P. VannucciUVSQ - Université de Versailles et Saint-Quentin-en-Yvelines

The polar method in optimal design of laminates

Università di PisaFacoltà di Ingegneria

13 luglio 2007

2

Foreword

This seminar deals with some results obtained in optimal design of laminates by the use of the polar method.

The advantages given by the polar method in this field are essentially the fact that the rotation formulae are expressed in a simple way and that the material characteristics appear through invariants expressing the elastic symmetries.

For these reasons, the polar method has proven to be rather effective in all those problems concerning the elastic design of a laminate.

The originality of these researches consists in having considered the design of the elastic symmetries as a part of the design phase.

This is usually discarded by other authors, who search the optimal solution in a class of laminates automatically giving some desired elastic symmetries (for instance balanced and symmetric sequences).

3

Foreword Unfortunately, this classical approach tightens so much the design

space that almost every time the solutions so found are not true optimal solutions.

Our approach can be distinguished into three phases: research of as much as possible exact solutions; research of a general formulation for the optimal design of laminates; research of a numerical strategy for the search of the solutions.

This presentation will briefly show these phases in the order.

All what will be said concerns laminate made of identical plies; this is a necessary assumption to have general solutions.

4

Content

Recall of the Classical Lamination Theory

Some exact solutions to simple design problems

A general statement for the optimal design of laminates

Numerical strategy for the search of solutions

Conclusions and perspectives

An unconventional historical note

5

Recall of the Classical Lamination Theory The Classical Lamination Theory provides the constitutive law for a

thin laminate under extension and bending actions:

,

χ

ε

DB

BA

M

N

z

p

z

p

h/2

h/2

zk-1

zk

0

1

k

-1

-k-p

h/2

h/2

zk-1

zk

1

k

-1

-k-p

n=2p+1 n=2p

6

Recall of the Classical Lamination Theory

The normalized tensors are also useful:

A laminate is said uncoupled if B=O and quasi-homogeneous if, in addition, also the homogeneity tensor

C=A*D*=O.

When translated in polar form, the previous formulae give, in the case of n identical plies,

./12*,/2*,/* 32 hhh DDBBAA

.

,)( )(1

,,

for 3 , for 2 , for 1

1

DBA

QDBA

m

p

pkmk

mkkk zz

m

7

Recall of the Classical Lamination Theory

;

,

,

,:* tensor

22121

44040

11

00

11

00

p

pkiii

p

pkiii

k

k

een

ReR

een

ReR

TT

TT

A

; ˆ

, ˆ

,0ˆ

,0ˆ:* tensor

2221ˆ2

1

4420ˆ4

0

1

0

11

00

p

pki

kii

p

pki

kii

k

k

eben

ReR

eben

ReR

T

T

B

8

Recall of the Classical Lamination Theory

. ~

, ~

,~

,~:* tensor

2231

~21

4430

~40

11

00

11

00

p

pki

kii

p

pki

kii

k

k

eden

ReR

eden

ReR

TT

TT

D

; 1

, 1

,0

,0: tensor

2213

21

4403

40

1

0

11

00

p

pki

kii

p

pki

kii

k

k

eceRn

eR

eceRn

eR

T

T

C

9

Recall of the Classical Lamination Theory The coefficients bk, ck and dk are

It is of some importance to remark that the bk's are odd, while the ck's and dk's are even.

;2 if0,1334

,12 if)3(4

022

22

pnckkp

pnkppck

;2 if0,2

,12 if 2

0 pnbkk

k

pnkbk

.2 if0,41212

,12 if112

02

2

pndkk

pnkdk

10

Recall of the Classical Lamination Theory It is important to notice that for laminates with identical plies, only

the anisotropic behavior can be designed: so, you have only two polar equations for each tensor.

Quasi-homogeneous laminates are not only uncoupled, but they show the same elastic behavior in extension and in bending in each direction.

So, the polar equations of quasi-homogeneity are

The uncoupling problem is ruled by only the two equations at left.

.0,0

,0,0

22

44

p

pki

kp

pki

k

p

pki

kp

pki

k

kk

kk

eceb

eceb

11

Recall of the Classical Lamination Theory These are 8 real equations; in fact, 4 equations, concerning the

isotropic part of B and C, are identically satisfied.

This is immediately recognized in polar, but not with Cartesian coordinates, when 12 equations, with only 8 independent, are to be solved.

The above equations have not, in the general case, a complete analytical solution.

Another point deserves attention: elastic quasi-homogeneous solutions are also thermo- and hygro-elastic quasi-homogeneous.

