stringer cross-section types

8/2/2019 Stringer Cross-section Types

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Optimum topological design of simply supported compositestiffened panels via genetic algorithms

Luigi Iuspa a, Eugenio Ruocco b,*

a Aerospace and Mechanical Engineering Department, Second University of Naples, Via Roma 29, 81031 Aversa (CE), Italyb Civil Engineering Department, Second University of Naples, Via Roma 29, 81031 Aversa (CE), Italy

Received 3 September 2007; accepted 14 February 2008Available online 14 April 2008

Abstract

In the present paper the topological optimal design of isotropic/orthotropic thin structures performed via genetic algorithms is shown.Examples involving structural weight minimization under compressive load or buckling load maximization are presented.

A modified finite strip method was developed and used to analyze parametric structures arranged in form of plates or stiffened panelswith almost arbitrary cross-section shapes. Specific design variables were defined to assure a robust control over geometrical and topo-logical features. In particular, a semi-analytical formulation for the determination of eigenvalues and eigenvectors was adopted in orderto reduce computational efforts requested by the optimization task.

A mesh-independent solver, involving a reduced number of degrees of freedom, was implemented and interfaced with a genetic opti-mizer for the purpose. The optimization procedure was based on a specific bit-masking oriented genetic algorithm, able to handle inparallel different genetic operators expressly conceived to process with proper metrics discrete and continuous design variables. As pre-liminary example, the buckling load maximization of a metallic plate with an arbitrary grid-shaped cross-section is described first. Then atopological optimization concerning the weight minimization of a composite stiffened panel subject to constraint about buckling load isillustrated and discussed in detail about parametric model definition and genetic procedure. 2008 Elsevier Ltd. All rights reserved.

Keywords: Composite panels; Buckling load; Semi-analytical methods; Optimization; Genetic algorithms; Bit-masking method

1. Introduction

The use of high-performance composite materials in sev-eral branches of aerospace and mechanical engineering has

brought about the development of several numerical proce-dures for the optimal design of composite stiffened panels.The main advantage of employing stiffened panels in aero-space structures lies in achieving an effective, lightweightdesign; while stiffeners add negligible weight to the overallstructure, their influence on strength and stability is veryconsistent. However, composite panels still remain thin/slender structures subject to buckling phenomena under a

compressive load; consequently, optimisation proceduresconcerning buckling load maximisation or weight minimi-sation with buckling load constraints, have received con-siderable attention in the last years, as demonstrated by a

number of publications focused on this topics [15].Structural optimisation procedures based on geneticalgorithms (GAs) are computationally expensive, and amajor task is to reduce their computational cost: in Nagen-dra et al. [6] the effects of various modifications applied to abasic genetic algorithm are analysed, in order to improveboth reliability and computational efficiency for the mini-mum weight design of stiffened panels subject to compres-sive load. Kogiso et al. [7,8] considered informationretained from past design to construct a meta-functionapproximation for the buckling load. Although effective

0045-7949/$ - see front matter 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2008.02.001

* Corresponding author. Tel.: +39 081 5010266; fax: +39 081 5037370.E-mail address: [email protected] (E. Ruocco).

www.elsevier.com/locate/compstruc

Available online at www.sciencedirect.com

Computers and Structures 86 (2008) 17181737
mailto:[email protected]:[email protected]


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about computational effort reduction, this approach high-lighted, on the other hand, a reduced capability to find iso-lated singular optima.

Dimou and Koumousis [9] introduced competitionamong the populations of a number of GAs having differ-ent sets of parameters. The co-evolution of embedded sub-

populations (metapopulation) improves the attitude of thealgorithm to find global optima with respect to a standardGAs. The authors demonstrated the efficiency of the pro-posed method with examples related to the optimal designof simple truss structures. In [10] a non-linear Finite Ele-ment Method (FEM) code is used to analyze the bucklingand post-buckling behaviour of composite structures foroptimisation purposes. To counteract the resulting largeamount of computing time, a parallel processing techniqueand a reduced number of design variables was adopted. In[11] the layout of fiber-reinforced laminates is decomposedinto reduced sub-structures with constrained local bound-ary conditions. However, the resulting stacking sequence

highlights some manufacturing difficulties due to localoptima convergence, and the need for a more robust, glob-ally blended solution.

In the present paper the structural optimization per-formed via genetic algorithms of composite stiffened pan-els/plates involving buckling load is presented. Bothtopological and geometrical stiffness-sensitive features ascross-section type, bay length, thickness, ply compositionand orientation are considered as design variables.

According to the previously described guidelines, mainlyfocused on procedure efficiency, a special kind of geneticalgorithm, able to handle properly continuous and discrete

design set with separate and dedicated operators, was usedhere [12].

Besides genetic algorithm implementation and set-up,another critical aspect of the optimisation procedure ana-lysed here is the definition of an efficient calculus enginefor the prediction of the buckling load and related modaldisplacements. Analytical methods become very difficultto apply successfully when stiffeners are introduced in thedesign; consequently a number of numerical procedureshave been proposed to achieve a satisfactory solution ofthe problem. A numerical approach FEM based is a verycommon technique that allows a wide arbitrariness aboutgeometry and boundary conditions. However, accurate,non-linear FEM-based buckling analysis exhibits someintrinsic numerical instability and almost unacceptablecomputational costs. When the geometry of structurebecomes regular (i.e. open ruled surface), more efficienttechniques can be successfully adopted; the finite stripmethod, based on the discretization of the structure alongthe transverse direction only, is systematically employedin buckling analysis with satisfactory results [1316].

