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Multiconstrained optimization of laminated and sandwich plates using evolutionary algorithms and higher-order plate theories
Composite Structures 59 (2003) 149–154 www.elsevier.com/locate/compstruct
Multiconstrained optimization of laminated and sandwich plates using evolutionary algorithms and higher-order plate theories M. Di Sciuva *, M. Gherlone, D. Lomario Department of Aeronautical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Abstract The optimization of laminated and sandwich plates with respect to buckling load and thickness has been performed, using different sets of constraints such as the fundamental frequency, the maximum deflection under transverse uniform distributed load, the mass and the buckling load. Two different evolutionary algorithms have been employed (genetic algorithm and simulated annealing (SA)) together with two plate models (classical plate theory and cubic zig-zag model). The performed analyses show that the two evolutionary algorithms provide almost the same results though the SA procedure is less time consuming; furthermore, results of the two displacement theories are the closer to each other the higher the side-to-thickness ratio is. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Multiconstrained optimization; Composite plates; Genetic algorithm; Simulated annealing; Penalty factors; Kirchhoff plate theory; Cubic zig-zag plate theory
1. Introduction In Mechanics, Aerospace, and other branches of engineering, composite materials are increasingly used due to their high stiffness/weight and strength/weight ratios, and their ease of tailoring. As a result, the need for developing numerical procedures for the mass minimization of laminated composite and sandwich plates under stiffness, buckling and frequency constraints is a topic of primary concern. Open literature deals with two problems: (i) the minimization of the mass, constrained, for example, by the buckling load [1], (ii) the maximization of the buckling load with constraints on the mass [2]. Other constraints are on the strength and on the lay-up [3]. Until now, the problem has mainly been addressed by using gradient-based optimization techniques. Recently, it has been studied by using evolutionary algorithms, which are optimization tools that do not make use of the gradient of the objective function. Genetic algorithm (GA) and simulated annealing (SA) are the most com* Corresponding author. Tel.: +39-0115-646826; fax: +39-0115646899. E-mail addresses: [email protected] (M. Di Sciuva), [email protected] (M. Gherlone), [email protected] (D. Lomario).
mon choices treated in literature. GA is an implementation of the rules stated by the DarwinÕs theory, while SA mimics the thermodynamic of a melted metallic mass during annealing. These two simple algorithms deal with discrete optimization problems, and become particularly interesting when the objective function contains many local optima and also discontinuities in the domain, non-derivable points, etc. In such situations, classical gradient-based methods can be unable to find the global optimum, and can easily find the local optima nearer to the starting point in the domain. Evolutionary algorithms can skip these difficulties, providing near-optimal global solutions. On the other side, these algorithms require a greater amount of calculations than classical algorithms [4,5]. The use of advanced composite materials involves incorporating a number of non-classical effects, among which we recall: (i) the transverse shear flexibility (especially for thick plates), (ii) the need for fulfilling the continuity requirement of transverse shear stresses at the layer interfaces (particularly when large variations in transverse shear moduli from layer to layer are experienced), (iii) interfacial bonding damages between the constituent laminae of the composite structures (as resulting from manufacturing processes and/or operating conditions). It is evident that the simple classical laminate theory (CLT) [6] is not capable to capture these
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non-classical effects, and therefore higher-order theories [7] must be used. In this work, the optimizations of the lay-up of laminated composite and sandwich flat plates with maximum buckling load and minimum mass as objectives, and transverse stiffness, mass, and frequency as constraints, is investigated by using GA and SA. The constraints are variously combined. Moreover, we use the CLT and the cubic zig-zag one (CZZ) proposed by Di Sciuva [7]. The calculations reported in this paper are aimed mainly to compare the two evolutionary algorithms (GA and SA) and the two plate theories (CLT and CZZ).
2.3. Simulated annealing The scheme here used for the SA is exactly the same proposed in [9]. The annealing schedule is the exponential decrease given by T ðtÞ ¼ T0 egt where T0 is the initial temperature, g is a coefficient that rules the decrease ratio, and t is the time. The design space is explored by means of a single point: this corresponds to following the path of a single molecule of the melted mass during the annealing.
3. The cubic zig-zag displacement model 2. Description of the evolutionary algorithms 2.1. Generalities The most evident differences between these new methods and the classical gradient-based ones can be identified in the rules used to explore the optimization space and in the number of solutions reached. Classical methods are algorithms that, starting from a single point in the problem variables space, employ a gradient method to reach the optimum of the objective function [8]. Each step of these algorithms is directed towards the nearest optimum, hence they find the optimum nearer to the starting point. Evolutionary algorithms, instead, sometimes allow steps in the opposite direction, so giving the possibility to skip local optima and reach the global one. For this reason they are often indicated as ‘‘Global’’ optimizers. Furthermore, GA uses a group of points for the exploration, and therefore provides a number of solutions at each run and increments the power of the global exploration. A presentation of evolutionary algorithms is given in [4,9].
In order to study the effects of transverse shear deformability and of other non-classical effects on the optimization techniques, we introduce a higher-order displacement field that will be used in the following beside the CLT [6]. The points of the 3-D plate space are referred to a set of orthogonal system of coordinates x, y and z; the dimensions of the plate are, respectively, a, b and h, while the displacement components are u, v and w. t is the region occupied by the plate in the undeformed (reference) configuration and X its reference surface with boundary curve C. For sake of generality, the location of the plate reference surface X is chosen to be given by gh apart from the bottom face, with 0 6 g 6 1. The CZZ theory, developed by Di Sciuva [7], may be summarized as follows fV ðx; y; z; tÞg ¼ V ð0Þ ðx; y; tÞ z rwð0Þ ðx; y; tÞ þ ½F ðzÞ fwðx; y; tÞg
ð1Þ
wðx; y; z; tÞ ¼ wð0Þ ðx; y; tÞ where
ð0Þ ð0Þ u u ; V ; ¼ v vð0Þ ( ) wð0Þ ð0Þ h ;x Dw ; fwg ¼ ¼ ð0Þ u w;y fV g ¼
2.2. Genetic algorithm The GA here used employs the natural codification of the problem variables, which means that the chromosomes contain the true values of the variables [10], and the genetic operators are applied in a deterministic way. The reproduction scheme is the roulette rule: the probability for each element of the population of participating to the creation of the next one is proportional to its fitness [4]. The crossover operators are one-point, two-point, arithmetic and heuristic, while the mutations are boundary, multi-non-uniform, non-uniform, uniform, permutation and new permutation. All the operators but permutation and new permutation have been taken from Houck [11,12], while permutation and new permutation have been taken from Le Riche et al. [1].
