Title: Two-level optimization of composite wing structures based on panel genetic optimization
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Title: Two-level optimization of composite wing structures based on panel genetic optimization
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Language: English
Creator: Liu, Boyang, 1966-
Publisher: University of Florida
Place of Publication: Gainesville Fla
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Publication Date: 2001
Copyright Date: 2001
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Subject: Aerospace Engineering, Mechanics, and Engineering Science thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF   ( lcsh )
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Summary: ABSTRACT: The design of complex composite structures used in aerospace or automotive vehicles presents a major challenge in terms of computational cost. Discrete choices for ply thicknesses and ply angles leads to a combinatorial optimization problem that is too expensive to solve with presently available computational resources. We developed the following methodology for handling this problem for wing structural design: we used a two-level optimization approach with response-surface approximations to optimize panel failure loads for the upper-level wing optimization. We tailored efficient permutation genetic algorithms to the panel stacking sequence design on the lower level. We also developed approach for improving continuity of ply stacking sequences among adjacent panels. The decomposition approach led to a lower-level optimization of stacking sequence with a given number of plies in each orientation. An efficient permutation genetic algorithm (GA) was developed for handling this problem. We demonstrated through examples that the permutation GAs are more efficient for stacking sequence optimization than a standard GA. Repair strategies for standard GA and the permutation GAs for dealing with constraints were also developed. The repair strategies can significantly reduce computation costs for both standard GA and permutation GA. A two-level optimization procedure for composite wing design subject to strength and buckling constraints is presented. At wing-level design, continuous optimization of ply thicknesses with orientations of 0 degrees, 90 degrees, and plus or minus 45 degrees is performed to minimize weight.
Summary: ABSTRACT (cont.): At the panel level, the number of plies of each orientation (rounded to integers) and inplane loads are specified, and a permutation genetic algorithm is used to optimize the stacking sequence. The process begins with many panel genetic optimizations for a range of loads and numbers of plies of each orientation. Next, a cubic polynomial response surface is fitted to the optimum buckling load. The resulting response surface is used for wing-level optimization. In general, complex composite structures consist of several laminates. A common problem in the design of such structures is that some plies in the adjacent laminates terminate in the boundary between the laminates. These discontinuities may cause stress concentrations and may increase manufacturing difficulty and cost. We developed measures of continuity of two adjacent laminates. We studied tradeoffs between weight and continuity through a simple composite wing design. Finally, we compared the two-level optimization to a single-level optimization based on flexural lamination parameters. The single-level optimization is efficient and feasible for a wing consisting of unstiffened panels.
Summary: KEYWORDS: two-level optimization, composite wing, genetic optimization, buckling loads, stacking sequence, continuity, flexural lamination parameters
Thesis: Thesis (Ph. D.)--University of Florida, 2001.
Bibliography: Includes bibliographical references (p. 131-138).
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Statement of Responsibility: by Boyang Liu.
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TWO-LEVEL OPTIMIZATION OF COMPOSITE WING STRUCTURES BASED ON
PANEL GENETIC OPTIMIZATION

















By

BOYANG LIU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2001













ACKNOWLEDGMENTS

First I thank Distinguished Professor Raphael T. Haftka, chairman of my advisory

committee, for much-needed guidance during my research. He provided the funding

necessary to complete my doctoral studies and constantly encouraged me to attend

conferences and to publish my work in scientific journals. His help extended beyond

problems encountered in the academic world. I sometimes wondered where he found the

patience he always had with me. Without it, this work would not have been possible.

I would like to thank my wife, Peining Chen, for the support, patience and

encouragement that she gave me during these years.

I would also like to thank the members of my committee: Professors Bhavani V.

Sankar, Loc Vu-Quoc, Gale E. Neville and Panagote M. Pardalos. I am grateful for their

willingness to serve on my committee, their help whenever required, their involvement

with my oral qualifying examination, and their review of this dissertation.

I would also like to thank professors Mehmet A. Akguin from Middle East

Technical University, Turkey; Akira Todoroki, from Tokyo Institute of Technology, Tokyo,

Japan; Fred Van Keulen, from Delft University of Technology, Delft, The Netherlands and

Philippe Trompette, University of Lyon, France. I enjoyed collaborating with them during

their stay at the University of Florida.

My colleagues in the Structural and Multidisciplinary Optimization Group at the

University of Florida also deserve thanks for their help, and for their many fruitful

discussions.
















TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ............... .............. ii
ACKNLIST OF TABLEDGM EN..... .............. ..................................................................... ii

LIST OF FIGURES............ ...... .... ................................. vi
L IS T O F F IG U R E S ............................................................................................................ ix

A B S T R A C T ........................................................................................................x i

INTRODUCTION .................. ............................ ...... .......................... ...

Introduction ................................................................ .. ......... ............. .
Two-level Optimization for Composite Wing Structures ..............................................3
Genetic Algorithm s for Panel Optim ization............................................... ...................5
Material Composition Continuity Constraints Between Adjacent Panels ...................6
Single-Level Optimization of Composite Wing Based on Flexural Lamination
Param eters. ....................................................... 7
O objectives ................................................................ . . 8
C o n te n ts ................................................................................. 9

BACKGROUND ................................................. ....................... ..... 11

M ultilevel Structural O ptim ization ...................................... ...................... .. ......... .11
Two-level Multidisciplinary Optimization............................. .............. 16
Composite Wing Structural Design by Two-level Optimization Using Response
S u rfa c e s ........................ ......... ...... .... ... .... .... ..... ... ....... ................ . 2 2
Stacking Sequence Optimization by Genetic Algorithms ...........................................25

PERMUTATION GENETIC ALGORITHM FOR STACKING SEQUENCE
OPTIMIZATION OF COMPOSITE LAMINATES ................................................28

Introdu action .................................... .... ................................ ................ 2 8
Composite Laminate Analysis and Optimization...................................................30
Buckling Load Analysis of Composite Laminate ................................................30
Normal Buckling Load Analysis................................................. ............... 31
Shear B uckling L oad A analysis ...................................................................... .... 31
Com bined Buckling Load Analysis.................................... ....................... 32
Statement of Stacking Sequence Optimization .....................................................33
G genetic A lgorithm s ........................ .................... .. .. .... ........ ......... 34









Standard G enetic A lgorithm ............................................. ............................ 34
Perm utation Genetic Algorithm s ........................................ .......................... 37
G ene-R ank C rossover.................................................... ................................. 37
Comparison of Efficiency of Three GAs ................... .... ....................................... 41
Baldwinian Repair for Number of Plies and Continuity Constraint..............................48
Summary and Concluding Remarks.................................................................... 52

TWO-LEVEL COMPOSITE WING STRUCTURAL OPTIMIZATION USING
RESPON SE SURFACES ................................................................ ............... 54

Intro du ctio n ......... ...... ..... .. . ...... .. ............................................... 54
Tw o-level Optim ization A pproach......................................... .......................... 55
Two-Level Optimization Procedure .............. ................................. 55
Panel-Level Optimization and Response Surface........... ....................................58
W ing-Level Optimization ......... .............. ................. ............... 58
Exam ple Problem D escription............................. .. .........................................59
Response Surface Approximation of Optimal Buckling Loads ............................... 60
Response Surface Approxim ation............................................ ............ ............... 60
Normalized Buckling Load Response Surface.................................. ... ..................61
Results of Response Surfaces of Optimal Buckling Loads .................................62
Results of Composite W ing Box Structure Design................................... ............... 64
Six-variable D esign Problem .......................................................................65
18-variable D esign Problem ............................ ...................... .............. .... 66
54-variable Design Problem........................... ................... ..... ............ ... 70
C including R em arks ............ ... .................................................. ...........................72

COMPOSITE WING STRUCTURAL DESIGN OPTIMIZATION WITH CONTINUITY
CONSTRAINTS ............................... ... ...... ... .................. 73

Introduction.......................................................................................... .............. 73
Material and stacking sequence continuity measures for symmetric laminates .............75
M material Com position Continuity......................................... ......................... 75
Stacking sequence continuity ............................................................................. 77
Examples ............ ....... ...... ... ..... .... .... ..... ................. 78
Minimization of composite wing weight with continuity constraints ............................80
One-sided Continuity Constraints for Multiple Composite Panels .........................82
Weight-continuity tradeoffs ........... ..... ............... ............... 84
Stacking Sequence Design ......... .................................... ................ ............... 85
Concluding Remarks and Future Work.................... .................... ...............89

SINGLE-LEVEL COMPOSITE WING OPTIMIZATION BASED ON FLEXURAL
LAM INATION PARAM ETERS...................................................................... .... 91

Introduction............. .... .... .......... ............ ....... ......................... 91
Maximization of Buckling Loads of Composite Laminates With Given Number of
Plies of Each O rientation............ ........... .... .................... ................. ............... 93
Bending Lam nation Param eters ........................................ .......................... 93









Domain of Flexural Lamination Parameters for Specified Amount of Plies ............95
Maximization of Buckling Loads Using Continuous Optimization..........................98
Comparison of Laminate Designs ........................... ....................99
M inimization of W eight of Composite W ing............................................ .............. 103
Minimization of Weight of Composite Wing by Continuous Variable Algorithm
Based on Design Variables of Flexural Lamination Parameters ......................... 104
Comparison of Single-level and Two-level Approaches for Composite Wing
D e sig n ................................................................................................... 1 0 6
S ix -v ariab le C ase .............................................................. ........... ... ....... 10 6
18-variable C ase .................................................................. 108
54-v ariab le C ase ...................................................... 110
C including R em arks .................119............................................

CON CLU D IN G REM ARK S ................................................................. 120

TUNING GENETIC PARAM ETERS .................................................................. ..... 122

CONSTITUTIVE RELATIONS FOR ORTHOTROPIC LAMINA (HAFTKA AND
GURDAL 1993).............................................. ......... 124

DOMAIN OF VARIATION OF FLEXURAL LAMINATION PARAMETERS FOR
GIVEN AM OUNTS OF PLIES ............................................................. .. ............ 127

LIST OF REFEREN CES .......................................................................... ..... 131

BIOGRAPHICAL SKETCH........................................ 139
















LIST OF TABLES


Table Page

3-1: Coefficient pI for shear buckling load factor (Whitney 1985)................................... 32

3-2: Material properties of graphite-epoxy T300/5208.....................................................34

3-3: Weighted gene-rank values of child averaging gene-rank values of two parents........... 39

3-4: Optimum number of stacks of the three orientations for five load cases using
sequential quadratic programming (? and k, are the shear buckling load and
combined buckling load factors, respectively) .............. ............. ............... 42

3-5: Comparison of computational efficiency of the three GAs................. ............. .....42

3-6: T three tick test lam inates............................................................................... ..... .. 47

3-7: Comparison of computational efficiency of the three GAs for three thick laminates.....47

3-8: Optimum lay-up for the three thick laminates........................... ...............49

3-9: Computational cost of laminate repair and chromosome repair..............................52

4-1: Allowable strains and safety factor ................ .. ......... ...................... 59

4-2: Statistics of three optimal buckling load response surfaces.................................63

4-3: GENESIS and rounded optimal design with six variables ............ .......................65

4-4: Rounded and adjusted upper panel design with six variables................................. 65

4-5: GENESIS and rounded optimal design with 18 variables............................ .........67

4-6: Rounded and adjusted lower skin panels with 18 variables ................................67

4-7: Rounded and adjusted upper skin panels with 18 variables................................68

4-8: GENESIS and rounded optimal design for 54 variables ...... ..................................69

4-9: Comparison of rounded and adjusted designs for lower skin panels, 54 variables.......70









4-10: Comparison of rounded and adjusted designs for upper skin panels, 54 variables .....71

4-11: Stacking sequences of wing-level panels for the adjusted design ............................ 71

5-1: Definition of layer type and its layer code for examples.............................................78

5-2: Composition and stacking sequence continuity indices for two laminate examples ...... 80

5-3: Minimum weight of composite wing versus required average (over all panels)
c o n tin u ity ...................................................................... ............... 8 3

5-4: GENESIS, rounded and adjusted optima for comp position continuity requirement of
5 0 % ......................................................... ......... ....................................... .... 8 6

5-5: Stacking sequence and average continuity of nine upper skin panels for composition
continuity requirement X=50% ........................ ..................87

5-6: GENESIS, rounded and adjusted optima for composition continuity requirement
X=85% ............. .......... .......... ......... ............................... 88

5-7: Stacking sequence and average continuity of nine upper skin panels for composition
continuity requirement X=85% ........................ ..................89

6-1: Definition of medium and thick laminates and applied loads .............. ................. 100

6-2: Comparison of optimal buckling loads by continuous optimization based on W,*, W3
and by GA for square laminates defined in Table 6-1 ................................ 101

6-3: Comparison of maximized buckling loads for various aspect ratios for no=5, n45=5,
n90=5; loads are N,=2000 lb/in, Ny=500 lb/in, Ny=1001b/in, a=24 in............. 102

6-4: Comparison of maximized buckling loads of continuous variable approach based on
W{, W3 with GA for 4-stack laminates........................................................... 104

6-5: Comparison of wing design results between two-level RS approach, and single-level
method with lamination parameters. Upper skin is one laminate and lower skin
is one lam inate ............................................................................ ..... 107

6-6: Adjusted designs for two-level RS approach, and single-level method with lamination
parameters. Upper skin is one laminate and lower skin is one laminate........... 108

6-7: Comparison of the results of GENESIS and rounded optimal design between the two-
level RS approach and the single-level method with lamination parameters. The
upper skin has three laminates and the lower skin has three laminates ............ 110

6-8: Comparison of results of rounded and adjusted lower-skin panels between two-level
RS approach and single-level method with lamination parameters. The lower
skin has three lam inmates ......................... ........................ ... .......... 111









6-9: Comparison of results of rounded and adjusted upper panels between two-level RS
and single-level method with lamination parameters. The upper skin has three
la m in ate s..................................................... ............... 1 12

6-10: Summary of rounded and adjusted optima of the two approaches. The upper skin has
three laminates and the lower skin has three laminates.............................. 113

6-11: Comparison of the results of GENESIS and rounded optimal design between two-
level RS approach and the single-level method with lamination parameters. The
upper skin has nine panels and the lower skin has nine panels .................... 113

6-12: Comparison of rounded and adjusted designs for lower-skin panels between the two-
level RS approach and the single-level method with lamination parameters. The
lower skin has nine panels.................................................. ....... ................ 114

6-13: Comparison of the results of rounded and adjusted designs for nine upper-skin panels
between the two-level RS approach and the single-level method with
lamination parameters. The upper skin has nine panels ................................... 115

6-14: Summary of rounded and adjusted optima of the two approaches ...........................16

6-15: Comparison of stacking sequences of nine upper-skin panels for the rounded design
for the two-level RS approach and the single-level method with lamination
parameters. The upper skin has nine panels........... ... ................................. .. 117

6-16: Comparison of stacking sequences of the nine upper-skin panels for the adjusted
design between the two-level RS approach and the single-level method with
lamination parameters. The upper skin has nine panels.............. ........... 118
















LIST OF FIGURES


Figure Page

2-1: M ultilevel approach ............................... ..... ....... .. ........... .......................... ... 14

2-2: Tw o-level M D O problem ............................................................ ........................... 17

3-1: Composite laminate plate geometry and loads ....................................................... 30

3-2: Reliability versus number of generations for five loading cases: Case (1) and Case
(2 ) ....................................................................................... 4 4

3-2: Reliability versus number of generations for five loading cases: Case (3) and Case
(4 ) ....................................................................................... 4 5

3-2: Reliability versus number of generations for five loading cases: Case (5)................... 46

3-3: Reliability versus number of generations for the three thick laminates: Case (6).......... 46

3-3: Reliability versus number of generations for the three thick laminates: Case (7)............ 47

3-3: Reliability versus number of generations for the three thick laminates: Case (8).......... 48

4-1: Response surface interface of two-level optimization................... .............................. 56

4-2: Flowchart of two-level optimization procedure .................................................... 56

4-3: W ing box structure ..................................................... ........... ............... 60

4-3: History of the objective function and maximum violation of normalized constraints for
six-variable case. .................... ................. .................. .............. 63

4-5: History of the objective function and maximum violation of normalized constraints for
18-variable case. ...................................................... .............. 66

4-6: History of the objective function and maximum violation of normalized constraints for
54-variable case. ...................................................... .............. 68

5-1: Com m on layers of two lam inmates ................................................................. .... 74









5-2: Count of the number of continuous layers of two laminates................... ........ ...... 76

5-3: Details of stacking sequence continuity: (A) Case 1; (B) Case 2............................. 79

5-4: Low er skin panels .................. .................. ................. ........... ....... ....... 82

5-5: Average abscissaa) and required (numbers on graph) material continuity vs. minimum
num ber of stacks. ................................ ................................ 84

6-1: Ply geom etry in a lam inate ........................... .. ................ ................................. ....... ... 93

6-2: Bending lamination parameter domain............................... ................................. 95

6-3: Laminates with all plies of the same orientation stacked together............................. 96

6-4: Six laminates corresponding to the six vertices of a hexagonal domain...................... 96

6-5: Hexagonal domain of variation of flexural lamination parameters when the number of
plies of each orientation is specified ............................. ............................. ... 99

A-1: Tune GA operator parameters: (A) shows the effect of population size...................... 122

A-1: Tune GA operator parameters: (B) displays effect of probability of crossover........ 123

A-1: Tune GA operator parameters: (C) shows effect of probability of mutation. ............... 123

B-1: An Orthotropic lamina with off-axis principal material directions ........................... 124

C-l: Ply geometry of laminate [(Oi)ni/(ej)nj/(Ok)nk]s ...................... ................. ................. 127















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

TWO-LEVEL OPTIMIZATION OF COMPOSITE WING STRUCTURES BASED ON
PANEL GENETIC OPTIMIZATION


By

Boyang Liu

May 20001

Chairman: Professor Raphael T. Haftka
Major Department: Aerospace Engineering, Mechanics, and Engineering Science

The design of complex composite structures used in aerospace or automotive

vehicles presents a major challenge in terms of computational cost. Discrete choices for

ply thicknesses and ply angles leads to a combinatorial optimization problem that is too

expensive to solve with presently available computational resources. We developed the

following methodology for handling this problem for wing structural design: we used a

two-level optimization approach with response-surface approximations to optimize panel

failure loads for the upper-level wing optimization. We tailored efficient permutation

genetic algorithms to the panel stacking sequence design on the lower level. We also

developed approach for improving continuity of ply stacking sequences among adjacent

panels.