This is easily recognized if one considers the laminates constitutive law considering also the thermal effects (for the moisture absorption results are similar):

12

Recall of the Classical Lamination Theory

with the normalized thermo-elastic tensors given by

The thermo-elastic homogeneity tensor can be also introduced:

,

W

V

V

U

κ

εDB

BA

M

N

ht

to

o

.1

*~,*~:12*

;1

*ˆ,0*ˆ:2*

; *,*:*

223

~23

222

ˆ22

222

p

pkiδ

kii

p

pkiδ

kii

p

pkiii

k

k

k

edeRn

eRTTh

ebeRn

eRTh

eenR

eRTTh

WW

VV

UU

. 1

,0: 223

2 p

pki

kii keceR

neRT**

WUZ

13

Recall of the Classical Lamination Theory It is suddenly recognized that the conditions for thermo-elastic

quasi-homogeneity, i.e. to have a laminate that has the thermal expansion coefficients identical in extension and bending in each direction, is to have

These two complex equations are just a part of those giving elastic quasi-homogeneity.

This means that a quasi-homogeneous laminate for the elastic properties is also quasi-homogeneous for the thermo-elastic behavior, but the converse is not, generally speaking, true.

.0

,0

2

2

p

pki

k

p

pkiδ

k

k

k

ec

eb

OZ

OV

14

Some exact solutions to simple design problems Some simple problems concerning the design of laminates for

different purposes can be solved analytically. Some of them are shown here.

Laminates composed by R0 or R1- orthotropic materials

In this case the problem is simpler, car one polar equation is identically satisfied.

It is worth noticing also that if a laminate is composed by R0- or R1-orthotropic layers (also different) it will be automatically R0- or R1-orthotropic, for all the tensors.

Some complete solutions, analytical or numerical, concern laminates designed to be uncoupled or quasi-homogeneous, with a small number of layers (4, 5 or 6).

15

Some exact solutions to simple design problems 6-layers designed to be quasi-homogeneous, composed by R1-

orthotropic layers (complete solution found numerically).

)/2

-30

-20

-10

0

10

20

30

-15 -10 -5 0 5 10 15

( 3 - -3

-2

-1

1

2 3 + -3 =0

;eeeeee

,eeeeeeiδiδiδiδiδiδ

iδiδiδiδiδiδ

05544

05533332211

332211

444444

444444

16

Some exact solutions to simple design problems A special class of laminates: the quasi-trivial solutions

Quasi-trivial solutions are a particular class of uncoupled or quasi-homogeneous laminates.

A quasi-trivial solution has the particularity that the solution is exact and depends only on the stacking sequence but not on the orientations: this is rather useful when other properties (stiffness, strength and so on) must be optimized.

Actually, though the general problem of solving the quasi-homogeneity (or simply the uncoupling) equations has not a unique analytical solution, a particular class of laminates satisfying these equations can be found exploiting a fundamental property of the coefficients bk and ck: their sum is null.

So, a quick glance at the quasi-homogeneity equations

17

Some exact solutions to simple design problems

show that a sufficient condition to have a solution is to dispose groups of layers with the same orientation, no matter of its value, in such a way that the sum of the coefficients for each group is zero.

Such groups are called saturated and the solutions quasi-trivial, because they are obtained without solving explicitly the previous equations.

As coefficients bk and ck are integer, the solutions so found are exact.

It is worth noting that the very well known symmetric solutions for uncoupling are just a subset of the quasi-trivial solution to the problem B=O.

.0,0

,0,0

22

44

p

pki

kp

pki

k

p

pki

kp

pki

k

kk

kk

eceb

eceb

18

Some exact solutions to simple design problems An example: an 18-layers laminate: q-h, q-t solution (unsymmetric!)

Two questions arise: how much q-t solutions do exist? when they can be useful?

The first question: just a look at the diagram below, showing the number of q-h q-t solutions as a function of the ply number (in brackets: the symmetric solutions).

The number of q-t is rapidly increasing with the ply number and gives a practically unlimited quantity of different possibilities for applications.

k -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9group 0 1 2 0 1 2 2 2 1 1 0 0 1 0 0 2 2 1

bk -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17

ck -136 -88 -46 -10 20 44 62 74 80 80 74 62 44 20 -10 -46 -88 -136

k -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9group 0 1 2 0 1 2 2 2 1 1 0 0 1 0 0 2 2 1

bk -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17

ck -136 -88 -46 -10 20 44 62 74 80 80 74 62 44 20 -10 -46 -88 -136

19

Some exact solutions to simple design problems

For the second question, some applications of q-t solutions, among the possible ones, are shown here.

