TWOLEVEL 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 muchneeded 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 VuQuoc, 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 ................................................................ .. ......... ............. .
Twolevel Optimization for Composite Wing Structures ..............................................3
Genetic Algorithm s for Panel Optim ization............................................... ...................5
Material Composition Continuity Constraints Between Adjacent Panels ...................6
SingleLevel 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
Twolevel Multidisciplinary Optimization............................. .............. 16
Composite Wing Structural Design by Twolevel 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 eneR 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
TWOLEVEL COMPOSITE WING STRUCTURAL OPTIMIZATION USING
RESPON SE SURFACES ................................................................ ............... 54
Intro du ctio n ......... ...... ..... .. . ...... .. ............................................... 54
Tw olevel Optim ization A pproach......................................... .......................... 55
TwoLevel Optimization Procedure .............. ................................. 55
PanelLevel Optimization and Response Surface........... ....................................58
W ingLevel 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
Sixvariable D esign Problem .......................................................................65
18variable D esign Problem ............................ ...................... .............. .... 66
54variable 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
Onesided Continuity Constraints for Multiple Composite Panels .........................82
Weightcontinuity tradeoffs ........... ..... ............... ............... 84
Stacking Sequence Design ......... .................................... ................ ............... 85
Concluding Remarks and Future Work.................... .................... ...............89
SINGLELEVEL 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 Singlelevel and Twolevel Approaches for Composite Wing
D e sig n ................................................................................................... 1 0 6
S ix v ariab le C ase .............................................................. ........... ... ....... 10 6
18variable C ase .................................................................. 108
54v 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
31: Coefficient pI for shear buckling load factor (Whitney 1985)................................... 32
32: Material properties of graphiteepoxy T300/5208.....................................................34
33: Weighted generank values of child averaging generank values of two parents........... 39
34: 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
35: Comparison of computational efficiency of the three GAs................. ............. .....42
36: T three tick test lam inates............................................................................... ..... .. 47
37: Comparison of computational efficiency of the three GAs for three thick laminates.....47
38: Optimum layup for the three thick laminates........................... ...............49
39: Computational cost of laminate repair and chromosome repair..............................52
41: Allowable strains and safety factor ................ .. ......... ...................... 59
42: Statistics of three optimal buckling load response surfaces.................................63
43: GENESIS and rounded optimal design with six variables ............ .......................65
44: Rounded and adjusted upper panel design with six variables................................. 65
45: GENESIS and rounded optimal design with 18 variables............................ .........67
46: Rounded and adjusted lower skin panels with 18 variables ................................67
47: Rounded and adjusted upper skin panels with 18 variables................................68
48: GENESIS and rounded optimal design for 54 variables ...... ..................................69
49: Comparison of rounded and adjusted designs for lower skin panels, 54 variables.......70
410: Comparison of rounded and adjusted designs for upper skin panels, 54 variables .....71
411: Stacking sequences of winglevel panels for the adjusted design ............................ 71
51: Definition of layer type and its layer code for examples.............................................78
52: Composition and stacking sequence continuity indices for two laminate examples ...... 80
53: Minimum weight of composite wing versus required average (over all panels)
c o n tin u ity ...................................................................... ............... 8 3
54: GENESIS, rounded and adjusted optima for comp position continuity requirement of
5 0 % ......................................................... ......... ....................................... .... 8 6
55: Stacking sequence and average continuity of nine upper skin panels for composition
continuity requirement X=50% ........................ ..................87
56: GENESIS, rounded and adjusted optima for composition continuity requirement
X=85% ............. .......... .......... ......... ............................... 88
57: Stacking sequence and average continuity of nine upper skin panels for composition
continuity requirement X=85% ........................ ..................89
61: Definition of medium and thick laminates and applied loads .............. ................. 100
62: Comparison of optimal buckling loads by continuous optimization based on W,*, W3
and by GA for square laminates defined in Table 61 ................................ 101
63: 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
64: Comparison of maximized buckling loads of continuous variable approach based on
W{, W3 with GA for 4stack laminates........................................................... 104
65: Comparison of wing design results between twolevel RS approach, and singlelevel
method with lamination parameters. Upper skin is one laminate and lower skin
is one lam inate ............................................................................ ..... 107
66: Adjusted designs for twolevel RS approach, and singlelevel method with lamination
parameters. Upper skin is one laminate and lower skin is one laminate........... 108
67: Comparison of the results of GENESIS and rounded optimal design between the two