In this context a more efficient, semi-analytical formula-tion, described in a detailed form in [17,18], was used. Thisprocedure has proved to predict global and local bucklingload of composite stiffened plates in good agreement with

theoretical solutions. The semi-analytical nature of the pro-

posed procedure returns exact critical load and modal dis-placements of structures with complex geometry using avery coarse mesh. On the other hand, as the model is basedon analytical solutions with related assumptions, its appli-cability is limited to specific boundary conditions (simplysupported plates) and material lay-up (symmetric ply orien-

tation). Fortunately these aspects represent a minor limita-tion because such a type of boundary conditions and lay-upsequences are typical of a wide variety of structural schemesinvolved in practical problems. The proposed method isvery attractive because of its computational efficiency, andthis aspect is extremely remarkable for numerical-intensivetasks as the present topological structural optimization.

In the next section the proposed method, developed forthe analysis of plates and panels, is described first. Guide-lines and theoretical bases are highlighted and the govern-ing differential equations are derived. Section 3 providessome background hints on GAs, where Section 4 outlinesthe specific approach designed by bit-masking oriented

genetic algorithm (BMOGA).In Section 5 two application examples are illustrated in

order to demonstrate applicability and efficiency of theproposed methodologies. First example is a preliminarytest-case related to the optimization of an isotropic platewith a grid-shaped cross-section. Next application concernsthe topological optimisation of a stiffened composite panel.Conclusions are finally reported in the last section.

2. Bifurcation analysis

Fiber-reinforced composites are manufactured in the

form of thin sheets, called laminae, or layers (see Fig. 1).In most application, the laminate thickness is much smallerthan planar dimensions, so that two-dimensional theoriescan be applied.

Under the classical thin plate hypothesis (KirchhoffLoves theory and von Karman straindisplacementrelationship), the following in-plane and out-of-planeequilibrium equations can be obtained:

Nx;x Nxy;y 0

Nyx;x Ny;y 0

Mx;xx 2Mxy;xy My;yy Nxw;x Nxyw;xy Nyw;y

1

For sake of clarity, the definition of internal forces intro-duced in (1) are shown in Fig. 2.

Constitutive relations for filamentary unidirectionallaminae may be established assuming homogeneous ortho-tropic materials and plane stress condition [19]. The result-ing constitutive matrix is generally fully populated, withelastic constants depending on four main terms. Conse-quently, it can be written in the following form:

N

M

A B

BT D

e

v

2

where the N M T and e v T vectors represent Kirch-

hoffs plate forces and related dual kinematical quantities,

L. Iuspa, E. Ruocco / Computers and Structures 86 (2008) 17181737 1719


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respectively. The terms A, B and C present in the constitu-tive matrix are expressed by the following expressions:

A Tksk; B Tks

2k

2

; D Tks

3k

3

3

with s thickness of the kth lamina and

Tk

Ex1mxymyx

mxyEy1mxymyx

0

myxEx

1mxymyx

Ey

1mxymyx0

0 0 l

2664

3775

k

cos2 h sin2 h 2cos h sin h

sin2 h cos2 h 2cos h sin h

cos h sin h cos h sin h cos2 h sin2 h

264

375

k

where h represents the deviation angle between the current

lamina principal axis and laminate reference axis.

When ply orientations are restricted to a discrete set ofangles and the symmetry with respect to the z axis isimposed, constitutive equations (2) are uncoupled about:(i) bending, twisting and stretching (i.e. the Bij terms are

equal to zero); (ii) normal stress and shear in the middlesurface (i.e. A13 = A23 = 0; D13 and D23 negligible). Theseconditions yield to the following three partial differentialequations expressed in terms of generalised displacements,referred to the local coordinate system represented inFig. 3:

A11u;xx A12v;xy A33u;yy v;xy 0

A12u;xy A22v;yy A33u;xy v;xx 0

D11w;xxxx 2D12 2D33w;xxyy D22w;yyyy Nxw;xx

Nyw;yy 2Nxyw;xy 0

5

Fig. 1. Typical lay-up for a fiber-reinforced plate.

Ny

yxN

xN

Nxy

NxxyN

yN

NyxxM

Mxy

Mx xyM

yxMyM

M y

Myx

x

yz

Fig. 2. Generalised forces in equilibrium equations.

1720 L. Iuspa, E. Ruocco / Computers and Structures 86 (2008) 17181737


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Eq. (5) do not allow a closed-form solution; however, usinga standard separation-of-variable technique, the systemcan be reduced to a set of one-dimensional ordinary differ-ential equations, suitable for analytical solution. Assumingthe following displacement field:

ux;y UY sinX

vx;y VY cosX X xp

k; Y

yp

k

wx;y WY cosX

6

with k half-wavelength, Ny = Nxy = 0 and Nx = NL (i.e.uniaxial load), Eq. (5) can be rewritten in the followingform:

sinXA33U00 A11 NLU A12 A33V

0 0

cosXA22V00 A33 NLV A12 A33U 0

cosX D22W0000

4D12 2D33W00 D11

NLk2

p2

W

0

7

where for sake of brevity, primes denotes differentiationwith respect to Y.