The decomposition approach led to a lower-level optimization of stacking

sequence with a given number of plies in each orientation. An efficient permutation genetic









algorithm (GA) was developed for handling this problem. We demonstrated through

examples that the permutation GAs are more efficient for stacking sequence optimization

than a standard GA. Repair strategies for standard GA and the permutation GAs for dealing

with constraints were also developed. The repair strategies can significantly reduce

computation costs for both standard GA and permutation GA.

A two-level optimization procedure for composite wing design subject to strength

and buckling constraints is presented. At wing-level design, continuous optimization of ply

thicknesses with orientations of 00, 900, and +450 is performed to minimize weight. At the

panel level, the number of plies of each orientation (rounded to integers) and inplane loads

are specified, and a permutation genetic algorithm is used to optimize the stacking

sequence. The process begins with many panel genetic optimizations for a range of loads

and numbers of plies of each orientation. Next, a cubic polynomial response surface is

fitted to the optimum buckling load. The resulting response surface is used for wing-level

optimization.

In general, complex composite structures consist of several laminates. A common

problem in the design of such structures is that some plies in the adjacent laminates

terminate in the boundary between the laminates. These discontinuities may cause stress

concentrations and may increase manufacturing difficulty and cost. We developed measures

of continuity of two adjacent laminates. We studied tradeoffs between weight and

continuity through a simple composite wing design.

Finally, we compared the two-level optimization to a single-level optimization

based on flexural lamination parameters. The single-level optimization is efficient and

feasible for a wing consisting ofunstiffened panels.














CHAPTER 1
INTRODUCTION

Introduction

Because of higher stiffness-to-weight or strength-to-weight ratios compared to

isotropic materials, composite laminates are becoming more popular. Composite structures

typically consist of laminates stacked from layers with different fiber orientation angles.

The layer thickness is normally fixed, and fiber orientation angles are often limited to a

discrete set such as 0, +45, and 900. This leads to an expensive combinatorial

optimization for designing composite structures. In addition, design of complex and large

aircraft structures, like a wing, requires sizing of local details of stiffened panels in the

wing (rib and skin panels). Details include stiffener shape and dimensions, stiffener

spacing, and choice of laminates for each part. The design of each panel requires a

substantial number of variables to describe geometry, ply composition, and stacking

sequence. Designing all the panels simultaneously constitutes a complex optimization

problem that requires detailed structural modeling of the entire wing, and appears to be

beyond present computational capabilities.

One approach to reducing the complexity of the optimization problem is to

decompose it to smaller optimization problems in a process of multilevel optimization.

Early studies developed multilevel formulations to wing structural design (Giles 1971;

Schmit & Ramannathan 1978, Schmit & Mehrinfar 1982; Sobieszczanski-Sobieski &

Leondorf 1972, Sobieszczanski-Sobieski et al. 1985). Multilevel structural optimization in









general consists of lower-level optimization for substructures and coordination

optimization, which exchanges information among the lower-level optimizations.

Similar decomposition techniques are naturally applied in multidisciplinary design,

which became a popular research area in the last twenty years. Multidisciplinary

optimization (MDO) problems involve several disciplines, and often have no clear

hierarchy between disciplines. Two popular approaches for MDO problems are concurrent

subspace optimization (CSSO) developed by Sobieszczanski-Sobieski (1988) and

collaborative optimization (CO) developed by Kroo (1995).

In Schmit's approach for multilevel structural optimization and in CSSO and CO

for MDO, equality constraints are used explicitly or implicitly in the coordination process.

The use of equality constraints can lead to numerical difficulties (Thareja & Haftka, 1986)

and even invalidate some formulations, as recently noted by Alexandrov and Lewis (2000)

for the CO approach. Even without the difficulties associated with equality constraints,

multilevel optimization can introduce numerical difficulties because lower-level optima

are non-smooth and noisy functions of upper-level parameters.

Response surface approaches fit data in the design domain with simple functions,

usually polynomials. Response surfaces smooth out noise, are simple, and easily integrate

different application packages. The technique therefore seems a promising approach for

easing the numerical difficulties in multilevel structural design or MDO problems.

Complex composite structures, such as aircraft wing or fuselage structures or car

chassis structures, commonly consist of multiple laminates. Because of different fiber

orientation angles of layers, or different layer thickness, or different type of layer

materials, discontinuities between adjacent panels or other components are produced. The









discontinuities can cause stress concentration, reduce strength and increase manufacturing

difficulty. Even though in many cases discontinuity among adjacent panels is inevitable, we

may be able to reduce discontinuity if discontinuity measures are included in the design

process.


Two-level Optimization for Composite Wing Structures

Because of the high computational cost of single-level optimization for composite

wing box design, current practice is to design composite wing structures at two levels. At

the wing level, individual panels are modeled without much detail, and few design

variables are assigned to each. Global constraints such as aeroelastic stability are enforced

along with simple stress or strain limits using finite element based structural optimization

programs such as NASTRAN (CSA/NASTRAN 1995), GENESIS (Vanderplaats 1997) or

ADOP (Sharma et al. 1993). The internal load distribution obtained by wing optimization

is then used as input to panel-design optimization programs, such as PANDA2 (Bushnell

1987) or PASCO (Stroud & Anderson 1981). These programs obtain detailed geometry

and ply compositions for individual panels that satisfy buckling and various local strength

failure constraints.

For complex configurations, coordination of the local-level (panel) and upper-level

(wing) optimization problems is currently handled largely by ad-hoc methods that are

cumbersome and sub optimal. As noted in the previous section, two-level optimization can

be ill-conditioned because the lower-level (panel) optima are not smooth functions of

upper-level (wing) design variables.

Response surface (RS) techniques offer an attractive way of overcoming both

implementation and smoothness difficulties. The RS approach to a two-level optimization









is to perform a large number of lower-level optimizations for different values of the global

design variables and loads. Then the results of the lower-level optimizations are fitted with

response surfaces, typically low-order polynomials. Finally, upper-level optimization is

performed with RS substituting for lower-level optimization. This approach eases

implementation problems associated with software integration, as lower-level optimization

program can be run independently of the upper-level analysis and optimization. In addition,

the RS smoothes out the lower-level optima.

Ragon et al. (1997) demonstrated this approach by fitting a response surface to the

optimum weights of stiffened panels designed using PASCO (Stroud et al. 1981) as a

function of panel loading and inplane-stiffness constraints specified by the wing-level

optimizer. The resulting response surface was used by the wing optimizer, ADOP (Sharma

et al. 1993). A similar approach was also used by Balabanov et al. (1999) for a two-level

aerodynamic-structural optimization. There the wing structural weight was fitted as a

function of aerodynamic shape parameters. The present work extends this technique for the

case in which the panel-level design involves discrete or combinatorial optimization, such

as stacking sequence design of composite laminates. In addition, Ragon et al. (1997) had

trouble with their approach because for some combinations of stiffness constraints, there

were no feasible solutions for the panel-optimization problem. Here we overcome this

problem by maximizing the load-carrying capacity of the panel for a fixed-weight budget.

We demonstrate the methodology for design of a simple wing structure, where a

permutation genetic algorithm performs the panel design.









Genetic Algorithms for Panel Optimization

Stacking sequence design of composite laminates is a local problem that is strongly

coupled to the overall design of complex composite structures. In complex composite

structures, the overall design imposes constraints on individual panel design. The

optimization of the overall structure often specifies number of 00, 45, and 900 plies, and

inplane loads on each panel. Therefore, the stacking-sequence design is often limited to

permutations of a given set of plies. For given inplane loads, and a given number of plies

of 00, 450 and 900 directions, stacking-sequence optimization is a combinatorial design

problem.

Genetic algorithms (GAs) are a popular technique for solving integer and

combinatorial optimization problems. In structural optimizations, Genetic algorithms have

been applied to integer problems. (Furuya & Haftka 93, Hajela 90, Hajela 1991, Hajela &

Lin 1992, Rao 1990, Watabe & Okino 1993). Of course, GAs have been used in composite

laminate design (Le Riche & Haftka 1993, 1995; Nagendra et al. 1993).

However, Genetic algorithms are often too expensive when analysis of one

candidate solution is computationally expensive, and they do not deal efficiently with

constraints. When applied to panel stacking sequence design, constraints on the number of

plies come from the wing design. It is possible to use a conventional genetic algorithm.

However, available permutation GAs developed mostly for scheduling problems

(Michalewicz 1992) are more efficient in the search for an optimal permutation because

they reduce the dimensionality of the design space.

Permutation GAs were mostly developed for the traveling salesman problem,

which seeks to minimize travel costs for a given list of towns and is insensitive to where









the sequence starts, so that cyclical permutations do not matter. In stacking sequence

design, a cyclical permutation moves the outermost ply into the innermost position, and thus

greatly influences the bending properties of the laminate.

Therefore, this work seeks to develop permutation GAs more tailored for the

stacking sequence design problem. GAs usually handle violations of constraints by penalty

functions which are added to the objective function. However, penalty-function approaches

often slow down the convergence speed of GAs. In this study, we tested the use of repair

strategies that deal with violations of constraints by repairing the laminate to satisfy the

constraints (Todoroki & Haftka 1998).


Material Composition Continuity Constraints Between Adjacent Panels

In the design of complex structures, it is customary to divide the structure into

panels or regions that may be designed independently or semi-independently (Liu et al.

2000, Ragon et al. 1997, Schmit & Mehrinfar 1982). This is done not only for

computational convenience, but because the loads vary from one part of the structure to

another, so that structural efficiency dictates variation in structural properties. With this

design approach, adjacent laminates may have different total thicknesses, and some plies

may terminate at the laminate boundaries. These discontinuities can cause stress

concentrations and increase manufacturing difficulty and cost.

While some discontinuities are inevitable if structural efficiency is to be

maintained, it is desirable to minimize these discontinuities. It is therefore desirable to add

continuity constraints for adjacent laminates in the design process, or to include a measure

of continuity in evaluating competing designs. It may be expected that optimization

including continuity constraints may lead to designs with fewer and smaller discontinuities.









Kristinsdottir et al. (2001) recently developed a concept named blending rule to

measure ply continuity in adjacent composite panels. The basic idea is to follow plies from

the region they originate until they terminate. Each ply stems from its key region (the most

heavily loaded region) and may cover any number of regions as long as they are adjacent to

one another. A ply is allowed to be dropped (discontinued) and is allowed to build up

plies at load concentrations. The fiber angle of a ply is held fixed for the entire coverage of

that ply. Dropping plies in a consistent manner means that once a ply is dropped, the ply is

not added back into the panel.

Comparing results with blending measure and results without blending measure,

design with blending is heavier than design without blending but design with blending is

easier to manufacture.

The blending approach of Kristinsdottir et al. (2001) is focused on a design

approach that limits discontinuities in a prescribed manner. Here the emphasis instead is

on measuring the degree of discontinuity so that it can be incorporated as a measure of

design performance. One objective of the present work is to develop such measures. Two

continuity measures are defined in terms of material composition and stacking sequence.

The continuity measures are applied to a simple wing design problem, and trade-off studies

between weight and continuity are performed


Single-Level Optimization of Composite Wing Based on Flexural Lamination Parameters.

While the two-level optimization approach makes intuitive sense, we do not have

any proof that it will converge to optimal or near optimal designs. Therefore, it is

desirable to find an alternative optimization that will check the optimality. We develop a

continuous optimization approach based on flexural lamination parameters for this purpose.









This approach, which disregards the discreteness of the problem and some of the

constraints, provides a lower bound for the optimal weight.

Lamination parameters, consisting of inplane and flexural lamination parameters

(Tsai et al., 1980), provide a compact representation of the stiffness properties of

composite laminates. They allow efficient approximate optimization of laminates for

desired stiffness properties. Miki (1986) developed a graphical procedure for the design

optimization. Miki and Sugiyama (1991, 1993), Fukunaga and Sekine (1992, 1994)

graphically solved stacking sequence design problems for stiffness and strength

maximization of symmetric laminates using lamination parameters. The simple graphical

approach also allows us to see that for many problems the optimal design lies on the

boundary of the lamination parameter space, corresponding to angle ply designs

(Grenestedt & Gudmandson 1993). Nagendra et al. (1996), Todoroki & Haftka (1998),

Yamazaki (1996) demonstrated the use of lamination parameters in applications of

composite laminate design.

The primary objective for introducing lamination parameters is to provide a single-

level continuous optimization to verify optimality of the two-level optimization approach.

However, we can also apply ply stacking sequence continuity constraints for multiple

adjacent panels based on these lamination parameters.


Objectives

This dissertation develops an approach for design of complex composite structures

such as a composite wing based on available technologies. First, efficient and robust

permutation GAs characterized random search nature were developed for stacking

sequence optimization for composite laminates to maximize buckling failure load. Next,









two-level optimization approach was used for overall wing design, with response surfaces

generated from panel optima used as an interface to couple with wing-level design. Third,

we developed continuity constraints among adjacent panels to cope with discontinuity,

stress concentration and high manufacturing cost. Finally, we developed continuous

variable optimization algorithms based on flexural lamination parameters for panel

optimization and wing optimization. We constructed response surfaces of continuity

constraints of ply stacking sequence based on flexural parameters and checked the

optimization results of the two-level approach. The four objectives of the dissertation are

as follows:

1) Develop efficient and robust permutation GAs for stacking sequence optimization
of composite laminates.
2) Develop a two-level optimization approach for composite wing design based on
panel genetic optima.
3) Develop two types of continuity constraints between two adjacent panels and study
tradeoff between weight and continuity.
4) Develop a continuous-optimization procedure based on flexural lamination
parameters and use them to test the optimality of the two-level optimization results.

Contents

Chapter 2 provides a literature review of two-level optimization and GAs applied

in structural design. Chapter 3 discusses the development of an efficient permutation

genetic algorithm for stacking sequence design of composite laminates. Chapter 4

describes an algorithm for optimizing an entire composite wing structure by integrating

wing optimization with panel optimization via response surface techniques. Chapter 5

describes how to design all panels' stacking sequence together by including material

composition continuity constraints among adjacent panels and study tradeoff between

weight and continuity. Chapter 6 describes a continuous variable optimization based on






10


flexural lamination parameters for panel and wing optimization. Chapter 7 provides

concluding remarks and recommendations for future work.