level RS approach and the singlelevel method with lamination parameters. The
upper skin has three laminates and the lower skin has three laminates ............ 110
68: Comparison of results of rounded and adjusted lowerskin panels between twolevel
RS approach and singlelevel method with lamination parameters. The lower
skin has three lam inmates ......................... ........................ ... .......... 111
69: Comparison of results of rounded and adjusted upper panels between twolevel RS
and singlelevel method with lamination parameters. The upper skin has three
la m in ate s..................................................... ............... 1 12
610: Summary of rounded and adjusted optima of the two approaches. The upper skin has
three laminates and the lower skin has three laminates.............................. 113
611: Comparison of the results of GENESIS and rounded optimal design between two
level RS approach and the singlelevel method with lamination parameters. The
upper skin has nine panels and the lower skin has nine panels .................... 113
612: Comparison of rounded and adjusted designs for lowerskin panels between the two
level RS approach and the singlelevel method with lamination parameters. The
lower skin has nine panels.................................................. ....... ................ 114
613: Comparison of the results of rounded and adjusted designs for nine upperskin panels
between the twolevel RS approach and the singlelevel method with
lamination parameters. The upper skin has nine panels ................................... 115
614: Summary of rounded and adjusted optima of the two approaches ...........................16
615: Comparison of stacking sequences of nine upperskin panels for the rounded design
for the twolevel RS approach and the singlelevel method with lamination
parameters. The upper skin has nine panels........... ... ................................. .. 117
616: Comparison of stacking sequences of the nine upperskin panels for the adjusted
design between the twolevel RS approach and the singlelevel method with
lamination parameters. The upper skin has nine panels.............. ........... 118
LIST OF FIGURES
Figure Page
21: M ultilevel approach ............................... ..... ....... .. ........... .......................... ... 14
22: Tw olevel M D O problem ............................................................ ........................... 17
31: Composite laminate plate geometry and loads ....................................................... 30
32: Reliability versus number of generations for five loading cases: Case (1) and Case
(2 ) ....................................................................................... 4 4
32: Reliability versus number of generations for five loading cases: Case (3) and Case
(4 ) ....................................................................................... 4 5
32: Reliability versus number of generations for five loading cases: Case (5)................... 46
33: Reliability versus number of generations for the three thick laminates: Case (6).......... 46
33: Reliability versus number of generations for the three thick laminates: Case (7)............ 47
33: Reliability versus number of generations for the three thick laminates: Case (8).......... 48
41: Response surface interface of twolevel optimization................... .............................. 56
42: Flowchart of twolevel optimization procedure .................................................... 56
43: W ing box structure ..................................................... ........... ............... 60
43: History of the objective function and maximum violation of normalized constraints for
sixvariable case. .................... ................. .................. .............. 63
45: History of the objective function and maximum violation of normalized constraints for
18variable case. ...................................................... .............. 66
46: History of the objective function and maximum violation of normalized constraints for
54variable case. ...................................................... .............. 68
51: Com m on layers of two lam inmates ................................................................. .... 74
52: Count of the number of continuous layers of two laminates................... ........ ...... 76
53: Details of stacking sequence continuity: (A) Case 1; (B) Case 2............................. 79
54: Low er skin panels .................. .................. ................. ........... ....... ....... 82
55: Average abscissaa) and required (numbers on graph) material continuity vs. minimum
num ber of stacks. ................................ ................................ 84
61: Ply geom etry in a lam inate ........................... .. ................ ................................. ....... ... 93
62: Bending lamination parameter domain............................... ................................. 95
63: Laminates with all plies of the same orientation stacked together............................. 96
64: Six laminates corresponding to the six vertices of a hexagonal domain...................... 96
65: Hexagonal domain of variation of flexural lamination parameters when the number of
plies of each orientation is specified ............................. ............................. ... 99
A1: Tune GA operator parameters: (A) shows the effect of population size...................... 122
A1: Tune GA operator parameters: (B) displays effect of probability of crossover........ 123
A1: Tune GA operator parameters: (C) shows effect of probability of mutation. ............... 123
B1: An Orthotropic lamina with offaxis principal material directions ........................... 124
Cl: 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
TWOLEVEL 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
twolevel optimization approach with responsesurface approximations to optimize panel
failure loads for the upperlevel 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 lowerlevel 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 twolevel optimization procedure for composite wing design subject to strength
and buckling constraints is presented. At winglevel 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 winglevel
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 twolevel optimization to a singlelevel optimization
based on flexural lamination parameters. The singlelevel optimization is efficient and
feasible for a wing consisting ofunstiffened panels.