1 1 1 1 3 1 3 2 4 8 23 5 52 40 44 130 59

416

723

5214

9512

8259

02 6146

4544

1

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Nombre de couches

(1)

(2)

(1)

(1) (2

) (3) (1

)

(7)

(1)

(3) (3)

1 1 1 1 3 1 3 2 4 8 23 5 52 40 44 130 59

416

723

5214

9512

8259

02 6146

4544

1

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Nombre de couches

(1)

(2)

(1)

(1) (2

) (3) (1

)

(7)

(1)

(3) (3)

20

Some exact solutions to simple design problems Generally speaking, working on the set of q-t solutions of q-h type

allows the designer to look for optimal solutions of bending properties working on the extension properties, which is much simpler, and with a lower number of unknowns, the orientation of the saturated groups in place of those of the layers.

This strategy can be applied to a number of different problems; some examples are shown here.

Fully orthotropic laminates

A fully orthotropic laminate is orthotropic in extension and in bending and is uncoupled.

Unlike extension orthotropy, rather easy to be obtained, bending orthotropy is very difficult to be obtained, so that in most researches a laminate is considered orthotropic in bending also if it is not!

21

Some exact solutions to simple design problems If layers form an anti-symmetric sequence, i.e. if

then extension and bending orthotropy are assured, but not B=O, in general.

A strategy consists in looking for anti-symmetric sequences that are also uncoupled.

It is not difficult to show that uncoupling polar equations, for anti-symmetric laminates, reduce to

This equation holds the problem of finding uncoupled anti-symmetric orthotropic laminates. For a small number of layers, solutions can be found analytically or completely described numerically and traced on a graph.

kkn 1

.02cos2sin2sin2sin2

2

21

2

2

21

4

2

p

kkkk

p

kkk

p

kkk bbbbb

22

Some exact solutions to simple design problems The figure shows the geometrical locus of the anti-symmetric

solutions in the space (2, 3, 4) for 9-ply laminates; in this case the previous equation becomes

Some plane sections of the surface in the figure aside:

a) Planes  4 = 0° and 90° ; b) Plane  4 = 30° ; c) Plane  4 = 45°

.0)2cos2sin2cos2sin22cos2sin3(16

)2sin2sin22sin3(16)2sin2sin22sin3(2

443322

2432

4432

23

Some exact solutions to simple design problems Another possibility is to

look for in-plane orthotropic solutions in the set of q-h, q-t laminates.

The results from 7 to 12 layers are presented in the table on the left.

These solutions can be used, for instance, in exact optimization of the buckling load of rectangular plates (all the solutions in the literature are approximated)

Ply number

2 3 4 7 8 9 10 11 12

7 plies 1 - - 0 - / / / / /

8 plies 2 - - - - / / / /

3 - - 0 - - / / /

9 plies 4 - 0 - 0 0 - / / /

5 0 - - 0 - 0 / / /

6 - - 0 0 - - / /

10 plies 7 - 0 - - 0 - / /

8 0 - - - 0 - / /

9 0 - - - - 0 / /

10 - - 0 0 0 - - /

11 - 0 - 0 - 0 - /

12 - 0 0 - 0 0 0 - /

11 plies 13 0 - - 0 0 0 0 - /

14 0 - - 0 - - 0 /

15 0 - 0 - 0 0 - 0 /

16 0 0 - - 0 - 0 0 /

17 - - - - - -

18 - - 0 0 0 0 - -

19 - 0 - 0 0 - 0 -

20 - 0 0 - - 0 0 -

12 plies 21 0 - - 0 - 0 0 -

22 0 - 0 - - 0 0 -

23 0 - - 0 0 - - 0

24 0 - 0 - - 0 - 0

25 0 0 - - - 0 - 0

26 0 0 - - - - 0 0

24

Some exact solutions to simple design problems Let us consider the case of a rectangular simply supported plate,

orthotropic in bending, with the axes of orthotropy parallel to the sides of the plate. If the plate is not subjected on its boundary to tangential loads, the expression of the buckling load multiplier, in polar form, is

with

,

2cos44cos612

12

2210

2221032

yx

p

pjj

p

pjj

K

NN

RRTTn

nh

Nx

Nx

Ny

Ny

y

xb

a.sinsin

,,22

byq

axp

cw

bq

ap

pq

25

Some exact solutions to simple design problems It is well known that the optimal solution to the above problem must

be looked for in the class of the angle-ply laminates (having only two possible orientations: and –). So, we have a problem with only one unknown, .

