A scatter search algorithm for stacking sequence optimisation of laminate composites

Composite Structures 70 (2005) 383–402 www.elsevier.com/locate/compstruct

A scatter search algorithm for stacking sequence optimisation of laminate composites A. Rama Mohan Rao *, N. Arvind Structural Engineering Research Centre, CSIR Campus, Taramani, Chennai 600 113, India

Abstract This paper explores the metaheuristic approach called scatter search for lay-up sequence optimisation of laminate composite panels. Scatter search is an evolutionary method that has recently been found to be promising for solving combinatorial optimisation problems. The scatter search framework is flexible and allows the development of alternative implementations with varying degree of sophistication. The main objective of this paper is to demonstrate the effectiveness of the proposed scatter search algorithm for the combinatorial problem like stacking sequence optimisation of laminate composite panels. Preliminary investigations have been carried out to compare the optimal stacking sequences obtained using scatter search algorithm for buckling load maximisation with the best known published results. Studies indicate that the optimal buckling load factors obtained using the proposed scatter search algorithm found to be either superior or comparable to the best known published results. Later, two case studies have been considered in this paper. Thermal buckling optimisation of laminated composite plates subjected to temperature rise is considered as the first case study. The results obtained are compared with an exact enumerative study conducted on the problem to demonstrate the effectiveness and performance of the proposed scatter search algorithm. The second case study is optimisation of hybrid laminate composite panels for weight and cost with frequency and buckling constraints. The two objectives are considered individually and also collectively to solve as multi-objective optimisation problem. Finally the computational efficiency of the proposed scatter search algorithm has been investigated by comparing the results with various implementations of genetic algorithm customised for laminate composites. It was shown in this paper through numerical experiments that the scatter search is capable of finding practical solutions for optimal lay-up sequence optimisation of composite laminates and results are comparable and sometimes even superior to genetic algorithms.
2004 Elsevier Ltd. All rights reserved. Keywords: Laminate composites; Optimal design; Scatter search; Evolutionary algorithm; Thermal buckling

1. Introduction In the last two decades, the use of fibre reinforced materials has become widespread not only because of their high strength-to-weight ratio but also because of the possibility of tailoring them to meet specific design requirements by selecting the fiber materials and their orientation. These laminate composite structures are being used extensively in the fields of aerospace, defense, *

Corresponding author. Tel.: +91 44 22549183; fax: +91 44 22541508. E-mail address: [email protected] (A. Rama Mohan Rao). 0263-8223/$ - see front matter
2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.09.031

marine and automotive industries. Structures composed of composite materials often contain components, which may be modelled as rectangular plates. A common type of composite plate is the symmetrically laminated angle ply configuration, which avoids strength reducing bending-stretching effects by virtue of mid-plane symmetry. These plates quite often are subjected to in-plane loads, which may cause buckling. In addition, vibration can be problematic when the excitation frequency coincides with the plateÕs resonance frequency. Similarly, in aerospace industry, the stability of these composite laminate panels, often subject to severe temperature variations during launching/re-entry is one of the most important

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design problem. Such loading occur at different times under in-service conditions, necessitating a design approach which is capable of taking into account these various loading conditions. These multi-layered laminate composite panels offer many opportunities for analysts and designers to optimise structures for a specific or even for multiple design criteria. This flexibility, however often results in a very large number of discrete design variables, including thickness, material type, and orientation of each layer, as well as the number and the lay-up sequence of layers. The discrete nature of the problem makes it even more difficult to solve, especially if there happens to be more than one possible design that meets or nearly meets the design criteria. Most practical laminate designs require integer or combinatorial optimisations because the ply orientations are usually restricted to a small set of discrete angles (0, 45, and 90) for the discrete ply thicknesses. For composite lay-up sequence optimisation, genetic algorithms (GA) has been the most popular method of tackling this problem. In the past few years, variety of GA based approaches for stacking sequence optimisation for different functional purposes have been devised and reported in the literature. The advantage of GAs is that they give the designer a family of near optimal designs with small variation in their performance index instead of a single design, while the global optimisation methods always search for a single global optimal value. Callahan and Weeks [1] used GA to maximize strength and stiffness of a laminate under in-plane and flexural loads. Woodson et al. [2] used GA to design composite frames for maximum energy absorption. Haftka, Gu¨rdal and their coworkers (LeRiche and Haftka [3], Gu¨rdal et al. [4], Kogiso et al. [5], Todoroki and Haftka [6] and Soremekun et al. [7]) have developed specialised GAs for stacking sequence optimisation under buckling and strength constraints using closed form solutions. Sargent et al. [8] compared GAs to other random search techniques for strength design of laminated plates. Potgieter and Stander [9] developed a GA for strain energy minimisation of laminated plates subject to lateral loads. LeRiche and Gaudin [10] developed a GA for selection of ply angles and stacking sequences for optimizing dimensional stability under thermal and hygral expansion. Similarly, several researchers (Lin and Lee [11], Muc and Gurba [12] and Park et al. [13]) proposed GA based stacking sequence optimisation techniques for laminate composite structures using detailed finite element analysis for evaluation of objective function. A detailed review on optimal design of laminate composite panels is given by Venkataraman and Haftka [14]. In this paper, a new algorithm for optimal design of composite laminates has been devised using the scatter search metaheuristic. Eventhough scatter search has been extensively used in the operation research applica-

tions and several other combinatorial optimisation problems, the authors are not aware of any work related to application of scatter search algorithm for composite laminate design.

2. Scatter search Scatter search [15,16] is a population based metaheuristic that uses a reference set to combine its solutions and constructs others. The method generates a reference set from a population of solutions. Then a subset is selected from this reference set. The selected solutions are combined to get starting solutions to run an improvement procedure. The result of the improvement can motivate the updating of the reference set and even updating of the population of solutions. The initial population must be wide sets of disperse solutions. However, it must also include good solutions. Several strategies can be applied to get a population with these properties. The solutions for the population can be created, for instance, by using a random procedure to achieve a certain level of diversity. Then a simple improvement heuristic procedure must be applied to these solutions in order to get better solutions. The initial population can also be obtained by a procedure that provides at the same time disperse and good solutions. A set of good representative solutions of the population is chosen to generate the reference set. The good solutions are not limited to those with the best objective. By good representative solutions we mean solutions with the best objective values as well as disperse solutions. Disperse solutions should reach different local minima by local search. A solution may be added to the reference set if the diversity of the set improves. So the reference set must consist of a set of disperse and good solutions selected from the populations. The criteria for updating the reference set, when necessary, must be based on comparisons and measures of diversity between the new solutions and existing solutions. The possible improvement solution methods applied to the solutions range from simplest local searches to a very specialised search. A very simple procedure is a local search based on basic moves consisting of selecting the best improving move or a first found improving move. The procedure must permit to use tools like recent or intermediate memory, variable neighborhoods, or hashing scanning methods of the neighborhood. Then the method could be a Tabu search [17,18], a variable neighborhood search [19] or any sophisticated hybrid heuristic search. The metaheuristic strategy includes the decision on how to update the reference set taking into account the state of search. The algorithm must also realise when the reference set does not change and seek to diversify the search by generating a new set of solutions for

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the population. In addition, the metaheuristic includes the stopping criterion for the whole search procedure. Then the best solutions provided by the method are the ones, which are finally left in the reference set after the stopping criterion for the whole search procedure is met. In addition to these six procedures, the metaheuristic involves three stopping procedures that implement the criteria to decide when to go to the subsequent step.

ence set. It may be noted that the diverse solutions generated during the initial population creation method are ensured to be feasible. Alternatively, they can as well be transformed into feasible solutions by applying suitable corrections to the design variables during the improvement method as a first step and then the feasible solutions are improved using the improvement method.

(i) Reference set criterion. This stopping criterion decides when to generate a new reference set from the population. (ii) New population criterion. This stopping criterion decides when to generate a new initial population. (iii) Termination criterion. This stopping criterion decides when to stop the whole search.

The diverse solutions obtained in the earlier step are improved by altering the design variable in a greedy fashion. The procedure stops when no further improvement is feasible without violating the constraints.