CHAPTER 1
INTRODUCTION
Introduction
Because of higher stiffnesstoweight or strengthtoweight 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; SobieszczanskiSobieski &
Leondorf 1972, SobieszczanskiSobieski et al. 1985). Multilevel structural optimization in
general consists of lowerlevel optimization for substructures and coordination
optimization, which exchanges information among the lowerlevel 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 SobieszczanskiSobieski (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 lowerlevel optima
are nonsmooth and noisy functions of upperlevel 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.
Twolevel Optimization for Composite Wing Structures
Because of the high computational cost of singlelevel 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 paneldesign 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 locallevel (panel) and upperlevel
(wing) optimization problems is currently handled largely by adhoc methods that are
cumbersome and sub optimal. As noted in the previous section, twolevel optimization can
be illconditioned because the lowerlevel (panel) optima are not smooth functions of
upperlevel (wing) design variables.
Response surface (RS) techniques offer an attractive way of overcoming both
implementation and smoothness difficulties. The RS approach to a twolevel optimization
is to perform a large number of lowerlevel optimizations for different values of the global
design variables and loads. Then the results of the lowerlevel optimizations are fitted with
response surfaces, typically loworder polynomials. Finally, upperlevel optimization is
performed with RS substituting for lowerlevel optimization. This approach eases
implementation problems associated with software integration, as lowerlevel optimization
program can be run independently of the upperlevel analysis and optimization. In addition,
the RS smoothes out the lowerlevel 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 inplanestiffness constraints specified by the winglevel
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 twolevel
aerodynamicstructural 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 panellevel 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 paneloptimization problem. Here we overcome this
problem by maximizing the loadcarrying capacity of the panel for a fixedweight 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 stackingsequence 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, stackingsequence 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, penaltyfunction 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 semiindependently (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 tradeoff studies
between weight and continuity are performed
SingleLevel Optimization of Composite Wing Based on Flexural Lamination Parameters.
While the twolevel 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 twolevel 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,
twolevel optimization approach was used for overall wing design, with response surfaces
generated from panel optima used as an interface to couple with winglevel 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 twolevel 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 twolevel 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 continuousoptimization procedure based on flexural lamination
parameters and use them to test the optimality of the twolevel optimization results.
Contents
Chapter 2 provides a literature review of twolevel 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 onelevel 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 designload 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 lumpedmodel components is then carried out by mathematical programming with sizing
design variables (e.g., skin thickness). This is essential by a twolevel approach.
SobieszczanskiSobieski and Leondorf (1972) developed a mixedoptimization 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 SobieszczanskiSobieski
(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
systemlevel and componentlevel 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 singlelevel optimization can be stated as follows:
Minimize W(S_)
such that Gq(S,) >O; qQ (
(21)
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 (21) is decomposed into a multilevel optimization problem
as follows:
At system level,
Minimize W(S)
(22)
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
(23)
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 (23) essentially was a quadratic penalty function to enforce
the equality constraints Kj (S) = K, (1i)
Figure 21: 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 hatstiffened
fibercomponent panels.
SobieszczanskiSobieski (1985) developed a more general multilevel approach.
Compared with Schmit's approach, SobieszczanskiSobieski 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 KreisselmeierSteinhauser (KS) function,
KS(g) =iln Iexp(pgj) (24)
P 1
where gj is a local constraint, p is userdefined 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 systemlevel 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
systemlevel constraints to assure satisfaction of all of the local constraints.
In both Schmit's and SobieszczanskiSobieski'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 singlelevel formulation without equality constraints. Second, equality
constraints were introduced to create a hierarchical structure, but the problem was still
solved as a singlelevel problem. Finally, a twolevel approach that took advantage of the
hierarchical structure was used. It was found that the twolevel formulation solution was
sensitive to the optimization parameters but the onelevel 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
twolevel optimization approach to multidisciplinary design in the next section.