ð2Þ
In the previous definitions we have introduced the five generalized displacements of the model, i.e. uð0Þ , vð0Þ and wð0Þ (which are the displacements of the reference surface in the x, y and z directions, respectively), h and u (which are representative of the rotations of the reference surface due to the transverse shear deformation). Moreover, the matrix function ½F ðzÞ , is a piecewise throughthe-thickness cubic function. The particular properties of this matrix function enable the zig-zag model to feature several non-classical behaviours of plates: 1. ½F ðzÞ may exhibit discontinuities at the interfaces of the laminate. In other words, the in-plane displace-
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ments may present jumps in their through-the-thickness distribution. In fact, as a result of manufacturing processes and/or operating conditions, interfacial bonding damage between the constituent laminae of the composite structures can occur. As a consequence, the plate may exhibit interlayer slips whose physical essence is that of tangential springs at each debonded interface; we suppose a linear relation between the in-plane displacement jump across a damaged interface (fDV g) and the transverse shear stress acting on it (fsg) fDV g ¼ ½R fsg
ð3Þ
where [R] is the matrix of the sliding constants between the two layers. Note that in addition to the cases of imperfectly bonded interfaces (Rab 6¼ 0; 1), Eq. (3) covers also the extreme situations corresponding to perfectly bonded interfaces (Rab ¼ 0 yielding DVa ¼ 0) and completely debonded interfaces (Rab ¼ 1 yielding sa3 ¼ 0). 2. The transverse shear stresses obtained with the constitutive equations of the model are through-thethickness continuous. 3. The same stresses vanish on the top and bottom plate surfaces.
4. Applications of the evolutionary algorithms 4.1. Generalities This work deals with problems concerning the design of rectangular flat, laminated and sandwich plates. The generic plate is always loaded in its mid-plane, therefore the main concern in the choice of the stacking sequence is to avoid the buckling. Another concern is the minimization of the mass, given that the plate must bear the assigned loads. In this work the design is tackled by varying the angular orientations of the fibres of the layers, and sometimes both the angular orientations and the thicknesses. The stacking sequence has been taken symmetric and balanced, i.e. of the kind ðþh1 ; h1 ; þh2 ; h2 ; . . .ÞS , and in the same fashion are arranged the thicknesses. As for the allowed values of angles and thicknesses, we have used of a limited discrete set of values (that, in the evolutionary algorithmsÕ language, is called the ‘‘alphabet’’). The problems solved are listed in Table 1. Problem 1 has been repeated for a sandwich plate with laminated faces, following Hwu et al. [13]. In this case the problem variables were the faces stacking sequences which were neither balanced nor symmetric. The angles could vary continuously in the range [)90°, 90°]. The calculations for both the laminated and sandwich plates (the core of the sandwich is substituted by an equivalent orthotropic layer, [14,15]) have been per-
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Table 1 Definition of the problems regarding laminated plates Problem no.
Objective
Constraints
Design variables
1 2 3 4 5 6 7
Max Nx;cr Max Nx;cr Min np Min np Min np Max Nx;cr Max Nx;cr
– u < ulim;1 Nx;cr > Nx w < wlim u > ulim;2 m < mlim u < ulim;2 , m < mlim
hi (fixed np ) hi (fixed np ) hi hi hi hi , si (fixed np ) hi , si (fixed np )
Nx;cr is the buckling load, w the displacement of the centre of the plate under transverse distributed load, np the number of plies and m the mass of the entire plate. Nx is the applied load and u the fundamental frequency. The design variables are the angles of the plies and their thicknesses, respectively indicated by hi and si . Limit values of the constraints: ulim;1 ¼ 1265 Hz; ulim;2 ¼ 791 Hz; Nx ¼ 500 N/mm, wlim ¼ 3 mm, mlim ¼ mmax 0:5 (mmax is the mass of the plate when each lamina is set to its highest allowed value).
formed by means of a Rayleigh-Ritz procedure based on the use of the Gram-Schmidt polynomials [16]. 4.2. Fitness functions The constrained optimization problems can be written in the following mathematical form: max F ðx1 ; x2 ; . . . xk Þ ð4Þ s:t: : gi ðx1 ; x2 ; . . . xk Þ P gi i ¼ 1; 2; . . . ; n where F is the objective, xi s are the problem variables and gi s are the constraints. The fitness functions have been written by means of the penalty factor approach [8], i.e. in the following form: ð1Þ
ðnÞ
f ¼ P0 F þ hð1Þ ðg1 ; Pi Þ þ þ hðnÞ ðgn ; Pi Þ; i ¼ 1; . . . ; n
ð5Þ
where the P1 ; P2 ; . . . ; Pn are the penalty factors, and the hi s are the corresponding penalty functions. Three kinds of penalty functions have been used, the equations of which are: hðjÞ ¼ P1 ðgj gj Þ ð6Þ ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjÞ ðjÞ if gj P gj P2 ½gj gj þ P3 hðjÞ ¼ ð7Þ ðjÞ ðjÞ 2 P4 ½gj gj P5 if gj < gj 8 ðjÞ ðjÞ h
i2 ðjÞ < P2 P3 g g þ P ðjÞ þ P3 if gj P gj j j 2 4 ðjÞ ðjÞ P h ¼ : 4 ðjÞ ðjÞ 2 P5 ½gj gj P6 if gj < gj ð8Þ Notice that the jth constraint is enhanced or penalized by adding or subtracting a quantity that depends upon the penalty factors and the distance of the gj from the
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u u 1 1 Du1 ¼ u 1
ð9Þ
When the constraint is on the buckling load, the degree of satisfaction is monitored by means of the following factor k¼
Nx;cr Nx
ð10Þ
where Nx is the external load, applied in the x-direction, and Nx;cr is the corresponding critical load. 5.1. Laminated plates
Fig. 1. Penalty functions given by Eqs. (7) and (8) for the GA (a and b) and for the SA (c and d).
limit value. Graphical representation of Eqs. (7) and (8) is given in Fig. 1 (a and b, respectively). Expression (8) has revealed itself the most useful of those used, because it penalizes those solutions that satisfy the constraint too much, together with those that do not satisfy the constraint at all. In fact, by properly ðjÞ ðjÞ choosing P2 and P3 , the curvature of the parable can be increased so much that the parable itself becomes negative for gj too higher than gj . This is a means to keep into reasonable limits the satisfaction of the constraints. All the fitnesses discussed above ave been employed in the GA. For the SA we used the symmetric of the same shapes (often with the same penalties) about the xaxis (Fig. 1c and d).