Relations (7) represent ordinary differential equations inthe Y variable having the general solution:

UY k1 cosh hY /k2 cosh/Y

k3 sinh hY /k4 sinh/Y

VY hk1 sinh hY k2 sinh/Y

hk3 cosh hY k4 cosh/Y

WY k5 sinh aY k6 cosh aY k7 sin bY k8 cosbY

8

depending on the following parameters:

h 2A33 A21 A11 2NL2A33 A12 A221

12

/ A12 A11 2NLA12 A221

12

a k1 k21 k2

12

12

b k21 k212 k1

12

k1 D33D122 k2 D

122 D11 NLk

2p2

9

Coefficients k1, . . . ,k8 are fixed imposing boundaryconditions.

Using relations (8) as shape functions of the displace-ment field, the local stiffness matrix of fiber-reinforcedstructures formed by plates rigidly connected at their edges,can be defined in terms of four unknown displacementmagnitudes for each side j:

dkj W W V U k

j j 1; 2 k 1; n plate 10

with W computed deriving the out-of-plane displacementW reported in (8) respect to the Y variable (see Fig. 4).

Well-consolidated, FEM-derived routines can be thenused to assemble the overall stiffness matrix. Finally, thesolution of the associated eigenproblem provides both thesmallest critical load and related normalized displacements.

Respect to the more usual finite strip method, this pro-posed procedure adopts an exact solution as shape function

in Y direction rather than a polynomial expression, whilethe same trigonometric approximation in X direction isused. Simply supported boundary conditions along theX-side result as consequence of the required approximationin X direction.

The model is mesh-independent and a solution involvinga reduced number of degrees of freedom can be easilydefined. For stiffened plates, the dimension of the systemis about four times only the number of stiffeners. InFig. 5, a comparison between standard finite strip methodFEM and the proposed semi-analytical method is shown interms of equivalent discretization for a typical stiffened

plate.

x,u

x,u

x,u

x,u

z,w

z,w

z,w

z,w

y,v

y,vy,v

y,v

Fig. 3. Generalised displacements and local coordinate system.



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Thin-wall structures with very complex and almost arbi-trary cross-section shapes can be easily handled by this

method (see Fig. 6). On the other hand, the computational

efficiency offered by this numerical approach can be advan-tageously capitalized by an optimization procedure based

on genetic algorithms.

U

V

W

A

B

C

D

Fig. 5. (AD) Finite strip approach for a typical stiffened plate; degree of freedoms (A), part decomposition (B) and discretization (C); equivalent F.E.M.mesh for comparison (D).

W

V0

x

y

z

U

1

2

1

1

1

1

1

1

1

1W

V

U 11

1

2

2

2

y x

0

z

2

2

W21

V1

2 U12

2

1

U2

2

2W 2

V22

Fig. 4. Unknown local displacement magnitudes.



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3. Basic concepts in genetic algorithms

In the 1950s and 1960s several computer scientists inde-pendently studied evolutionary systems with the idea thatsimulated evolution of biologic entities could be used asan optimization tool for engineering problems. In Gold-bergs short history of evolutionary computation ([20],chapter 4) the names of Box [21] and Friedman [22] areassociated with works containing the rudiments of evolu-tion in various forms. All had some kind of selection ofthe fittest, some had population-based schemes for selec-tion and variation, and some others, like many GAs, hadbinary strings as abstractions of biological chromosomes.GAs, as we know them now, were early described in the1960s and fully developed by Holland [23], where for thefirst time GAs were presented as abstraction of biologicalevolution with main theoretical frameworks. In geneticalgorithms, a particular design (A) is represented by a setof characteristics ai called phenotype and defined as realor integer numbers (Fig. 7).

For each phenotype it is possible to obtain the genotypeencoding each ai into a particular alphabet (usually short as

a binary/Gray code). The transformation of the phenotypestructure into a string of bits leads to the so-called chromo-somes that represent, like in natural systems, the totalgenetic prescription for the construction and operation ofsome organism.

A genetic algorithm operates on populations of stringsand progressively modifies their genotypes to get the betterperformance of their phenotype measured through a fitnessfunction. The adaptation process is based on the mecha-nism of natural selection and natural genetics. In geneticbased evolution simulations, phenotypes express theparameter sets of alternative solutions to the problem. Thisprocess is iterated over many time steps, each of themcalled a generation. After several generations, the result isa number of high-performance chromosomes in thepopulation.

4. Genetic algorithms implementation

Since the first appearance in the late 60s, genetic algo-rithms have been subject to many implementations andvariants due to their own nature of evolutionary simulation

Fig. 6. Examples of thin-walled structures with arbitrary cross-section shape.

a4n+m+1 pa

4n+1a a4n+m

a ......a1 n

an+1 a2n a2n+1

a4n

a3n+1

.

.

.

a3n

......

Fig. 7. Phenotype example in multi-layered plate design.



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methods. Although simplified, a well-established classifica-tion subdivides binary coded genetic algorithms (bGAs) intwo main branches: explicit or implicit binary formulation.The first method is the simplest way to translate geneticrepresentation of an artificial DNA into a computer datastructure using boolean arrays for the purpose. The second

approach codifies genetic information directly at achromosomal level by means of integer arrays. The mainadvantage of an implicit representation is an increasedcomputational efficiency, as no binary-to-decimal conver-sion and vice versa is required. On the other hand, somealgebraic manipulations involving integers must be appliedin order to perform crossover and mutation operators.