Twolevel 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 lowlevel 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 twodiscipline 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 22: Twolevel 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) (25)
Similarly,
a2 = A2(a(2,12,t ) (26)
Reformulation in terms of CO can be expressed as follows:
Systemlevel optimization which coordinates two disciplinary design as
Minimize f(s,t,,t,)
S l t2 (27)
subject to C(s, tl,t) = 0
where Cis called Ninterdisciplinary consistency constraints. C = {cl ,c2} is shown
below; tl and t2 are systemlevel targets of al and a2.
The systemlevel problem issues design targets (s, ti, t2) to the constituent
disciplines. In the lowerlevel problem, the disciplines must design to match these targets
as follow:
1 I 1
minimize c, + a1(a ,l1 ) t
Discipline 1: o,, 2 (28)
subject to g, (oi, 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 (29)
subject to g2(a2, l2, a2 (ao2,12,t )) > 0
where o2 is the target of s at discipline 2.
Another twolevel MDO approach, Current Subspace Optimization (CSSO)
introduced by SobieszcazanskiSobieski (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)
(SobieszcazanskiSobieski 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 nonhierarchic
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 systemlevel 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 systemsensitivity 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 optimumsensitivity 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 optimumsensitivity 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 SobieszcazanskiSobieski & 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 KarushKuhn
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 systemlevel problems that are nonlinear even when the original
problem is linear. These features make it difficult for conventional optimization algorithms
to solve the CO systemlevel 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 lowerlevel optima as function of
upperlevel 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 elementarystrength or elementstiffness 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 winggroup 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 upperlevel
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 twolevel
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 twolevel 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 Twolevel Optimization Using Response Surfaces
Thinwalled box beams are extremely efficient structures. The thinwalled 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
wingbox structure belongs to a generic and simplified prototype of this kind of structures.
Due to higher ratios of stiffnesstoweight or strengthtoweight, 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 multicell 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 panellevel design based on loads from winglevel design. Gerard
(1960) used orthotropic plate theory to analyze idealized longplate structures
longitudinal, transverse, and wafflegrid 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 plateweight equations or plateweight tables. These weight equations
or weight tables can then be used in the overall wing design.
In Gerard's approach for panellevel design, stiffness constraints for winglevel
design are not included. This does not provide any mechanism for winglevel 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 inplanestiffness parameters. The PASCO program (Stroud et al. 1981) was
used for panel structural optimization. Paneldesign 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 optimumpanel 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 (210)
g= 1.0 <0 (210)
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 inplanestiffness
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 inplanestiffness
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 integerprogramming 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 plyidentity 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 inplate 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, numberofply 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 inplate 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, numberofply 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 generank 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 31: 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 inplane
loads and specified total number of of 00, +450, and 900 plies. The loads and the specified
number of plies come from the overall winglevel 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
31) 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 graphiteepoxy laminate composed of 00, +45,
and 900 plies.
Because of symmetry, there is no extensionalflexural coupling. The prebuckling
deformations are hence purely inplane. The balance condition requires that for every ply
with a positive fiberorientation angle, there is a corresponding ply with the negative fiber
orientation angle. This implies no normalshear extensional couplings. In addition, the
laminate is assumed specially orthotropic (i.e. there will be no bendingtorsion coupling).
This is a common assumption in the analysis of balanced symmetric laminates for which
the bendingtorsion 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 (31)
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 (32)
4pJ1 D22(D12 +2D66
S2Nx, for 0
b2 ,
F ID22 (33)
D12 +2D66
and values of p1 are given in Table 31.
Table 31: 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 (34)
;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"") } (35)
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 plycontiguity constraints, the optimization problem can be
stated as follows:
Given three ply orientation choices (0, 450, and 900), applied inplane 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 24inch square graphiteepoxy plate with the following
properties shown in Table 32.
Table 32: Material properties of graphiteepoxy 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+1i)(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 twopoint 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 twoply 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 stackswap, 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 twoply 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 (36)
where Penalty is a parameter (set to 2.0 ) for violation of specified amounts of 900, 45,
and 00 plies, and
r=ror45 r90 (37)
no +1
if no < nog
og +1 (38)
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 (39)
Pcon, is penalty parameter (set to 1.05 here) for violation of the fourply limit on
contiguous plies of the same orientation, ncont is total number of sameorientation
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
GeneRank 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.