In such a case the previous equation simplifies to

min(p,q) must be maximized. This can be done easily and exactly if a q-h - q-t orthotropic solution is used: an in-plane orthotropic q-h - q-t solution can be easily selected; the laminate will be also orthotropic in bending and uncoupled, so the

only parameter to be chosen is the lamination angle ; this is determined as the solution of a 2nd degree equation, hence in

closed form.

.

2cos44cos612

12

2210

22210

32

yx

K

NN

RRTTn

nh

26

Some exact solutions to simple design problems

The figure shows the case of a plate with a/b= 2, Nx=Ny in carbon-epoxy (T0= 26.88 GPa, T1= 24.74 GPa, R0= 19.71 GPa, R1= 21.43 GPa, K=0 ).

opt

=70.65°

nh

12

32

27

Some exact solutions to simple design problems Fully isotropic laminates

The hard problem of finding fully isotropic laminates has been addressed by several authors.

For what concerns exact fully isotropic solutions, a strategy is to apply the Werren and Norris rule to q-h q-t solutions.

So, we have looked for q-h q-t laminates having at least 3 saturated groups with an equal number of layers in each group; if the groups are equally spaced, the solution is in-plane isotropic; quasi-homogeneity ensures also fully isotropy.

In this way we have found the exact fully isotropic solutions with the least number of layers: 5 unsymmetrical laminates with 18 layers (before, the number of layers was 36!).

In the next table, some solutions of exact fully isotropic laminates.

28

Some exact solutions to simple design problems

Number of plies

Orientations Stacking sequence

180= –60°

1= 0°2= 60°

0 1 2 0 1 2 2 2 1 1 0 0 1 0 0 2 2 10 1 2 2 0 1 1 2 0 1 2 0 2 0 1 0 1 20 1 2 2 0 1 1 2 0 2 1 0 1 0 2 0 2 10 1 2 2 1 0 2 0 1 1 0 2 1 2 0 0 2 10 1 1 2 2 2 0 0 2 1 0 0 1 1 1 2 2 0

24

0= –45°1= 0°

2= 45°3= 90°

0 1 2 3 2 3 1 3 0 2 0 1 0 1 3 1 2 0 2 3 2 3 0 1

270= –60°

1= 0°2= 60°

0 0 1 2 1 2 1 2 2 2 1 1 0 0 0 0 0 1 2 2 0 1 1 2 1 2 00 0 1 2 2 1 2 1 1 2 2 1 0 0 0 0 0 1 2 1 0 2 2 1 2 1 00 1 0 1 2 2 0 2 1 2 2 1 2 1 0 1 0 0 1 0 2 0 0 1 2 1 20 1 0 1 2 2 2 0 1 2 2 1 0 1 2 1 0 0 1 2 0 0 0 1 2 1 20 1 0 2 2 2 1 0 1 2 1 1 0 1 2 2 0 0 1 2 0 0 0 2 1 2 1

30

0= 0°1= 72°

2= 144°3= 216°4= 288°

0 1 2 3 3 0 4 4 1 2 4 4 2 2 3 1 1 3 1 0 0 0 0 2 3 3 4 4 2 10 1 2 3 4 3 0 2 1 4 4 2 4 1 3 1 3 0 2 0 0 3 2 4 1 0 1 2 3 40 1 2 3 4 4 0 2 1 3 3 4 2 1 3 1 2 0 4 0 0 4 3 2 1 0 1 2 4 30 1 2 3 4 3 0 2 1 4 4 2 4 3 1 1 3 0 2 0 0 1 2 4 3 0 3 2 1 40 1 2 3 4 4 1 0 3 2 2 4 3 3 0 2 1 1 4 0 1 4 3 0 0 2 2 1 4 30 1 2 3 4 3 4 0 1 2 2 4 1 3 0 2 4 1 3 0 3 4 1 0 0 2 2 1 3 40 1 2 3 4 4 3 0 1 2 2 4 1 3 0 2 3 1 4 0 3 4 1 0 0 2 2 1 4 30 1 2 3 4 4 2 0 3 1 3 1 4 2 0 4 3 2 1 0 1 3 2 0 0 4 4 2 1 30 1 2 3 4 4 2 0 3 3 1 4 1 2 0 1 3 2 4 0 1 4 2 0 0 3 3 2 4 10 1 2 3 3 4 4 0 1 2 2 4 1 4 0 3 3 1 2 2 0 0 1 3 4 0 3 1 4 2

29

Some exact solutions to simple design problems Optimal design of test specimens.

Experimental testing of composite materials and laminates is not as easy as for classical materials, because more mechanical properties are concerned and more mechanical effects must be accounted for.

