2700 0.1417 0.0980 0.1016 (8) 

0.5080 0.0127 0.0127 0.0254 (2) 

0.5080 0.0267 0.0127 0.0254 (2) 

0.5080 0.0757 0.0389 0.0509 (4) 

0.2540 0.0127 0.0127 0.0254 (2) 

0.7620 0.5151 0.5415 0.5334 (41) 

0.2540 0.1486 0.1184 0.1270 (10) 

0.2540 0.0127 0.0127 0.0254 (2) 

0.7620 0.2169 0.2273 0.2286 (18) 

0.2540 0.1143 0.1128 0.1270 (10) 

0.2540 0.0127 0.0127 0.0254’(2) 

0.2540 0.0724 

0.2540 0.0262* 

0.2540 0.0127’ 

37757.1 5475.74 

1624.6 635.84 

39381.7 6111.58 

--- 29 

it is att, ed in materia: 

0.0668 

0.0315’ 

0.0127* 

5253.96 

557.81 

5811.77 

15 

I tile bottom, 5) 

0.0762 (6) 

0.0254 (2) 

0.0254 (2) 

5456.33 

570.22 

6026.55 

180 
Table 27B. Initial and Final Designs for Problem 6 

Delta Wing Model 

,Tl c r...^C^... --.. .,..:L-\ 

\U.S. Lu’L”“aLy “IIILJ, 

Linked Design Fiber Initial NEXiSUMT DUAL 2 DUAL 1 

Variable Region Orientation Design continuous continuous mixed 

(in 1 case (in) case (in) case (in) 

1 00 0.15 0.0096 0.0078 0.01 (2) 

245" 0.15 0.0076 0.0050f 0.01 (2) 

900 0.15 0.0084* 0.0059* O.Olf (2) 

2 00 0.60 0.0281 0.0469 0.04 (8) 

+45" 0.50 0.0163* 0.0050f 0.01* (2) 

900 0.10 0.0063' 0.0050* 0.01* (2) 

3 00 1.5 0.1491 0.0908 

2450 

0.10 (20) 

1.0 0.0234* 0.0128* 0.02* (4) 

900 0.3 0.0050* 0.0050* 0.01' (2) 

4 00 1.5 0.9343 1.160 1.10 (220) 

2450 1.0 0.0354* 0.0239* 0.03* (6) 

900 0.3 0.0137 0.0080 0.01 (2) 

5 00 0.60 0.0135 0.0209 0.02 

245" 

(4) 

0.50 0.0098 0.0050 0.01 (2) 

90" 0.10 0.0050 0.0050* 0.01' (2) 

6 00 1.5 0.1138 0.0556 0.07 

Z45" 

(14) 

1.0 0.0316 0.0109 0.01 (2) 

900 0.3 0.0050 0.0050f 0.01' (2) 

7 00 1.5 0.7561 0.8293 0.83 (166) 

?45" 1.0 0.0338 0.0129 0.02 (4) 

900 0.3 0.0159 0.0118 0.01 (2) 

8 00 0.2 0.0069 0.0050 0.01 (2) 

Z45" 0.2 0.0050 0.0050 0.01 (2) 

900 0.1 0.0050 0.0050 0.01 (2) 

9 00 1.0 0.0809 0.0305 0.04 (8) 

_+45" 0.5 0.0406 0.0177 0.02 (4) 

900 0.2 0.0050 0.0050 0.01 (2) 

10 00 1.0 0.5800 0.6256 0.61 (122) 

+45a 0.5 0.0405 0.0251 0.03 (6) 

900 0.2 0.0050 0.0050 0.01 (2) 

11 00 1.0 0.0411 0.0133 0.02 (4) 

Z45" 0.5 0.0445 0.0224 0.03 (6) 

900 0.2 0.0050 0.0050 0.01 (2) 

12 00 1.0 0.3747 0.3989 0.39 (78) 

2450 0.5 0.0558 0.0386 0.04 (8: 

900 0.2 0.0050 0.0050 0.01 (2) 

13 00 0.2 0.0105 0.0050 0.01 (2) 

2450 0.2 0.0298 0.0153 0.02 (4) 

90" 0.1 0.0050 0.0050 0.01 (2) 

14 00 0.3 0.2028 0.2132 

Z45" 

0.21 (42) 

0.1 0.0585 0.0466 0.05 (10) 

900 0.1 0.0050 0.0050 0.01 (21 

15 00 0.3 0.0854 0.0895 0.09 (18) 

2450 0.1 0.0450 0.0444 0.05 (10) 

900 0.1 0.0050 0.0050 0.01' (2) 

16 00 0.1 0.0285 0.0263 0.03 (6) 

245" 0.1 o-0103* 0.0124* 0.01 (2) 

900 0.1 0.0050' 0.0050' 0.01 (2) 

Skin Mass (lbm) 83238.8 12071.74 11582.80 12028.94 

Web Mass (lbm) 3501.5 1401.76 1229.74 1257.10 

Total Structural Mass (lbm) 86820.4 13473.50 12812.54 13286.04 

NO. of Analyses --- 29 15 13 

l Transverse tension strain limit is attained in material of the bottom skin (within 5%). 