Generank 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 generank crossover
averages the two rankings. For example, consider the simple case with three contestants,
A, B, and C. The first judge ranked them as: A1, B2, C3, denoted in shorthand as
[A B C]. The second judge ranked them as: A2, B3,C1, 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 33: Weighted generank values of child averaging generank values of two parents
RankValue in RankValue in Weighted
Gene
Permutation 1 Permutation 2 RankValue
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 33), 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 GeneRank
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 GeneRank
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 substring between the two cut points from the second parent.
3. Take genes of the two outer substrings 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 twocut 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 substring, 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
costthe 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
(310)
r(1r)
n
So that for 100 runs and r=0.63, we obtain a standard deviation of about 0.048.
Table 34: 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)
NonRounded 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 35: Comparison of computational efficiency of the three GAs
Loading Given Number of Stacks Failure Number of Analyses Required
(Rounded From Table 34) 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: GeneRank 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 35 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 35 gives results for the three GAs in terms of number
of analyses required for 80% reliability.
From Table 35, we can also see that, as expected, thicker laminates are
computationally more expensive to optimize. All the laminates in Table 35 are quasi
isotropic or close to quasiisotropic. 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 36, 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 32: 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 32: Reliability versus number of generations for five loading cases: Case (3) and
Case (4)
From Table 35 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 32. From the figure, it is clear that the two permutation GAs
46
perform much better than the standard GA. The GeneRank 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 32: 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 33: Reliability versus number of generations for the three thick laminates: Case (6)
Case (5)
G RR
SSGA
SPMx
Table 36:
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 37: 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 33: Reliability versus number of generations for the three thick laminates: Case (7)
The results summarized in Table 37 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 38. 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 34 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 33: 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 38: Optimum layup 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 twoply 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 90stack because the number of decoded 90genes
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 0stack, because the number of required 0stacks is two. The
decoding procedure then tries to see if there are available stacks for a 45stack, and when
it finds that none are available, it puts a 90stack 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 39.
From Table 39, 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 39 to
Table 37, 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 39: 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 inplane loading was investigated using genetic algorithms. A new permutation GA,
which we called a generank 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 panellevel optimization for maximum
buckling load of composite laminates. The permutation GA and its corresponding
chromosomerepair 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 inplane 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
TWOLEVEL COMPOSITE WING STRUCTURAL OPTIMIZATION USING RESPONSE
SURFACES
Introduction
The objective of this chapter is to demonstrate use of a twolevel 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 winglevel and panellevel 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 twolevel optimization procedure and summarized
formulation of panel and wing optimization and coordination of twolevel optimization.
Second, we briefly reviewed response surface methodology and discussed normalized
response surface of normalized buckling load. Then, we presented results of 6variable,
18variable and 54variable cases of wing box structure. Finally, we summarized
concluding remarks of the twolevel optimization.
Twolevel Optimization Approach
TwoLevel 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 41: Response surface interface of twolevel optimization
Figure 42: Flowchart of twolevel optimization procedure
buckling loads are used as constraints in the upperlevel (wing ) optimization. The process
is shown schematically in figure 41.
The twolevel optimization process is described by the flow chart in Figure 42.
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 Doptimal 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 winglevel 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 winglevel optimization converges, the plynumber 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.
PanelLevel 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 panellevel 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.
WingLevel Optimization
The objective function for the winglevel 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) (41)
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 (42)
(Buckling) i' (n', n45, N, N ,N', ) >1.0, i= 1,,n (43)
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 41: 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 43.
All the panels are symmetric and balanced laminates made of graphiteepoxy
T300/5208, with material properties given in Table 32. The allowable strains and safety
factor used are given in Table 41. 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 43: 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 (44)
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, Doptimal design is widely
used. The Doptimality 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 Doptimal 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 (45)
(no + n45 + n90)3
The number of stacks and loads were also normalized.