So, the optimal design of test specimens has a great importance for obtaining good results from testing.

We have worked on two problems: the optimal design of a unique laminate specimen for the elastic testing

of the basic layer elastic properties, by tension, bending and anticlastic bending tests;

the optimal design of a specimen for the fracture propagation tests.

Let us briefly consider this last case: the goal of the experimental research was to measure some delamination parameters, as a function of the lamination angle, in the absence of parasite effects, such as twisting-bending coupling and change of the elastic properties in the separated sub-laminates.

30

Some exact solutions to simple design problems Properties imposed to the specimen (of angle-ply type):

uncoupling of the entire specimen and of the sub-laminates; same elastic properties for the entire laminate and the two sub

laminates; same behavior in extension and in bending; no coupling terms of the type D16 and D26; possibility of varying the lamination angle without altering the above

properties.

This problem has been solved using q-h q-t sequences:

2L= 120 mm

hh a

b

2L= 120 mm

hh a

b

16 plies: []s

26 plies: []s

31

Some exact solutions to simple design problems These laminates have the following properties:

they are obtained as the symmetric superposition of two anti-symmetric identical q-h q-t sequences;

they have the same number of plies at and – A orthotropic A16=A26=0

the laminate and the sub-laminates are quasi-homogeneous B=O and C=O D orthotropic D16=D26=0;

A*=D* for the laminate and the two sub-laminates, as they are quasi-homogeneous and the layers volume fraction is the same;

quasi-trivial solutions the lamination angle can be varied without altering the above properties.

In this way the delamination parameters, such as the fracture toughness, can be measured without parasite effects.

32

Some exact solutions to simple design problems Laminates with null piezo-electric deformations in some

directions.

Patches of piezo-electric actuators acts in the same way in each direction. In some cases it can be interesting to have a laminate which, under the action of a piezo-electric actuator, has in-plane and bending deformations null in one direction.

This problem can be solved in closed form for the in-plane strains, and for a standard laminate:

33

Some exact solutions to simple design problems It can be shown that the components of the in-plane piezo-electric

strains are (t1, r1 an 1 are some of the polar components of A-1)

So, the problem is reduced to posing x()=0, and this leads to

if t1= r1, x=0 for

if t1<r1, x=0 for and x<0 between these two directions;

if t1= 0, x=0 for the two orthogonal directions isochoric piezo-electric in-plane response;

if r1= 0, x= y and s=0 : isotropic piezo-electric in-plane response.

).(2sinˆ8)(

)],(2cos[ˆ4)(

)],(2cos[ˆ4)(

113

1113

1113

re

rte

rte

s

y

x

;21

1

11 arccos

21

2 r

t

;41

34

Some exact solutions to simple design problems Let us concentrate on the second case, the most interesting.

The corresponding condition can be stated using the stiffness polar components, and a closed-form solution can be found for the case of an angle-ply laminate:

An example: T300/5208 carbon-epoxy layers and PZT-4 actuators.

.4cos)1(2cos22

00

001A

MkMM

M

A

T

RTR

h

h

1 2max /40

Solutions are in 1 ( 14,14°)≤≤ 2 ( 39.05°). For =max ( 28.54°), = max= 0.146. So, if =max the highest quantity of piezoelectric layers can be used (the 14.6% in thickness) to maximize the piezoelectric action.

35

A general statement for the optimal design of laminates Further, we have looked for a general approach to the optimal

design of laminates, where the basic elastic properties, such as uncoupling, orthotropy and so on, are a part of the design process.

This means also that the search for a basic elastic property is seen itself as an optimization problem.

We have worked in two steps: in the first step, we have given a general formulation of all the design

problems of basic elastic properties and we have used this formulation to solve some design problems of laminates;

in the second step, we have given a completely and classical unified formulation of the optimal design of laminates with respect to a given objective function including basic elastic properties.

In all the cases, a suitable numerical approach is needed; this will be discussed in the next section.

36

A general statement for the optimal design of laminates In the framework of the polar method, it is possible to give a unified

formulation of all the problems concerning the design of laminates with respect to their basic elastic properties.

To this purpose, let us introduce the quadratic form of R18

P is the vector of all the polar parameters of the laminate (for A*, B* and D*), divided by a given factor M to work with non-dimensional quantities, for instance

,,18,...,1,,)( THHPHP jiPPHP jiijkI

.421

121

20

21

20 n

i iiii RRTTn

M

37

A general statement for the optimal design of laminates

The solutions are the minima of the quadratic form I(Pk).

The advantage is that the value of these minima is known a priori (usually it is zero)

The choice of the matrix H determines the problem to be solved.