Usual stopping conditions are based on allowing a total maximum computational time or a maximum computational effort since the last improvement. The computational effort is measured by the number of iterations, number of local searches or real time. The generalised framework of the scatter search algorithm is given in Fig. 1. From the above discussion, it is clear that the scatter search involves the following six procedures and three stopping criteria to solve an optimisation problem. 2.1. Initial population creation method This method employs control randomisation and frequency memory to generate a set of diverse solutions. The diversification generator is used at the beginning of the scatter search to generate a set of P solutions with Psize, the cardinality of the set, generally set at max(100, 5 * b) diverse solutions, where b is the size of the refer-

2.2. Improvement method

2.3. Reference set update method The reference set, Refset, is a collection of both high quality solutions and diverse solutions that are used to generate new solutions by way of applying the solution combination method. Specifically the reference set consists of the union of two subsets, Refset1 and Refset2, of size b1 and b2 respectively. That is, jRefsetj = b = b1 + b2. The construction of the initial reference set starts with the selection of the best b1 solutions from P. These solutions are added to Refset and deleted from P. For each improved solution in P, the minimum of the Euclidean distances to the solutions in Refset is computed. Then the solution with the maximum of these minimum distances is selected. This solution is added to RefSet and deleted from P and the minimum distances are updated. This process is repeated b2 times. The resulting reference set has b1 high quality solutions and b2 diverse solutions. 2.4. Subset generation method This method consists of generating the subsets that will be used for creating structured combinations in the next step. The method is typically designed to generate different type of subsets as given here.

Fig. 1. General framework of scatter search algorithm.

(i) All two element subsets. (ii) Three element subsets derived from the twoelement subsets by augmenting each two-element subset with the element corresponding to the best solution (as measured by the objective value). The two element subsets which already contain the best element will not be considered for the three element subset. (iii) Three element subsets are first generated with all possible combinations. Four element subsets are then derived from the three-element subsets by augmenting each three-element subset with the element corresponding to the best solution (as measured by

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the objective value) The three element subsets which already contain the best element will not be considered for the three element subset. (iv) The subsets consisting of the best i elements (as measured by the objective value), where i ranging from 5 to b.

2.5. Solution combination method This method uses the subsets generated in the previous step to combine the elements in each subset with the purpose of creating new trial solutions. The combination method typically problem-specific since it is directly related to solution representation. Depending on the specific form of the solution combination method, each subset can create one or more new solutions. It may be noted that the combination mechanism constructs solutions that may or may not be feasible. This does not represent a problem in general, since improvement method is always applied to each trial solution created after the application of the solution combination method to make it feasible and improve. It may be noted that the improvement method must be designed to deal with either feasible or infeasible starting solutions. 2.6. Reference set update method This method updates the reference set by deciding when and how the obtained improved solutions are included in the reference set, replacing some solutions already in it. A solution may become a member if its objective value is better than the objective value of any of the solutions in the high quality subset. Alternatively, if a new solution improves the diversity of the reference set, this solution can replace the one that is currently in the diverse set. If the reference set is modified, then the subset generation method, the solution combination method and the improvement method are applied in sequence. The application of the methods continues until the reference set converges (i.e., no elements in the set are replaced). At this point, diversification method can be applied again from a different seed and search continues, until termination criteria is satisfied.

3. Application of scatter search to the lay-up sequence optimisation of composites In this paper, scatter search technique has been applied for lay-up sequence optimisation of laminate composite plates. The lay-up problem may be stated as follows: For a laminate made up of n plies, where the number m of distinct plies of different orientations is be-

tween 1 and n, devise an approach or algorithm, which will generate a sequence of complete lay-ups to be evaluated for fitness for the specific purpose for which it is intended. The algorithm must use this information to find the best lay-ups to fulfill several different incommensurate criteria. Regardless of the search technique employed, it is necessary to specify a solution coding, which encodes alternative candidate solutions for manipulation. The choice of the coding that provides an efficient way of implementing the moves and evaluating the solutions is essential for the success of the search heuristic. For laminate lay-up sequence optimisation, each candidate solution represents a design, i.e. lay-up sequence of a composite laminate. Every solution is encoded by arranging all ply angles of the given composite laminate in an array v (vi : i = 1, 2, . . . , n), where vi is an encoded value corresponding to a ply angle and n stands for the number of plies in the laminate composite. The ply angles are encoded as 1, 2, 3 which stand for the three possible stacks, 02, ±45, 902 respectively. For example, the laminate [902, ±45, 902, 02]s is encoded as 3 2 3 1. The rightmost 1 corresponds to the layer closest to the laminate plane of symmetry. The leftmost 3 describe the outermost layer. 3.1. Initial population creation method The initial population creation method must be designed in order to get a random set of disperse and good solutions. In order to preserve the diversity of the created initial population, p, we employed two different approaches. They are: Selection type 1 (SM1) (i) Select N candidate solutions randomly. (ii) Improve the solutions and make them feasible so that they satisfy all the constraints. (iii) Compute the fitness function i.e. the objective function of each candidate solution. (iv) Sort the candidate solutions according to their objective values. (v) Partition the sorted feasible candidate solutions into five zones based on the objective function. (vi) Choose randomly equal number of solutions from each zone to form good and well disperse solutions of initial population. Selection type 2 (SM2): Steps (i)–(v) are same as selection type 1. (vi) Choose two candidate solutions from two of the extreme five zones and mark these two zones. (vii) Find the distances of the selected candidate solutions from the solutions in the unmarked zones.

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(viii) Find the minimum distance of each solution in the unmarked zone from the already selected candidate solutions and record the distance against the respective solution. (ix) Select the candidate solution, which has the maximum recorded distance among the solutions of the unmarked zone. (x) Mark the zone from which the candidate solution is selected. Repeat steps (vii)–(x), till all zones are marked. (xi) Unmark all zones and repeat step (vii)–(x) till the required number of candidate solutions of initial population, p are chosen. For evaluating the diversity among solutions, we need to consider a parameter called distance between the solutions. This distance can be defined as Distanceði; jÞ ¼

½fitðiÞ
fitðjÞ

2

ðmax fit
min fitÞ2

ð1Þ

where fit(i) and fit(j) represents the objective value of the two candidate solutions i and j respectively. max_fit and min_fit are the extreme objective values i.e. minimum and maximum objective values of the N feasible candidate solutions generated earlier. 3.2. Reference set generation method The reference set consists of a set of good and disperse solutions selected from the population. The generation of a reference set is done by selecting jRefset1j solutions from the best solutions and jRefset2j disperse solutions taking into account the already selected jRefset1j solutions (jRefsetj = jRefset1j + jRefset2j). After including the best jRefset1j solutions in Refset, we iteratively include in the Refset the farthest solution from the solutions already in Refset, repeating this procedure jRefset2j times. The methods employed for choosing Refset solutions are elaborated in the later part of this paper. 3.3. Improvement solution method The improvement solution method is the local search based on the base moves. A local search for combinatorial optimisation performs a series of moves in solution space, which improves each time, the value of the objective function until a local optimum is found. For composite laminate design problem we have implemented the following two improvement methods. 3.3.1. Method 1 (IM1) The local optimal value is arrived in this improvement solution method by changing one variable each time (say 1, 2 or 3) in each candidate solution and all

387

the possible combinations are generated. Among the several solutions derived from each existing candidate solution, the best solution based on objective value is chosen to compete as the replacement solution in Refset. 3.3.2. Method 2 (IM2) In this improvement solution method, a cut-off point is chosen and all the variables to the left of the cut-off point in the candidate solutions are considered for improvement. Series of solutions are then generated from each candidate solution, by using all possible combinations (i.e. 1, 2 or 3) for the chosen variables. The local optimal value obtained through the improvement (i.e. the new candidate solution derived from each of the existing solutions with maximum objective value) will replace the existing candidate solution in refset. Since the influence of extreme fibers in the composite laminate is significant in altering the buckling load, the variables to the left of cut-off point in the existing candidate solution are considered for swapping to obtain a local optimal value. 3.4. Subset generation method The selection of a subset for applying the combination usually consists of selecting all subsets of fixed size r. For laminate composite optimisation problem, we used r = 2 and r = 3. 3.4.1. Solution combination method (CM) Once the subsets of selected solutions from the reference set are chosen during subset generation method, the combination method tries to combine the elements in each subset in order to generate a new trial solution with good characteristics of the solutions in the subset. In order to generate new trial solutions, we calculate the score of each variable corresponding to the candidate solutions present in a typical combination. The score of each variable can be computed using the following equation: P j j2s fitnessðjÞ  vi j 
P  scorei ¼
P ð2Þ j j2S fitnessðjÞ  j2s vi  FACT where fitness(j) is the objective value of solution j and vji is the value of the ith variable in solution j. S is a set of candidate solutions present in a typical combination. FACT is a scaling factor, which is taken as 0.90 in our studies. Once the scores of each variable of the candidate solutions of a particular subset are obtained, then the score of each variable is computed as X scoreðiÞ ¼ scoreji ð3Þ j2S

The trial solution (vt) is then constructed by arriving at the score of each variable as given below, i.e.