181 
Table 28A. Final Webs Thicknesses for Problem 6 

Delta Wing Model 

(SI Units) 

Variable 

(continuous (continuotis 

12 

Mass 

(kg) 

0.0508 0.0508 0.0508 

-. 

635.84 557.81 570.22 

182 
Table 28B. Final Webs Thicknesses for Problem 6 

Delta Wing Model 

(U.S. Customary Units) 

Linked 

Design 

Variable 

Regioc 

NEWSUMT 

(continuous 

case) 

(in) 

DUAL2 

(continuous 

case) 

(in) 

DUAL1 

(mixed 

case) 

(in) 

1 0.02 0.02 0.02 

2 0.02 0.02 0.02 

3 0.03437 0.02917 0.03004 

4 0.02001 0.02 0.02 

5 0.02001 0.02 0.02 

6 0.02013 0.02 0.02 

7 0.02 0.02 0.02240 

8 0.02362 0.02 0.02 

9 0.1324 0.08657 0.1014 

10 0.2689 0.2670 0.2551 

11 0.1422 0.09635 0.1088 

12 0.02 0.02 0.02 

Mass 

(lhnn) 

1401.76 1229.74 1257.10 

183 
Table 29. Detailed Iteration History Data for Problem 6 

Delta Wing Model-Mixed Case (DUALl) 

No. of No. of No. of No. of No. of Lower Dual Final Upper 

Stage Retained Active Discon. Restarts O.D.M's Bound Function Weight Bound 

No. Constraints Constraints Planes Weight R* Weight 

QR QR-N P W W 

- 

1 127 3 3 4 138 10376 10412 10417 10444 

2 136 3 3 4 44 8519 8545 8583 8583 

3 119 6 6 3 19 7623 7650 7699 7739 

4 119 4 3 4 20 7454 7481 7493 7493 

5 94 6 6 2 15 7110 7175 7200 7227 

6 38 4 4 4 20 6976 7006 7041 7041 

7 34 5 4 2 10 6885 6915 6962 6962 

8 33 5 5 2 12 6683 6730 6762 6776 

9 35 6 6 2 19 6640 6706 6706 6760 

10 35 5 5 2 16 6608 6658 6687 6700 

11 36 8 7 2 17 6552 6613 6645 6672 

12 35 7 6 2 17 6551 6611 6643 6656 

13 35 7 '6 4 33 6539 6586 6643 6657 
0 CONSTRAINT DELETION (EQ. 2.14) 

fi l LlNKlNG(EQ.24) . . h 

0 EXPLICIT CONSTRAINTS (EQ 2 18) 

FIND “D such that 

gq (6) > 0: qeQ (2.2) 

AND 

M(6) + MIN (2.3) 

FIND 2 such that 

‘;$I (2’) > 0: qcQ1) (2.16) 

W(a’) -f MIN (2.15) 

Figure 1. Key to a Tractable Formulation 

185 
ELASTICITY MODULUS E = 7000 kg/mm2 

SPECIFIC WEIGHT p = 2.8 1O-6 kg/mm3 

STRESS LIMITS 

a = -25 kg/mm 2 

ZF = 50 kg/mm2 

Figure 2. Design Space For 3-Bar Truss 

186 
0.1t 

O.l( 

O.OI 

( - 

0 NEWSUMT (0.5 x 1) 

0 NEWSUMT (0.3 x 2) 

. 

I 

NEWSUMT (0.1 x 3) 

DUAL 2 

FEASIBLE 

REGION 

YA 

O3 

STAR I’ING 

POINT 

Figure 3. 3-Bar Truss - Trajectories in Reciprocal Space 

187 
0 NEWSUMT (0.5 x 1) 

0 NEWSUMT (0.3 x 2) 

NEWSUMT (0.1 x 3) 

DUAL 2 

2 4 6 8 10 

NUMBER OF ANALYSES 

Figure 4. 3-Bar Truss - Convermna of Weight 

188 
ZERO ORDER 

----me ---- 

(NORMAL PLANE) 

&& (P) 

2 - 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

=o 

=o 

h(x) EC2 

(U) 

- u2 = 

a2 NJ) 0 

Figure 5. Zero and First Order Approximations 

189 
. 