5. Numerical results We have applied the developed algorithms to optimization problems concerning laminated and sandwich plates. The goodness of the results can be stated on the basis of the degree of satisfaction of the constraints (in problems where constraints are present), and on the concordance of the results of different programs that run the same optimization problem, with different codifications and algorithms. In the case of frequency, deflection and mass constraints, the degree of satisfaction has been evaluated by means of the percent distance of the variable to be constrained from the imposed limit. If, for 1 the example, u1 is the fundamental frequency and u corresponding limit, the degree of satisfaction is defined as
In all the problems we have studied a laminated plate having dimensions a ¼ 508 mm and b ¼ 127 mm, simply supported on the four edges and loaded by in-plane distributed loads with a ratio r ¼ Ny =Nx ¼ 0:125. The mechanical properties of the constituent lamina are the following: E1 ¼ 158; 000 MPa, E2 ¼ 10; 000 MPa, E3 ¼ 10; 000 MPa, m12 ¼ 0:315, m13 ¼ 1 105 , m23 ¼ 1 106 , G12 ¼ 6000 MPa, G13 ¼ 6000 MPa, G23 ¼ 4000 MPa, q ¼ 1:6 106 kg/mm3 . In all the problems the lamina thickness has been kept to the value t ¼ 0:127 mm, exception made for those where also the thicknesses, together with the angles of the fibres, were problem variables. The discrete sets of values are [0°, 30°, 45°, 60°, 90°] for the angles, and [0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, 0.11, 0.12, 0.127] mm for the thicknesses. In problems 6 and 7 the objective is to maximize the buckling load and the pliesÕ thicknesses are design variables. In this case, a limit on the total mass is needed in order to avoid that all the thicknesses take their highest allowed value. In the buckling load maximization problems (1, 2, 6 and 7) the number of plies has always been 64, while in the mass minimization ones (3, 4 and 5) the value 64 was the maximum available. Table 2 contains the results obtained for all the problems, using both CLT and CZZ displacement theory, and the two employed evolutionary algorithm, GA and SA. The aim of the present study is both to compare the performances of different evolutionary algorithms on the same problems and the performances of the two mathematical models. At first, we have solved the seven problems (GA and SA) using the CLT, trying three different kinds of fitness function shapes. Once we have found, for each problem, the best choice for fitness shape and penalty factors, we have used the same choice in the corresponding problems solved by means of the CZZ model. The constraint was yet satisfied in all the cases but the one highlighted in Table 1 in bold. For this problem only we had to modify the penalty factors in order to obtain the shown results. Specifically, in the CZZ model it has been nec-
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Table 2 Results for the laminated plates Problem no.
Genetic algorithm
Simulated annealing
CLT
CZZ
CLT
CZZ
1
Nx;cr ¼ 8028 N/mm
Nx;cr ¼ 7184 N/mm
Nx;cr ¼ 8026 N/mm
Nx;cr ¼ 7179 N/mm
2
Nx;cr ¼ 6796 N/mm Du1 ¼ 0:01%
Nx;cr ¼ 6539 N/mm Du1 ¼ 0:13%
Nx;cr ¼ 6503 N/mm Du1 ¼ 0:35%
Nx;cr ¼ 6091 N/mm Du1 ¼ 3:50%
3
num: lay: ¼ 28 k ¼ 1:01
num: lay: ¼ 28 k ¼ 1:02
num: lay: ¼ 28 k ¼ 1:69
num: lay: ¼ 28 k ¼ 1:08
4
num: lay: ¼ 40 Dwc ¼ 4:92%
num: lay: ¼ 52 Dwc ¼ 2:19%
num: lay: ¼ 40 Dwc ¼ 2:05%
num: lay: ¼ 48 Dwc ¼ 1:29%
5
num: lay: ¼ 28 Du1 ¼ 0:29%
num: lay: ¼ 36 Du1 ¼ 4:96%
num: lay: ¼ 24 Du1 ¼ 0:84%
num: lay: ¼ 24 Du1 ¼ 1:76%
6
Nx;cr ¼ 2254 N/mm Dm ¼ 0:18%
Nx;cr ¼ 2213 N/mm Dm ¼ 0:18%
Nx;cr ¼ 1205 N/mm Dm ¼ 0:09%
Nx;cr ¼ 1138 N/mm Dm ¼ 0:18%
7
Nx;cr ¼ 1241 N/mm Du1 ¼ 0:85% Dm ¼ 0:18%
Nx;cr ¼ 1189 N/mm Du1 ¼ 2:08% Dm ¼ 0:55%
Nx;cr ¼ 1193 N/mm Du1 ¼ 0:08% Dm ¼ 0:31%
Nx;cr ¼ 1078 N/mm Du1 ¼ 1:16% Dm ¼ 0:82%
essary to give lower importance to the objective than to the constraint satisfaction. As expected, the algorithms involving the CZZ theory yield lower buckling loads and higher number of layers than those obtained using the CLT model. This is due to the improved capability of the CZZ model of capturing the behaviour of laminated plates, in particular when the side-to-thickness ratio is much lower than 30. In fact, if we consider the problems (1 and 2) in which we maximize the buckling load and the total thickness is fixed, we have that b=h is equal to 15.6 and the final laminations and the corresponding critical loads are significantly different; the stacking-sequences differ in particular for the inner plies, where the effect of the transverse shear deformability is stronger. For example the two optimal stacking sequences obtained for problem 1 using GA are ð452 ; 60; 45; 60; 457 ; 60; 453 ÞS and ð4511 ; 602 ; 45; 60; 90ÞS for CLT and CZZ respectively. Conversely, when the ratio is higher than 30, we expect that the two theories give quite the same results. This is confirmed in the mass minimization problems when the final number of plies is low. In order to confirm the conclusion here stated, we have performed a single run of problem 1 (GA), with a very high (100) side-to-thickness ratio obtaining the same stacking-sequence ð4515 ; 60ÞS and almost the same buckling load in both the theories (203.1 and 202.5 N/mm for CLT and CZZ respectively). Generally speaking, the results obtained from the same displacement model and from the two evolutionary algorithms are the same. Anyway, in some particular cases we found a relevant difference. For example, if we consider problem 6, solved by means of SA, it is readily seen that the buckling loads are almost the half of those
obtained using GA. This is probably due to the fact that the set of penalty factors chosen was such that the optimum search was more stressed on the frequency constraint satisfaction than on the buckling load maximization. A great difference in the runtime requested by the various algorithms has been detected, depending both on the evolutionary algorithm (GA, SA) and on the displacement model (CLT, CZZ). The solution by means of the CZZ has taken, for the GA (SA) algorithm, 30–60 (40–50) times the time requested for the corresponding problems solved by using the CLT. Therefore, from this point of view the CLT has revealed to be much more advantageous than the CZZ. The comparison between the times requested by GA and SA, using both the CLT and the CZZ theory, shows that the SA is 3.5–5.5 times quicker than the GA. For problems 4 and 5 we have observed that optimal solutions often contained a great number of consecutive layers having either 0° or 90°. In order to limited this phenomenon, we have added the contiguity constraint, which is a limit on the number of consecutive plies with either 0° or 90° orientation. 5.2. Sandwich plates In order to assess the performances of the CZZ for sandwich plates, we solved the same unconstrained buckling load maximization problem as that studied by Moh and Hwu [17]. The plateÕs dimensions are those indicated in Table 3. The plate is simply supported on the four edges and loaded by in-plane distributed loads with variable ratios r ¼ Ny =Nx . The faces are laminated plates, made of four laminae; the stacking-sequence is
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Table 3 Results for the sandwich plates a (mm)
b (mm)
r
Moh and Hwu Stack. Seq.