388

8 > <1 t vi ¼ 2 > : 3

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if scoreðiÞ 6 1 if 1 > scoreðiÞ 6 2

ð4Þ

if 2 > scoreðiÞ 6 3

For laminate composite optimisation, we prefer to use two subsets for solution combination as mentioned earlier. The subset sizes considered are 2(CM1) and 3(CM2). However, it should be mentioned here that there is an every possibility that the solution combination method illustrated here can construct a solution, which may be infeasible. In view of this, as already mentioned the improvement method should be designed to deal with a mechanism to convert infeasible solutions into feasible ones. For this purpose a correction operator has been built into the scatter search algorithm and will be discussed in the later part of the paper while dealing with numerical case studies. Apart from that, the improvement solution method improves the trial feasible solutions generated using the methods described earlier. 3.5. Reference set update method Once the improved trial solutions are obtained by the combined operation of solution combination and improvement on the reference set solutions, the reference set is updated. As already mentioned, reference set consists of two parts: Refset1 and Refset2. While Refset1 maintains the best trial solutions, Refset2 maintains diverse solutions. For composite laminate design problem, the reference set update method has been devised as follows: The feasible trial solutions obtained are sorted according to their fitness value. The best solutions (i.e. the trial solutions which has the highest objective values) are compared with the already available Refset1 values. If the trial solutions obtained are superior to already existing Refset1 values, then the Refset1 will be updated with the new best trial solutions. Refset2 should maintain diverse solutions. In order to maintain diversity in Refset2 values, the following two approaches have been devised. 3.5.1. Method 1 (RUM1) In this method, the distances of all the trial solutions and the solutions already existing in Refset2 are computed with respect to the updated Refset1 values. The minimum of all the distances of a trail solution (or Refset2 solution) computed with respect to all Refset1 values are recorded against each trial solution. The solutions with maximum recorded distance are chosen for updating the Refset2 values. This method ensures diversity of solutions in Refset2.

3.5.2. Method 2 (RUM2) In this method, the Refset2 values are updated incrementally. The procedure followed is as given below: (i) Mark all the Refset1 solutions. Similarly unmark the solutions in Refset2 and also all the remaining trail solutions obtained using combination and improvement. (ii) Compute the distances of all unmarked trail solutions with respect to the marked solutions. (iii) Record the minimum distance of each unmarked solution. (iv) Choose the solution whose recorded distance is maximum for inclusion in Refset2 and mark this solution. (v) Repeat steps (i)–(iv) till Refset2 is full.

3.6. Convergence The solution is assumed to have been converged if there is no update of Refset1 or Refset2. It should be mentioned here that in the proposed algorithm, we prefer to run scatter search with only one set of new population and only one generation of new reference set in order to maintain higher computational efficiency, without sacrificing quality. In view of this, the termination criteria given in Fig. 1 are not strictly followed. In the later part of this paper, we however have demonstrated that the proposed implementation of scatter search is quite effective in obtaining optimal solutions at high computational performance.

4. Laminate composite models for numerical studies In this paper, two laminate composite models are considered as case studies to demonstrate the effectiveness of the proposed lay-up sequence optimisation technique employing scatter search metaheuristic algorithm. The first case study is thermal buckling of laminate composite panel and the second one is optimisation of bi-material composite laminate panel. 4.1. Buckling of laminate composites A simply supported laminate composite plate of rectangular shape (a · b), with symmetrical stacking sequence is shown in Fig. 2. The laminated composite plate is composed of N layers, each of thickness k and subject to a uniform temperature variation DT through the P thickness. The total thickness of the laminate is h ¼ Nk¼1 tk . The laminate is assumed to be made up of 0, 90 and 45 plies. To enforce the symmetric condition, we consider only top half of the laminate. Similarly, balanced

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kb D11 ¼ p2

m4 a

 2  2  4 þ 2ðD12 þ 2D66 Þ ma nb þ D22 nb  2  2   N x ma þ N y nb þ N xy mn ab

derived from the displacement field XX mpx npy sin wðx; yÞ ¼ Amn sin a b m n Fig. 2. A simply supported laminate composite plate.

condition is enforced by considering stacks of two 0 and 90 plies, and +45/
45 pair plies. Using this lamination scheme, N/4 ply orientations are needed to define the stacking sequence of the laminate. The stress developed due to DT temperature rise in a laminate are given by Tsai and Hahn [20]: N x ¼ DT

N X

ðQ11 ax þ Q12 ay Þk ðhkþ1
hk Þ ¼ R1 DT

k¼1

N y ¼ DT

N X

ðQ12 ax þ Q22 ay Þk ðhkþ1
hk Þ ¼ R2 DT

ð5Þ

k¼1

N xy

N X ¼ DT ðQ66 axy Þk ðhkþ1
hk Þ ¼ R12 DT

ð7Þ

ð8Þ

that satisfies all imposed boundary conditions of the plate. The smallest value of kb is the critical buckling load kcb. The critical buckling load kcb is a function of (m, n) and varies with the plate aspect ratio, loading case and material properties. kcb can be evaluated from Eq. (7), assuming a reference value for Nx. Once kcb is obtained, the critical temperature rise Dtcr is calculated as Dtcr = kcb/R1. In order to compute the failure load of the laminate composite panels, the value of corresponding multiplier kcs must be evaluated. The principal strains in the generic ith layer are related to the loads by the following relations [3]: kN x ¼ A11 ex þ A12 ey ; kN xy ¼ A66 cxy

kN y ¼ A12 ex þ A22 ey

ð9Þ

and

k¼1

where Qij are the material stiffness coefficients of the single layer, hi is the ith layer with respect to mid plane and ax, ay and axy are the thermal coefficients of the ith layer. As fibers impose a mechanical restraint on the matrix in the longitudinal direction, coefficient ax is normally small whereas the matrix is forced to expand in the transverse direction, giving larger transverse coefficient ay. R1, R12 and R2 are the loads due to a unit temperature rise, and their sign will decide whether the buckling is caused by an increase or decrease in the temperature. For example, Kevlar/epoxy laminates buckle under cooling, whereas graphite/epoxy laminates buckle under heating [21]. From Eq. (5) it is possible to obtain the quantities Ny/Nx = R2/R1 and Nxy/Nx = R12/R1 for a given stacking sequence. Assuming that the plate is loaded, in the x–y plane, by the forces kNx, kNy, and kNxy, where k is a scalar amplitude parameter, the governing differential equation for the buckling behaviour of the plate, under the assumption of the classical plate theory [22] is o4 w o4 w o4 w þ 2ðD þ 2D Þ þ D 12 66 22 ox4 ox2 oy 2 oy 4  2 2 2  ow ow ow ¼ k N x 2 þ N y 2 þ N xy ox oy oxoy