CHAPTER 2
BACKGROUND

Multilevel Structural Optimization

Exorbitant computer resources are required for the design of realistic structures

carrying a large number of loading cases and having many components that need design

variables to describe detailed geometry, direct one-level optimization. The computational

resources required for the solution of an optimization problem typically increase with

dimensionality of the problem at a rate that is more than linear. That is, if we double the

number of design variables in a problem, the cost will typically more than double. One

obvious solution is to break up large optimization problems into smaller subproblems and

a coordination problem to preserve the couplings among these subproblems.

One important benefit of this approach is that it makes the big problem more

tractable and allows simultaneous work on different parts of the problem. The latter

advantage also matches the trend of computer technology development, where distributed

and parallel processing are becoming more popular. Moreover, to break a big problem into

several small problems is natural in engineering optimization because engineers tend to

work in teams concentrating on parts of a project in order to develop a broad work front in

order to shorten development time.

Early approaches to applying multilevel optimization to minimum weight design of

complex structures such as wing and fuselage were based on the fully stressed design

(FSD) method. The FSD method is an optimality criteria method. The FSD optimality

criterion is that for optimum design, each member of the structure that is not at its minimum









gage is fully stressed by at least one of the design-load conditions. The FSD technique

applies to structures that are subject to only stress and minimum gage constraints. The FSD

approach implies that we should remove material from members that are not fully stressed

unless minimum gage constraints are prevented. This method has been used extensively for

aerospace structures (Lansing et al. 1971, Giles 1971). In applying the FSD method, the

overall structure (such as fuselage or wing) is represented in a lumped model in which

stringers, rings, and skins are represented only in a coarse manner. The detailed design of

the lumped-model components is then carried out by mathematical programming with sizing

design variables (e.g., skin thickness). This is essential by a two-level approach.

Sobieszczanski-Sobieski and Leondorf (1972) developed a mixed-optimization method

combining FSD for structure system and mathematical programming for components for

preliminary design of fuselages. Giles (1971) developed an automated preliminary

program called Design of Aircraft Wing Structures (DAWNS) for wing design. In

DAWNS, the external shape, aerodynamics loads, structural geometry, internal loads, and

fuel mass were included to obtain optimal wing by a FSD method.

Schmit and Ramannathan (1978) found two main shortcomings to the above

approaches: the use of weight as the objective function at the component level and the use

of fully stressed type resizing algorithms at the system level. As Sobieszczanski-Sobieski

(1972) and Schmit et al. (1978, 1982) noted, the minimum weight structural system is not

necessarily made up from a collection of minimum weight components. To cope with this

problem, Schmit et al. (1978, 1982) later developed a multilevel approach wherein

system-level and component-level design phases are characterized as follows: 1) at the

system level, minimize the total structural weight subject to the system level constraints









such as displacements, system buckling, and strength, and 2) for each component, minimize

the change in equivalent system stiffness subject to local strength and buckling constraints.

We denote the system design variables as S and denote by L the concatenation of

l ,j=1.... M, the local design variable vector for all M components. A standard

formulation for single-level optimization can be stated as follows:

Minimize W(S_)
such that Gq(S,) >O; qQ (
(2-1)
and
g,(lj,S)>0, ljL; jeM

Where Wis the total system weight and objective function. Here, Gq are system

constraints which strongly depend on system design variables S, like displacement, stress,

system buckling constraints, Q represents the set of system level constraints. Component

constraints are gij that primarily depend on the detailed component design variable vector

/ Now the design problem (2-1) is decomposed into a multilevel optimization problem

as follows:

At system level,

Minimize W(S)
(2-2)
such that Gq(S, l ) >0; q EQ


where 1 is the detailed design variable vector, which does not change during a system

level design modification stage.

For thejth component level, the objective function is to minimize the discrepancy

between the component stiffness parameters Krj, j=1,...,R which may affect system

response. That is, the component level optimization problem is:









R
Minimize c, = Y (K (S ) K, (1 ))2
r=l
such that g; (l, S_( ) > 0; 1 e L


(2-3)


where S' is the value of the system design variables corresponding to thejth component at

the end of the foregoing system stage and is held invariant during the component design

modification stage. Equation (2-3) essentially was a quadratic penalty function to enforce

the equality constraints Kj (S) = K, (1i)


Figure 2-1: Multilevel approach


The basic idea of minimizing the change of stiffness is to reduce load redistribution

at the system level due to component level synthesis. Schmit et al. (Schmit & Ramannathan

1978; Schmit & Mehrinfar 1982) coworkers successfully applied this approach into

minimum weight design of truss and wing box structures with sandwich and hat-stiffened

fiber-component panels.









Sobieszczanski-Sobieski (1985) developed a more general multilevel approach.

Compared with Schmit's approach, Sobieszczanski-Sobieski used a cumulative constraint

concept which is a number used to measure the degree of constraint violation of all

constraints.

In Schmit's approach, the discrepancy between the system level and the component

level was measured by c, which is a quadratic function of the discrepancy. Sobieszczanski-

Sobieski (1985) used a cumulative constraint to replace both the equality constraints

between component and system level as well as the local constraints. The cumulative

constraint is the Kreisselmeier-Steinhauser (KS) function,


KS(g) =iln Iexp(pgj) (2-4)
P -1

where gj is a local constraint, p is user-defined constant, and m is the number of local

constraints. Compared to the quadratic penalty function, the KS function has the advantage

that it does not have zero derivative at the optimum (which, as we see later, can cause

numerical difficulties). This was a conceptual improvement, since in Schmit's approach

the system-level design problem does not receive any input from the local problem during

its optimization.

In the system level, cumulative constraint of the components was added to the

system-level constraints to assure satisfaction of all of the local constraints.

In both Schmit's and Sobieszczanski-Sobieski's approaches, equality constraints

are directly or indirectly to assure consistency between the system level and the component

level. However, Thareja and Haftka (1986) demonstrated via a portal frame example that

numerical difficulties are introduced by using equality constraints. First, the problem was

solved using a single-level formulation without equality constraints. Second, equality









constraints were introduced to create a hierarchical structure, but the problem was still

solved as a single-level problem. Finally, a two-level approach that took advantage of the

hierarchical structure was used. It was found that the two-level formulation solution was

sensitive to the optimization parameters but the one-level formulation solution was not.

Numerical difficulties were associated with the additional global variables and the

presence of equality constraints. These problems are further discussed in the context of a

two-level optimization approach to multidisciplinary design in the next section.


Two-level Multidisciplinary Optimization

Multidisciplinary design optimization (MDO) can be described as a methodology

for the design of systems where the interaction between several disciplines must be

considered, and where the designer is free to significantly affect system performance in

more than one discipline. For example, the design of aircraft involves significant

interaction among the disciplines of aerodynamics, structural analysis, propulsion, and

control.

Typically when beginning a multidisciplinary project, a project leader must

decompose the original problem and distribute the relevant parts among the existing

organizational groups. The multilevel optimization approach takes the same route.

Collaborative optimization (CO) is a popular example of this approach (Kroo 1995).

Collaborative optimization is based on the decomposition of the system problem along the

lines of the constituent disciplines.

Collaborative optimization seeks to formulate and solve the MDO problem in a

way that preserves the autonomy of disciplinary calculations by eliminating those local

variables to individual disciplinary subsystems from the system level using equality









constraints similar to those used in the multilevel optimization discussed previously. The

values of these constraints are obtained by solving distributed low-level optimization

subproblems whose objectives minimize the interdisciplinary inconsistency, subject to

satisfying the disciplinary design constraints.

Collaborative optimization was first proposed by Kroo (1995), and improved by

Kroo et al. (1995, 1996). The algorithms have been applied by researchers to a number of

different design problems since then. Braun et al. (1996a, 1996b) applied this approach to

the design of launch vehicles, and Sobieski and Kroo (1996) applied it to aircraft

configuration design.

An example of a two-discipline design (Alexandrov & Lewis 2000) is used to

describe CO formulation as follows:

The mathematical statement of the standard MDO formulation is


>, 11 Disciplinary al
Analysis 1 ul1
Analysis Minimize f(s, a,, a2)
subject to g, (s, a1) 0
S,12 Disciplinary g2 (s, 12, a2) <0
Analysis 2 2

Figure 2-2: Two-level MDO problem



where s is a system variable, 11 and 12 are local variables that belong to discipline 1 and

discipline 2 respectively; and g, and g2 are discipline scope constraints; al anda2 are

computed via the disciplinary analysis.

a1 = A (a,1,, 12) (2-5)

Similarly,









a2 = A2(a(2,12,t ) (2-6)

Reformulation in terms of CO can be expressed as follows:

System-level optimization which coordinates two disciplinary design as

Minimize f(s,t,,t,)
S l t2 (2-7)
subject to C(s, tl,t) = 0

where Cis called Ninterdisciplinary consistency constraints. C = {cl ,c2} is shown

below; tl and t2 are system-level targets of al and a2.

The system-level problem issues design targets (s, ti, t2) to the constituent

disciplines. In the lower-level problem, the disciplines must design to match these targets

as follow:

1 I 1
minimize c, + a1(a ,l1 ) t
Discipline 1: o,, 2 (2-8)
subject to g, (o-i, a, (o-, 1,, t, )) > 0

where o" is the target of s at discipline 1.


minimize c2 o s + a, (o2,,12,t)- t2
Discipline 2: 2,12 2 (2-9)
subject to g2(a2, l2, a2 (ao2,12,t )) > 0

where o2 is the target of s at discipline 2.

Another two-level MDO approach, Current Subspace Optimization (CSSO)

introduced by Sobieszcazanski-Sobieski (1988), allocates the design variables uniquely to

subspaces that correspond to engineering disciplines or subsystems. Each subspace

performs a separate optimization operating on its own unique subset of design variables.

The coordination problem is solved by using the Global Sensitivity Equation (GSE)









(Sobieszcazanski-Sobieski 1990) and optimum sensitivity derivatives with respect to

parameters.

The CSSO permits the decoupling of a large engineering system into smaller

subsystem modules in order to achieve concurrent optimization in each of these subspaces.

The CSSO eliminates the need for a full analysis in each subspace, thereby enabling their

simultaneous treatment. Subspaces are created on the basis of a non-hierarchic

decomposition (by discipline or design goals) and at any given time, each design variable

is considered active in one of the subsystems.

The system optimization procedure begins with a system-level analysis. The actual

assignment of design variables to a particular subspace is made on the basis of the ability

of variable to influence the goals within that subspace based on the sensitivity of the

objective function and cumulative constraints in each subspace. After system analysis, we

perform a system sensitivity analysis to compute the system-sensitivity derivatives. Each

derivative measures the influence of a particular design variable Ij on a particular behavior

variable ai.

After allocating design variables, temporarily decoupled optimization is performed

concurrently in each subspace. The goal of these subspace optimizations is to reduce the

violation of the cumulative constraints with the least increase of the system objective

function or greatest decrease of the cumulative constraints which are already satisfied.

After finishing all of the subspace optimizations, an optimum-sensitivity analysis is

performed in order to determine the sensitivity of objective function to cross influence

coefficients which measure how subspaces affect one another. The derivative information

obtained in optimum-sensitivity analysis will be now used in coordination optimization









problem in which the system function is minimized with respect to the coefficients.

Completion of coordination optimization yields new coefficients for use in the next

subspace optimization.

The advantages of the CSSO are modularity of the subsystems and the ability of to

incorporate human intervention and decision making. However, application of this

algorithm without move limits may lead to convergence problems (Shankar et al. 1993).

Bloebaum et al. (1992) reformulated the approach by using an expert system made up of

heuristic rules to adjust the move limits and other parameters that control the process. The

algorithm was extended to problems with discrete design variables, and there are several

variants of CSSO developed (see Sobieszcazanski-Sobieski & Haftka, 1997 for additional

references).

After numerical difficulties in multilevel structural optimization were found out by

Thareja and Haftka (1986), similar computational difficulties were also found in numerical

tests of CO (Alexandrov & Kodiyalamm 1998, Kodiyalam 1998). Alexandrov and Lewis

(2000) showed that the treatment of compatibility via quadratic penalties led to system-

level optimization problems that necessarily failed to satisfy the standard Karush-Kuhn-

Tucker optimality conditions, either because multipliers did not exist, or because the

constraint Jacobian was discontinuous at solutions. In addition, collaborative optimization

formulations lead to system-level problems that are nonlinear even when the original

problem is linear. These features make it difficult for conventional optimization algorithms

to solve the CO system-level problem reliably or efficiently.

The CSSO also used equality constraints, and indeed some numerical difficulties

were found in applying it to simple problems (Shankar et al. 1993).









An older approach, which avoids the numerical difficulties associated with

equality constraints is to find a simple approximation for lower-level optima as function of

upper-level design variables. This approach has been particularly successful when the

lower level involves structural design. Weight equations that predict the optimal structural

weight of wing and fuselage structures have been popular.

Shanley (1960) proposed to use weight prediction method for minimization of wing

weight based on elementary-strength or element-stiffness considerations, augmented by

experimental results and statistical data. A weight equation, which relies on a

computational procedure where the amount of material are required to resist bending and

torsion at a number of selected spanwise locations, is determined numerically. The primary

wing box weight is thus found from integration along the span. The weight equation

produced this way can then be used by the aircraft designer, obviously the need for

including the structural design in the aircraft configuration design.

Torenbeek (1992) presented a method for generating weight equations for wing

structures. The method makes use of elementary stress analysis combined with historical

data. The wing-group weight is expressed as the sum of the primary weight (top and bottom

covers, spars, ribs, and attachments) and the secondary weight comprised of the weight of

components in front of the front spar, components behind the rear spar, plus any

miscellaneous weight.

Initially, wing weight equations were based on historical data and simple beam

models. Later, large number of structural optimizations were performed and used to

generate these equations (McCullers 1984). These equations then allow upper-level









configuration design because they predict the optimum structural weight as function of the

aircraft configuration variables (McCullers 1997).

Instead of using weight equations that are applicable to all transports, Kaufman et

al. (1996) developed weight equations customized to a high speed civil transport (HSCT)

design by fitting large number of structural optimizations by a polynomial response

surfaces. A similar approach was also used by Balabanov et al. (1999) for a two-level

aerodynamic and structural optimization of the HSCT.

The advantages of using response surfaces to fit a large number of disciplinary

optima can be summarized as follows: 1) It allows the disciplinary optimizations to be

performed by specialists. 2) It provides a simple method for facilitating communication

between the specialists on the design team. 3) RS techniques smooth the discontinuities and

noise associated with two-level formulations. 4) Performing large number of optimizations

allows easy use of parallel computers (Burgee et al. 1996; Eldred & Schimel 1999,

Krasteva et al. 1999) and error deletion (Papila & Haftka, 2000; Kim et al. 2000).

However, response surfaces become increasingly expensive or inaccurate with increasing

dimensionality of the design space. This difficulty typically limits response surfaces to

problems under 30 design variables.


Composite Wing Structural Design by Two-level Optimization Using Response Surfaces

Thin-walled box beams are extremely efficient structures. The thin-walled box

beam in some forms has become a fundamental structural element in the construction of

aircraft, ships, offshore platforms, bridges and the cores of tall buildings. The advantage of

the hollow section is that the material is efficiently used both in flexure and torsion. The

wing-box structure belongs to a generic and simplified prototype of this kind of structures.









Due to higher ratios of stiffness-to-weight or strength-to-weight, complex

composite structures like wing box and fuselages consisting of composite laminates, have

attracted much industry attention. In general, composite structures are typically made of

laminates where distinct layers are stacked. Each layer is composed of fibers of a given

orientation embedded in a matrix of different materials. Fibrous composites are usually

manufactured in the form of layers of fixed thickness, and fiber orientation angles are

limited to a small set. Designing composite structures involves finding the number of layers

and the fiber orientation of each layer inside each laminate that maximize the performance

of the structures under requirements such as failure, geometry and cost. Compared with

wing box with isotropic material, design of composite wing box structures is more

complicated and more computationally expensive. Consequently, designing a composite

wing or fuselage structure at one level is not feasible computationally.

A natural decomposition of wing or fuselage structures is to deal with them as an

assembly of stiffened panels. Consequently, early approach to design wing box structures

was based on weight equations developed by Shanley (1960) and Gerard (1960) During

50's, a common form of multi-cell construction with relative thick covers supported by a

series of longitudinal webs was used. Efficiently using these structures with the

rearrangement and reduction of structure materials of the cover plates was the objective.