Nx;cr (N/mm)
Stack. Seq.
Nx;cr (N/mm)
1000 1000 1000
500 1000 1000
0 0.5 2
ð44; 44; 44; 44; cÞS ð45; 45; 45; 45; cÞS ð45; 45; 45; 45; cÞS
352.66 66.62 33.31
ð46; 42; 45; 37; cÞS ð41; 48; 40; 42; cÞS ð41; 49; 46; 48; cÞS
356.82 66.77 33.42
ðh1 ; h2 ; h3 ; h4 ; cÞ, i.e. there are no constraints on the lamination. The laminaeÕs mechanical properties are the following: E1 ¼ 181; 000 MPa, E2 ¼ 10; 300 GPa, G12 ¼ 7170 Mpa, m12 ¼ 0:28; the coreÕs mechanical properties are: G13 ¼ 146 MPa, G23 ¼ 90:4 Mpa. 1 In all the problems the lamina thickness has been kept to the value t ¼ 0:125 mm; the coreÕs thickness has been, in all the cases, c ¼ 10 mm. Moh and Hwu solved the problem by means of the PowellÕs zero-order conjugate method. The authors reported only one optimal configuration, which is (h; h; h; h; c). For the same problem, we have obtained a group of solutions, by means of SA and CZZ model, with almost the same value of buckling load (Table 3). In other words, there exists a family of optimal configurations for all the studied cases. This family contains the stacking sequences (h1 ; h2 ; h3 ; h4 ; c), (h1 ; h2 ; h3 ; h4 ; c), (h1 ; h2 ; h3 ; h4 ; c), with 35° < hj < 50°.
6. Conclusions We have solved several optimization problems, testing two structural models (CLT and CZZ) and two evolutionary algorithms (GA and SA). The objectives have been the buckling load and the mass, which are two of the most important tasks in Aerospace design. Results show that in all the treated cases there is a good agreement between the GA and the SA methods. We also found that the SA is less time-consuming, therefore it appears to be more suitable for those problems in which complex numerical models (for instance, FEM models) have to be dealt with. The fitness functions and the evolutionary operators have been able to produce optimal solutions with a small degree of satisfaction of the constraints. As expected, the two displacement models have yielded the same results in terms of optimal stacking sequences and objectives when the plate side-to-thickness ratio was high; the CLT requests, in a general case, a
1 The remaining mechanical properties of the core have been determined by means of simplified models based on the honeycomb cells supposed geometry [14,15].
Present
shorter computing time than the CZZ theory but for thick plates it gives coarser results. The results of the performed investigations show that the evolutionary methods are simple to implement, and give good results in all the studied problems. Moreover, it seems that the presence of different kinds of constraints heavily influences the best lamination for a given problem. References [1] Le Riche R, Haftka RT. Improved genetic algorithm for minimum thickness composite laminate design. Compos Eng 1995;2(5):143–61. [2] Le Riche R, Haftka RT. Optimization of Laminate Stacking Sequence for Buckling Load Maximization by Genetic Algorithm. AIAA/ASME/AHS/ASCE/ASC 33rd Structures, Structural Dynamics and Materials Conference, Dallas, TX. AIAA J 1993;(31)5:951–6. [3] Di Sciuva M, Lomario D, Voyat JP. Evolutionary Algorithms. Their Use in Multiconstrained Optimization of Laminated and Sandwich Plates. Proceedings of the European Conference on Computational Mechanics (ECCM), CTS, Cracow, 26–29 June 2001. [4] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading (MA): Addison-Wesley; 1989. [5] Haftka RT, Gurdal Z. Elements of structural optimization. Kluwer Academic Publishers; 1996. [6] Jones RM. Mechanics of composite materials. Kogakusha, Tokyo: McGraw-Hill; 1975. [7] Di Sciuva M. A geometrically nonlinear theory of multilayered plates with interlayer slips. AIAA J 1997;11(35):1753–9. [8] Gottfried BS, Weisman J. Introduction to optimization theory. Englewood Cliffs, NJ: Prentice Hall Inc.; 1973. [9] Jang JSR, Sun CT, Mizutani E. Neuro-fuzzy and soft computing: A computational approach to learning and machine intelligence. Upper Saddle River: Prentice Hall; 1997. [10] G€ urdal Z, Haftka RT, Hajela P. Design and optimization of laminated composite materials. NY: Wiley; 1999. [11] Houck CR, Joines JA, Kay MG. Binary and Real-Valued Simulation Evolution for MATLAB. 1996. [12] Houck CR, Joines J, Kay M. A Genetic Algorithm for Function Optimization: a MATLAB Implementation. ACM Transactions on Mathematical Software, 1996. [13] Hwu C, Hu JS. Buckling and postbuckling of the laminated composite sandwich beams. AIAA J 1992;7(30):1901–9. [14] Becker W. The in-plane stiffnesses of a honeycomb core including the thickness effect. Arch Appl Mech 1998;(68):334–41. [15] Grediac M. A finite element study of the transverse shear in honeycomb cores. Int J Solids Struct 1993;13(30):1777–88. [16] Arfken G. Mathematical methods for physicists. Orlando, FL: Academic Press; 1985. [17] Moh JS, Hwu C. Optimization for buckling of composites sandwich plates. AIAA J 1997;5(35):863–8.