389

D11

ð6Þ

where Dij are the bending stiffness coefficients and w is the vertical displacement. The laminate will buckle into m and n half-waves in the x and y directions when the amplitude parameter reaches a value kb given by [22]

ei1 ¼ cos2 hi ex þ sin2 hi ey
2 sin hi cos hi cxy ei2 ¼ sin2 hi ex þ cos2 hi ey
2 sin hi cos hi cxy ci12 ¼ sin hi cos hi ex
sin hi cos hi ey þ ðcos2 hi
sin2 hi Þcxy ð10Þ where Aij are the extensional stiffness matrix coefficients, hi are ply orientations, ei and cij are the strains considered. The strain failure load kcs corresponds to the smallest load factor k in Eq. (9) such that, in at least one of the layers, one of the principal strains reaches or exceeds its allowable value, calculated from the ultiua ua mate allowable values eua 1 ; e2 ; c12 , using a safety factor equal to 1.5. In order to avoid some undesirable stiffness coupling effects in laminated composite structure, angle plies must be balanced. Apart from that, the maximum number of contiguous plies of the same orientation is constrained to four, to alleviate large scale matrix cracking and to provide damage tolerant structures [3]. Therefore, the optimisation problem is the buckling load maximisation by changing their ply orientations in the stacking sequence of the plate subjected to strain constraints with the ply contiguity and angle ply balance constraints. The constraints are usually treated in evolutionary algorithms by a penalty approach [3]. However, in scatter search, we prefer to deviate from this and a correction operator is devised to convert infeasible designs (due to constraint violation) into feasible ones by correcting (or rather repairing) the laminate ply sequences.

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4.1.1. Evaluation of scatter search algorithm with the published results The scatter search metaheuristic algorithm presented in this paper is first evaluated by solving the numerical examples given in the literature and comparing the results with the best known published results. For this purpose, simply supported composite panels with 48 (0, 45 and 90) and 64 plies (0, 45 and 90) are considered. The plies considered here are made of graphite epoxy with thickness, t = 0.005 in. (0.0127 cm). The material properties are same as given by LeRiche and Haftka [3] and are as follows: E1 = 18.50E6 psi (127.59 GPa); E2 = 1.89E6 psi (13.03 GPa); G12 = 0.93E6 psi (6.41 GPa); m12 = 0.30. The ultimate allowaua ua ble strains are eua 1 ¼ 0:008, e2 ¼ 0:029, c12 ¼ 0:015. For evaluation purpose, four load cases as given in Table 1 are considered. Table 1 gives the details of composite panel dimensions (a and b), number of plies and also Nx and Ny values for each load case. These four load cases have been solved in the literature [3,5,7] for the optimum buckling and failure loads using genetic algorithms. The buckling and failure loads of the simply supported composite panels can be computed using Eqs. (7)–(10). Several optimal designs exist (multi-modal solutions) for all these load cases, which are close to optimal buckling and failure loads. The best known buckling and failure load factors for all the four load cases are compiled from the literature [3,5,7] and are given in Table 2. To evaluate the quality of the solutions obtained using the proposed scatter search algorithm, the concept of practical reliability [3] is used. Practical reliability is given by the percentage of solutions obtained using the

In the present case study the constraints include strain constraints, ply contiguity and angle ply constraints. The strain constraints if violated, will reduce the buckling/failure load multiplier and therefore will get eliminated during the improvement method in scatter search. The angle ply balance constraint is automatically taken care of as we consider only one symmetric half of the laminate during the optimisation process with ply angles in stacks of two like 02, ±45, 902 in the laminate sequence. The ply contiguity constraint however requires special attention. In order to correct the laminate sequence with more than four contiguous plies with the same orientation, the laminate sequence is scanned from the outermost ply, and if five contiguous plies of the same orientation are encountered, the innermost ply code is changed either by decrementing or incrementing the value of the code. The option of decrementing or incrementing is decided based on the quality of solution. Similarly near plane of symmetry, if the laminate sequence encounters more than two contiguous plies of the same orientation, the code value (i.e. the value of the ply immediate to the plane of symmetry) is incremented or decremented as discussed earlier. Table 1 Details of the load cases considered for evaluation Load case

Number of plies

a (mm)

b (mm)

Nx (N/cm)

Ny (N/cm)

1 2 3 4

48 48 48 64

50.8 50.8 50.8 50.8

12.7 12.7 12.7 25.4

1.75 1.75 1.75 1.75

0.22 0.44 0.88 1.75

Table 2 Optimal stacking sequences for buckling load factor maximisation using scatter search Load case

Best known results published in the literature [3,5,7]

Scatter search results

Buckling load factor

Failure load factor

Stacking sequence

1

14618.12 14134.76 14013.71 13662.61

13518.66 13518.66 13518.66 13518.66

[2 [2 [2 [2

2]s 2]s 1]s 1]s

16120.38 15982.86 16008.29 15974.94

5099.60 8088.92 8467.68 8563.07

2

12725.26 12698.40 12743.45

12678.77 12678.77 12678.78

[2 2 3 2 2 2 2 2 2 2 2 3]s [2 2 3 2 2 2 2 2 2 3 2 3]s [2 2 2 2 3 3 2 2 2 3 2 3]s

13432.89 13418.40 13399.34

7880.62 8283.52 8424.34

3

9998.19 9997.60 9976.58 –

10394.81 10187.93 10187.93 –

[3 [3 [3 [3

2 2 2 3

2 2 2 2

3 2 2 2

2 3 3 2

3 2 2 2

2 3 3 2

2 3 3 2

2 2 2 2

2 3 3 3

2 2 2 2

2]s 3]s 2]s 2]s

9998.70 9998.66 9996.50 9995.20

10403.75 9282.54 9802.62 10403.74

4

3973.01 3973.01 3973.01 3973.01 3973.01

14205.18 8935.74 8935.74 8935.74 14205.18

[3 [3 [2 [3 [3

3 3 3 3 2

2 2 3 2 3

3 3 3 2 3

3 3 3 3 3

2 2 3 3 2

3 3 2 3 3

3 3 3 3 3

2 2 3 3 3

3 3 3 3 3

3 3 3 3 2

2 2 2 3 3

3957.22a 3953.01a 3977.12 3977.12 3977.12

10733.33a 11620.92a 8934.13 8934.13 8934.13

a

Obtained with four ply contiguity constraint.

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 3

2 1 1 1

Buckling load factor 2 1 3 3

3 3 3 2 3

3 2 3 3 3

2 3 3 3 3

3]s 2]s 3]s 3]s 3]s

Failure load factor

A. Rama Mohan Rao, N. Arvind / Composite Structures 70 (2005) 383–402

proposed scatter search algorithm with the buckling and failure loads greater than or equal to 99.9% of the optimal solutions. The practical reliability is obtained for each load case by running 100 different instances of scatter search and determines the number of final solutions that satisfy the above requirement. It may however be noted that the scatter search maintains perfect reliability as there are no probabilistic moves unlike in genetic algorithms. This is also ascertained here through the numerical experiments. Another measure used in the present numerical experiments to measure the effectiveness of the proposed scatter search algorithm is the normalised price of each run [3]. The price is defined as the number of function evaluations during one complete run of scatter search algorithm. The normalised price is defined as the ratio of the average price of the runs to the practical reliability. The optimal stacking sequences obtained using the proposed scatter search for maximisation of buckling load and their corresponding buckling and failure load factors are shown in Table 2. In all the tables, the laminate sequences are shown in a coded string form for convenience. It may be noted that Ô1Õ in coded string corresponds to Ô02Õ and similarly Ô2Õ and Ô3Õ in the string corresponds to Ô±45 and Ô902Õ respectively. A close look at the results presented in Table 2 indicate that the optimal sequences obtained using the proposed scatter search algorithm for buckling load maximisation infact generate solutions which are far superior to the best known bucking load factors from the literature. It may be noted that the first two stacking sequences given in Table 2 for the fourth load case (64 ply laminate) have been obtained with ply contiguity constraint. However, the results reported in the literature [3,7] are without ply contiguity constraint. In view of this, the buckling load factor for the 64 ply laminate (4th load case) is marginally lower than the best known values for the first two stacking sequences given in Table 2. The rest of the stacking sequences for the 64-ply laminate given in Ta-