,~,~,~12

,~12

,~12

,~12

,ˆ,ˆ,ˆ2

,ˆ2

,ˆ2

,ˆ2

,,,,,,

11801731

1630

1531

1430

13

11201121

1020

921

820

7

16051

40

31

20

1

PPMh

RP

Mh

RP

Mh

TP

Mh

TP

PPMh

RP

Mh

RP

Mh

TP

Mh

TP

PPMh

RP

Mh

RP

Mh

TP

Mh

TP

38

A general statement for the optimal design of laminates

11

11

11

11

11

11

11

11

11

11

11

11

C= O

06

1

212

i ii PP

A-A A-B A-D

A-A A-B A-D

B-A B-B B-D

D-A D-B D-D

A-A A-B A-D

A-A A-B A-D

B-A B-B B-D

D-A D-B D-D

1

1

1

1

1

1

1

1

B= O

010

7

2 Pi i

0

0

24

1

P

R

2

39

A general, unified and totally free from simplifying assumptions formulation for the optimal design of laminates with respect to current and important objectives (minimum weight, highest stiffness and/or strength, highest buckling load and so on) can also be obtained in the framework of the polar method.

In this case, the previous general approach to the basic elastic properties must be integrated into a more general formulation of an optimum problem, becoming in this framework a constraint condition.

What is new, is the fact that in this way a completely general approach to the optimum design of laminates can be obtained.

A general statement for the optimal design of laminates

40

A general statement for the optimal design of laminates In fact, let us suppose that a laminate must be designed to minimise

a certain objective function f, but with some basic elastic properties to be respected, e.g. uncoupling and membrane orthotropy.

Then, the problem can be stated as follows:

Here, x is the vector of design variables (orientations, thicknesses etc.).

find x such that f(x)= min

with I[Pk(x)]=0

and H corresponding to the

desired elastic symmetries

41

A general statement for the optimal design of laminates By the technique of Lagrange multipliers, this problem can be put in

the form of an unconstrained optimization problem:

This is the most general form of the mono-objective design problem of a laminate.

The challenge, in this case, concerns much more the construction of an effective algorithm for the solution of hard constrained and multi-purpose optimization problems

find x such that f(x)+ I [Pk(x)]= min

with H corresponding to the desired elastic symmetries

42

Numerical strategy for the search of solutions The search of the solutions is a delicate

point: generally speaking the problems formulated in the previous section are non-convex, highly multimodal and with non-isolated minima.

In addition, the number of the design variables is often rather great (at least n-1, n being the number of layers).

Also, the design variables can be of different type: continuous, discrete, grouped (i.e., representing more than one quantity).

For these reasons, we decided to use a genetic algorithm (Holland, 1965).

The general structure of a classical genetic algorithm is sketched in the following figure.

43

Numerical strategy for the search of solutions

Fitnessoperator

Stop condition

Output: best individual, mean fitness of the population

Input : population of n

individuals

no

yes

1000110111001001

1110010101100010

0111001001

0101100010

111001

100011

2 parents 2 children

point of cross-over

1110010101100010

Original gene Mutated gene

1110110101100010

position of mutation

0 1

Cross-over

Mutation

Elitism

Selection operator

Cross-over

MutationNew generation

44

Numerical strategy for the search of solutions We have made a genetic algorithm specially conceived for laminate

problems, BIANCA (BIo ANalyse de Composites Assemblés).

Characteristics haploid structure multi-chromosome elitism gene-based cross-over Boolean operators

n

k

4

3

2

1

n layersn

k

4

3

2

1

n layers

chr. 1

chr. 2

chr. 3

chr. 4

chr. k

chr. n

gen

ome

wit

h n

chro

mos

omes

c hr o

mo s

o me

k

gene of the material

gene of the orientation

6 genes of components

10010

10111

10010100

45

Numerical strategy for the search of solutions Some examples.

A 12-plies designed to be quasi-homogeneous and square symmetric.

Solution Orientations (°) I (Pk) residualBIANCA [0/62.46/- 53.44/81.56/-15.80/- 75.75/66.59/0/-0.54/46.07/-28.12/-88.94] 2.27 x 10

-5

BIANCA approximated

[0/62/-53/82/-16/-76/67/0/-1/46/-28/-89] 7.84 x 10-5

[0/61.7640/- 52.1221/82.6706/-18.2096/-78.3146/64.6143/1.0953/- 2.5155/44.6293/-29.8974/-89.6532]

Gradientapproximated

[0/62/-52/83/-18/-78/65/1/-2 /45/-30/90] 8.56 x 10-5

Gradient 1.09 x 10-13

EA and EDEA and ED

Global objectiveSquare symmetryB= OC= O

10-8

10-6

10-4

10-2

1

0 2000 4000

f

p

46

Numerical strategy for the search of solutions In the table, some results obtained by the code BIANCA.