Multiconstrained optimization of laminated and sandwich plates using evolutionary algorithms and higher-order plate theories M. Di Sciuva *, M. Gherlone, D. Lomario Department of Aeronautical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Abstract The optimization of laminated and sandwich plates with respect to buckling load and thickness has been performed, using different sets of constraints such as the fundamental frequency, the maximum deflection under transverse uniform distributed load, the mass and the buckling load. Two different evolutionary algorithms have been employed (genetic algorithm and simulated annealing (SA)) together with two plate models (classical plate theory and cubic zig-zag model). The performed analyses show that the two evolutionary algorithms provide almost the same results though the SA procedure is less time consuming; furthermore, results of the two displacement theories are the closer to each other the higher the side-to-thickness ratio is. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Multiconstrained optimization; Composite plates; Genetic algorithm; Simulated annealing; Penalty factors; Kirchhoff plate theory; Cubic zig-zag plate theory
1. Introduction In Mechanics, Aerospace, and other branches of engineering, composite materials are increasingly used due to their high stiffness/weight and strength/weight ratios, and their ease of tailoring. As a result, the need for developing numerical procedures for the mass minimization of laminated composite and sandwich plates under stiffness, buckling and frequency constraints is a topic of primary concern. Open literature deals with two problems: (i) the minimization of the mass, constrained, for example, by the buckling load [1], (ii) the maximization of the buckling load with constraints on the mass [2]. Other constraints are on the strength and on the lay-up [3]. Until now, the problem has mainly been addressed by using gradient-based optimization techniques. Recently, it has been studied by using evolutionary algorithms, which are optimization tools that do not make use of the gradient of the objective function. Genetic algorithm (GA) and simulated annealing (SA) are the most com* Corresponding author. Tel.: +39-0115-646826; fax: +39-0115646899. E-mail addresses: [email protected] (M. Di Sciuva), [email protected] (M. Gherlone), [email protected] (D. Lomario).
mon choices treated in literature. GA is an implementation of the rules stated by the DarwinÕs theory, while SA mimics the thermodynamic of a melted metallic mass during annealing. These two simple algorithms deal with discrete optimization problems, and become particularly interesting when the objective function contains many local optima and also discontinuities in the domain, non-derivable points, etc. In such situations, classical gradient-based methods can be unable to find the global optimum, and can easily find the local optima nearer to the starting point in the domain. Evolutionary algorithms can skip these difficulties, providing near-optimal global solutions. On the other side, these algorithms require a greater amount of calculations than classical algorithms [4,5]. The use of advanced composite materials involves incorporating a number of non-classical effects, among which we recall: (i) the transverse shear flexibility (especially for thick plates), (ii) the need for fulfilling the continuity requirement of transverse shear stresses at the layer interfaces (particularly when large variations in transverse shear moduli from layer to layer are experienced), (iii) interfacial bonding damages between the constituent laminae of the composite structures (as resulting from manufacturing processes and/or operating conditions). It is evident that the simple classical laminate theory (CLT) [6] is not capable to capture these
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non-classical effects, and therefore higher-order theories [7] must be used. In this work, the optimizations of the lay-up of laminated composite and sandwich flat plates with maximum buckling load and minimum mass as objectives, and transverse stiffness, mass, and frequency as constraints, is investigated by using GA and SA. The constraints are variously combined. Moreover, we use the CLT and the cubic zig-zag one (CZZ) proposed by Di Sciuva [7]. The calculations reported in this paper are aimed mainly to compare the two evolutionary algorithms (GA and SA) and the two plate theories (CLT and CZZ).
2.3. Simulated annealing The scheme here used for the SA is exactly the same proposed in [9]. The annealing schedule is the exponential decrease given by T ðtÞ ¼ T0 egt where T0 is the initial temperature, g is a coefficient that rules the decrease ratio, and t is the time. The design space is explored by means of a single point: this corresponds to following the path of a single molecule of the melted mass during the annealing.
3. The cubic zig-zag displacement model 2. Description of the evolutionary algorithms 2.1. Generalities The most evident differences between these new methods and the classical gradient-based ones can be identified in the rules used to explore the optimization space and in the number of solutions reached. Classical methods are algorithms that, starting from a single point in the problem variables space, employ a gradient method to reach the optimum of the objective function [8]. Each step of these algorithms is directed towards the nearest optimum, hence they find the optimum nearer to the starting point. Evolutionary algorithms, instead, sometimes allow steps in the opposite direction, so giving the possibility to skip local optima and reach the global one. For this reason they are often indicated as ‘‘Global’’ optimizers. Furthermore, GA uses a group of points for the exploration, and therefore provides a number of solutions at each run and increments the power of the global exploration. A presentation of evolutionary algorithms is given in [4,9].
In order to study the effects of transverse shear deformability and of other non-classical effects on the optimization techniques, we introduce a higher-order displacement field that will be used in the following beside the CLT [6]. The points of the 3-D plate space are referred to a set of orthogonal system of coordinates x, y and z; the dimensions of the plate are, respectively, a, b and h, while the displacement components are u, v and w. t is the region occupied by the plate in the undeformed (reference) configuration and X its reference surface with boundary curve C. For sake of generality, the location of the plate reference surface X is chosen to be given by gh apart from the bottom face, with 0 6 g 6 1. The CZZ theory, developed by Di Sciuva [7], may be summarized as follows fV ðx; y; z; tÞg ¼ V ð0Þ ðx; y; tÞ z rwð0Þ ðx; y; tÞ þ ½F ðzÞ fwðx; y; tÞg
ð1Þ
wðx; y; z; tÞ ¼ wð0Þ ðx; y; tÞ where
ð0Þ ð0Þ u u ; V ; ¼ v vð0Þ ( ) wð0Þ ð0Þ h ;x Dw ; fwg ¼ ¼ ð0Þ u w;y fV g ¼
2.2. Genetic algorithm The GA here used employs the natural codification of the problem variables, which means that the chromosomes contain the true values of the variables [10], and the genetic operators are applied in a deterministic way. The reproduction scheme is the roulette rule: the probability for each element of the population of participating to the creation of the next one is proportional to its fitness [4]. The crossover operators are one-point, two-point, arithmetic and heuristic, while the mutations are boundary, multi-non-uniform, non-uniform, uniform, permutation and new permutation. All the operators but permutation and new permutation have been taken from Houck [11,12], while permutation and new permutation have been taken from Le Riche et al. [1].