391

ble 2 are obtained without imposing the ply contiguity constraint and are therefore comparable with the best known results. One can observe from the results presented in Table 2 that even though the buckling load factor obtained using scatter search is far higher, the failure load factor is much lower than the best known results given in the table. It may also be noted that the critical load for the first two load cases is governed by the failure load, while the last two load cases are governed by the buckling load. The reason for lesser failure load factors for the optimal sequences generated by scatter search can be explained as follows: The buckling load depends on the stacking sequence, while the failure load does not depend on the stacking sequence, but just on the thickness and orientation of the plies in the laminate. The scatter search explores the search space to locate the optimal sequences for the objective chosen and does not bother to capture the best values for other parameters like failure load. Since the buckling load depends on the stacking sequence, it can be concluded from the results given in Table 2 that the proposed scatter search algorithm explores the search space effectively in identifying the optimal sequences for the given objective function. Table 3 shows optimal stacking sequences obtained using scatter search with maximisation of failure load as objective function. A close look at the results given in Table 3 indicate that failure load factors obtained using scatter search are far more superior to the best known results given in Table 2 for all load cases. However, the critical load for these stacking sequences is governed by the buckling load factor, which is far lower than the best known results given in Table 2. The explanation given in the earlier context while discussing the results given in Table 2 holds good for this case also as scatter search aims at maximisation of the given objective function and do not bother about the relative merit of other parameters. The practical reliability is found to be 1.00 for all the load cases.

Table 3 Optimal stacking sequences for failure load factor maximisation using scatter search Load case

Scatter search results

Critical

Stacking sequence 1

[1 [1 [1 [1 [1 [1 [3 [3 [3 [3

2 3 4

a

1 1 2 3 1 1 3 2 3 3

2 2 1 1 3 2 2 3 3 3

1 1 2 2 1 1 3 2 3 3

1 3 1 1 1 3 3 3 2 3

2 1 1 3 2 1 2 2 3 2

1 1 2 1 1 1 3 3 2 2

2 2 1 1 1 2 3 2 3 3

1 1 2 2 3 1 2 3 2 2

2 2 1 1 1 3 3 2 3 2

1 1 2 2 3 1 3 3 2 2

2]s 2]s 1]s 1]s 1]s 2]s 23 23 22 32

3 3 2 2

Obtained with four ply contiguity constraint.

2 2 2 2

3]s 3]s 2]s 2]s

Buckling load factor

Failure load factor

8421.42 9432.72 6702.59 8775.02 4657.54 4231.11 3957.22a 3721.45a 3977.12 3977.12

16908.20 16727.10 18364.01 16979.45 18326.54 18289.75 10733.33a 12498.01a 14208.95 14208.94

Buckling Buckling Buckling Buckling Buckling Buckling Buckling Buckling Buckling Buckling

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A. Rama Mohan Rao, N. Arvind / Composite Structures 70 (2005) 383–402

As stated earlier, the critical load for the first two load cases given in Table 2 is constrained by the lower failure load factor yielded by the optimal stacking sequences generated by scatter search. In order to improve critical load for the first two load cases, the objective function has been modified such that the new objective function is a weighted sum of buckling and failure load and is given by kC ¼ kCB þ bkCS

ð11Þ

where kCB is the buckling load factor and kCS is the failure load factor and b is the weight factor whose values range from 0 to 1.0. In the present studies, the value of b is varied from 0.0 to 0.40 and the optimal stacking sequences obtained using scatter search for maximisation of kC for all the four load cases are presented in Table 4. A close look at the results presented in Table 4 indicates that for all the load cases, the buckling and failure load factors obtained using the proposed scatter search algorithm are either comparable or superior to the best known results given in Table 2. As stated earlier, the practical reliability remains as 1.0 for all the numerical experiments carried out using scatter search. Finally the price performance of the proposed scatter search algorithm has been evaluated by considering 64ply composite laminate panel for which normalised price for various GA implementations is given in the literature [7]. Fig. 3 shows the normalised price of various GA implementations and the proposed scatter search algorithm. It may be noted that scatter search can run with combinations of various operators for selection, improvement, reference set update etc. Here, we prefer to use the combination of (SM1 + IM1 + CM1 + RUM1) for scatter search. It can be easily verified from

Fig. 3. Comparative performance of scatter search algorithm with GA published in the literature [7].

Fig. 3 that the proposed algorithm is relatively faster than genetic algorithm. 4.1.2. Optimal stacking sequences for thermal buckling The first case study considered in this paper is thermal buckling of composite laminate panels. For this purpose, two simply supported composite panels with 48 plies (0, ±45, 90) and 64 plies (0, ±45, 90) respectively are considered to generate optimal stacking sequences for thermal buckling using the proposed scatter search algorithm. The material properties of graphite–epoxy composite lamina (AS/3501) are [23]: E1 = 20.0E06 psi (138 GPa) E2 = 1.30E06 psi (8.96 GPa) G12 = 1.03E06 psi (7.10 GPa), m12 = 0.3, t = 0.005 in. (0.127 mm). The ultimate allowable strains are taken as eua 1 ¼ 0:008,

Table 4 Optimal stacking sequences for combined buckling and failure load factor maximisation using scatter search Load case

Scatter search results

Critical Buckling load factor

Failure load factor

1

Stacking sequence [2 [2 [2 [2

2 2 2 2

2 2 2 2

2 3 2 2

2 2 1 1

1 1 2 3

1 1 1 1

2 2 2 2

1 1 1 1

1 1 1 1

3 2 3 2

1]s 1]s 1]s 1]s

14659.46 14470.19 14437.30 14329.15

13518.67 13518.67 13518.67 13518.67

Failure Failure Failure Failure

2

[2 [2 [2 [3

2 3 3 2

3 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

1 1 1 1

2 2 2 2

1 1 1 1

1 1 1 1

2 2 1 2

1]s 1]s 2]s 1]s

12746.93 12733.54 12706.66 12683.17

12689.08 12689.08 12689.08 12689.08

Failure Failure Failure Buckling

3

[3 [3 [3 [3

2 2 3 2

2 2 2 3

3 3 2 2

2 2 2 2

3 2 2 2

2 3 2 2

2 2 2 3

2 3 2 2

2 2 3 3

2 2 2 2

2]s 2]s 2]s 2]s

9998.70 9998.11 9995.20 9994.61

10403.75 10192.33 10403.74 10192.33

Buckling Buckling Buckling Buckling

4

[3 [3 [3 [3

3 3 3 3

2 2 3 3

3 3 3 3

3 3 3 2

2 2 2 3

3 3 2 2

3 3 3 3

2 2 2 2

3 3 2 3

3 3 2 2

2 2 3 2

3957.22a 3953.01a 3977.12 3977.12

10733.33a 11620.92a 14208.94 14208.94

Buckling Buckling Buckling Buckling

a

3 3 2 2

3 2 2 2

2 3 2 2

Obtained with four ply contiguity constraint.