Solution (°)

[4.5735/0/0/1.5364/6.1092]

[0/-17.16/-2.69/5.05/-14.60/-5.97/ -14.45/5.34/1.65/-12.95]

[0/44.67/15.70/-39.98/-25.46/ -37.21/59.04/54.28/36.92/-38.16/18.58/-5.23]

[0/10/40/-40/-20/50/-20/30/-40/30/10/-10]

[0/75.99/-31.45/-67.48/37.97/31.97/-38.49/ -76.87/57.69/89.31/14.15/-23.88]

[0/60/70/10/-60/-50/-60/-50/60/0/10/70]

[0/62.46/-53.44/81.56/-15.80/-75.75/66.59/0/ -0.54/46.07/-28.12/-88.94]

[0/51.58/-51.49/85.83/-51.34/85.04/24.09/ -19.08/30.94/-11.16/63.28/-65.21]

f

0

2.53 x 10-5

4.74 x 10-5

5.08 x 10-4

9.40 x 10-6

1.13 x 10-4

2.27 x 10-5

3.46 x 10-4

t

7

5

5

3

7

7

70

100

p

200

500

500

300

500

500

4000

3000

N

200

200

200

200

200

200

1000

2000

n

5

10

12

12

12

12

12

12

Ensemble de définition des orientations

]-45°, 45°]

]-90°, 90°]

]-90°, 90°]

]-90°, 90°], pangle=10°

]-90°, 90°]

]-90°, 90°], pangle=10°

]-90°, 90°]

]-90°, 90°]

Type de couche

R1=0

quelconque

quelconque

quelconque

quelconque

quelconque

quelconque

quelconque

Type de problème

Découplage

Orthotropie K=0 pour A et D, avec axes coïncidents et

découplage

Orthotropie K=1 pour A et K=0 pour D, avec axes

coïncidents et découplage

Orthotropie K=1 pour A et K=0 pour D, avec axes

coïncidents et découplage

Isotropie de A, orthotropie K=0 pour D, découplage

Isotropie de A, orthotropie K=0 pour D, découplage

quasi-homogénéité avec symétrie carrée

isotropie totale

Numéro du problème

23

28

28

28

13+14+27

13+14+27

24

18

1

2

3

4

5

6

7

8

47

Numerical strategy for the search of solutions An example of practical design (constrained optimization): a 12-

plies laminate made of carbone-epoxy T300-5208, designed to have B=O, A orthotropic and such that:

Emmax≥100 GPa (0.55 E1);

Emmin≥40 GPa (3.88 E2);

orientations {0°, 15°, 30°, 45°, 60°, 75°, 90° etc.}.

A solution found by BIANCA[0°/30°/–15°/15°/90°/–75°/0°/45°/–75°/0°/–15°/15°].

-80

0

80

-160 -80 0 80 160

Ef()

Em()

0 10 20 30 40 50

generation

Emax

EminA

A

0

50

100

150

0

50

100

150

0 10 20 30 40 50

Emax

Emin

GPa

A

AmEm

max

Emmin

48

Numerical strategy for the search of solutions A 12-plies T300/5208 carbone-epoxy laminate designed to have

R1=0 in extension and bending, B=O and isotropic piezoelectric response (PZT4 actuators).

Solution found by BIANCA:

[0/90/44.98/-41.80/-74.53/40.47/0/-71.92/34.36/-45/-1.86/85.08]

Elastic properties In-plane, tensor A

Bending, tensor D

Coupling, tensor B

(MPa) 27218 27805 0

(MPa) 25240 26100 0

(MPa) 3550 6781 224

(MPa) 122 138 564

(°) 16 -1 =

(°) = 0 =

Piezoelastic criteria In-plane, tensor a Bending, tensor d  

(V-1) 1.88·10-7 1.25·10-8  

(V-1) 1.97·10-9 1.47·10-10  

-100000

-50000

0

50000

100000

-100000 -50000 0 50000 100000Direction 0° (GPa)

A11

B11

D11

-100000

-50000

0

50000

100000

-100000 -50000 0 50000 100000Direction 0° (GPa)

A11

B11

D11

Variation des rigidités pour la membrane, la flexion et le couplage

-0.0000002

-0.0000001

0

0.0000001

0.0000002

-0.0000002 -0.0000001 0 0.0000001 0.0000002Direction 0° (GPa)

1

6 = 0 et 6 = 0

1

49

Numerical strategy for the search of solutions A 12-plies T300/5208 carbone-epoxy laminate designed to be

isotropic in extension, K=0 orthotropic in bending, B=O, with isotropic in-plane thermo-elastic response and one direction of zero bending thermal coefficient due to a temperature gradient through the thickness.