GlVENXq, ; q&R 

2 -r IDENTIFY SET OF q VALUES 

M= q I’Xqt>O : qCOR FROM SET M 

t 

3 

av - 

EVALUATEdhqlhMI ; qfM 

YI FIND LARGEST I 

+I:,,>0 ; qcQR 

q 

DENOTE CORRESPONDING 

INDEX AS .+ q+ 

Figure 6. Dual 2 Algorithm - Block Diagram. 

190 
I 

STARTGIVEN Xqt >O ; qcM 

t 

COMPUTE $- IX,) - : qcM 4 

q 

I 

4 

TEST IVV(X,j I C i XM .- x, 

5 

- 

1 

d Min ‘qf 

max . 

- 

sq,<o Sqt : q(M: fwq’ 

I I 

t 

7b 7a 

d, - 1 dt . %ilax 

I 1 

Figure 7. Seek Max of 1 CT; 1 in Subspace M 

s 

8 

A 

q,t+1 

. Xqt+dtsqt : qtM 

- COMPUTE AND STORE Q+, VIA EQS. (2.39-2.41) 

t 

AND i, SAME DOMAIN? 

i.e., SAME SET OF FREE VARIABLES AT 

191 
A. Discrete Points in Primal Space 

2.BAR TRUSS 

B. Regions in Dual Space 

t A2 

C. Contours of Dual Function 

Figure 8. Simple 2 D Example - Pure Discrete ProMem 
Of,=E, 

\ 

P a 

/ 

Figure 9. Projxting Multiple Gradients Into Discontinuity Plane 

193 

l- 
,SEE EQS 2.39-2.41 AND 4.81 

1 

* 

COMPUTE DlRECTlON OF CONSTRAINED ASCENT i, 

EXE EQS. 4.42.A AND El BASE PLANE7 

Figure 10. Dual 1 Algorithm - Block Diagram. 

194 
d* 

6. Pure Discrete Variable Care. 

C. Mixed Discrete-Continuous Variable Case. 

Cl. care1 c2. case2 

Figure 11. One Dimensional tvlaximization Scheme (Dual 1). 
PREPROCESSOR 

t 

r ------ 9 

I 

DESIGN PROCESS CONTROL DESIGN PROCESS . CONVERGENCE TEST 

I 1 t 

i 

PRIMAL APPROXIMATE APPROXIMATE EXPLICIT NEW SET OF 

TRIAL PRIMAL PRIMAL PROBLEM + PRIMAL DESIGN 

DESIGN PROBLEM EQS. 2.19-2.22 INITIAL TRIAL VALUES 

I A 

1 I I 

APPROXIMATE PROBLEM GENERATOR I OPTIMIZATION ALGORITHM(S) 

STRUCTURAL ANALYSIS 

CONSTRAINT DE LETION 

SENSITIVITY ANALYSIS 

CONSTRUCT EXPLICIT 

CONSTRAINT APPROXIMATIONS 

FORM EXPLICIT DUAL PROBLEM I 

(SEE EQS. 2.36-2.41) I 

OTHER 

OPTIMIZATION 

----me------- 

FIND NUMERICAL SOLUTION OF 

ALGORITHMS 

r 

CURRENT DUAL PROBLEMx* I 

------ ------ 

DUAL 1 

-IDUAL 2 

I 

NEWSUMT 

----- ------ 

I 

PRIMAL 2 

EVALUATE CORRESPONDING PRIMAL 

SOLUTION 3 I 

(SEE EDS. 2.39-2.41 AND 4.8) I ETC. 

Figure 12. Bsic Organization of ACCESS 3. 
-R----w .- 

R = 914.4 cm (360 in.) 

Figure 13. Planar Ten-Bar Cantilever Truss (Problem 1). 

R 

- I

197 
I I. 

f2 

N 

0.75 1 

Wl = 3807 kg (8393 lbm) 

r El NEWSUMT 

0 DUAL 2 0.70 

0.65 

0.60 

5 10 

NUMBER OF ANALYSES 

Figure 14. Iteration History for Problem 1 (Case B) 

Ten-Bar Cantilever Truss. 

198 
a = 63.5 cm (25 im) 

Y 

Figure 7s. 25-~ 

ar Spa27 Truss (Pr,,blem 2, 

199 
0.7c 

, - 

ISlbm) 

\ 

A Ref. 13 (Gellatly-Berke) 

I L I 1 I I ) 

2 4 6 

NUMBER OF ANALYSES 

Figure 16. Iteration Hisory for Problem 2 (Case A) 

25.Bar Space Truss. 