3]s 2]s 2]s 2]s

A. Rama Mohan Rao, N. Arvind / Composite Structures 70 (2005) 383–402 ua eua 2 ¼ 0:029 and m12 ¼ 0:015. The coefficients of thermal expansion, ax = 0.03E
06 C
1 and ay = 28.1E
06 C
1. A safety factor of 1.5 has been used to reduce these values to allowable values. The plate considered in the numerical studies is simply supported on all edges, with dimensions a = 20 in. (0.508 m) and b = 5 in. (0.127 m). In order to demonstrate the performance of the proposed scatter search algorithm for lay-up sequence optimisation, the best objective values are first identified for 48 and 64 ply laminates through an enumerative study. This enumerative study has been conducted by taking all possible feasible combinations for 48 and 64 ply laminates. Since the laminates are considered to be symmetric and balanced, only one quarter of the discreet variables need to be considered for enumerative study. For example, in 48 laminate only 12 variables (N/4) need to be considered and a total of 312 possible designs exist. The enumerative study has become possible here, as the function evaluation is inexpensive because Classical Laminate Theory (CLT) has been used instead of a detailed FEM solution for buckling load evaluation. The feasible best solutions obtained using enumerative study for 48 ply laminate and 64 ply laminate are presented in Tables 5 and 6 respectively. The optimal stacking sequences obtained for 48 ply laminate employing the proposed scatter search algorithm are shown in Tables 7 and 8 respectively for the two reference set update techniques proposed in this paper. The optimal values presented in Tables 7 and 8 are obtained using

393

various selection, improvement, and solution combination methods discussed in the earlier sections. Similarly the results obtained for 64-ply laminate are shown in Tables 9 and 10. The following observations can be made based on the results presented in Tables 5–10. A close look at the results indicates that the proposed scatter search algorithm is extremely effective in capturing the optimal stacking sequences of laminate composites. This fact is clearly evident from the comparison of the best results obtained through enumerative study and given in Tables 5 and 6 and the stacking sequences obtained using scatter search and presented in Tables 7–10. It can be observed that the improvement method-1 presented in this paper is more effective in exploring the search space when compared to the second method as the results obtained using this method are consistently good for all the evaluations given in the tables. The reasons are quite obvious as the improvement method-2 try to improve solution by varying only the variables to the left of the cut off point chosen and there by limits the exploratory characteristics of the improvement method. It can also be noticed that many practical optimal designs exist, for this problem where a practical design can be defined as a design giving an objective function very close to the optimal solution. The scatter search is as effective as genetic algorithms in capturing the multiple optimal solutions. This is clearly evident from the multiple optimal solutions presented in Tables 7–10.

Table 5 Optimal ply sequences for 48 ply composite laminate using enumeration S. No.

Sequence

1 2 3 4 5 6 7 8

[3 [3 [3 [3 [3 [3 [3 [3

3 3 3 3 1 3 1 1

2 2 2 2 3 1 3 3

3 3 3 3 3 1 3 1

3 3 3 3 1 3 1 1

2 2 2 2 3 1 1 3

3 3 3 3 1 3 3 3

3 3 3 3 3 3 1 1

2 2 2 2 1 1 3 1

3 3 3 3 1 1 1 3

3 3 2 2 3 3 1 1

2]s 1]s 3]s 2]s 1]s 1]s 3]s 3]s

Buckling load multiplier k

Failure load multiplier k

Temperature (C)

Criteria for optimisation

7460.99 7459.31 7443.17 7440.20 5744.30 5632.76 5437.56 4935.63

7565.97 5739.80 7565.97 8616.26 14220.40 14220.40 14220.40 14220.40

635.67 598.65 635.67 607.71 5332.60 5332.60 5332.60 5332.60

Buckling Buckling Buckling Buckling Failure Failure Failure Failure

Table 6 Optimal ply sequences for 64 ply composite laminate using enumeration S. No.

Sequence

1 2 3 4 5 6 7 8

[3 [3 [3 [3 [3 [3 [3 [1

3 3 3 3 1 1 1 3

2 2 2 2 3 3 3 1

3 3 3 3 1 3 1 1

3 3 3 3 3 1 3 3

2 2 2 2 1 3 1 2

3 3 3 3 1 1 3 3

3 3 3 3 3 3 1 3

2 2 2 2 1 1 3 2

3 3 3 3 3 3 1 1

3 3 3 3 1 1 3 3

2 2 2 2 1 3 1 1

3 3 3 3 3 1 3 3

3 3 2 2 3 1 1 1

2 2 3 2 1 3 3 1

3]s 2]s 2]s 1]s 3]s 1]s 1]s 3]s

Buckling load multiplier kcb Failure load multiplier kcs Temperature (C)

Criteria for optimisation

17443.21 17440.24 17404.60 17382.13 11320.24 13244.31 12128.91 10202.40

Buckling Buckling Buckling Buckling Failure Failure Failure Failure

9734.40 10791.12 10791.12 8322.03 18960.53 18960.53 18960.53 18958.63

646.52 621.03 621.03 479.24 5305.09 5305.09 5305.09 2308.76

394

A. Rama Mohan Rao, N. Arvind / Composite Structures 70 (2005) 383–402

Table 7 Performance scatter search algorithm for 48 ply laminate—refupdate-1 S. No.

Laminate sequence

Buckling load multiplier kcb

Failure load multiplier kcs

Temperature (C)

Selection M-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

[3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 1 3

2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 2

3 3 3 3 3 3 3 3 3 2 2 2 3 2 2 2 3 3 3 3 3 3 3 3 3 2 3 1 3 2 3 1

3 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3 3 3 3 3 3 3 3 3 2 2 2 3 2 2 2 3

2 2 2 2 2 2 2 2 1 3 2 1 1 3 2 1 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1

3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 2 3 3 3 2 3 3

3 2 2 2 3 2 2 2 2 2 3 1 2 2 3 1 3 2 2 2 3 2 2 2 1 1 1 1 1 1 1 1

2]s 3]s 2]s 2]s 2]s 3]s 2]s 2]s 2]s 1]s 2]s 3]s 2]s 1]s 2]s 3]s 2]s 3]s 2]s 2]s 2]s 3]s 2]s 2]s 3]s 2]s 2]s 2]s 3]s 2]s 2]s 2]s

7460.99 7443.17 7440.20 7383.77 7460.99 7443.17 7440.20 7383.77 7378.13 7096.98 7065.98 7042.66 7378.13 7096.98 7065.98 7042.66 7460.99 7443.17 7440.20 7383.77 7460.99 7443.17 7440.20 7383.77 7188.19 7030.48 6948.93 6943.46 7188.19 7030.48 6948.93 6943.46

7565.97 7565.97 8616.26 9645.81 7565.97 7565.97 8616.26 9645.81 6464.96 7242.02 8616.26 8881.96 6464.96 7242.02 8616.26 8881.96 7565.97 7565.97 8616.26 9645.81 7565.97 7565.97 8616.26 9645.81 8029.68 8095.59 8029.68 10403.03 8029.68 8095.59 8029.68 10403.03

In order to evaluate the computational efficiency of scatter search algorithm, the number of evaluations employing scatter search with various combinations of improvement, selection and combination is compared. The results are presented in Figs. 4 and 5 respectively for 48 and 64 ply laminates. A close look at the figures indicate that the scatter search with the combination of (SM1 + IM1 + CM1+ RUM1) found to be effective both interms of obtaining better solution and also in computational efficiency. In view of this the results given in the rest of the paper employs only the combination of (SM1 + IM1 + CM1 + RUM1) for scatter search. The optimal lay-up sequences for maximisation of failure load obtained employing scatter search algorithm are given in Tables 11 and 12 respectively for 48 ply and 64 ply laminate composite panels. A close look at the results presented in Tables 11 and 12 indicate that different stacking sequences yield same values of critical failure load multiplier. It can also be observed that the stacking sequences given in Tables 11 and 12 yields different buckling load multiplier values and critical temperature rise values for the same failure load multiplier.

638.01 638.01 607.71 584.68 638.01 638.01 607.71 584.68 545.69 511.19 607.71 750.42 545.69 511.19 607.71 750.42 638.01 638.01 607.71 584.68 638.01 638.01 607.71 584.68 843.77 491.05 843.77 1094.46 843.77 491.05 843.77 1094.46

M-2

X X X X X X X X X X X X X X X X

Improvement

Combination

M-1

M-1

M-2

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X

M-2

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X

X X X X X X X X

Further, it can also be observed that the failure load multiplier values obtained using scatter search for 48 and 64 ply laminates (Tables 11 and 12) are comparing very well with results obtained through enumerative study and shown in Tables 5 and 6. However, one can notice that the stacking sequences obtained through enumerative study are different from the scatter search results. This can be explained as follows. While computing the best stacking sequences through enumerative study, the laminate sequences which have the highest failure load multiplier are compiled and best among them which are comparatively higher in buckling load multiplier and temperature rise are chosen and tabulated. However, in the scatter search, the objective function is defined as maximisation of the failure load multiplier. Hence, scatter search could not capture the laminate sequences with higher buckling load multiplier and temperature rise. In order to capture these best laminate sequences, one has to invariably resort to multi-criteria optimisation by considering all three objectives (i.e. buckling, failure and temperature) [25] simultaneously.