In the present work a special, bit-masking oriented ver-sion of an implicit binary coded GA BMOGA was used.This novel approach keeps the computational efficiencyof an integer data structure adding some powerful features.Without going into details, classical procedural geneticoperators are replaced by a unique synthetic operator

engine able to emulate all the traditional crossover andmutation schemes, plus many others suggested by specificrequests, simply varying the arguments of a special booleanfunction. This way complex procedural schemes as multi-level crossovers operating in parallel on partitions of chro-mosomal strings can be easily obtained by the final usersimply defining an appropriate input file. An in-depthdescription and further details about BMOGA implemen-tation can be found in Ref. [24]. In this context some basicfeatures about the handling of sub-strings connected tointeger/real design variables were adopted, in order toincrease the convergence speed of optimization tasks based

on limited calculus resources (small populations and/orreduced generations).

5. Application examples

5.1. Topological optimization of a grid structure

As first example, the analysis of a metallic plate with agrid-shaped cross-section is shown. The analyzed structureis constrained to stay inside a bounding box of assigned

depth, width and height dimensions (respectively: 200,180 and 20 mm). The plate is subject to the previouslydefined natural boundary conditions applied on bordersand free edges and optimized to maximize the bucklingload.

In Fig. 8 a pictorial scheme of the plate is shown. The

shape of the plate cross-section is topologically-free

,but expressly conditioned to form anyway an orthogonalgrid. The section comes with a variable number of cellsarranged in rows and columns. Topological design param-eters n and m define the number of cells in x and y direc-tions, while geometrical dimensions of each row/columnare defined by further independent parameters.

Let Dx and Dy be, respectively, the overall dimensionsin x and y directions of the plate cross-section, the follow-ing conditions must be meet:

Xni1

lx;i Dx;Xmj1

ly;j Dy 1 6 n 6 nmax; 1 6 m 6 mmax;

11

where lx,i and ly,j denote ith and jth parametric cell-lengthalong x and y directions and nmax, mmax are the maximumnumber of allowed rows and columns in the section.

Analyzing relations (11) a clear difficulty appears whenone tries to introduce an independent set of design vari-ables that directly model cells dimensions because: (i) thenumber of rows and columns does not remain constantin the parametric model; (ii) rows and columns representan ordered set and consequently an implicit hierarchyexists about grid coordinates in both directions.

In order to satisfy relations (11) developing at the sametime a robust formulations for geometrical design vari-ables, two new sets Yx and Yy of independent, normalizedparameters have been introduced. Aim of these design vec-tors is to decouple geometrical features by topology. Afunctional relation based on natural cubic splines is usedfor this scope.

Let (xk, yk), k= 0,1, . . . ,p be a set of p + 1 pointsbelonging to an orthogonal domain [x0, xp] [0, 1] andequally spaced in x. The following relations will result:

X

Y Z

1 2 n

1

m

2

Dx

Dy

tki

Fig. 8. Plate cross-section layout.



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xk x0 kh; yk 2 0; 1; h xp x0=p;

k 0; 1; . . . ;p 12

a natural cubic spline s(x) interpolating these points can bedefined by means of the following relations:

sx akx xk13 bkx xk12 ckx xk1 dk;

xk1 6 x 6 xk

s0x 3akx xk12

2bkx xk1 ck

s00x 6akx xk1 2bk

13

where ak, bk, ck, dk k= 1, p are 4p unknown coefficients.Imposing on s(x) a C2 class membership, a number of4p 2 conditions can be given:

sxk1 dk yk1 k 1;p

sxk akh3 bkh2 ckh dk yk k 1;p

s0xk 3akh2 2bkh ck ck1 k 1;p 1

s00xk 6akh 2bk 2bk1 k 1;p 1

8>>>>>:

14

Adding two more conditions (i.e. zero slope at externalpoints), a complete set of 4p equations is defined:

s00x0 2b1 0

s00xp 6aph 2bp 0

15

By solving Eqs. (14) and (15) a generic cubic spline-basedrelation is now available in the form:

y sX; Y;x whereX 0; 1=p; 2=p; . . . ; p 1=p;1

Y y0;y1;y2; . . . ;yp;

x0 6x6xp 16

Using function (16) a link can be established between p + 1design parameters represented by the yp coordinates of thespline control points and the normalized values returned byevaluation of(16) at some reference points. In order to ap-ply this approach to the definition of the grid geometricalproperties, the following steps are executed: (1) a normal-ized, equally spaced grid in both x and y directions is first

considered; (2) two separate splines, applied on x and ycoordinate are defined by means of design vectors Yx andYy; (3) for each coordinate values xi (i= 1, n 1) and yj(j= 1, m 1) related to internal grid points, the followingdouble spline evaluation is performed:

yx;i s1Xx; Yx;xx;i i 1; n 1

yy;j s2Xy; Yy;xy;j j 1; m 117

Normalized, non dimensional values obtained by relations(17) are finally used to generate dimensional coordinate ofgrid intersection points with the following recursiveexpressions:

lx;i lx;i1 Dx lx;i1 yx;i; lx;0 0; i 1; n 1

ly;j ly;j1 Dy ly;j1 yy;j; ly;0 0; j 1; m 1

18

Fig. 9AD shows how the dimensional attribution works.In this example, design vectors Yx and Yy are supposed

to consist respectively of six and three control points, whilethe actual topology assumes n = 4, m = 5.