Solution found by BIANCA:

[0/-29.97/44.3/-61.88/89.3/61.83/31.56/-89.12/33.4/-71.72/-11.6/-28.13]

Elastic properties In-plane, tensor A Bending, tensor D Coupling, tensor B

(MPa) 26880 26880 0

(MPa) 24743 24743 0

(MPa) 97 5370 427

(MPa) 243 11227 119

(°) = -18.27 =

(°) = -18.19 =

Thermal properties In-plane, tensor uThermal

propertiesBending, tensor w

(°C-1) 1.56·10-6 (°C-1) 2.74·10-6

(°C-1) 3.19·10-8 (°C-1) 2.58·10-6

(°) = (°) 71.5

50

Numerical strategy for the search of solutions A 10-plies T300/5208 carbon-epoxy laminate designed to maximize

the Young's modulus along an orthotropy axis, K=1 in-plane orthotropic and uncoupled.

Solution found by BIANCA (residual: 4.6210-4):

[0./54.50/-44.67/87.19/-33.75/87.19/26.90/16.60/78.75/-31.31]

Average in-plane Young's modulus

generation

E

Gxy

51

Numerical strategy for the search of solutions Some final considerations about the use of genetic algorithms in the

mechanical context.

Bio-inspired metaheuristics mark the entrance of biological laws in various sectors of knowledge, also in hard sciences like mechanics.

This, in a sense, is the recognition that laws of the living world have a wider validity than that they have in their own biological context.

La loi de l’évolution est la plus importante de toutes les loi du monde; elle a présidé à notre naissance, a régi notre passé et, dans une large mesure, contrôle notre avenir.

Y. Coppens

52

Numerical strategy for the search of solutions A mathematical interpretation can be given to these laws (see The

simple genetic algorithm, by M. D. Vose) but, actually, the proof of their effectiveness is a matter of fact.

Genetic algorithms are able to well manage complexity; in the treatment of some inverse problems, the organization and management of complexity are, sometimes, a way to success.

The basic question is: when to simplify is the good choice?

Actually, in the nature, it is complexity which dominates biological systems (sexual reproduction, diploids and dominance, redundancy in the stocking of genetic data etc.).

In a sense, it is just the way we have followed with BIANCA, which is a genetic algorithm completely different from those used in laminates optimization.

53

Conclusions and perspectives The use of the polar method has proven to be rather effective in

some laminate design problems.

When coupled with a genetic algorithm, some hard problems can be solved with a sufficient accuracy.

A further step will be the inclusion of the ply number among the design variables. This will allow for weight optimization.

We believe that this can be done by some special genetic operations, i.e., we think that there must be a genetic way for the optimal design of the ply number (still to be verified: work in progress…!).

54

Conclusions and perspectives A promising way of action is the use of another metaheuristic, the

PSO (Particle Swarm Optimization, Eberhart & Kennedy, 1995).

This seems to be a very effective and robust numerical method for

the solution of non convex optimization problems in Rn.

An example: the search of an in-plane isotropic 4-ply laminate, formulated with the unified polar approach.

The algorithm finds quickly one of the Werren and Norris solutions.

fmean

step

55

An unconventional historical note The use of bio-inspired metaheuristics in the solution of hard

numerical problems in mechanics are only the last of a long sequel of points of contact between these two sciences, and demonstrates once more the usefulness of transversal knowledge.

Actually, it is a little bit curious to know that at the origin of modern mechanics the contacts with biology have been in the mind and in the work of three great scientists.

G. Galilei, in Discorsi e dimostrazioni matematiche intorno a due nuove scienze… (Leiden, 1638) makes some speculations of strength of materials concerning biological structures.

56

An unconventional historical note

P. L. M. de Maupertuis, in Venus Physica (Paris, 1745) rigorously demonstrates the genetic transmission of characters from the father and the mother. In De universali naturae systematae (Erlangen, 1751) he is the first to make the hypothesis that mutation is a cause of biodiversity. He published also some papers about his naturalistic observations.

R. Hooke, in Micrographia (London, 1665) is the first to publish systematic observations of biological tissues made by himself with a microscopy that he fabricated. He is the father of the word cell, that he proposed after the observation of the texture of the cork.

Thank you very much for your attention.