This is prototype of panel-level design based on loads from wing-level design. Gerard

(1960) used orthotropic plate theory to analyze idealized long-plate structures

-longitudinal, transverse, and waffle-grid stiffening system for flat plates, which keep the

significant details of the actual structure and yet are sufficiently simplified to permit broad

conclusion to be drawn as to the optimum stress and configuration of the minimum weight









plate. A comparison of the optimum plate of each type in terms of common weight and

loading parameters can then be effected to establish the ranges of efficient application of

each type. So minimum weight is obtained for a given set of loads and geometric

parameters by using plate-weight equations or plate-weight tables. These weight equations

or weight tables can then be used in the overall wing design.

In Gerard's approach for panel-level design, stiffness constraints for wing-level

design are not included. This does not provide any mechanism for wing-level stiffness

constraints to affect panel design. In addition, the effect of panel design on load

redistribution is not considered.

An approach that provides for including global stiffness constraints on panel-

weight equations was developed by Ragon et al. (1997). At the global level, design

variables are stringer areas and skin thicknesses. Global constraints included required

stress constraints for all elements, global stiffness constraints and weight constraints using

ADOP program (Sharma et al. 1993). A weight equation for optimal panel weight was

fitted as a RS to a large number of panel structural optima as a function of the loads on the

panel and of inplane-stiffness parameters. The PASCO program (Stroud et al. 1981) was

used for panel structural optimization. Panel-design variables described ply thicknesses of

elements of the stiffened panel and constraints included buckling, strength, and

manufacturing constraints as well as constraints on the inplane stiffness All and A66.

Response surface of the optimum-panel weight is a function of the inplane loads Nx, Ny, and

Nxy, and inplane stiffness All and A66. This response surface is used in the weight

constraints in the global optimizer.


w1.0 (2-10)
g= 1.0- <0 (2-10)
W1









where Wg is panel weight in the overall design, and wt is the local panel weight (response

surface). This constraint requires the overall optimizer to allocate to each panel enough

weight to satisfy the local constraints under the specified loads and inplane-stiffness

requirements.

While Ragon et al. (1997) obtained good designs, they ran into one problem in the

generation of the response surface. For many combinations of loads and inplane-stiffness

parameters, no feasible panel design could be found. The work in this dissertation

eliminates this problem by using the optimizations to create a response surface of an

optimum load instead of an optimum weight. This is described in Chapter 4.


Stacking Sequence Optimization by Genetic Algorithms

Genetic algorithms (GAs) are search techniques based on a simulation of the

Darwinian concept of survival of the fittest and natural reproduction genetics operating on

a population of designs. These algorithms belong to the class of probabilistic search

methods. Compared with traditional search algorithms such as gradient based continuous

variable methods or enumerative integer-programming techniques, probabilistic search

methods sample the design space based on probabilistic rules, and they are of global scope

because they have a nonzero probability of eventually reaching any point of the design

space, and also they are not sensitive to the problem nonconvexities and nonlinearities.

Holland (1975) pioneered the implementation and theoretical analysis of genetic

algorithms. DeJong (1975) then applied GAs to optimization. Since then, many people

applied GAs to many fields. Application includes artificial neutral networks (Fullmer

1991), geophysics (Gallagher 1992), social science (Greene 1987), control (Kristinsson

1992), biology (Lucasuis 1991), and diagnosis (Potter 1990).









Goldberg and Santani (1987) pioneered the application of GAs to structural

optimization. Since then, GAs have been applied to numerous structural optimization

applications (Furuya & Haftka 1993, Hajela 1990, Hajela 1991, Hajela & Lin 1992;

Powell & Skolnick 1993, Shoenauer & Xanthakis 1993, Watabe & Okino 1993).

The design of composite laminates is often formulated as a continuous optimization

problem with ply thickness and ply orientation angles used as design variables. Schmit and

Farshi (1977) first formulated the design of composite laminates as a continuous

optimization problem with ply thickness used as design variables. However for many

practical problems, the ply thickness is fixed, and ply orientation angles are limited to a

small set such as 0, +450, and 900. Thus, the design problem becomes a combinatorial

problem of choosing the fiber direction from a permissible set for each ply.

Mesquita and Kamat (1987) optimized the stacking sequence of laminates with the

number of plies of given orientation used as the design variables by integer programming.

Haftka and Walsh (1992) used ply-identity design variable to maximize buckling load

using linear integer programming. However, when strength constraints are also considered,

the problem becomes nonlinear and has been solved as a sequence of linearized integer

programming subproblems (Nagendra et al. 1992).

GAs has been used extensively to solve this combinatorial problem (Le Riche &

Haftka 1993, 1995; Kogiso et al., 94a, 94b; Nagendra et al., 93a, 93b). GAs are well

suited for stacking sequence optimization, and because of their random nature, they easily

produce alternative optimum in repeated runs. This latter property is particularly important

in stacking sequence optimization, because widely different stacking sequences can have

very similar performance (Shin et al. 1989).









Stacking sequence design of composite panels is a local design problem that is

often strongly coupled to the overall design of a structure. In wing structural optimization,

the overall wing structural design imposes constraints on individual panel designs. The

optimization of the overall wing structure often specifies number of 00, +450, and 900 plies

and in-plate loads of each panel. The stacking sequence design is then limited to

permutations of given plies, but not to changes in the number of plies of each orientation.

It is possible to solve this problem by using a conventional GA with additional

constraints imposed on the design. However, permutation GAs, developed mostly for

solving scheduling problems (Michalewicz 1992), handle more efficiently the search for an

optimal permutation, because they reduce the dimensionality of the design space.

Permutation GAs mostly were developed for the traveling salesman problem, which seeks

to minimize travel cost for a given list of towns, and is insensitive to where the sequence

starts, so that cyclical permutations do not matter. In stacking sequence design, in contrast,

a cyclical permutation will move the outermost ply into the innermost position, and thus

greatly influence the bending properties of the laminate.

Aside from the use of permutation GAs, number-of-ply constraints may be handled

by repair strategies. Such repair strategies may also be useful for dealing with another

constraint common to a laminate design a limit on the number of contiguous plies of the

same orientation (Todoroki & Haftka 1998).

The next chapter describes a permutation GA and repair strategy developed for

laminate stacking sequence design.














CHAPTER 3
PERMUTATION GENETIC ALGORITHM FOR STACKING SEQUENCE
OPTIMIZATION OF COMPOSITE LAMINATES

Introduction

Stacking sequence design of composite panels is a local design problem that is

often strongly coupled to the overall design of a structure. In wing structural optimization,

the overall wing structural design imposes constraints on individual panel designs. The

optimization of the overall wing structure often specifies the number of 00, +450, and 900

plies and in-plate loads of each panel. The stacking sequence design is then limited to

permutations of given plies, but not to changes in the number of plies of each orientation.

It is possible to solve this problem by using a conventional genetic algorithm (GA)

with additional constraints imposed on the design. However, permutation GAs, developed

mostly for solving scheduling problems (Michalewicz 1992), handle more efficiently the

search for an optimal permutation, because they reduce the dimensionality of the design

space. Permutation GAs were mostly developed for the traveling salesman problem, which

seeks to minimize travel cost for a given list of towns, and is insensitive to where the

sequence starts, so that cyclical permutations do not matter. In stacking sequence design, in

contrast, a cyclical permutation will move the outermost ply into the innermost position,

and thus greatly influence the bending properties of the laminate.

Aside from using permutation GAs, number-of-ply constraints may be handled by

repair strategies. Such repair strategies may also be useful for dealing with another









constraint common to a laminate design a limit on the number of contiguous plies of the

same orientation.

This chapter presents a permutation GA that is better suited to stacking sequence

design. We compare the permutation algorithm to a standard permutation GA, Partially

Mapped GA (Goldberg & Lingle 1985), as well as to a standard genetic algorithm. The

new algorithm shares some properties with Bean's Random Keys algorithm (Bean 1994)

and therefore the two algorithms are compared. In addition, the use of a repair strategy for

the standard GA and the permutation GA based on a Baldwinian repair strategy is

introduced (Todoroki & Haftka 1998). We compare the algorithm for maximization of the

buckling load of a laminate with specified number of 00, 45, and 900 plies.

Genetic algorithms are random in nature, and therefore comparing the efficiencies

of alternative algorithms requires averaging many runs. For this reason, a simply supported

unstiffened panel is selected since its closed form solutions are available. We can thus

perform the millions of analyses required for a thorough comparison of the efficiency of the

various genetic algorithms. We measure efficiency of the algorithms in terms of number of

analyses required for high reliability in finding the optimal design. Computation times are

not given because they are dominated by GA operations, while they will be dominated by

structural analyses in more realistic problems.

The rest of the chapter starts by describing the physical model of the composite

laminates and a standard formulation of optimization of a composite laminate. A new

permutation GA, which we call a gene-rank crossover GA, suited for stacking sequence

optimization is developed, and the standard GA and a permutation GA based on partially

mapped crossover are reviewed and implemented. The computational efficiency of the











three GAs are then compared under various load cases. The effect on performance of a

contiguity constraint limiting the number of identical adjacent ply orientations to four, is


Y

Ply 90
A Ny Sequence 90
b 45o
Innermost +450 X
.a iNx, Stack of +45 Laminate
22 Plies Plane
45Plane of
S\> 9o Symmetry

-- Outermost 0o
Stack of o
2 Plies
INxy i Ny

a) Laminate Plate Geometry and Applied Loading b) Ply Sequence Location
Figure 3-1: Composite laminate plate geometry and loads




investigated. Two repair strategies, chromosome repair and laminate repair, for

permutation violating the contiguity constraint are discussed.



Composite Laminate Analysis and Optimization

Buckling Load Analysis of Composite Laminate

This chapter deals with the optimization of symmetric and balanced stacking

sequences of composite wing panels. Usually, a panel is to be designed for given in-plane


loads and specified total number of of 00, +450, and 900 plies. The loads and the specified

number of plies come from the overall wing-level optimization. Here the panel is designed

to maximize the buckling load subject to a constraint on the number of contiguous plies of

the same orientation.

An unstiffened, simply supported, laminated panel with dimensions a and b (Figure

3-1) is subjected to normal loads per unit length Nx and Ny, and a shear load per unit length









Nx,. It is made of a symmetric and balanced graphite-epoxy laminate composed of 00, +45,

and 900 plies.

Because of symmetry, there is no extensional-flexural coupling. The pre-buckling

deformations are hence purely in-plane. The balance condition requires that for every ply

with a positive fiber-orientation angle, there is a corresponding ply with the negative fiber-

orientation angle. This implies no normal-shear extensional couplings. In addition, the

laminate is assumed specially orthotropic (i.e. there will be no bending-torsion coupling).

This is a common assumption in the analysis of balanced symmetric laminates for which

the bending-torsion coupling terms are usually very small and negligible.

Normal Buckling Load Analysis

Under biaxial loading, the laminate can buckle into m and n half waves in the x and

y directions, respectively, when the load amplitude (a factor multiplying the applied loads)

reaches a value ;,,m,"', which depends on flexural stiffness D, and loads Nx and Ny.

;"n) D, (m/a)4 +2(D12 +2D )(m/a)2 (n/b)2 +D2(n/b)4 (3-1)
72 (m/a)2 N, +(n/b)2 Ny

The pair (m, n) that yields the smallest value of mn"', which is the critical buckling

load cb, varies with the loading conditions, total number of plies considered, material,

and the plate aspect ratio.

Shear Buckling Load Analysis

A second mechanism is buckling due to shear loading. Modeling of this buckling

mode for a finite plate is computationally expensive. Instead, the plate is assumed to have

an infinite length, and analytical solutions available for a plate of infinite length in the x









direction are used as approximations (Whitney 1985). The critical shear buckling load

factor ~, is given in Whitney (1985) as a function of the variable r


4P (DD 221/4
2,, for l b2N fr (3-2)
4pJ1 D22(D12 +2D66
S2Nx, for 0 b2 ,

F- ID22 (3-3)
D12 +2D66

and values of p1 are given in Table 3-1.



Table 3-1: Coefficient 31 for shear buckling load factor (Whitney 1985)


0.0 11.71

0.2 11.80

0.5 12.20

1.0 13.17

2.0 10.80

3.0 9.95

5.0 9.25

10.0 8.70

20.0 8.40

40.0 8.25

co 8.13



Combined Buckling Load Analysis

When normal and shear loads are applied simultaneously to the panel their

interaction is approximated by the following interaction equation (Lekhnitskii 1968)

1 1 1 (3-4)
;If""' ;~"" --









where x3"- and k, are the critical load amplitudes under normal and shear loads,

respectively. The combined buckling load factor 2"" is always more critical than the

normal buckling load factor M).

To prevent buckling, 2) and k, must be greater than one. Shear buckling occurs

independently of the sign of the shear load. So buckling load ? is taken to be the minimum

of the load factors.

S= =min({|il,|,o"") } (3-5)

In addition, to reduce problems with matrix cracking, we do not allow more than

four contiguous plies with the same orientation. This is referred to as the contiguity

constraint.

Statement of Stacking Sequence Optimization

For maximizing the buckling load of composite laminates for given total number of

0, 450, and 900 plies, and ply-contiguity constraints, the optimization problem can be

stated as follows:

Given three ply orientation choices (0, 450, and 900), applied in-plane normal

and shear loading Nx, Ny, and Nx, and the total number of 0, 450, and 900 plies.

Optimize a symmetric and balanced stacking sequence in order to maximize the

buckling load X (that is the panel will buckle under loads kNx, )Ny, and kNxy).

Subject to the constraints that there be no more than four contiguous plies of the

same orientation, and number of 00, 450, and 900 plies be equal to given total number of

0, 450, and 900 plies.

Results were obtained for a 24-inch square graphite-epoxy plate with the following

properties shown in Table 3-2.









Table 3-2: Material properties of graphite-epoxy T300/5208
El 18.5x106 psi (128 GPa)
E2 1.89x106 psi (13.0 GPa)
G12 0.93x 106psi (6.4 GPa)

V12 0.3

tply 0.005 in (0.0127 cm)


Genetic Algorithms

A genetic algorithm is a guided random search technique that works on a population

of designs. Each individual in the population represents a design, i.e. a stacking sequence,

coded in the form of a bit string. The genetic algorithm begins with the random generation

of a population of design alternatives. Designs are processed by means of genetic

operators to create a new population, which combines the desirable characteristics of the

old population, and then the old population is replaced by the new one. Herein the best

design of each generation is always copied into the next generation, which we call an

elitist strategy. The process is repeated for a fixed number of generations or for a fixed

number of analyses resulting in no improvement in the best design.

A genetic search changes the population of strings by mimicking evolution. The

individual strings are mated to create child designs. Each individual has a fitness value that

determines its probability of being chosen as parents. Here the fitness is based on a rank in

terms of objective function in the population. The fitness assigned to the ith best individual

of n designs is then equal to [2(n+1-i)(n2 +n)], so that the sum of all witnesses is equal to 1.

Standard Genetic Algorithm

For the standard GA, a laminate is coded using the standard stacking sequence

notation. Because of the symmetry of the laminate and its balance, only one quarter of the









plies is encoded. This is done by adding the requirement that the laminate is composed of

pairs of 00 plies, pairs of 900 plies, or a stack of 450 plies. For example, the laminate

[02 / 45 /902 /902 / 45/02 1 is encoded as [0 /45/90] the latter being the chromosome

for the laminate. The rightmost gene corresponds to the stack closest to the laminate

midplane. The leftmost position in the chromosome describes the outermost stack of two

plies. A two-point crossover is used.

Mutation is applied with a small probability by randomly switching a stack

orientation (0, 450, 900) to one of the other two choices available. Since the total

numbers of 00, 450, and 900 two-ply stacks are fixed, the mutation is biased to promote

compliance with this constraint. The mutation is biased so that a 00 stack will mutate only

if the number of 0 plies is not equal to the allocated amount. This rule also applies to

+450 and 900 stacks. The mutation operator hence uses the problem information and acts as

a partial repair operator. Besides the regular mutation, there is also an interchange

mutation operator called stack-swap, which allows two stacks to exchange their genes with

a given probability.

The objective function for maximizing the failure of the composite laminate is equal

to the failure load X penalized for violations of the given number of plies and the limit of

no more than four contiguous plies of the same orientation. We denote the number of 00,

+450, and 900 two-ply stack in the string by no, n45, 90o respectively, and denote the

specified total number as n,. n45g, respectively. Then the objective function is given

as follows:

p= r Penalh (3-6)









where Penalty is a parameter (set to 2.0 ) for violation of specified amounts of 900, 45,

and 00 plies, and

r=ror45 r90 (3-7)

no +1
if no < nog
og +1 (3-8)
ro =1 if no =nog

if no > ng
no +1

with similar definitions for r45 and r90.