A. Rama Mohan Rao, N. Arvind / Composite Structures 70 (2005) 383–402

395

Table 8 Performance scatter search algorithm for 48 ply laminate—refupdate-2 S. No.

Laminate sequence

Buckling load multiplier kcb

Failure load multiplier kcs

Temperature (C)

Selection M-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

[3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 2 3

2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 2

3 3 3 2 3 3 3 3 3 2 2 2 3 2 2 2 3 3 3 3 3 3 3 3 3 2 3 1 3 3 3 2

3 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3 3 3 3 3 3 3 3 3 2 2 2 3 2 2 2 2

2 2 2 1 2 2 2 2 2 3 2 1 1 3 2 1 2 2 2 2 2 2 2 2 1 3 3 1 2 1 3 3

3 3 3 2 3 3 3 2 2 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 2 3 3 2 3 2 2

3 2 2 1 3 2 2 2 2 2 3 1 2 2 3 1 3 2 2 2 3 2 2 2 1 1 1 1 2 1 2 1

2]s 3]s 2]s 3]s 2]s 3]s 2]s 2]s 2]s 1]s 2]s 3]s 2]s 1]s 2]s 3]s 2]s 3]s 2]s 2]s 2]s 3]s 2]s 2]s 3]s 2]s 2]s 2]s 2]s 3]s 2]s 2]s

7460.99 7443.17 7440.20 7042.66 7460.99 7443.17 7440.20 7383.77 7383.77 7096.98 7065.98 7042.66 7378.13 7096.98 7065.98 7042.66 7460.99 7443.17 7440.20 7383.77 7460.99 7443.17 7440.20 7383.77 7188.19 7030.48 6948.93 6943.46 7202.60 7188.19 7187.75 7030.48

7565.97 7565.97 8616.26 8881.96 7565.97 7565.97 8616.26 9645.81 9645.81 7242.02 8616.26 8881.96 6464.96 7242.02 8616.26 8881.96 7565.97 7565.97 8616.26 9645.81 7565.97 7565.97 8616.26 9645.81 8029.68 8095.59 8029.68 10403.03 10646.81 8029.68 9645.81 8095.59

Finally, comparisons have been made with genetic algorithm. For this purpose an elitist genetic algorithm [26] customised for composite lay-up sequence optimisation problems is considered. Several GA implementations like simple GA, elitist GA, multiple elitist GA are considered for comparison purposes. In the present study, tournament selection with two-point crossover with crossover probability of 0.60 is considered. Mutation, which is a combination of simple invert and permutation, is considered with probability as 0.05 for the present study. Tables 13 and 14 show the comparative performances of the proposed scatter search algorithm and genetic algorithm for 48 ply and 64 ply laminate composites. A close look at the results indicates that the proposed scatter search algorithm provides solutions, which are comparable with various GA implementations. It can also be observed that for 64-ply laminate, the solutions obtained using the scatter search algorithm are marginally superior to genetic algorithms. However, the scatter search algorithm is found to be superior in terms of computational performance i.e. total number of function evaluations.

638.01 638.01 607.71 750.42 638.01 638.01 607.71 584.68 584.68 511.19 607.71 750.42 545.69 511.19 607.71 750.42 638.01 638.01 607.71 584.68 638.01 638.01 607.71 584.68 843.77 491.05 843.77 1094.46 565.80 843.77 584.68 491.05

M-2

X X X X X X X X X X X X X X X X

Improvement

Combination

M-1

M-1

M-2

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X

M-2

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X

X X X X X X X X

4.2. Optimisation of hybrid laminate composite panels This numerical study is concerned with the design of a hybrid laminate composite panel for optimum layer thicknesses, weight and ply angle sequence. Hybrid laminates incorporating two or more fibre composite materials in their construction can offer improved designs and better tailoring capabilities, as it is possible to combine the desirable properties of the two materials. In the present example, we considered two different materials. One is graphite–epoxy, which is expensive, but has high stiffness properties and the other is the glass–epoxy, which is not as stiff but is relatively cheaper. The objective of the composite laminate stacking sequence optimisation problem is to determine the number of plies, their orientation and material properties of each ply, that gives the desired properties: stiffness, strength, frequency, buckling load etc. The strength of composite materials is their very high stiffness-to-weight or strength-to-weight ratios, but the cost of those materials increases very fast with performance. Therefore, it can be advantageous to use a combination of efficient

396

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Table 9 Performance scatter search algorithm for 64 ply laminate—refupdate-1 S. No.

Laminate sequence

Buckling load multiplier kcb

Failure load multiplier kcs

Temperature (C)

Selection

M-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

[3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 1 3 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3

2 2 2 2 2 2 2 2 2 2 3 1 2 2 2 3 2 2 2 2 2 2 2 2 2 3 2 1 2 3 2 1

3 3 3 3 3 3 3 3 3 3 3 1 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 2 3 3 3 3 3 3 3 3 2 2 3 2 2 2 3 2

2 2 2 2 2 2 2 2 2 2 3 3 2 2 2 3 2 2 2 2 2 2 2 2 3 3 2 3 3 3 2 3

3 3 3 3 3 3 3 3 1 2 3 2 1 2 2 3 3 3 3 3 3 3 3 3 3 2 1 1 3 2 1 1

3 3 3 3 3 3 3 3 3 2 1 3 3 2 2 1 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3 3

2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 2 2

3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 1 3 3 3 3 3 3 3 3 3 2 3 2 3 2 3 2

3 3 2 2 3 3 2 2 1 1 2 2 1 1 1 2 3 3 2 2 3 3 2 2 2 3 1 3 2 3 1 3

2 2 3 2 2 2 3 2 1 2 2 1 1 2 1 2 2 2 2 2 2 2 3 2 1 1 1 3 1 1 1 3

3]s 2]s 2]s 1]s 3]s 2]s 2]s 2]s 3]s 1]s 1]s 3]s 3]s 1]s 2]s 1]s 3]s 2]s 2]s 1]s 3]s 2]s 2]s 2]s 3]s 3]s 3]s 1]s 3]s 3]s 3]s 1]s

17443.21 17440.24 17404.60 17382.13 17443.21 17440.24 17404.60 17383.81 16752.93 16702.80 15782.68 15582.69 16752.93 16702.80 15887.88 15782.68 17443.21 17440.24 17383.81 17382.13 17443.21 17440.24 17404.60 17383.81 16962.21 16786.98 16752.93 16061.39 16962.21 16786.98 16752.93 16061.39

9734.40 10791.12 10791.12 8322.03 9734.40 10791.12 10791.12 11834.46 11485.19 11660.36 14887.78 12944.60 11485.19 11660.36 12644.11 14887.78 9734.40 10791.12 11834.46 8322.03 9734.40 10791.12 10791.12 11834.46 8322.03 7597.62 11485.19 11485.18 8322.03 7597.62 11485.19 11485.18

but expensive graphite–epoxy as surface layers and less expensive, glass–epoxy, which has low stiffness as core layers. This way we can reduce the cost while ensuring a high level of performance. In this section, several studies have been carried out to optimise weight and cost of the hybrid laminates by assuming frequency and buckling constraints. Finally a multi criterion optimisation of the hybrid rectangular laminate panel is attempted for simultaneously optimising weight and cost. A rectangular laminate plate of length, 92 cm and width, 75 cm is considered for the studies carried out in this section. The first natural frequency, f of a rectangular plate is given by the following expression [24]: p f ðm; nÞ ¼ pffiffiffiffiffiffi 2 qh rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m4
m2
n2
n4 þ 2ðD12 þ 2D66 Þ þ D22  D11 a a b a

ð12Þ

where D is the flexural stiffness matrix calculated using the classical Lamination theory, q is the average density

646.52 621.03 621.03 479.24 646.52 621.03 621.03 600.87 904.12 592.71 991.77 1247.99 904.12 592.71 574.93 991.77 646.52 621.03 600.87 479.24 646.52 621.03 621.03 600.87 479.24 504.98 904.12 904.12 479.24 504.98 904.12 904.12