ð2Þ
In the previous definitions we have introduced the five generalized displacements of the model, i.e. uð0Þ , vð0Þ and wð0Þ (which are the displacements of the reference surface in the x, y and z directions, respectively), h and u (which are representative of the rotations of the reference surface due to the transverse shear deformation). Moreover, the matrix function ½F ðzÞ , is a piecewise throughthe-thickness cubic function. The particular properties of this matrix function enable the zig-zag model to feature several non-classical behaviours of plates: 1. ½F ðzÞ may exhibit discontinuities at the interfaces of the laminate. In other words, the in-plane displace-
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ments may present jumps in their through-the-thickness distribution. In fact, as a result of manufacturing processes and/or operating conditions, interfacial bonding damage between the constituent laminae of the composite structures can occur. As a consequence, the plate may exhibit interlayer slips whose physical essence is that of tangential springs at each debonded interface; we suppose a linear relation between the in-plane displacement jump across a damaged interface (fDV g) and the transverse shear stress acting on it (fsg) fDV g ¼ ½R fsg
ð3Þ
where [R] is the matrix of the sliding constants between the two layers. Note that in addition to the cases of imperfectly bonded interfaces (Rab 6¼ 0; 1), Eq. (3) covers also the extreme situations corresponding to perfectly bonded interfaces (Rab ¼ 0 yielding DVa ¼ 0) and completely debonded interfaces (Rab ¼ 1 yielding sa3 ¼ 0). 2. The transverse shear stresses obtained with the constitutive equations of the model are through-thethickness continuous. 3. The same stresses vanish on the top and bottom plate surfaces.
4. Applications of the evolutionary algorithms 4.1. Generalities This work deals with problems concerning the design of rectangular flat, laminated and sandwich plates. The generic plate is always loaded in its mid-plane, therefore the main concern in the choice of the stacking sequence is to avoid the buckling. Another concern is the minimization of the mass, given that the plate must bear the assigned loads. In this work the design is tackled by varying the angular orientations of the fibres of the layers, and sometimes both the angular orientations and the thicknesses. The stacking sequence has been taken symmetric and balanced, i.e. of the kind ðþh1 ; h1 ; þh2 ; h2 ; . . .ÞS , and in the same fashion are arranged the thicknesses. As for the allowed values of angles and thicknesses, we have used of a limited discrete set of values (that, in the evolutionary algorithmsÕ language, is called the ‘‘alphabet’’). The problems solved are listed in Table 1. Problem 1 has been repeated for a sandwich plate with laminated faces, following Hwu et al. [13]. In this case the problem variables were the faces stacking sequences which were neither balanced nor symmetric. The angles could vary continuously in the range [)90°, 90°]. The calculations for both the laminated and sandwich plates (the core of the sandwich is substituted by an equivalent orthotropic layer, [14,15]) have been per-
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Table 1 Definition of the problems regarding laminated plates Problem no.
Objective
Constraints
Design variables
1 2 3 4 5 6 7
Max Nx;cr Max Nx;cr Min np Min np Min np Max Nx;cr Max Nx;cr
– u < ulim;1 Nx;cr > Nx w < wlim u > ulim;2 m < mlim u < ulim;2 , m < mlim
hi (fixed np ) hi (fixed np ) hi hi hi hi , si (fixed np ) hi , si (fixed np )
Nx;cr is the buckling load, w the displacement of the centre of the plate under transverse distributed load, np the number of plies and m the mass of the entire plate. Nx is the applied load and u the fundamental frequency. The design variables are the angles of the plies and their thicknesses, respectively indicated by hi and si . Limit values of the constraints: ulim;1 ¼ 1265 Hz; ulim;2 ¼ 791 Hz; Nx ¼ 500 N/mm, wlim ¼ 3 mm, mlim ¼ mmax 0:5 (mmax is the mass of the plate when each lamina is set to its highest allowed value).
formed by means of a Rayleigh-Ritz procedure based on the use of the Gram-Schmidt polynomials [16]. 4.2. Fitness functions The constrained optimization problems can be written in the following mathematical form: max F ðx1 ; x2 ; . . . xk Þ ð4Þ s:t: : gi ðx1 ; x2 ; . . . xk Þ P gi i ¼ 1; 2; . . . ; n where F is the objective, xi s are the problem variables and gi s are the constraints. The fitness functions have been written by means of the penalty factor approach [8], i.e. in the following form: ð1Þ
ðnÞ
f ¼ P0 F þ hð1Þ ðg1 ; Pi Þ þ þ hðnÞ ðgn ; Pi Þ; i ¼ 1; . . . ; n
ð5Þ
where the P1 ; P2 ; . . . ; Pn are the penalty factors, and the hi s are the corresponding penalty functions. Three kinds of penalty functions have been used, the equations of which are: hðjÞ ¼ P1 ðgj gj Þ ð6Þ ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjÞ ðjÞ if gj P gj P2 ½gj gj þ P3 hðjÞ ¼ ð7Þ ðjÞ ðjÞ 2 P4 ½gj gj P5 if gj < gj 8 ðjÞ ðjÞ h
i2 ðjÞ < P2 P3 g g þ P ðjÞ þ P3 if gj P gj j j 2 4 ðjÞ ðjÞ P h ¼ : 4 ðjÞ ðjÞ 2 P5 ½gj gj P6 if gj < gj ð8Þ Notice that the jth constraint is enhanced or penalized by adding or subtracting a quantity that depends upon the penalty factors and the distance of the gj from the
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u u 1 1 Du1 ¼ u 1
ð9Þ
When the constraint is on the buckling load, the degree of satisfaction is monitored by means of the following factor k¼
Nx;cr Nx
ð10Þ
where Nx is the external load, applied in the x-direction, and Nx;cr is the corresponding critical load. 5.1. Laminated plates
Fig. 1. Penalty functions given by Eqs. (7) and (8) for the GA (a and b) and for the SA (c and d).
limit value. Graphical representation of Eqs. (7) and (8) is given in Fig. 1 (a and b, respectively). Expression (8) has revealed itself the most useful of those used, because it penalizes those solutions that satisfy the constraint too much, together with those that do not satisfy the constraint at all. In fact, by properly ðjÞ ðjÞ choosing P2 and P3 , the curvature of the parable can be increased so much that the parable itself becomes negative for gj too higher than gj . This is a means to keep into reasonable limits the satisfaction of the constraints. All the fitnesses discussed above ave been employed in the GA. For the SA we used the symmetric of the same shapes (often with the same penalties) about the xaxis (Fig. 1c and d).