Initially the normalized grid is represented equallyspaced in both x and y direction (Fig. 9C); the resultingxx,i coordinates of the intersection points are evaluatedthrough the s1 spline (Fig. 9A) and the related outputvalues yx,i are used to indent hierarchically physical coor-dinates lx,i along the x direction. The same procedure isreplied in y direction processing the xy,j coordinatesthrough the s2 spline (Fig. 9B). The resulting modifiedgeometry is shown in Fig. 9D. Deliberately, no implicitsymmetries were imposed.

The geometrical model is finally completed by two fur-ther continuous parameters tkx and tky, which define uni-form wall thickness in x and y directions in terms ofpercent values referred to the longest cell wall present inthe actual grid.

5.1.1. Mathematical modelling of design variables

In order to furnish a better understanding of the gridparametric construction, let consider the following numer-ical example based on five-point splines (p = 4) for both xand y coordinates.

Let n = 7, Yx = (0.1, 0.1, 0.18, 0.6, 0.65)T, m = 4,

Yy = (0.2, 0.15, 0.23, 0.52, 0.35)T

, tkx = 0.01, tky = 0.02be the values of the actual design set, and Dx = 180,Dy = 20 the fixed overall dimensions of the grid cross-section.

Step (1): A normalized 7 4 cell grid, having five equallyspaced spline control points along both x and ydirections, is defined into a unit square with ori-gin placed on the top-left corner; i.e.Xx = Xy = (0.0, 0.25, 0.5, 0.75, 1.0)

T.Step (2): Eqs. (13) and (14) are solved for the x-spanned

spline; a piecewise cubic interpolation evaluation

is then performed for each x coordinate of then 1 internal crossing points of the grid, i.e:

SplineXx; Yx; 0:1429; 0:2857; 0:4286; 0:5714;

0:7143; 0:8571T

0:1195; 0:0948; 0:1197; 0:2822; 0:5404; 0:7223T

19

where, for sake of brevity, Spline operator meanscubic interpolation over a set of given points.

Step (3): Non dimensional values obtained by (19) areconverted into dimensional ones via recursiveexpression (18-1):



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Xx5Xx0 Xx1 Xx2 Xx40

1 32

yx,3

yx,1

yx,2

1

A

D

Xx3

lx,1 = Dx*yx,1

lx,2 = lx,1 +(Dx-lx,1)*yx,2

lx,3 = lx,2 +(Dx-lx,2)*yx,3

s1(Yx)

0Xy0

Xy1

Xy2

1

yy,1

ly,1ly,2

ly,3ly,4

s2(Yy)

2

3

4

2 3

B

Yx2Yx3

Yx4

Yx0

Yy0

Yy1

Yy2 xx,1 xx,2 xx,3

xy,1

C

Fig. 9. (AD) Control splines with related geometrical properties attribution.

0 20 40 60 80 100 120 140 160 180

-60

-40

-20

0

20

40

60

80

Fig. 10. Grid structure numerical example; topological configuration (wall thickness not reported).



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lx;1 180 0:1195 21:5100

lx;2 21:5100 180 21:5100 0:0948 36:5349

lx;3 36:5349 180 36:5349 0:1197 53:7077

lx;4 53:7077 180 53:7077 0:2822 89:3474

lx;5 89:3474 180 89:3474 0:5404 138:3361

lx;6 138:3361 180 138:3361 0:7223 168:4299lx;7 180

20

Step (4): Each cell-length is computed as lx,i lx, i1 (i= 1,n; lx,0 = 0) i.e.: (21.5100, 15.0249, 17.1728,35.6397, 48.9887, 30.0938, 11.5701); uniform wallthickness along x direction is obtained multiply-ing the longest cell wall by tkx design variable:

Max21:5100; 15:0249; 17:1728; 35:6397;

48:9887; 30:0938; 11:5701 0:01 0:4899 21

Steps (2), (3) (expression (18-2)) and (4) are replied for ydirection; briefly:

SplineXy; Yy; 0:25; 0:50:75T 0:15; 0:23; 0:52T 22

ly;1 20 0:1500 3:0000

ly;2 3:0000 20 3:0000 0:2300 6:9100

ly;3 6:9100 20 6:9100 0:5200 13:7168

ly;4 20

23

Max3:0000; 3:9100; 6:8068; 6:2832 0:02 0:1257 24

Fig. 10 shows the resulting arrangement of the cross-section.

5.1.2. Grid structure procedure set-up and optimization

results

In order to perform the optimization tasks hereindescribed, a specific tool was used. The ProGenie programdeveloped by the first author is a general-purpose

Table 1Design parameters, ranges, type and dynamic resolution for the grid structure

DV Type Min Max Resolution (bit) DV Type Min Max Resolution (bit)

Nsx Integer 2 9 3 3 3 3 3 Yy1 Real 0.05 0.2 8 8 12 12 15

Yx1 Real 0.05 0.2 8 8 12 12 15 Yy2 Real 0.05 0.2 8 8 12 12 15Yx2 Real 0.05 0.2 8 8 12 12 15 Yy3 Real 0.1 0.3 8 8 12 12 15Yx3 Real 0.1 0.3 8 8 12 12 15 Yy4 Real 0.1 0.4 8 8 12 12 15Yx4 Real 0.1 0.4 8 8 12 12 15 Yy5 Real 0.1 0.8 8 8 12 12 15

Yx5 Real 0.1 0.8 8 8 12 12 15 Tkx Real 0.1 0.12 8 8 8 8 8Nsy Integer 2 5 2 2 2 2 2 Tky Real 0.1 0.12 8 8 8 8 8

0 2 4 6 8 10 12 14 16 18 202.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7

x 104

Generations

BucklingLoad[N]

Fig. 11. Grid structure optimisation; buckling load (N).