200 
T 

b 

-L 

t 

b 

+ 

b 

i 

b 

Note: For the sake of clarity, not all elements are drawn in this figure. 

Figure 17. 72-Bar Space Truss (Problem 3). 
Wl = 387.0 kg (853.1 lbm) 

0 NEWSUMT (0.5 x 

n NEWSUMT (0.3 x 

V NEWSUMT (0.1 x 

1) 

2) 

3) 

Ref. 15 (Taig-Kerr) 

Ref. 17 (Berke-Khot) 

Ref. 30 (Fleury-Sander) 

, 

I I I I I I I ) 

2 4 6 8 

NUMBER OF ANALYSES 

Figure 18. Iteration History for Problem 3 

72-Bar Space Truss. 

202 
Note: For the sake of clarity, not all elements are drawn in the figure. 

[See Table 15 for nodal coordinate data1 

Figure 19. 63-Bar Space Truss (Problem 4). 

203 
0.25 - 

A 

z 0.20 - 

z 

P 

9 

N 

Y 

P 

0.15 - 

0.10 - 

W, = 30,222 kg (66,628 lbm) 

17 NEWSUMT (0.5 x 1) 

A NEWSUMT (0.5 x 2) 

0 DUAL 2 

I 1 I I I I I I I I I I I I I I) 

5 10 15 

NUMBER OF ANALYSES 

Figure 20. Iteration History for Problem 4 

63.Bar Space Truss. 

204 
[See Table 19 for nodal coordinate data] 

-- X 

IN PANELS (60) 

Figure. 21. Swept Wing Analysis Model (Problem 5). 

205 
206 
0.8 

0.6 

cl 

h = 2249 kg (4959 lbm) 

0 NEWSUMT 

0 DUAL 2 

I I 1 I 1 I I I I I * 

2 4 6 8 10 

NUMBER OF ANALYSES 

Figure 23. Iteration History for Problem 5 

Swept Wing Model, 

207 
c = 2438 cm (960 in) 

s = 1854 cm (730 in,) 

d = 213 cm (84 in.> 

Figure 24. Delta Wing Analysis Model (Problem 6). 

208 
5 Web 

6 

2 

Figure 25. Delta Wing Design Model (Problem 6). 

209 
0.6 

0.4 

0.2 

Cl NEWSUMT (CONTINUOUS VARIABLES) 

0 DUAL 1 (DISCRETE VARIABLES) 

Wl = 39,382 kg (86,820 lbm) 

I I I I I I II I I I I I I I I II I I I II I I I) 

5 10 15 20 25 

NUMBER OF ANALYSES 

Figure 26. Iteration History for Problem 6 

Delta Wing Model. 

210 
s 

. . 

1. Report No. 2. Government Accession No. 

NASA CR-3226 

4. Title and Subtitle 

DUAL METHODS AND APPROXIMATION CONCEPTS IN 

STRUCTURAL SYNTHESIS 

7. Author(s) 

Claude Fleury and Lucien A. Schmit, Jr. 

9. Performing Organization Name and Address 

3. Recipient’s Catalog No. 

5. Report Date 

December 1980 

6. Performing Organization Code 

6. Performing Orgamzation Report No. 

10. Work Unit No. 

University of California, Los Angeles 

Los Angeles, CA 90024 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546 

11. Contract or Grant No. 

NSG-1490 

- 

13. Type of Repon and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

505-33-63-02 

5. Supplementary Notes 

Langley Technical Monitor: J. Sobieski 

Progress Report 

6. Abstract 

Approximation concepts and dual method algorithms are combined to create a new 

method for minimum weight design of structural systems. Approximation concepts 

convert the basic mathematical programming statement of the structural synthesis 

problem into a sequence of explicit primal problems of separable form. These 

problems are solved by constructing explicit dual functions, which are maximized 

subject to nonnegativity constraints on the dual variables. It is shown that the 

joining together of approximation concepts and dual methods can be viewed as a 

generalized optimality criteria approach. The dual method is successfully extended 

to deal with pure discrete and mixed continuous-discrete design variable problems. 

The power of the method presented is illustrated with numerical results for example 

problems, including a metallic swept wing and a thin delta wing with fiber composite 

skins. 

7. Key Words (Suggested by Author(s)) 

Optimization 

Structures 

Numerical methods 

16. Distribution Statement 

Unclassified - Unlimited 

Subject Category 39 

3. Security Classif. (of this report] 20. Security Classif. (of this page) 

Unclassified Unclassified 

21. No. of Pages 

222 

22. Price 

A10 

For sale by the National Technical Information Service, Springfield, Virginia 22161 

NASA-Langley, 1980