This form of the penalty function and penalty parameters were selected according to

previous studies with similar constraints (Kogiso et al. 1994a, 1994b; Le Riche & Haftka

93, 95). That is

0 = yp (3-9)


Pcon, is penalty parameter (set to 1.05 here) for violation of the four-ply limit on

contiguous plies of the same orientation, ncont is total number of same-orientation

contiguous plies in excess of four. Note that the contiguity constraint is applied only to 00

and 90 plies. The 45 plies alternate between 450 and -45 directions, and so do not

have any contiguity problem, no matter how many 450 stacks are contiguous.

Using a penalty function to incorporate the limits on the number of plies slows

down the progress of the optimization. This justifies using permutation based GAs, which

do not need these constraints.

Permutation problems seek the optimal arrangement of a list of items, in our case,

the given 0, 450, and 900 stacks. Natural coding with the orientation angles 00, 45, and

+900 is not well suited for representing permutations since it will tend to generate









duplicate or missing allele values. A permutation encoding is represented by a list of

distinct integer values, such as 1, 2, 3, ..., coding the orderings of 00, 450, and 900 stacks

referenced to a baseline laminate. We selected the baseline laminate to have all the

specified 900 stacks on the outside, followed by the 450 stacks and then the 00 stacks. So

the baseline laminate looks like [902 /902 / .. / 45 /+ 45 / ... /0 /0, and it is

coded as [1/2/... /n/./n1/ +n45+ 1/ .. ./n+n45+n90]. A baseline laminate [902/452/02] for

example, is coded into the permutation [1/2/3], while the laminate [902/02/452], is coded

[1/3/2] by reference to the baseline laminate.

Permutation coding has the advantage, compared to the traditional coding, that the

specified amounts of 00, 45, and 900 stacks are always met. However, traditional

crossover and mutation do not work well for permutation coding because they tend to

produce infeasible children from feasible parents. Specific permutation crossovers have

been developed for the travelling salesman problem (TSP). In this work we use the

partially mapped crossover, developed by Goldberg and Lingle (1985). We also

developed a crossover suited for the design of composite laminates that we call a Gene-

Rank crossover. Mutation for permutation coding is performed by randomly selecting two

genes, and then swapping them with a given probability.

Permutation Genetic Algorithms

Gene-Rank Crossover

In a composite laminate, the outermost plies, hence leftmost genes, affect flexural

stiffnesses more than the inner plies. This is in contrast with TSP, where the chromosome

may be viewed as a ring, where the absolute position of a gene does not matter. A









chromosome for coding a stacking sequence in contrast may be viewed as a directed linear

segment.

Gene-rank crossover is based on imitating the process used to average the rankings

that two judges give a group of contestants with plies playing the role of contestants. Each

laminate can then be viewed as a ranking of the set of plies, and gene-rank crossover

averages the two rankings. For example, consider the simple case with three contestants,

A, B, and C. The first judge ranked them as: A-1, B-2, C-3, denoted in shorthand as

[A B C]. The second judge ranked them as: A-2, B-3,C-1, or [C, A, B]. We associate

weights W1 and W2 with the two judges, representing their relative influence (with W1

+W2=1). In the implementation of the crossover, W1 is a uniformly distributed random

number in [0., 1.] selected anew for each pair of parents for each generation. The final

ranking is then obtained as the weighted rank of each individual

A: (1) (W)+(2) (W2)
B: (2) (W)+(3) (W)
C: (3) (W)+(1) (W2)
For example, with W1 =0.4, W2 =0.6, we get [1.6, 2.6, 1.8] for the weighted

averages, corresponding to a composite ranking of [A, C, B].

Consider next, for example, the stacking sequence of the baseline laminate

[902/902/902/45/45/02]s, with its permutation being defined by the chromosome

[1/2/3/4/5/6]. If two permutations of the laminate are:

Permutation 1 (Parent 1) [2/5/4/3/6/1]
Laminate [902/45/45/902/02/902]s
Permutation 2 (Parent 2) [1/2/4/5/3/6]
Laminate [902/902/45/45/902/02]s









For W1= 0.4634 and W2=0.5366, the average rank of each gene of the child design

is shown in the table below. For example, the average rank of gene 1 is equal to 6Wi+W2

since gene 1 is ranked the sixth and the first in the two permutations, respectively.




Table 3-3: Weighted gene-rank values of child averaging gene-rank values of two parents
Rank-Value in Rank-Value in Weighted
Gene
Permutation 1 Permutation 2 Rank-Value
1 6 1 3.32

2 1 2 1.53

3 4 5 4.54

4 3 3 3.00

5 2 4 3.07

6 5 6 5.54


Sorting genes by their average weighted ranks (Table 3-3), the permutation of the

child is

Permutation (Child) [2/4/5/1/3/6]
Laminate [902/45/45/902/902/02]s
Besides the uniformly distributed random weight, W1, we also experimented with a

random variable biased to be close to one or zero, so that one of the parent laminates

dominates. However, we did not find a distinct advantage to that variant. The Gene-Rank

GA has some similarities with Bean's Random Keys algorithm (Bean 1994). The Random

Keys algorithm uses a chromosome with numbers in [0, 1.], with their order determining

the permutation. For example, the chromosome [.46/ .91/ .33/ .75/ .51] corresponds to the

permutation [3/1/5/4/2]. The advantage of this form of coding is that standard crossover

and mutation can be used. This coding tends to preserve rank more than the partially









mapped crossover discussed next, but it is not as conscious of rank as the Gene-Rank

algorithm. For example, consider two parents that are both identical with the baseline

laminate, so that in permutation coding they will both be coded as [1/2/3/4]. Any Gene-

Rank crossover will produce a child design identical to the parents. On the other hand,

with the Random Keys algorithm, one parent may be coded as [.1/.2/.3/.4], and the other

parent may be coded as [.5/.6/.7./.8]. Some of the child designs obtained by crossover are

very different. For example, with a cut in the middle of the chromosome, one child design

is [.5/.6./.3/.4], which corresponds to a permutation of [3/4/1/2].

Partially Mapped Crossover

The partially mapped crossover, developed by Goldberg and Lingle (1985) for the

TSP, employs the following four steps:

1. Define two break points randomly.
2. Use the middle sub-string between the two cut points from the second parent.
3. Take genes of the two outer sub-strings from the first parent when they do not
conflict with the genes taken from the second parent.
4. Define the map relationship of genes in conflict, and fill genes in conflict by a map
relationship.
The mechanism of the crossover is illustrated through an example of a laminate

with a nominal stacking sequence of 8 stacks corresponding to 32 plies. The stacking

sequence of the baseline laminate is [902/902/45/45/45/45/02/02]s, its gene code is

defined as [1/2/3/4/5/6/7/8]. Two permutations of the laminate are listed as follows:

Permutation 1 (Parent 1) [3/6/4/2/7/5/8/1]
Laminate [45/45/+45/+45/902/02/02/902]s
Permutation 2 (Parent 2) [3/7/5/1/6/8/2/4]
Laminate [45/02/45/902/45/02/902/45]s
The random cut points are 2 and 5, so the segment between two-cut points of the child design is
Child permutation [*/7/5/1/6/*/*/*]









where the asterisk denotes presently unknown. Then we fill positions of the genes, which

are not in conflict with these genes,

Child permutation [3/7/5/2/1/6/*/8/*]

Two genes from Parent 1 in Positions 6 and 8 of the permutation conflict with genes

in the middle sub-string, which come from Parent 2. The conflicting gene in position 6 is 5,

and the corresponding gene in Parent 2 was in same position as gene 4 from Parent 1.

(Since the gene will not conflict with any genes from same parent, we need to go back to

Parent 2 to find the corresponding gene of Parent 2). We check whether the mapped gene 4

conflicts with genes previously filled in the child. We find that it does not conflict with

any. So the conflicting gene 5 from parent 1 in position 6 in the child's permutation is

replaced by gene 4. Similarly, we find that conflicting gene 1 from parent 1 in position 8 of

the child's permutation mapped gene 2 from parent 2. We fill genes 4 and 2 into position's

5 and 8 of the child's permutation to obtain

Child permutation [3/7/5/1/6/4/8/2]
Laminate [45/02/+45/902/45/+45/02/902]s

Comparison of Efficiency of Three GAs


The efficiency of the three GAs is discussed here in terms of the computational

cost-the average of number of analyses required for obtaining a given level of reliability

in finding the global optimum. The reliability is calculated here by performing 100

optimization runs each for 4000 analyses and checking how many of the 100 runs reached

the optimum at any given point. For example, if 63 runs reached the global optimum after

500 analyses, then the reliability of the algorithm is estimated to be 0.63 after 500









analyses. Of course, this is only an estimate, but it is easy to check that the standard

deviation of a value r of the reliability estimated from n runs is


(3-10)


r(1-r)
n


So that for 100 runs and r=0.63, we obtain a standard deviation of about 0.048.




Table 3-4: Optimum number of stacks of the three orientations for five load cases using
sequential quadratic programming (ks and k, are the shear buckling load and combined


buckling load factors, respectively)
Loading (lb/in)


Non-Rounded Optimization Results


Case Nx Ny Nx no n45 n90 total s c
1 -20000 -2000 1000 9.18 18.32 9.18 36.65 26.56 1.0
2 -15000 -2000 1000 8.41 16.82 8.41 33.63 21.24 1.0
3 -10000 -2000 1000 7.49 14.98 7.49 30.00 15.09 1.0
4 -5000 -2000 1000 6.28 12.52 6.28 25.10 8.87 1.0
5 0 -2000 1000 4.31 8.62 4.31 17.24 2.87 1.0


Table 3-5: Comparison of computational efficiency of the three GAs

Loading Given Number of Stacks Failure Number of Analyses Required
(Rounded From Table 3-4) Load For 80% Reliability
Case nog n45g n90g k SGA GR PMX

1 9 18 9 0.948 10432 1184 1328

2 8 17 8 0.948 8600 856 1224

3 7 15 7 0.909 5216 776 1024

4 6 12 6 0.870 3304 608 824

5 4 8 4 0.778 1672 408 560


Standard GA
Partially Mapped Crossover


GR: Gene-Rank Crossover


Refer appendix for selection of the genetic parameters.

Because reaching the global optimum is often very time consuming, the requirement

is often relaxed, and replaced by a practical global optimum, which is defined to be


SGA:
PMX:









within a specified fraction of the optimum. In the present work, a design was considered to

be a practical optimum if the failure load was within 0.5% of the global optimum.

In general, the loads and number of plies used in the panel optimization come from

the overall wing design. Here, in order to generate test cases, we selected some

representative load cases, and used continuous optimization to find reasonable required

number of plies. For the continuous optimization, we used nine ply thicknesses as

continuous design variables t,, i=1 ...9 and sequential quadratic programming (SQP) as

implemented in the DOT program (Vanderplaats et al. 1995). The stacking sequence was

set as [909, / 45,8 / 07 / 90;6 / 45'5 / 04 /90,3 / 45;2 / 0l,] The results are given in

Table 3-5 in terms of number of plies of a given orientation (for ply thickness of 0.005 in)

rather than the detailed stacking sequence. Next, the number of plies was rounded into

integers, and the rounded numbers were used as the specified set for the genetic algorithms.

The results for the three algorithms were obtained with a population size of 8 and

with the probabilities of mutation and crossover set to 1. The appendix discusses choice of

three parameters. For the mutation operation, one gene is changed to one of two other

alleles available in each child design for the standard GA, and for the permutation GAs,

two genes are swapped for each child design. The number of multiple runs is 100, and the

number of generations is 500. Table 3-5 gives results for the three GAs in terms of number

of analyses required for 80% reliability.

From Table 3-5, we can also see that, as expected, thicker laminates are

computationally more expensive to optimize. All the laminates in Table 3-5 are quasi-

isotropic or close to quasi-isotropic. The small number of 00 and 900 plies in such








44



laminates makes the contiguity constraint easy to satisfy. To explore the performance for


more general and thicker laminates, three new cases, defined in Table 3-6, were selected.


Case (1)


Generations


1

0.9

0.8

0.7

0.6

S0.5

0.4

0.3

0.2

0.1

n


0 100 200 300 400 500
Generations

Figure 3-2: Reliability versus number of generations for five loading cases: Case (1) and
Case (2)


Case (2)

0 GR
31e S GA
__ MX





















































n
(.
al)


45


Case (3)

o GR
0.9 --- x SGA
PMX
0.8

0.7

0.6

- 0.5

0.4

0.3

0.2

0.1

0
0 100 200 300 400 50(
Generations



Case (4)

Jo GR
0.9 x SGA
'/* PMX .
0.8

0.7

0.6___

0.5

0.4

0.3

0.2

0.1

0
0 100 200 300 400 500
Generations


Figure 3-2: Reliability versus number of generations for five loading cases: Case (3) and
Case (4)





From Table 3-5 we see that for the first three load cases, the reliability of the


standard GA did not reach 80% for 4000 analyses. The reliability is shown versus number


of generations in Figure 3-2. From the figure, it is clear that the two permutation GAs








46



perform much better than the standard GA. The Gene-Rank Crossover generally has the


highest reliability except occasionally for low number of generations.


1

0.9

0.8

0.7

0.6

| 0.5

0.4

0.3

0.2

0.1

0


0 100 200 300 400 500
Generations

Figure 3-2: Reliability versus number of generations for five loading cases: Case (5)





Case (6)


0.9- GR
x SGA
0.8 -I* PMX

0.7
0.7 ------ ~ f-----------------

0.6

. 0.5

0.4

0.3

0.2

0.1


0 100 200 300 400 500
Generations

Figure 3-3: Reliability versus number of generations for the three thick laminates: Case (6)


Case (5)


G RR
SSGA
SPMx










Table 3-6:
Case
No.
6
7
8


Three tick test laminates
Nx Ny
(lb/in) (lb/in)
0 -16000
15980 -14764
-16657 1963


Nxy
:lb/in)
8000
10160
828


total

32
30
35


Table 3-7: Comparison of computational efficiency of the three GAs for three thick
laminates

Case No. Number of Analyses Required For 80% Reliability

SGA GR PMX
With Without With Without With Without
6 7984 5112 1328 480 1480 848

7 23544 2176 11840 360 5784 336
8 26320 5024 2216 296 2504 840


Case (7)
0.7
o GR
x SGA
0.6 PMX

0.5

0.4

S0.3
Q; 0.3 ---- --- ------ ---_ __-_-- -

0.2 _




0
0.1 _______ ____

0 __--- ---------- X_ __fX- -< --------
100 150 200 250 300 350 400 450 500
Generations
Figure 3-3: Reliability versus number of generations for the three thick laminates: Case (7)



The results summarized in Table 3-7 show that for the three thicker laminates, the

contiguity constraint dominate the search for the optimum. Comparing the three thick











laminates above with contiguity constraints and without contiguity constraint, we can easily

see that case 7 has the most difficult constraints. This is explained by examining the

optimum laminates shown in Table 3-8. For case 6 and case 8, the outermost plies in the


optimum design are +450, so that the contiguity constraint affects only the less important


inner plies, while for case 7 it affects the critical outer plies. Figure 3-4 shows the

reliability versus number of generations of the three thick laminates. We also inspected the

various solutions and found that for the cases we optimized here, the optimum design was

unique, so that the number of analyses needed for 80% reliability is a good indicator of the

efficiency of the algorithm.


Case (8)

0.9

0.8 0r o GR
x SGA
0.7 --* PMX

0.6
> 0.6 ----------- T --------------

S0.5

0.4

0.3

0.2

0.1

0 X!
0 100 200 300 400 500
Generations
Figure 3-3: Reliability versus number of generations for the three thick laminates: Case (8)




Baldwinian Repair for Number of Plies and Continuity Constraint

The previous results demonstrate the high cost of dealing with constraints via

penalty function. An alternate approach is to repair laminates that violate constraints.

Todoroki & Haftka (1998) introduced a Baldwinian repair strategy, which they called









recessive repair, for dealing with contiguity constraints for standard GA. Here the strategy

is extended to the permutation GAs. Additionally, a similar repair approach is used for

enforcing the required number of plies of given orientations for the standard GA.

The key concept of Baldwinian repair is to repair the stacking sequence without

repairing the chromosome. Repairing the chromosome is known as Lamarckian repair. The

advantages of Baldwinian repair have been noted before, for example, by Hinton &

Nowlan (1987). There may be also an advantage to repairing a small percentage of the

chromosomes (Orvosh & Davis 1994).