62
ro = no
no + n45 + n90
Sn45 (46)
no +n45 +n90
r9 = 1.0 r r45
 2Nx Nxmax .xmin
2Ny Nymax Nymin (47)
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) (48)
Results of Response Surfaces of Optimal Buckling Loads
For the wing shown in Figure 43, 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 (410)
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 (411)
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 (412)
5 n,, 35 (412)
5 < n,0 20
Table 42: 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 (49), (410), (411), and (412), and then 180 Doptimal 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 42, where R and Ra are the coefficients of multiple determination and its
adjusted value, respectively.
0 2 4 6 8 10 12 14
Figure 43: History of the objective function and maximum violation of normalized
constraints for sixvariable 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 twolevel optimization procedure is demonstrated through
sixvariable, 18variable, and 54variable design problems.
Sixvariable Design Problem
For this case, all upperskin panels are the same and all lowerskin panels are the
same, so that the design variables are no, n45, and ngo for the lower skin and the upper skin.
Table 43 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 44 shows the history of the objective
function and the maximum violation of normalized constraints during the winglevel
optimization.
Table 43: 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 44: 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 45: History of the objective function and maximum violation of normalized
constraints for 18variable case.
At the optimum, upper root panel 16 is active in buckling, and one strain constraint
is active at lower root panel 7. Table 43 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 reoptimized 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 44 shows the buckling load
and stacking sequence after manual adjustment. This adjustment increased the total number
of stacks to 477.
18variable Design Problem
For the 18variable design, each wing skin is divided into three regions: root
panels, intermediate panels, and tip panels. Each region has three stack design variables.
Table 45: 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 45 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 46 and 47.
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 46: 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 46: History of the objective function and maximum violation of normalized
constraints for 54variable case.
Table 47: 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 48: 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
54variable Design Problem
For the 54variable design case, each of the panels was permitted to have its own
three design variables. Table 48 shows the GENESIS and rounded designs, and indicates
that one strain constraint and one buckling load constraint are violated. Tables 49 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 411 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 49: 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 410: 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 411: Stacking sequences of winglevel 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 twolevel 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 winglevel 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 semiindependently (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 tradeoff studies between weight and continuity are performed.
Common materials
Figure 51: 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 51.
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)} (51)
Then, a onesided composition continuity measure, referred to the first panel is
defined as
cc sum{min{n,,m,}} (52)
1>2 H2
H2
Similarly, the same measure, referred to the second panel is
cc sum{min{n,,mi}} (53)
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 52: Count of the number of continuous layers of two laminates
I
1
I
*L
In contrast to the onesided composition continuity measure, a twosided
composition continuity measure is defined as the fraction of common layers to the thicker
laminate:
= sum{min{n,,mi}} (54)
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 52 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 52, 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 (55)
and the onesided stacking sequence continuity indices are calculated as
1>2 co
Hl (56)
Cs hcont
2>1 (57)
H2 (57)
The twosided stacking sequence continuity measure is calculated as
C, =ont (58)
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 51.
Table 51: 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 52. Detailed stacking sequences of two laminates are shown in Figure 53.
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 53: Details of stacking sequence continuity: (A) Case 1; (B) Case 2
Table 52: 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 Onesided Twosided Onesided Twosided 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 43. 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 graphiteepoxy T300/5208
whose material properties are shown in Table 32. 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 twolevel
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) (59)
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 (510)
Buckling load constraints: A;i (nn ,no, N', N'.N) > 1 (511)
Continuity constraints: cpj > x% (512)
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 twosided continuity constraints are nonsmooth, only one
sided continuity constraints are considered.
Figure 54: Lower skin panels
Onesided 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}} (513)
1>2 H,
su mf m f Panel 1 Panel 6 ""
C >6= sum{min{n[ ,m 6}} (514)
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, C56 and C5>8
Panel 6: C6>1, C6>5 and C6>7
Panel 7: C76 and C7>8
Panel 8: C8>5, C_>7 and C_>9
Panel 9: C9>4 and C9>s
Table 53: 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 55: Average abscissaa) and required (numbers on graph) material continuity vs.
minimum number of stacks.
Weightcontinuity 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 sixdesign variable problem. For this sixvariable 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. (59). 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 53 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 55.
From Table 53 and Figure 55, 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 54: 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 55: 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 54 shows GENESIS optima
(continuous optima), rounded optima and manual design. Table 55 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 56, 57 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 twosided
stacking sequence continuity.
Table 56: 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%