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0 2 4 6 8 10 12 14 16 18 202

3

4

5

6

7

8

9

Generations

TopologyPars.[ad]

n

m

Fig. 12. Grid structure optimisation; topological parameters (ad).

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Generations

SplineYxCoords[ad]

Yx1

Yx2

Yx3

Yx4

Yx5

Fig. 13. Grid structure optimisation; s1 spline control points (ad).



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optimization environment with a unique bit-masking ori-ented genetic engine allowing high performances and flexi-

bility for the final user. Via its programmable interface toexternal codes, ProGenie was coupled with a home-made

0 2 4 6 8 10 12 14 16 18 200.1

0.105

0.11

0.115

0.12

0.125

Generations

Thick

ness[ad]

Tkx

Tky

Fig. 15. Grid structure optimisation; s2 wall thickness parameters (ad).

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Generations

SplineYyCoords[ad]

Yy1Yy2

Yy3

Yy4

Yy5

Fig. 14. Grid structure optimisation; s2 spline control points (ad).



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program implementing the structural calculus engine forthe semi-analytical method. For the optimization proce-dure several runs were preliminary performed, modifyingmain genetic parameters as population size, couples andgenerations. For sake of brevity, the faster procedure, per-formed with a small population of 30 individuals per 20

generations, is herein shown. To increase the convergencespeed, a five-point time envelope table for bit resolution,crossover composition, mutation rate and other minorparameters was defined. Using layer-composition featureoffered by BMOGA, a hybrid one-cut/bit-to-bit crossoverwas used during initial generations. In Table 1 designparameters with related ranges and resolutions are summa-rized. Cross-section dimensions Dx and Dy were set to 180and 20 mm, respectively and a steel material(E= 210,000 N/mm2; m = 0.3) was considered.

In Figs. 1116 the optimization results are reported astime-histories of values referred to best configurations. InFig. 11 the buckling load is shown. Topological parameters

are reported in Fig. 12, while Figs. 13 and 14 collect the ycoordinate values ofs1 and s2 splines control points respec-tively. Wall thickness attribution is represented in Fig. 15.Finally in Fig. 16 the reached best design is shown as topo-logical layout. Best design set (design variables and Objfunction values) is also listed in Table 2.

5.2. Weight minimisation of a stiffened composite panel

In this second example the optimisation of a compositepanel provided with both geometrical and topologicaldesign variables is illustrated. The proposed structure is

very similar to other ones already described in literature

as test-case for buckling analysis in composite stiffenedpanels [25]. It consists of a simply supported stiffened panelwith fixed length and depth dimensions (900 600 mm). InFig. 17 a pictorial view of the panel is shown. The panel isoptimised for minimum weight and subject to a criticalload not below 250 N mm1.

Carbonepoxy composite material (E1 = 92600 N/mm2

;E2 = 7730 N/mm2; m12 = 0.36; G12 = 3820 N/mm

2; q =1.431E6 kg/mm2) with a constant 0.07 mm ply thicknesswas defined.

A variable number of equally spaced composite stiffen-ers are disposed along the shorter way and controlled bythe integer design variable Nst. For symmetry purposes,this variable defines the number of stiffeners on the half-length of the panel. Stiffeners present a fixed height(30 mm) and a common cross-section that can be arrangedin several fashions: T-shaped, hat-shaped, I-shaped and C-shaped. An integer four-valued design variable, named Stp,makes the choice of the geometrical shape. In addition, a

dimensional design variable Cle is used to refine the shapein each cross-section type: i.e. the horizontal cap extensionin T, hat and C-shaped sections. In order to satisfy coher-ence requirements about design variables definition, alsothe I-shaped section is arranged with this auxiliary param-eter. This way a double-thickness vertical flange (web pluscap) is also allowed.

Each panel element (plate, web and cap) is formed by anexternal skin and a core component. The skin component,surrounding plate, web and cap parts, is the same for theentire panel and consists of two symmetric and balancedquartets chosen from a list of eight preassembled effective

staking sequences, each one obtained combining 0,

0 20 40 60 80 100 120 140 160 180

-60

-40

-20

0

20

40

60

80

Fig. 16. Grid structure optimisation; cross-section geometricaltopological configuration.



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+45, 45 and 90 plies properly. The integer design var-iable Qls in the range (18) is used to make the choice. Platecore is organised in a more complex way as it consists of avariable number of couples of layers. Analogously to the

skin assembly procedure, each couple is chosen from apre-compiled list; in this case six combinations of two plies,extracted by the same basic orientations previouslyreported.

In Fig. 18AD geometrical and topological features ofthe composite panel are summarized. In Fig. 18A the class

of stiffener cross-section type, formed by four differentshapes, is represented; for each cross-section shape, the cor-responding Stp value is reported. In Fig. 18B a detail of theexternal skin with related staking sequence options isshown. Finally, Fig. 18C and D shows a pictorial view ofthe variable staking sequence for plate, web and cap cores,with a list reporting the allowed couples.