Table 3-8: Optimum lay-up for the three thick laminates
Without
Case 6 Contiguity [(145)1/(902)8/(02),8s
Case 6 Contiguity
With
Contiguity [(45)16/(902)2/02/902/02/(902)2/(02)2/(9

Without
WCase [(902)8/(45)/902/(45)2/(902)/(145)2/(
Case 7 Contiguity

With [(902)/(45)/(902)2/(45)/(902)/(902/+
Contiguity (902)/(02)2/(45)/(02)2/(45)/(02)/(902)

Without
without [(145)7/(902)15/(02)131s
Case 8 Contiguity

With [(45)/(902/45)2/(902)2/(45)/(902)/(
Contiguity (02)2/(902/02)3]s


02)/(02)2/(902)/(02)2/(902)]s


902)2/(45)/902/(45)/(02)9/(45)]s

15)3/(902)2/(45)/(02)/(902)2/(02)2/
/(02)]s




02/902)4/(02)/(902/02)2/(02)/(902)/


The process is explained first for enforcing the required number of plies of given

orientation for the standard GA. The decoding of a chromosome proceeds from the

outermost plies to the innermost ones, one two-ply stack at a time. As long as the number of


I









decoded stacks of any given orientation does not exceed the prescribed number of stacks,

the decoding proceeds normally. However, once the number of stacks of any given

orientation reaches the prescribed number, subsequent genes that indicate that orientation

will be translated to the next available orientation (in a circular 0/45/90 order). For

example, consider a laminate with no=2, n45=0, and n90=1. When a chromosome [0/90/90]

is decoded, the first two genes are decoded normally, but when the third gene is

encountered, it cannot be decoded into a 90-stack because the number of decoded 90-genes

already reached the target of n90=1, so it is decoded as a zero ply. Similarly, when a

chromosome [0/0/0] is decoded, the first two genes are decoded normally. The third gene

cannot be decoded into a 0-stack, because the number of required 0-stacks is two. The

decoding procedure then tries to see if there are available stacks for a 45-stack, and when

it finds that none are available, it puts a 90-stack in the innermost position. It should be

noted that the circular order chosen for the orientation used in repair may introduce some

bias, and a random selection of the orientation may be a good alternative.

The repair of the stacking sequence without changing the chromosome allows a

sequence of mutations needed to achieve a good design to complete successfully even if the

intermediate steps are infeasible designs. For example, consider the evolution of a design

defined by [0/0/90] chromosome when the optimum is defined by [0/90/0] chromosome

(that is stacking sequences of [04/902]s to [02/902/02]s, respectively). Without repair we

have to depend on hitting the single permutation that will exchange the second and third

genes. With the repair strategy described above, we can also go through the intermediate

step of [0/90/90], which is decoded into [02/902/02]s, or through the intermediate step of

[0/0/0], which is decoded into [04/902]s. Then another mutation can transform either









intermediate step into the optimum. The last gene in both alternatives acts like a recessive

gene, in that it is unexpressed due to the decoding scheme, but it will become expressed

following the mutation of another gene.

The repair of violations of contiguity constraints follows the similar approach of

repairing only the laminate, and of trying to apply the repair to the innermost plies, which

have the least effect on the buckling load. Details may be found in (Todoroki & Haftka,

1998).

For the permutation GA, the constraints of number of plies are incorporated into

gene coding, and only contiguity constraints may be violated. To repair contiguity

violations, it is desirable to interchange the closest couple of genes with different

orientation angles since this minimizes the change in bending properties. The following

example illustrates the repair operator.

For the laminate
[02/02/902/902/902/+45]s

Three contiguous 90 stacks violate the contiguity constraint. Two candidate couples of

stacks can be swapped: the rightmost 900 with the 450, or the leftmost 900 with its

neighouring 00 stack. The first option is selected because the inner plies influence laminate

stiffness less than the outer plies.


[02/02/902/902/45/90 s

In order to demonstrate the advantage of recessive repair, it is compared to direct

repair of the chromosome in Table 3-9.

From Table 3-9, we can see that the Baldwinian repair (laminate only) is more

efficient than repairing the chromosome (Lamarckian repair). The advantage is most









pronounced for the repair strategy helps the standard GA achieve similar efficiencies to

that of the permutation GAs, except for the most difficult case (7). Comparing Table 3-9 to

Table 3-7, we see that the combined use of permutation and repair is to reduce the cost of

the standard GA by one to two orders of magnitude.




Table 3-9: Computational cost of laminate repair and chromosome repair
Case No. Number of Analyses Required for 80% Reliability
GR PMX SGA
Chromosome Laminate Chromosome Laminate Chromosome Laminate
Repair Repair Repair Repair Repair Repair

1 1048 456 944 792 368 672

2 952 400 808 792 400 536

3 832 352 784 658 384 368

4 680 304 560 496 280 224

5 304 184 272 272 128 80

6 744 416 688 552 416 400

7 480 288 416 336 3936 3512

8 728 352 744 696 56 48



Summary and Concluding Remarks

In this chapter, maximization of the buckling load of composite laminates via

stacking sequence optimization for a given number of 00, +450, and 900 plies and for a

given in-plane loading was investigated using genetic algorithms. A new permutation GA,

which we called a gene-rank crossover GA, was developed and implemented along with

two other GAs, a standard GA and a permutation GA based on partially mapped crossover.

Computational efficiency of these GAs were compared under eight load cases in terms of









the number of analyses required to reach a certain reliability. The effect on performance of

a contiguity constraint, which limits the number of identical adjacent ply orientations to

four, was investigated and two repair strategies for dealing with violation of this constraint

were implemented.

Stacking sequence design for given number of plies is a combinatorial problem

consisting of seeking an optimal permutation. It was demonstrated that the two genetic

algorithms based on permutation are much more efficient and more reliable for solving this

problem than standard genetic algorithms. Furthermore, a genetic algorithm developed for

stacking sequence design showed an advantage over an algorithm developed originally for

the traveling salesman problem. Repair developed for overcoming violation of constraints

can significantly reduce the computational cost for both the standard GA and the

permutation GAs, and with repair the difference between the standard GA and permutation

GA is smaller.

The permutation GAs and the repair strategy developed can be easily tailored for

application to more complicated structures with more constraints by coding these

constraints into gene coding or through repair.

The scope of this chapter research work is panel-level optimization for maximum

buckling load of composite laminates. The permutation GA and its corresponding

chromosome-repair technique were used in a large number of stacking sequence

optimization runs for a range of loads and number of plies. Based on these optima, a cubic

polynomial response surface was fitted as a function of in-plane loads and number of 0,

+450, and 900 plies. The response surface was then used in a wing box optimization that is

described chapter four.














CHAPTER 4
TWO-LEVEL COMPOSITE WING STRUCTURAL OPTIMIZATION USING RESPONSE
SURFACES

Introduction

The objective of this chapter is to demonstrate use of a two-level optimization

technique for wing panels when the design involves discrete or combinatorial optimization.

A wing structure is composed of a large number of panels that must be designed

simultaneously to obtain an optimum structural design. Composite stiffened panels often

have complex geometries and failure modes. The design of each panel requires a

substantial number of variables to describe geometry, ply composition, and stacking

sequence. Designing all the panels simultaneously constitutes a complex optimization

problem that requires detailed structural modeling of the entire wing, and appears to be

beyond present computational capabilities.

This chapter demonstrates use of response surface for maximal panel buckling

loads, which involves for coordinating wing-level and panel-level optimization. The

methodology is demonstrated for design of a simple wing structure, where the panel design

is performed by a genetic algorithm.

First, we described a two-level optimization procedure and summarized

formulation of panel and wing optimization and coordination of two-level optimization.

Second, we briefly reviewed response surface methodology and discussed normalized

response surface of normalized buckling load. Then, we presented results of 6-variable,









18-variable and 54-variable cases of wing box structure. Finally, we summarized

concluding remarks of the two-level optimization.


Two-level Optimization Approach

Two-Level Optimization Procedure

In this work, the wing is assumed to consist of n unstiffened composite panels. Ply

orientations are limited to 0 90 and 45 It is also assumed that wing depth is much

greater than skin thickness, so that the stresses in the skin are influenced by the number of

plies of each orientation rather than their arrangement in the stacking sequence.

Consequently, the design process will have the overall wing design determine the amount

of plies of each orientation, while the panel design will determine the stacking sequence.

The two design processes must be coordinated in order to assure the optimality of the

process and insure that the wing design optimization takes into account the effect of its

decisions on the panel design.

Here, the two design processes are coordinated through an equation that predicts

the buckling load multiplier that a panel can attain with the best stacking sequence. This

optimal load equation is a function of the number of plies of each orientation and the loads

on the panel. The equation is obtained as a response surface fitted to a large number of

panel stacking sequence optimizations for various combinations of numbers of plies and

loads.Internal loads and number of 00, +450, and 900 plies which completely determines

panel stiffness parameters are used as input parameters for subsystem (panel) optimization.

That is, the response surface for optimal buckling load depends on loads Nx, Ny, Nx, and

no, n45, n90 to output approximate buckling loads. These approximated







56
Global Optimization
(Minimization of
Wing Weight)

N, Ny, Ny
nn. nA. no Buckling Load


Response Surface Interface





Local Optimization
(Maximization of
Buckling Load)

Figure 4-1: Response surface interface of two-level optimization


Figure 4-2: Flowchart of two-level optimization procedure









buckling loads are used as constraints in the upper-level (wing ) optimization. The process

is shown schematically in figure 4-1.

The two-level optimization process is described by the flow chart in Figure 4-2.

The process starts with the creation of the response surface shown on the right side of the

flow chart. First, a set of design points in specified ranges of loads and number of plies is

created. Then a subset of these points is selected by a design of experiments procedure

known as D-optimal design. A genetic optimization (GA) of the stacking sequence is

carried out at all points at that set, and a response surface for the optimal buckling load is

fitted to the results.

The wing-level optimization is carried out by the GENESIS (Vanderplaats et al.

1997) program using the response surface optimization results. Following a finite element

analysis of a candidate design, strain constraints are calculated directly by GENESIS, and

the buckling load constraint is calculated from the response surface by using the panel

loads obtained from the finite element analysis. GENESIS iterates to find the optimum

design, using, as continuous design variables, the number of plies in each direction for

each panel.

Finally, when the wing-level optimization converges, the ply-number design

variables have to be rounded to the nearest integer, then each panel is redesigned by the

GA. Rounding and errors in the response surface usually cause some panels to be

infeasible. For these panels, the last part of the process requires some adjustment in the

number of plies to satisfy buckling constraints.









Panel-Level Optimization and Response Surface

In the panel level optimization, the number of 02, + 45', and 902 stacks, no, n45,

and n90, and the inplane loads on the panel, Nx, Ny, and Nx are specified. Thus, the design

problem becomes a combinatorial problem of choosing the optimal stacking sequence for

given amounts of plies in each direction so as to maximize the buckling load factor )b (that

is, the loads that the optimized panel can carry are bNx, lbNy, and )AbNy). This naturally

forms a permutation problem. The stacking sequence is optimized subject to a limit of four

contiguous plies of the same orientation (applied to reduce the chance of matrix cracking).

A permutation genetic algorithm (GA) developed by the authors (Chapter 3) is used for the

stacking sequence design. Buckling analysis is described in Chapter 3.

The panel-level optimization is repeated for a large number of load and ply number

combinations and the optimum buckling loads )y are fitted by a cubic response surface as

a function of no, n45, n90, Nx, Ny, and Ny.

Wing-Level Optimization

The objective function for the wing-level optimization is structure weight. Design

variables are the thicknesses of upper and lower skin panels. The ply orientations are

limited to 0 90 and 45 and each panel has three design variables describing the

number of plies of each orientation (with the balance condition, the number of + 450 and

- 450 plies is the same). Strain and buckling constraints are applied.

The numbers of stacks per panel, no, n45, and n9o are treated as continuous design

variables. Each stack consists of two plies: o2, 90, and 450. Minimizing wing weight is

equivalent to minimize the total number of plies. GENESIS is used to perform the overall

optimization subject to strain and buckling constraints in all panels.









Wing level optimization is formulated as follows:

n
Minimize (no + n5 + niO) (4-1)
i= 1

where i is the panel number.

By changing no ,n's, n$0 i= ...,n


Subject to: Laminates are symmetric and balanced

(Strain) X >1.0, i = 1, *, n (4-2)


(Buckling) i' (n', n45, N, N ,N', ) >1.0, i= 1,-,n (4-3)


where X, indicates the load factor (failure load over applied load) for strain

constraints, as calculated by GENESIS. The buckling load factor X is calculated using the

response surface approximation fitted to the optimum buckling load factor of the panels.




Table 4-1: Allowable strains and safety factor
sla 0.008
62a 0.029
Y12 0.015
Safety Factor 1.5




Example Problem Description

The wing structure considered here is an unswept, untapered, wing box with four

spars and three ribs with a total of 18 panels. The wing box is clamped at the root and

subjected at the tip to the applied load distribution shown in Figure 4-3.










All the panels are symmetric and balanced laminates made of graphite-epoxy

T300/5208, with material properties given in Table 3-2. The allowable strains and safety

factor used are given in Table 4-1. Each panel is assumed to be simply supported.




P, = 85467 lbs














88.2 in
------ --- -- ---- --- --- -------- P = 20235 Ibs













10 \ 15 16

11 \ 14 \ 17


12 \ 13 \ 18





Figure 4-3: Wing box structure



Response Surface Approximation of Optimal Buckling Loads

Response Surface Approximation

Response surfaces are used to obtain an approximate relationship between the

response of a system and its control variables. The response function is denoted as Yand it









is assumed that it can be approximated as a function of the control variable vector Xand a

vector of ng parameters 3; that is,


Y = Y(X, f)+e (4-4)


where Y represents the approximation, and g is the error. Least square fit is generally used

to estimate the values of the unknown linear regression coefficients 3.

Selecting points in the design space where numerical experiments are to be

performed is possibly the most important part of obtaining a good approximation to a

response function. Several standard designs are available. One example is the central

composite design. Standard designs are easy to use, but they may only be applied to a

regularly shaped design domain. For more general domains, D-optimal design is widely

used. The D-optimality criterion minimizes variance associated with the estimation of the

unknown coefficients in the response model. In the present work, the JMP software (SAS

1995) was used to select a D-optimal set of points.

Normalized Buckling Load Response Surface

Because the buckling load is proportional to the cube of the thickness, its magnitude

varies greatly from thin laminates to thick ones. This large variation can reduce accuracy of

the response surface. To overcome this problem, a buckling load is divided by the cube of

the number of stacks and normalized to be order of one (0(1)) as shown below.

= 10000A (4-5)
(no + n45 + n90)3


The number of stacks and loads were also normalized.







62



ro = no
no + n45 + n90

Sn45 (4-6)
no +n45 +n90
r9 = 1.0 r r45


- 2Nx -Nxmax .-xmin

2Ny -Nymax -Nymin (4-7)
NV y yma ym-------------
N -N
ymax Nymin
- 2N -N -N
N = xy xymax xymmn
N -N
Nymax ymin


In the above expressions, "max" and "min" denote the ranges of variables and load

components.

Because ro+r45+r0=1, the normalized buckling load can be expressed as a function

of five control parameters.


= I(ror45,Nx ,N y Nxy) (4-8)

Results of Response Surfaces of Optimal Buckling Loads

For the wing shown in Figure 4-3, all panels have the same dimensions. Lower skin

panels are subjected mainly to shear load Nxy and tensile loads Nx, Ny. Upper skin panels

are mostly subjected to compressive loads Nx, Ny, and shear load Ny. Therefore, buckling

constraints are applied only for upper skin panels. Since the load ranges are very different,

in order to construct high accuracy response surfaces, three critical buckling load response

surfaces are fitted, one for root panels, one for intermediate panels, and one for tip panels.


Load ranges for root panels are


-15000 b / in > Nx > -20000 1b / in
- 1000 lb/in > Ny > -4000 lb / in (49)
0 Nx 30001bl in









Load ranges for intermediate panels are

-10000 lb / in N 2 -15000 Ib / in
- 1000 lb /in Ny -4000 lb/ in (4-10)
0 < Ny, <3000 lb /in


Load ranges for tip panels are

-2000 lb/ in > N > -5000 lb /in
- 1000 lb / in 2 Ny -2000 lb / in (4-11)
0 N, < 3000 lb/ in


For all panels, the following ranges of number of 0 +450, and 900 stacks are used.