5. Numerical results We have applied the developed algorithms to optimization problems concerning laminated and sandwich plates. The goodness of the results can be stated on the basis of the degree of satisfaction of the constraints (in problems where constraints are present), and on the concordance of the results of different programs that run the same optimization problem, with different codifications and algorithms. In the case of frequency, deflection and mass constraints, the degree of satisfaction has been evaluated by means of the percent distance of the variable to be constrained from the imposed limit. If, for 1 the example, u1 is the fundamental frequency and u corresponding limit, the degree of satisfaction is defined as
In all the problems we have studied a laminated plate having dimensions a ¼ 508 mm and b ¼ 127 mm, simply supported on the four edges and loaded by in-plane distributed loads with a ratio r ¼ Ny =Nx ¼ 0:125. The mechanical properties of the constituent lamina are the following: E1 ¼ 158; 000 MPa, E2 ¼ 10; 000 MPa, E3 ¼ 10; 000 MPa, m12 ¼ 0:315, m13 ¼ 1 105 , m23 ¼ 1 106 , G12 ¼ 6000 MPa, G13 ¼ 6000 MPa, G23 ¼ 4000 MPa, q ¼ 1:6 106 kg/mm3 . In all the problems the lamina thickness has been kept to the value t ¼ 0:127 mm, exception made for those where also the thicknesses, together with the angles of the fibres, were problem variables. The discrete sets of values are [0°, 30°, 45°, 60°, 90°] for the angles, and [0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, 0.11, 0.12, 0.127] mm for the thicknesses. In problems 6 and 7 the objective is to maximize the buckling load and the pliesÕ thicknesses are design variables. In this case, a limit on the total mass is needed in order to avoid that all the thicknesses take their highest allowed value. In the buckling load maximization problems (1, 2, 6 and 7) the number of plies has always been 64, while in the mass minimization ones (3, 4 and 5) the value 64 was the maximum available. Table 2 contains the results obtained for all the problems, using both CLT and CZZ displacement theory, and the two employed evolutionary algorithm, GA and SA. The aim of the present study is both to compare the performances of different evolutionary algorithms on the same problems and the performances of the two mathematical models. At first, we have solved the seven problems (GA and SA) using the CLT, trying three different kinds of fitness function shapes. Once we have found, for each problem, the best choice for fitness shape and penalty factors, we have used the same choice in the corresponding problems solved by means of the CZZ model. The constraint was yet satisfied in all the cases but the one highlighted in Table 1 in bold. For this problem only we had to modify the penalty factors in order to obtain the shown results. Specifically, in the CZZ model it has been nec-
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Table 2 Results for the laminated plates Problem no.
Genetic algorithm
Simulated annealing
CLT
CZZ
CLT
CZZ
1
Nx;cr ¼ 8028 N/mm
Nx;cr ¼ 7184 N/mm
Nx;cr ¼ 8026 N/mm
Nx;cr ¼ 7179 N/mm
2
Nx;cr ¼ 6796 N/mm Du1 ¼ 0:01%
Nx;cr ¼ 6539 N/mm Du1 ¼ 0:13%
Nx;cr ¼ 6503 N/mm Du1 ¼ 0:35%
Nx;cr ¼ 6091 N/mm Du1 ¼ 3:50%
3
num: lay: ¼ 28 k ¼ 1:01
num: lay: ¼ 28 k ¼ 1:02
num: lay: ¼ 28 k ¼ 1:69
num: lay: ¼ 28 k ¼ 1:08
4
num: lay: ¼ 40 Dwc ¼ 4:92%
num: lay: ¼ 52 Dwc ¼ 2:19%
num: lay: ¼ 40 Dwc ¼ 2:05%
num: lay: ¼ 48 Dwc ¼ 1:29%
5
num: lay: ¼ 28 Du1 ¼ 0:29%
num: lay: ¼ 36 Du1 ¼ 4:96%
num: lay: ¼ 24 Du1 ¼ 0:84%
num: lay: ¼ 24 Du1 ¼ 1:76%
6
Nx;cr ¼ 2254 N/mm Dm ¼ 0:18%
Nx;cr ¼ 2213 N/mm Dm ¼ 0:18%
Nx;cr ¼ 1205 N/mm Dm ¼ 0:09%
Nx;cr ¼ 1138 N/mm Dm ¼ 0:18%
7
Nx;cr ¼ 1241 N/mm Du1 ¼ 0:85% Dm ¼ 0:18%
Nx;cr ¼ 1189 N/mm Du1 ¼ 2:08% Dm ¼ 0:55%
Nx;cr ¼ 1193 N/mm Du1 ¼ 0:08% Dm ¼ 0:31%
Nx;cr ¼ 1078 N/mm Du1 ¼ 1:16% Dm ¼ 0:82%
essary to give lower importance to the objective than to the constraint satisfaction. As expected, the algorithms involving the CZZ theory yield lower buckling loads and higher number of layers than those obtained using the CLT model. This is due to the improved capability of the CZZ model of capturing the behaviour of laminated plates, in particular when the side-to-thickness ratio is much lower than 30. In fact, if we consider the problems (1 and 2) in which we maximize the buckling load and the total thickness is fixed, we have that b=h is equal to 15.6 and the final laminations and the corresponding critical loads are significantly different; the stacking-sequences differ in particular for the inner plies, where the effect of the transverse shear deformability is stronger. For example the two optimal stacking sequences obtained for problem 1 using GA are ð452 ; 60; 45; 60; 457 ; 60; 453 ÞS and ð4511 ; 602 ; 45; 60; 90ÞS for CLT and CZZ respectively. Conversely, when the ratio is higher than 30, we expect that the two theories give quite the same results. This is confirmed in the mass minimization problems when the final number of plies is low. In order to confirm the conclusion here stated, we have performed a single run of problem 1 (GA), with a very high (100) side-to-thickness ratio obtaining the same stacking-sequence ð4515 ; 60ÞS and almost the same buckling load in both the theories (203.1 and 202.5 N/mm for CLT and CZZ respectively). Generally speaking, the results obtained from the same displacement model and from the two evolutionary algorithms are the same. Anyway, in some particular cases we found a relevant difference. For example, if we consider problem 6, solved by means of SA, it is readily seen that the buckling loads are almost the half of those
obtained using GA. This is probably due to the fact that the set of penalty factors chosen was such that the optimum search was more stressed on the frequency constraint satisfaction than on the buckling load maximization. A great difference in the runtime requested by the various algorithms has been detected, depending both on the evolutionary algorithm (GA, SA) and on the displacement model (CLT, CZZ). The solution by means of the CZZ has taken, for the GA (SA) algorithm, 30–60 (40–50) times the time requested for the corresponding problems solved by using the CLT. Therefore, from this point of view the CLT has revealed to be much more advantageous than the CZZ. The comparison between the times requested by GA and SA, using both the CLT and the CZZ theory, shows that the SA is 3.5–5.5 times quicker than the GA. For problems 4 and 5 we have observed that optimal solutions often contained a great number of consecutive layers having either 0° or 90°. In order to limited this phenomenon, we have added the contiguity constraint, which is a limit on the number of consecutive plies with either 0° or 90° orientation. 5.2. Sandwich plates In order to assess the performances of the CZZ for sandwich plates, we solved the same unconstrained buckling load maximization problem as that studied by Moh and Hwu [17]. The plateÕs dimensions are those indicated in Table 3. The plate is simply supported on the four edges and loaded by in-plane distributed loads with variable ratios r ¼ Ny =Nx . The faces are laminated plates, made of four laminae; the stacking-sequence is
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Table 3 Results for the sandwich plates a (mm)
b (mm)
r
Moh and Hwu Stack. Seq.