The integer design parameter Npc, defined on interval(1, 8), selects the actual number of ply couples in thehalf-thickness of the plate. Unlike ply assignment forthe skin component, a direct coding of the staking

Table 2Grid structure best design set (design variables and obj function)

DV Type Value DV Type Value

Nsx Integer 9 Yy1 Real 0.18887143Yx1 Real 0.06992248 Yy2 Real 0.09874416

Yx2 Real 0.14610127 Yy3 Real 0.29082614Yx3 Real 0.21485947 Yy4 Real 0.31915647

Yx4 Real 0.34317148 Yy5 Real 0.33520615Yx5 Real 0.23546252 Tkx Real 0.10290197Nsy Integer 2 Tky Real 0.11749019

Bkl Obj 2.65656445E + 4 Vol Sva 727.545532

a Vol function only monitored, not constrained.

lskin

skinh

lweb

capl

1 3 stiffn2

Fig. 17. Pictorial view of the stiffened composite panel.

Cap core, 4 n 8

Web core, 4 n 8

Plate core, 4 n 32

Skin , n = 16

External skin stack sequence assigment

Spline Value Staking sequence

1 [0; +45; -45; 90]

2 [0; 90; +45; -45]

3 [+45; -45; 0; 90]

4 [+45; -45; 90; 0]

5 [90; +45; -45; 0]

6 [90; 0; +45; -45]

7 [+45; 0; 90; -45]

8 [+45; 90; 0; -45]

plyext

S

S

S

S

S

S

S

S

Qls parameter Staking sequence

1 [0; 0]

2 [0; 90]

3 [90; 0]

4 [90; 90]

5 [+45; -45]

6 [-45; +45]

Internal skin, web and cap stack sequence

assigment

plyweb

skinply

plycap

Ply ranges for skin, web and cap in the full tikness

123

n - 2plyn - 1nply

ply

A

BC

D

Stp=4

Stp=3

Stp=2

Stp=1

Fig. 18. (AD) Skin, cap, web variables and stiffeners cross-section type.



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sequence for the plate core via integer pointer is notallowed, due to its intrinsic variability in number; i.e. awell-defined optimisation problem requires a fixed num-ber of design variables. An indirect coding is thereforeperformed using again a spline-based attribution law,spanned over the core half-thickness and conditioned to

return integer values in the interval (1, 6). Eight equallyspaced spline control points Pi (i= 1, 8) are defined for

the purpose, and the related independent coordinatesPy,i are used as design variables. For each ply couplepresent in the actual configuration, the middle-planecoordinate (i.e. the interface plane coordinate) is usedto evaluate the spline. The resulting value is then trans-formed into a pointer to a table containing ply combina-

tions. Fig. 19 shows this spline-based procedure for thestaking sequence assignment in the plate core.

External skin

Couple 1

Couple 2

CoupleN

..

..

1 23 4

5 6Table pointer

Symmetry plane

Py,1

Py,8

Py,2

Fig. 19. Spline-based ply assignment for plate core.

Table 3Design parameters, description, ranges, type and dynamic resolution for the stiffened panel

DV Description Type Min Max Envelope resolution (bit)

0 1 2 3 4

Qls Skin quartet sequence pointer Integer 1 8 3 3 3 3 3

Npc Ply couple number (plate, half-thickness) Integer 1 8 3 3 3 3 3

Py,18 Spline control points coordinates (plate) Real[s] 0 1 8 8 12 12 15Nst Stiffeners number (half-quantity) Integer 1 4 2 2 2 2 2Stp Stiffener type Integer 1 4 2 2 2 2 2

Nwc Ply couple number (web, half-thickness) Integer 1 2 1 1 1 1 1Wy,12 Spline control points coordinates (web) Real[s] 0 1 8 8 12 12 15Cle Cap length Real 2 5 12 12 12 12 12Ncc Ply couple number (cap, half-thickness) Integer 1 2 1 1 1 1 1

Cy,12 Spline control points coordinates (cap) Real[s] 0 1 8 8 12 12 15

Table 4Stiffened panel optimization; envelope parameters

Individuals = 50; couples = 25; generations = 50

Envelope 0 1 2 3 4

Generation range 010 1120 2130 3140 4150Selection Random walk Random walk Roulette wheel Roulette wheel Roulette wheelScaling Linear Linear LinearMutation 0.02 0.02 0.04 0.04 0.06Crossovera 1* + 2* 1 1 1 1

a

(1) One-cut crossover; (2) bit-by-bit crossover. (1*

) Applied on real related sub-strings; (2*

) applied on integer-related sub-strings.



16/20

The web core is assembled similarly than the plate core,but with a smaller number of allowable couples (max 2 inthe half-thickness). Therefore, instead of a cubic spline, a

straight line with only two control points Wy1,2 is used tomake this ply assignment. The same procedure, with spe-cific control points is also applied to the cap core ( Cy1,2).

0 5 10 15 20 25 30 35 40 45 50

2.7

2.75

2.8

2.85

2.9

2.95

3

3.05

Generations

Weight(kg)

Fig. 20. Stiffened panel optimisation; structural weight.

0 5 10 15 20 25 30 35 40 45 501

2

3

4

5

6

7

8

Generations

Qls(Skinquartetseq)[ad]

&Npc(Plycouplenumber)[ad]

Qls

Npc

Fig. 21. Stiffened panel optimisation; skin quartet sequence and core ply couple number.



17/20

Table 3 summarizes design variables, with range, type anda short description.

Every skin or core component is arranged to return any-way a balanced and symmetric laminate in the full-thickness.