5 5 n.20 (4-12)
5 n,, 35 (4-12)
5 < n,0 20




Table 4-2: Statistics of three optimal buckling load response surfaces
Statistics Root Panels Middle Panels Tip Panels
R 0.9969 0.9976 0.9956
R, 0.9955 0.9966 0.9936

Root Mean Square Error I 0.0020 0.0027 0.0122
Mean of Response I 0.1312 0.1705 0.4706
RMS Error/Mean (%) 1.51% 1.56% 2.02%

Average Absolute Error of AL 0.0111 0.0147 0.0613
Average Value A 1.2411 1.7108 4.7981
Absolute Error of ) /Average Value 0.89% 0.86% 1.28%



More than 30,000 points were randomly generated for each of the three domains

defined in (4-9), (4-10), (4-11), and (4-12), and then 180 D-optimal design points were

selected from each domain. Stacking sequence GA optimizations were performed at each

point. A cubic response surface was fitted to the normalized optimal buckling load X in










terms of ro, r45, and Nx Ny Nxy. The statistics of the three response surfaces are

given in Table 4-2, where R and Ra are the coefficients of multiple determination and its

adjusted value, respectively.


0 2 4 6 8 10 12 14

Figure 4-3: History of the objective function and maximum violation of normalized
constraints for six-variable case.



The results indicate that the response surfaces have average errors below 2.1% for

the normalized optimal buckling load, and average errors below 1.3% for optimal buckling

load.


Results of Composite Wing Box Structure Design

The performance of the two-level optimization procedure is demonstrated through

six-variable, 18-variable, and 54-variable design problems.









Six-variable Design Problem

For this case, all upper-skin panels are the same and all lower-skin panels are the

same, so that the design variables are no, n45, and ngo for the lower skin and the upper skin.

Table 4-3 shows the final design, including the number of 00, +45 and 90 stacks for

lower skin panels and upper skin panels, the total number of stacks for all the panels, and

the type of active constraints at the optimum. Figure 4-4 shows the history of the objective

function and the maximum violation of normalized constraints during the wing-level

optimization.




Table 4-3: GENESIS and rounded optimal design with six variables
Active no/n45/n90 no/n45/n90 Failure Load
Constraints (GENESIS) (Rounded) Factor k
Strain
Lower Skin Panels tra 8.69/1.76/0.04 9/2/0 1.04127
(Panel #7)
Buckling
Upper Skin Panels Buckling 15.33/12.44/13.92 15/12/14 0.9664
(Panel #16)
Objective Function 469.70 468
(Total Number of Stacks)





Table 4-4: Rounded and adjusted upper panel design with six variables
Panel # GENESIS Rounded k (Rounded Adjusted k (Adjusted
Design Design Design) Design Design)
Objective 469.70 468 477
Function
no/n45/n90 n0/n45/n90 no/n45/n90
15.33/12.44 CHAPTER 21
16 53 44 C0.9664 16/12/14 1.0326
/13.92 5/12/14
Stacking Rounded Design [(45)12/904/04/(904/02)2/902/02/(02/902)3/(902/04)3/902/02]s
Sequence [(+45)12/(904/02)2/(902/04)2/902/02/904/02/(02/902)2/
(Panel #16) Adjusted Design (04/902)3]











40
S+ Objective Function (X 100)
x Violation of Normalized Constraints( %
35 -

S30 -
30

z 25
0




S10-
2
0




0 2 4 6 8 10 12 14 16 18
No.of Cycles
Figure 4-5: History of the objective function and maximum violation of normalized
constraints for 18-variable case.




At the optimum, upper root panel 16 is active in buckling, and one strain constraint

is active at lower root panel 7. Table 4-3 shows that upper skin panels are thicker than

lower skin panels because of the buckling constraints. After the wing level optimization

was completed, the continuous design variables obtained by GENESIS were rounded to the

nearest integer, and each panel was re-optimized by permutation genetic algorithm. The

buckling constraint was violated after rounding as can be expected, because the objective

function was reduced to 468 from the optimal 469.7. Table 4-4 shows the buckling load

and stacking sequence after manual adjustment. This adjustment increased the total number

of stacks to 477.

18-variable Design Problem

For the 18-variable design, each wing skin is divided into three regions: root

panels, intermediate panels, and tip panels. Each region has three stack design variables.









Table 4-5: GENESIS and rounded optimal design with 18 variables
Lower Skin Panels /n45/n90 n/n45/n90 Failure Load
(GENESIS) (Rounded)) k
(Panel #7) 9.78/0/0 10/0/0 1.0064

Panel #8 5.42/0.29/0 5/0/0 0.8601

Panel #4 0.87/2.04/0 1/2/0 1.0767

no/n45/n90 non45/n90 Failure Load
Upper Skin Panels (GENESIS) (Rounded)) k

(Panel #16) 14.20/13.29/14.13 14/13/14 0.9557

Panel #14 4.60/21.33/7.06 5/21/7 1.0161

Panel #18 3.70/16.80/2.84 4/17/3 1.0583

Objective Function 3
(Total Number of stacks)


Table 4-5 shows the continuous optimum obtained by GENESIS and the rounded

design, and also reveals that after rounding one strain constraint and one buckling

constraint are violated. The manually adjusted designs are shown in Tables 4-6 and 4-7.

This time most of the lower skin is made of unidirectional material, which is not feasible.

In actual design, a limit on the maximum percentage of zero plies must be added.


Table 4-6: Rounded and adjusted lower skin panels with 18 variables
GENESIS Rounded k (Rounded Adjusted k (Adjusted
Design Design Design) Design Design)
no/n45/n90 no/n45/n90 no/n45/n90

Panel #7 9.78/0/0 10/0/0 1.0064 10/0/0 1.0183

Panel #8 5.42/0.29/0 5/0/0 0.8601 7/0/0 1.0919

Panel #4 0.87/2.04/0 1/2/0 1.0767 3/1/0 1.0884











40
+ Objective Function (X 100)
35 x Violation of Normalized Constraints( %

30

25

20

15

10

5

01
0 2 4 6 8 10 12 14 16 18

Figure 4-6: History of the objective function and maximum violation of normalized
constraints for 54-variable case.


Table 4-7: Rounded and adjusted upper skin panels with 18 variables
Pal # Unrounded Rounded X (Rounded Adjusted X (Adjusted


16

14

18


Stacking
Sequence of
Rounded
Design


Stacking
Sequence of
Adjusted
Design


Design

no/n45/n90
14.02/13.29/14.
13
4.60/21.33/7.06

3.70/16.80/2.84

Panel #16

Panel #14

Panel #18

Panel #16

Panel #14

Panel #18


Design Design) Design Design)

no/n45/n90 no/n45/n90

14/13/14 0.9557 15/13/14 1.0361

5/21/7 1.0161 5/21/7 1.0186

4/17/3 1.0583 4/17/3 1.0241

[(45)13/04/(902/02)2/02//902/(902/02)2/904/04/(904/02)3/02/
902/02]s

[(+45)15/902/(+45)4/902/(902/+45)2/(902/04)2/902/02]s

[(45)17/04/(902/02)2/902]

[(45)13/(902/04)4/902/(902/02)2/902/04/(904/02)2/902/02]s

[(45) 5/904/(45)3/902/45/904/02/(02/902/02)2]s

[(45)7/04/902/(02/902)2]s









The total number of stacks for the continuous design was 349.22. It was reduced to

348 for the rounded design, and increased to 360 after adjustment necessary to satisfy the

constraints.


Table 4-8: GENESIS and rounded optimal design for 54 variables
Lower Skin Panels no/n45/n90 no/n45/n90 Failure Load
(GENESIS) (Rounded) k


1
2
3
4
5
6
7
8
9

Skin Panels


6.09/0.66/0.17
3.37/0/0.12
0.77/0/0
0.61/1.72/0.0001
4.73/2.28/0.54
6.9/0.68/0
11.16/0.69/1.10
6.09/0.75/0.70
1.18/1.59/0

no/n45/n9o
(GENESIS)

11.67/11.61/11.83
6.62/16.37/6.53
4.15/12.07/5.40
5.03/12.01/5.40
5.84/17.32/8.96
14.67/13.85/11.67
12.06/18/13.3
5.91/20.52/6.92
4.82/14.58/5.46


Upper


6/1/0
3/0/0
1/0/0
1/2/0
5/2/1
7/1/0
11/1/1
6/1/1
1/2/0

no/n45/n90
(Rounded)

12/12/12
7/16/7
4/12/5
5/13/5
6/17/9
15/14/12
12/18/13
6/21/7
5/15/5


1.0444
0.9388
1.2893
1.2399
1.0376
1.0111
1.0013
1.0037
1.0335

Constraints
(Buckling)

1.0824
1.1200
1.1535
1.0307
0.9747
1.0833
1.0524
1.0211
1.0310









54-variable Design Problem

For the 54-variable design case, each of the panels was permitted to have its own

three design variables. Table 4-8 shows the GENESIS and rounded designs, and indicates

that one strain constraint and one buckling load constraint are violated. Tables 4-9 and 4-

10 compare the strain failure loads and the buckling loads of the rounded design and

manually adjusted design for the lower and upper panels, respectively. Table 4-11 shows

the stacking sequences of the manually adjusted design. The total number of stacks for the

continuous design, increased from 335.44 to 340 for the rounded design, and reduced to

338 after manual adjustment to satisfy all the strain and buckling constraints. The objective

function was reduced from 477 to 360 to 338 by increasing the number of design variables




Table 4-9: Comparison of rounded and adjusted designs for lower skin panels, 54
variables
Panel # GENESIS Rounded k (Rounded Adjusted k (Adjusted
Design Design Design) Design Design)
no/n45/n90 no/n45/n90 no/n45/n90

1 6.09/0.66/0.17 6/1/0 1.0444 6/1/0 1.0332

2 3.37/0/0.12 3/0/0 0.9388 4/0/0 1.0789

3 0.77/0/0 1/0/0 1.2893 1/0/0 1.0949

4 0.61/1.72/0.0001 1/2/0 1.2399 1/2/0 1.1171

5 4.73/2.28/0.54 5/2/1 1.0376 4/2/1 1.0600

6 6.9/0.68/0 7/1/0 1.0111 7/1/0 1.0423

7 11.16/0.69/1.10 11/1/1 1.0013 11/1/1 0.9928

8 6.09/0.75/0.70 6/1/1 1.0037 7/1/1 1.0507

9 1.18/1.59/0 1/2/0 1.0335 2/2/0 1.2442









Table 4-10: Comparison of rounded and adjusted designs for upper skin panels, 54
variables
Panel #GENESIS Design Rounded k (Rounded Adjusted k (Adjusted
Design Design) Design Design)
no/n45/n90 no/n45/n90 no/n45/n90

10 11.67/11.60/11.8 12/12/12 1.0824 11/12/12 1.02384
11 6.62/16.37/6.53 7/16/7 1.1200 6/16/7 1.0895
12 4.15/12.07/4.82 4/12/5 1.1535 3/12/5 1.0921
13 5.03/12.61/5.40 5/13/5 1.0307 5/13/5 1.0782
14 5.84/17.32/8.96 6/17/9 0.9747 7/17/9 1.0003
15 14.67/13.85/11.7 15/14/12 1.0833 15/14/12 1.0236
16 12.06/18.01/13.0 12/18/13 1.0524 11/18/13 0.9915
17 5.91/20.52/6.92 6/21/7 1.0211 6/21/7 1.0134
18 4.82/14.58/5.46 5/15/5 1.0310 5/15/5 1.0051


Table 4-11: Stacking sequences of wing-level panels for the adjusted design
Panel Real Number Integer Number Stacking Sequence Buckling


Number
no 1145 n90
10 11.6711.6011.83


6.62 16.37 6.53
4.15 12.07 4.82
5.03 12.61 5.40
5.84 17.32 8.96
14.6713.8511.67


16 12.0618.0113.30 11


5.91 20.52 6.92
4.82 14.58 5.46


n45 N90
12 12 [(45)12/904/02/(902/02)5/
(02/904)2/04/902/02]s
16 7 [(45)15/02/45/04/(904/02)3/902]s
12 5 [(45)12/(02/904)2/02/902]s
13 5 [(45)12/02/45/04/(904/02)2/902]s
17 9 [(45)16/02/45/02/(02/904)4/02/902]s
14 12 [(45)11/902/(45)2/902/
45/902/(902/02)3/(02/902/02)6s
18 13 [(45)16/(902/45)/(904/02)3/
(902/02)2/(02/902/02)3] s
21 7 [(45)2o/02/45/04/(904/02)3/902]s
15 5 [(45)15/04/(902/02)2/904/02/902]s


Load


1.02384

1.0895
1.0921
1.0782
1.0003
1.0236

0.9915

1.0134
1.0051









Concluding Remarks

A two-level wing design optimization was developed and demonstrated using a

simple wing example. The procedure is based on continuous optimization at the wing level

using a finite element model, and genetic optimization at the panel level. A response

surface of optimal panel buckling load is used for communication between the two levels.

It was shown that a cubic response surface can fit accurately the buckling load of

the optimal panel stacking sequence as a function of the loading on the panel and the given

number of plies in each orientation. It was also shown that the response surface could be

used effectively to allow the wing-level optimization to find a near optimal wing design.

The use of continuous variables at the wing level allowed for inexpensive

optimization and use of the commercial GENESIS software program. Some constraint

violations occurred when the number of plies was rounded off and the stacking sequence

was optimized to find the actual design. However, it was possible to manually adjust

thicknesses to correct violations with very small increases in total weight.














CHAPTER 5
COMPOSITE WING STRUCTURAL DESIGN OPTIMIZATION WITH CONTINUITY
CONSTRAINTS

Introduction

Because of efficiency for structure weight compared to traditional structures

consisting of isotropic materials, industry now is paying much more attention to the use of

composite structures. Complex composite structures, such as aircraft wing or fuselage

structures or car chassis structures, commonly consist of multiple laminates. Composite

laminates consist of layers of one or more materials stacked at different orientation angles.

The layer thickness for each material is usually fixed and fiber orientation angles are often

limited to a discrete set such as 0, +450, and 90.

In the design of complex structures, it is customary to divide the structures into

panels or regions that may be designed independently or semi-independently (Schmit &

Mehrinfar 1982; Ragon et al. 1997; Liu et al. 2000). This is done not only for

computational convenience, but also because the loads vary from one part of the structure

to another, so that structural efficiency dictates variation in structural properties. With this

design approach, adjacent laminates may have different total thicknesses, and some plies

may terminate at the laminate boundaries. These discontinuities can cause stress

concentrations and increase manufacturing difficulty and cost.

While some discontinuities are inevitable if structural efficiency is to be

maintained, it is desirable to minimize these discontinuities. It is therefore desirable to add

continuity constraints for adjacent laminates in the design process, or include a measure of









continuity in evaluating competing designs. It may be expected that optimization including

continuity constraints may lead to designs with fewer and smaller discontinuities.

Kristinsdottir et al. (2001) recently developed the concept of blending rule to

measure ply continuity in adjacent composite panels. Two ways of specifying the blending

rules in optimal design formulation are set forth and compared. Comparing results with

blending measure and results without blending measure, design with blending is heavier

than design without blending but design with blending is easier to manufacture.

The first step in incorporating continuity in the design process is to develop

measures of continuity between adjacent panels. The objective of this chapter is to develop

such measures. Two continuity measures are defined in terms of material composition and

stacking sequence. The continuity measures are applied to a simple wing design problem,

and trade-off studies between weight and continuity are performed.


Common materials


Figure 5-1: Common layers of two laminates









Material and stacking sequence continuity measures for symmetric laminates

Algorithms for designing complex composite structures often design panels at two

levels (see chapter 4). At the global level the material composition of each laminate is

determined. For example, at that level it may be decided based on overall stiffness

considerations that a laminate is made of 20% 00 tape plies, 30% +450 tape plies, and

50% 900 cloth plies. At the local level, the stacking sequence of the plies is decided.

Consequently, it may be useful to develop two measures of continuity: one measure of

material composition continuity for the global design, and another measure of stacking

sequence continuity for the local level design.

Material Composition Continuity

We assume that there are Npossible layer types. These layer types may differ in

material properties, thickness, or fiber orientation angles. Layer type li has material mi,

fiber orientation angle ai, and thickness ti. Of course, it is possible that all the materials

and thicknesses are the same, in which case we just have a problem of continuity of ply

orientations.

Given two composite laminates, our first measure of continuity, composition

continuity Cm, is the fraction of common layers of the two laminates to the total thickness of

one of the two laminates, used as a reference. This is depicted schematically in Figure 5-1.