Nx;cr (N/mm)
Stack. Seq.
Nx;cr (N/mm)
1000 1000 1000
500 1000 1000
0 0.5 2
ð44; 44; 44; 44; cÞS ð45; 45; 45; 45; cÞS ð45; 45; 45; 45; cÞS
352.66 66.62 33.31
ð46; 42; 45; 37; cÞS ð41; 48; 40; 42; cÞS ð41; 49; 46; 48; cÞS
356.82 66.77 33.42
ðh1 ; h2 ; h3 ; h4 ; cÞ, i.e. there are no constraints on the lamination. The laminaeÕs mechanical properties are the following: E1 ¼ 181; 000 MPa, E2 ¼ 10; 300 GPa, G12 ¼ 7170 Mpa, m12 ¼ 0:28; the coreÕs mechanical properties are: G13 ¼ 146 MPa, G23 ¼ 90:4 Mpa. 1 In all the problems the lamina thickness has been kept to the value t ¼ 0:125 mm; the coreÕs thickness has been, in all the cases, c ¼ 10 mm. Moh and Hwu solved the problem by means of the PowellÕs zero-order conjugate method. The authors reported only one optimal configuration, which is (h; h; h; h; c). For the same problem, we have obtained a group of solutions, by means of SA and CZZ model, with almost the same value of buckling load (Table 3). In other words, there exists a family of optimal configurations for all the studied cases. This family contains the stacking sequences (h1 ; h2 ; h3 ; h4 ; c), (h1 ; h2 ; h3 ; h4 ; c), (h1 ; h2 ; h3 ; h4 ; c), with 35° < hj < 50°.
6. Conclusions We have solved several optimization problems, testing two structural models (CLT and CZZ) and two evolutionary algorithms (GA and SA). The objectives have been the buckling load and the mass, which are two of the most important tasks in Aerospace design. Results show that in all the treated cases there is a good agreement between the GA and the SA methods. We also found that the SA is less time-consuming, therefore it appears to be more suitable for those problems in which complex numerical models (for instance, FEM models) have to be dealt with. The fitness functions and the evolutionary operators have been able to produce optimal solutions with a small degree of satisfaction of the constraints. As expected, the two displacement models have yielded the same results in terms of optimal stacking sequences and objectives when the plate side-to-thickness ratio was high; the CLT requests, in a general case, a
1 The remaining mechanical properties of the core have been determined by means of simplified models based on the honeycomb cells supposed geometry [14,15].
Present
shorter computing time than the CZZ theory but for thick plates it gives coarser results. The results of the performed investigations show that the evolutionary methods are simple to implement, and give good results in all the studied problems. Moreover, it seems that the presence of different kinds of constraints heavily influences the best lamination for a given problem. References [1] Le Riche R, Haftka RT. Improved genetic algorithm for minimum thickness composite laminate design. Compos Eng 1995;2(5):143–61. [2] Le Riche R, Haftka RT. Optimization of Laminate Stacking Sequence for Buckling Load Maximization by Genetic Algorithm. AIAA/ASME/AHS/ASCE/ASC 33rd Structures, Structural Dynamics and Materials Conference, Dallas, TX. AIAA J 1993;(31)5:951–6. [3] Di Sciuva M, Lomario D, Voyat JP. Evolutionary Algorithms. Their Use in Multiconstrained Optimization of Laminated and Sandwich Plates. Proceedings of the European Conference on Computational Mechanics (ECCM), CTS, Cracow, 26–29 June 2001. [4] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading (MA): Addison-Wesley; 1989. [5] Haftka RT, Gurdal Z. Elements of structural optimization. Kluwer Academic Publishers; 1996. [6] Jones RM. Mechanics of composite materials. Kogakusha, Tokyo: McGraw-Hill; 1975. [7] Di Sciuva M. A geometrically nonlinear theory of multilayered plates with interlayer slips. AIAA J 1997;11(35):1753–9. [8] Gottfried BS, Weisman J. Introduction to optimization theory. Englewood Cliffs, NJ: Prentice Hall Inc.; 1973. [9] Jang JSR, Sun CT, Mizutani E. Neuro-fuzzy and soft computing: A computational approach to learning and machine intelligence. Upper Saddle River: Prentice Hall; 1997. [10] G€ urdal Z, Haftka RT, Hajela P. Design and optimization of laminated composite materials. NY: Wiley; 1999. [11] Houck CR, Joines JA, Kay MG. Binary and Real-Valued Simulation Evolution for MATLAB. 1996. [12] Houck CR, Joines J, Kay M. A Genetic Algorithm for Function Optimization: a MATLAB Implementation. ACM Transactions on Mathematical Software, 1996. [13] Hwu C, Hu JS. Buckling and postbuckling of the laminated composite sandwich beams. AIAA J 1992;7(30):1901–9. [14] Becker W. The in-plane stiffnesses of a honeycomb core including the thickness effect. Arch Appl Mech 1998;(68):334–41. [15] Grediac M. A finite element study of the transverse shear in honeycomb cores. Int J Solids Struct 1993;13(30):1777–88. [16] Arfken G. Mathematical methods for physicists. Orlando, FL: Academic Press; 1985. [17] Moh JS, Hwu C. Optimization for buckling of composites sandwich plates. AIAA J 1997;5(35):863–8.