5.2.1. Procedure set-up and optimisation results for the

stiffened panel

Likewise preliminary example, also for the stiffenedpanel optimisation, the BMOGA evolution engine hasbeen set to perform a dynamic, multi-layer crossover; in

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Generations

Py,1...Ps,8SplineC

ontrolpointscoordinates[ad]

Py1

Py2

Py3

Py4

Py5

Py6

Py7

Py8

Fig. 22. Stiffened panel optimisation; Py,1, . . ., Py,8 spline control points for core staking assignment.

0 5 10 15 20 25 30 35 40 45 501

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Generations

Plycouplenumber(Webandcap)[ad]

Web

Cap

Fig. 23. Stiffened panel optimisation; ply couple number in web and cap core.



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this case to process in parallel specific partitions of thegenetic string with dedicated rules. When a direct binaryrepresentation for integer design variables is used, sub-strings codifying integer values exhibit a much shorter

bit length than sub-strings related to real values. As conse-quence, using a standard crossover much more efforts arerequired to perform a satisfying bit shuffling that couldinvolve integer variables during the initial generations.

0 5 10 15 20 25 30 35 40 45 5011

12

13

14

15

16

17

18

19

20

Generations

Cle(Caplength)[mm]

Fig. 24. Stiffened panel optimisation; cap length.

0 5 10 15 20 25 30 35 40 45 50210

220

230

240

250

260

270

Generations

Buckling

load(N/mm)

Infeasible region

Fig. 25. Stiffened panel optimisation; buckling load.



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To increase convergence speed, in the first ten generationsthe layer-composition feature for genetic operators offeredby BMOGA has been used to set: (i) a bit-by-bit crossoveroperating only on integer-related sub-strings; (ii) a simpleone-cut crossover applied to the remaining ones. In subse-quent generations a canonical simple one-cut crossoverwas applied to the whole binary string. In Table 4 thedynamic envelope of main genetic parameters used in theoptimisation task is summarised; every envelope pointtakes ten generations.

In Fig. 20 the time-history of the structural weight isreported; Figs. 2124 show the progress of some majordesign variables, in particular: skin quartet sequence, num-ber of ply couples in the plate core (half-thickness), splinecontrol points for ply sequence assignment in plate core,number of ply couples in web and cap cores and cap length,Time-histories of web and cap spline control points wereomitted for sake of brevity. In Fig. 25 the constrainedbuckling load with the assigned lower bound is reported.

In Fig. 26 the sketch of the resulting best design, com-plete of lay-up composition for skin and core parts (plate,

web and cap), is shown.

According to the procedure shown in Fig. 18, plysequences were obtained by decoding spline interpolationsevaluated at each ply couple interface plane first, and thenusing the resulting pointers on the pre-compiled list ofFig. 18B. Spline evaluation returned the following pointers:(1, 6, 3, 1, 5)T; (1); (1, 1)T for plate, web and cap,respectively.

The best design set in detailed form, including objectand state functions, is anyway reported in Table 5.

6. Conclusions

In the present paper the topological optimization ofarbitrary isotropic/orthotropic thin structures, arrangedin grid-based configurations or stiffened panels was shown.The maximum buckling load and the structural weightwere involved as constraints and/or objective functions intwo optimization examples. In order to achieve computa-tionally efficient tasks, a bit-masking oriented genetic pro-cedure was used and coupled with a expressly developedmethod for equilibrium stability analysis. This methodhas proved to be effective and accurate in the predictionof the arising buckling phenomena and very efficient froma computational point of view, as it resulted considerablyfaster than any equivalent buckling analysis performedvia finite element method. The main drawback experiencedusing this numerical approach was the need for assuring aconsistent geometrical slenderness (width to thicknessratio) for each finite strip constituting the numerical model.This problem is somewhat increased in any optimizationprocedure that makes substantial topological modificationswith sudden changes in shape and dimensions. In the gridstructure optimization, a near fixed-slenderness controlwas successfully experimented.

About genetic procedure, the use of BMOGA has sim-plified an efficient definition of dedicated genetic operators

applied on specific partitions of the genetic string.

Plate lamination sequence

Skin lamination sequence

Cap lamination sequence

Web lamination sequence[00]s

[0; 0; -45; 45; 90; 0; 0; 0; 45; -45]s

[0; 90; 45; -45]s

[0; 0; 0; 0]s

Fig. 26. Stiffened panel optimisation; best design.

Table 5Stiffened panel best design set (design variables, object and state functions)

DV Type Value DV Type Value

Qls Integer 2 Nst Integer 4

Npc Integer 5 Stp Integer 3Py,1 Real 0.00392157 Nwc Integer 1Py,2 Real 0.19607843 Wy,1 Real 0.19215687Py,3 Real 0.99607843 Wy,2 Real 0.04705882

Py,4 Real 0.09019608 Cle Real 11.7533579Py,5 Real 0.72549021 Ncc Integer 2Py,6 Real 0.11372549 Cy,1 Real 0.04313726Py,7 Real 0.57254905 Cy,2 Real 0.09803922

Py,8 Real 0.23529412

Wgt Obj 2.66494012 Bkl SV 251.013138



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Consequently, optimization tasks with a limited amount ofresources in terms of population size and generations weresuccessfully performed. About the preliminary test-case,the spline based approach for the grid-shaped cross-sectionhas proved to be simple to implement and very effective forthe definition of a consistent arrangement of the topologi-

cal features. A similar technique was also applied to thecomposite panel problem for the staking sequence defini-tion; specifically to define a robust rule for ply assignmentwith a variable number of layers using a fixed-length designvector.

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