We will describe a laminate using the notation [11/13/12/14/.. 12]s, where li denotes a

layer of type i. For example, two laminates are given as follows:

Laminate 1 [1l / /i2 ./in
Laminate 2 [1Ij1 /j2 jml







76


where n and m are the total number of lamina in the first and second laminate,

respectively. We denote by hi(1) the thickness of layers with type 1, in Laminate 1, and by

h2(l) the thickness of layers with type 1, in Laminate 2.

The thickness of common layers hc(l) of type 1, is then

hc(l) = Minimum{h,(l,), h2(li)} (5-1)

Then, a one-sided composition continuity measure, referred to the first panel is

defined as


cc sum{min{n,,m,}} (5-2)
1->2 H2
H2
Similarly, the same measure, referred to the second panel is


cc sum{min{n,,mi}} (5-3)
2->1 H2

where H, is the total thickness of Laminate 1, and H2 is the total thickness of Laminate 2.


12

13

11

11

13

11

11

14

15

12


Midplane


12

13

11

16

14

11


Laminate 1 Laminate 2

Figure 5-2: Count of the number of continuous layers of two laminates


I


1


I




*L









In contrast to the one-sided composition continuity measure, a two-sided

composition continuity measure is defined as the fraction of common layers to the thicker

laminate:

= sum{min{n,,mi}} (5-4)
max imum {H1, H2}

Stacking sequence continuity

Stacking sequence continuity is a measure of the number of layers that can be

continuous between two adjacent laminates. That is, a ply in one laminate can continue to

the next laminate if both layers are of the same type 1, and if they are separated in the

thickness direction by a small number of terminated layers. In the present work, we assume

that this separation must not exceed one layer. For example, given the two laminates

Laminate 1: s[12 15/14/ l/ 13/ 1 1/ 1/ 13/ 12]
Laminate 2: s [l/ 14/I 11/ 13/ 12]
Figure 5-2 shows how we count the number of continuous plies by assuming that the

symmetric laminates share their midplane. Note that the outermost plies of Laminate 1 and

Laminate 2, of type 12, are assumed to be terminated because they are separated by three

truncated plies.

From Figure 5-2, we observe that there are three continuous layers, and these layer

with thicknesses hi, h3 and h4. (h, is the layer thickness with layer type 1,). So the total

thickness of the continuous layers hcon, is

hcont=hl+h3+h4 (5-5)

and the one-sided stacking sequence continuity indices are calculated as


1->2 co
Hl (5-6)

Cs hcont
2->1 (5-7)
H2 (5-7)









The two-sided stacking sequence continuity measure is calculated as

C, =ont (5-8)
max imum{H1,H2}
Examples

In order to demonstrate the continuity measures, two cases are selected. The fiber

orientation angle set is {0, 45, 90}, the material property set is {mi, m2}, and the ply

thickness set is {0.01,0.02, 0.03 (in). The notation (a, I mj I tk)represents fiber orientation

angle a,, material property m,, and ply thickness tk, ai e {0,45,90), mj e {m1,m2},

tk e {0.01,0.02,0.03}. The total number of layer types is 18, see Table 5-1.




Table 5-1: Definition of layer type and its layer code for examples
Layer code Layer type Layer code Layer type Layer code Layer type

11 (0/mi/0.01) 17 (450/mi/0.01) 113 (900/mi/0.01)

12 (00/mi/0.02) 18 (450/mi/0.02) 114 (900/mi/0.03)

13 (00/mi/0.03) 19 (450/mi/0.03) 115 (900/mi/0.03)

14 (00/m2/0.01) lio (450/m2/0.01) 116 (900/m2/0.01)

15 (00/m2/0.02) 111 (450/m2/0.02) 117 (900/m2/0.02)

16 (00/m2/0.03) 112 (450/m2/0.03) 118 (900/m2/0.03)


Material composition continuity and stacking sequence continuity indices are listed

in Table 5-2. Detailed stacking sequences of two laminates are shown in Figure 5-3.

Case 1

Laminate 1 s[1121 15\1171 15\1 16]
Laminate 2 s [1412\1156]
Case 2










Laminate 1
Laminate 2


116
----li7 ------
117
115
112


s[i 17\l 0\113\ 4\1 16 11 14 101 17\ 4\4\113\ 18\12]
s[li1812\12\1 18\15V11\ 8\116\1 13 1A13]

Midplane


V


16
-15
----if-------
12
14


-_ S I I


Laminate 1


12
118
- i- -3

14
14
117
110
114
----1---------

116
14
113
110
117


Laminate 2


- -
---hi


13

113
110
116
18
---111

15
118
12
12
S1


Midplane


Laminate 1 Laminate 2
(B)
Figure 5-3: Details of stacking sequence continuity: (A) Case 1; (B) Case 2









Table 5-2: Composition and stacking sequence continuity indices for two laminate
examples
Number of Number of Number of
Common
Case No. Layers/ Layers/ Materials continuous
Thickness Thickness layers/Thickness of
(Thickness)
(Laminate 1) (Laminate 2 ) iecontinuous layers
1 5/0.12 4/0.09 0.03 1/0.03
2 14/0.21 11/0.20 0.11 2/0.03
Type of One-sided Two-sided One-sided Two-sided stacking
continuity composition composition stacking sequence
sequence
1 C1>2=25% 25% C1->2=33.33% 25%
C2->1=33.33% C2->1=25%
2 C1>2=52.4% 52.4% C21>2=15% 14.3%
C2->1=55% C2->1=14.3%


Minimization of composite wing weight with continuity constraints

The composite wing structure considered here is an unswept and untapered wing

box with four spars and three ribs, with a total of 18 skin panels, shown in Figure 4-3. The

wing box is clamped at the root and subject to the tip load distribution shown in Figure 4-

3. All the panels are symmetric and balanced laminates made of graphite-epoxy T300/5208

whose material properties are shown in Table 3-2. Ply thickness is fixed at 0.005 in, and

the fiber orientation angle is selected from a small set {0, 45o,90}. So continuity of

laminates is calculated only considering fiber orientation angles.

The optimization of the composite wing is performed using a two-level

optimization procedure using response surfaces for communication between the two levels

(Chapter 4). The upper skin panels are substantially thicker than the lower skin panels due

to buckling constraints. After an overall wing design is obtained that defines the number of

0, 450 and 900 plies for each panel, a genetic algorithm is used to obtain the stacking

sequence of each panel.









The formulation of the minimization of the wing weight with continuity constraints

is expressed as follows:

18
Minimize X(n + n" + no) (5-9)
i=1

where i is the panel number, by changing no, nj n45 i=1,..., n, subject to:

Laminates are symmetric and balanced

Strain failure load constraints: X, > 1 (5-10)

Buckling load constraints: A;i (nn ,no, N', N'.N) > 1 (5-11)

Continuity constraints: cpj > x% (5-12)

The design variables are the number of 00, 450 and 900 stacks (no, n45, and n90) in

each panel. The objective function is the total weight of 18 composite panels that is

proportional to the sum of the number of 00, 450 and 900 stacks in all the panels.

The strain failure load is calculated by a finite element (FE) analysis using

GENESIS (Vanderplaats 1997). Buckling loads are approximated through response

surfaces fitted to the results of multiple panel optimizations that maximize the buckling load

by changing the stacking sequence of the panels. Continuity constraints for multiple panels

Ci_>j are calculated for given amounts of 00, 450, and 900 stacks of two laminates.

Additional details about the continuity constraints are given in following sections for the

wing box problem. Because the two-sided continuity constraints are non-smooth, only one-

sided continuity constraints are considered.
























Figure 5-4: Lower skin panels



One-sided Continuity Constraints for Multiple Composite Panels

Continuity constraints are applied to each pair of adjacent panels in the wing. For

the nine lower skin panels, the total number of continuity constraints is 24, and the same

numbers of constraints are used for the nine upper skin panels. For example, Panel 1 has

two neighbors: Panels 2 and Panel 6. So the two continuity constraints for Panel 1 are:

su mf m Panel 1 Panle2
C >2 sum{mnin{n 'l,m anl}} (5-13)
1->2 H,
su mf m f Panel 1 Panel 6 ""
C >6= sum{min{n[ ,m 6}} (5-14)

Similarly,

Panel 2: C2->, C2->5 and C2->3

Panel 3:C3->2 and C3->4

Panel 4: C>3, C4->5 and C4>9

Panel 5: C5->2, C5->4, C5-6 and C5->8

Panel 6: C6->1, C6->5 and C6->7

Panel 7: C7-6 and C7->8

Panel 8: C8->5, C_->7 and C_->9









Panel 9: C9->4 and C9->s




Table 5-3: Minimum weight of composite wing versus required average (over all panels)
continuity
Continuity
Rquir nt X Weight Average continuity
Requirement X
50% 323.64 78.87%

55% 324.70 79.77%

60% 326.40 80.94%

65% 330.68 83.22%

70% 336.23 85.95%

75% 342.83 87.36%

80% 358.23 89.27%

85% 378.32 91.80%

90% 405.81 95.27%

95% 432.08 97.77%

98% 454.19 99.00%

99% 462.43 99.40%




Similarly for the 9 upper skin panels,

Panel 10: Clo->11 and Clo->15

Panel 11: Cn->1o, C11->12 and C11->14

Panel 12: C12->11, C12->13

Panel 13: C13->12, C3->14 and C13->18










Panel 14: C14->11,4->13 ,C14->15 and C14->17

Panel 15: C15->1o,C15->14 and C15->16

Panel 16: C16->15 and C16->17

Panel 17: C17->16, C17->14 and C17->1s

Panel 18: C18->17 and C18->13


110

S105

S100-
W 9999".,
o 6 <5"o 98"1,
C 5
0








minimum number of stacks.
0- 85-- --71-- 0 ---

0
| ) 8 0 657011------------------
C1 4 -55114
S75

70-
315 365 415 465
Minimum number of stacks ( Proportional to weight)

Figure 5-5: Average abscissaa) and required (numbers on graph) material continuity vs.
minimum number of stacks.



Weight-continuity tradeoffs

The design variables of this problem are the numbers of 00, +450 and 900 stacks for

each of the 18 wing panels. So the number of design variables is 54. If we constrain all

nine lower skin panels to have the same laminate and all nine upper skin panels to have

same laminate, we obtain a six-design variable problem. For this six-variable problem,

there are no discontinuities between adjacent laminates, so that the continuity indices are









100%. The weight of the wing provides one extreme to the tradeoff between weight and

continuity for X=100% in Eq. (5-9). The weight of the wing with 54 design variables and

no continuity constraints provides another extreme to this tradeoff. As will be shown

below, this design can be obtained without continuity constraints, or also with X=50%.

Table 5-3 shows the average composition continuity and minimum weight for

different continuity requirement value X The average continuity is taken over all the panels

in the wing. The information is illustrated graphically in Figure 5-5.

From Table 5-3 and Figure 5-5, we observe that increasing the required continuity

up to 70% requires only about 4% increase in weight, increasing it from 70% to 80%

requires about an additional 7% increase in weight, and increasing it from 80% to 90% an

additional 13%. Thus it appears that substantial increases in continuity are available at

little additional weight, but beyond a certain point it becomes very expensive.

Stacking Sequence Design

After the global level optimization, continuous global optima need to be rounded to

integer optima and GAs are employed to obtain the stacking sequence design of individual

panel. Stacking sequence continuity constraints may be included in panel level design. In

the present work, these constraints are not included, and instead we simply observe what

level of stacking sequence continuity is achieved without these constraints. The procedure

is to first round the continuous optima of numbers of 00, 450, and 900 stacks and obtain

the stacking sequence from the GA. The rounding of ply stack numbers will normally cause

internal panel load redistribution and cause some of the buckling and strain constraints to

be violated. So, manual adjustment of integer optima is necessary, see Chapter 4.










Table 5-4: GENESIS, rounded and adjusted optima for composition continuity requirement
of 50%


Panel No.
Lower skin

1

2

3

4

5

6

7

8

9

Upper skin panels
10

11

12

13

14

15

16

17

18

Optima


Genesis optima

no/n45/n90

4.61/0.00/0.00

2.93/0.00/0.04

1.47/0.00/0.01

2.11/0.00/0.00

4.10/0.00/0.00

6.97/0.00/0.00

12.97/0.00/0.00

8.04/0.18/0.001

3.93/0.28/0.00

no/n45/n9o

0.52/18.79/9.42

0.35/26.72/0.77

0.03/18.74/0.42

16.83/20.20/0.75

0.01/26.21/6.55

19.42/15.18/8.81

12.35/17.15/12.8

0.79/31.93/3.26

15.07/23.30/0.00

323.64


Rounded optima

no/n45/n90

5/0/0

3/0/0

1/0/0

2/0/0

4/0/0

7/0/0

13/0/0

8/0/0

4/0/0

no/n45/n9o

1/19/9

0/27/1

0/19/0

0/20/0

0/26/7

19//15/9

12/17/13

1/32/3

2/23/0

323


Adjusted optima

no/n45/n90

5/0/0

3/0/0

1/0/0

2/0/0

4/0/0

7/0/0

13/0/0

8/0/0

4/0/0

no/n45/n9o

3/19/9

2/23/1

2/17/0

0/19/1

0/25/7

19/15/19

12/17/13

1/33/3

2/24/0

324


Average
Composition
Continuity


78.87%


78.10%


77.79%









Table 5-5: Stacking sequence and average continuity of nine upper skin panels for
composition continuity requirement X=50%
Panel No. Stacking Sequence
10 [02/45/02/904/(45)2/902/(902/45)2/45/(+45/902)2/
(+45)7/(45/902)2/(45)3]
11 s [02/45/02/(45)14/902/(45)9]
12 s[02/(45)18]
13 s[902/(45)19]
14 s[902/(45/904)2/(45)2/902/(45)4/902/(45)17]
15 [02/902/04/902/04/45/(04/902)4/(04/45)2/04/45/04/(45)9/
902/(45)3]
16 s [902/02/(904/02)5//)/(902/04)2/45/04/(45)16]
17 s [02/(45)3/902/(45)13/902/(45)16/902/45]
18 s[02/45/02/(45)23]
Average Stacking 56.52%
Sequence Continuity


For each panel, the average material and stacking sequence continuity measures are

computed. The average is taken over all neighboring panels. Two design cases are selected

here for comparison purposes. One is the material continuity requirement at 50%. The

other is the material continuity requirement at 85%. Table 5-4 shows GENESIS optima

(continuous optima), rounded optima and manual design. Table 5-5 lists the detail stacking

sequence of 9 upper skin panels at material continuity 50% and gives the average stacking

continuity for 9 upper skin panels.

Table 5-6, 5-7 show results of GENESIS optima, rounded optima, adjusted optima

and corresponding to stacking sequence, average stacking sequence continuity at material

continuity requirement 85%. Here, the stacking sequence continuity mentioned is two-sided

stacking sequence continuity.










Table 5-6: GENESIS, rounded and adjusted optima for composition continuity requirement
X=85%


Panel No.


Lower skin panels

1

2

3

4

5

6

7

8

9

Upper skin panels
10

11

12

13

14

15

16

17

18

Optima

Average Composition
Continuity


Genesis optima

no/n45/n9o

7.56/0.00/0.00

6.41/0.07/0.00

5.43/0.07/0.00

6.41/0.07/0.00

7.55/0.07/0.00

8.91/0.07/0.00

10.49/0.06/0.00

8.90/0.07/0.00

7.55/0.07/0.00

no/n45/n90
12.44/6.36/10.25

10.40/6.37/7.00

10.44/4.21/6.82

10.40/4.21/11.30

10.37/6.36/12.28

16.60/6.33/12.28

11.57/7.50/15.96

6.48/7.51/14.70

6.51/5.99/12.35

378.32


91.80% 91.23%


Rounded
optima
no/n45/n9o
8/0/0

6/0/0

5/0/0

6/0/0

8/0/0

9/0/0

10/0/0

9/0/0

8/0/0

no/n45/n90
12/13/10

10/13/7

10/8/7

10/8/11

10/13/12

7/13/12

12/15/16

6/15/15

7/12/12

375


Adjusted
optima
no/n45/n90
8/0/0

6/0/0

5/0/0

6/0/0

8/0/0

9/0/0

11/0/0

9/0/0

8/0/0

no/n45/n90
11/13/11

10/13/7

10/8/7

10/8/11

10/13/12

17/13/12

12/15/16

6/15/15

7/12/12

376


90.89%




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