M-2

X X X X X X X X X X X X X X X X

Improvement

Combination

M-1

M-1

M-2

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X

M-2

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X

X X X X X X X X

and h the total thickness of the plate. f(m, n) are various frequencies correspond to different mode shapes (different values of m and n in Eq. (12)). The fundamental frequency is obtained when m and n are both one. The material properties of the two composite materials are given in Table 15. The stiffness-to-weight ratio of graphite–epoxy is about four times higher than that of glass–epoxy, with Eq1 ¼ 345 against Eq1 ¼ 87:5. However, it is also more expensive, with a cost per unit weight is 8 times higher than that of glass–epoxy. The ply orientations are considered as 0, ±45 and 90. The laminates are considered to be symmetric and balanced. Similar to the earlier case study, one string of individuals represents one half of the symmetrically laminated composite plate. Each element in the string is an integer between 0 and 6, where 0 represents a pair of empty plies, 1, 2 and 3 represents 02, ±45 and 902 plies of graphite epoxy. Similarly 4, 5 and 6 represents 02, ±45 and 902 plies of glass epoxy. We have to introduce 0 to represent empty plies in the string as the number of

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397

Table 10 Performance scatter search algorithm for 64 ply laminate—refupdate-2 S. No.

Laminate sequence

Buckling load multiplier kcb

Failure load multiplier kcs

Temperature (C)

Selection

M-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

[3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3 [3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3

2 2 2 2 2 2 2 2 2 2 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 1 2 3 2 1

3 3 3 3 3 3 3 3 3 3 3 1 3 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 3 2 2 2 3 2

2 2 2 2 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 3 2 3 3 3 2 3

3 3 3 3 3 3 3 3 1 2 3 2 1 2 2 2 3 3 3 3 3 3 3 3 3 2 1 1 3 2 1 1

3 3 3 3 3 3 3 3 3 2 1 3 3 2 2 2 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3 3

2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 2 2

3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 2 3 2 3 2

3 3 2 2 3 3 2 2 1 1 2 2 1 1 1 1 3 3 2 2 3 3 2 2 2 3 1 3 2 3 1 3

2 2 3 2 2 2 2 2 1 2 2 1 1 2 2 1 2 2 2 2 2 2 3 2 1 1 1 3 1 1 1 3

3]s 2]s 2]s 2]s 3]s 2]s 2]s 1]s 3]s 1]s 1]s 3]s 3]s 1]s 2]s 2]s 3]s 2]s 2]s 1]s 3]s 2]s 2]s 3]s 3]s 3]s 3]s 1]s 3]s 3]s 3]s 1]s

17443.21 17440.24 17404.60 17383.81 17443.21 17440.24 17383.81 17382.13 16752.93 16702.80 15782.68 15582.69 16752.93 16702.80 15899.62 15887.88 17443.21 17440.24 17383.81 17382.13 17443.21 17440.24 17404.60 17386.78 16962.21 16786.98 16752.93 16061.39 16962.21 16786.98 16752.93 16061.39

9734.40 10791.12 10791.12 11834.46 9734.40 10791.12 11834.46 8322.03 11485.19 11660.36 14887.78 12944.60 11485.19 11660.36 10786.92 12644.11 9734.40 10791.12 11834.46 8322.03 9734.40 10791.12 10791.12 10791.12 8322.03 7597.62 11485.19 11485.18 8322.03 7597.62 11485.19 11485.18

646.52 621.03 621.03 600.87 646.52 621.03 600.87 479.24 904.12 592.71 991.77 1247.99 904.12 592.71 443.48 574.93 646.52 621.03 600.87 479.24 646.52 621.03 621.03 621.03 479.24 504.98 904.12 904.12 479.24 504.98 904.12 904.12

individuals in a string are constant where as the number of plies in a laminate is not. First, an attempt has been made to optimise with a single objective and later multi-objective situation is considered. Table 16 shows the optimal weight of the hybrid laminate composite with a constraint on fundamental frequency. Similar to the earlier case study, the

M-2

X X X X X X X X X X X X X X X X

Improvement

Combination

M-1

M-1

M-2

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X

X X X X X X X X X X X X X X X X

X X X X X X X X

X X X X

Number of evaluations

S1= SM1+IM1+CM1+RUM1 S2= SM1+IM1+CM2+RUM1 S3= SM2+IM1+CM1+RUM1 S4= SM2+IM1+CM2+RUM1 S5= SM1+IM1+CM1+RUM2 S6= SM1+IM1+CM2+RUM2 S7= SM2+IM1+CM1+RUM2 S8= SM2+IM1+CM2+RUM2

5000 4000 3000 2000 1000 0 S2

S3

S4

S5

X X X X

laminate stacking sequences in all the tables (i.e. Tables 16–21) are shown in coded string form. It may be noted that Ô1Õ, Ô2Õ and Ô3Õ of the coded strings in the tables indicate graphite–epoxy plies of orientation Ô02Õ, Ô±45Õ and Ô902Õ respectively. Similarly, Ô4Õ, Ô5Õ and Ô6Õ refers to glass–epoxy plies of orientation Ô02Õ, Ô±45Õ and Ô902Õ respectively.

6000

S1

M-2

S6

S7

S8

Fig. 4. Performance of scatter search technique with various options for lay-up sequence optimisation of 48 ply laminate.

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Number of evaluations

30000

S1= SM1+IM1+CM1+RUM1 S2= SM1+IM1+CM2+RUM1 S3= SM2+IM1+CM1+RUM1 S4= SM2+IM1+CM2+RUM1 S5= SM1+IM1+CM1+RUM2 S6= SM1+IM1+CM2+RUM2 S7= SM2+IM1+CM1+RUM2 S8= SM2+IM1+CM2+RUM2

25000 20000 15000 10000 5000 0 S1

S2

S3

S4

S5

S6

S7

S8

Fig. 5. Performance of scatter search technique with various options for lay-up sequence optimisation of 64 ply laminate.

Table 11 Optimal ply sequences of 48 ply laminate for maximal failure load using scatter search S. No.

Sequence

1 2 3 4 5 6

[2 [3 [1 [3 [3 [3

2 1 3 1 3 1

2 3 1 3 1 3

2 1 3 1 3 1

2 3 1 3 1 3

2 1 3 1 3 1

2 3 1 3 1 3

2 1 3 1 1 1

2 3 1 3 2 3

2 1 3 1 2 1

2 3 1 3 1 3

2]s 1]s 3]s 1]s 3]s 1]s

Buckling load multiplier kcb

Failure load multiplier kcs

Temperature (C)

Criteria for optimisation

3623.79 5242.36 4238.50 5242.36 5805.71 5242.36

14220.42 14220.40 14220.40 14220.40 14218.42 14220.40

475.16 5332.6 5338.24 5332.6 1945.84 5332.6

Failure Failure Failure Failure Failure Failure

Table 12 Optimal ply sequences of 64 ply laminate for maximal failure load using scatter search S. No.

Sequence

1 2 3 4 5

[2 [1 [1 [3 [3

2 3 3 3 3

2 1 1 2 1

2 3 1 3 3

2 1 3 2 1

Buckling load multiplier kcb Failure load multiplier kcs Temperature (C) Criteria for optimisation 2 3 3 3 1

2 1 1 1 3

2 3 3 3 1

2 1 2 1 1

2 3 1 1 3

2 1 3 2 3

2 3 3 1 1

2 1 2 3 3

2 3 1 1 1

2 1 1 1 3

2]s 3]s 3]s 2]s 1]s

8589.7 10344.2 10099.5 14638.3 12658.7

18960.50 18960.50 18958.60 18956.20 18960.50

472.6 5309.2 2308.7 1474.6 5305.1

Failure Failure Failure Failure Failure

Table 13 Comparative performance of GA and scatter search algorithm for 48 ply laminate S. No.

Algorithm

Stacking sequence

1 2 3 4 5

Enumeration Scatter search GA without elitism GA with elitism GA with multiple elitist strategy

[3 [3 [3 [3 [3

3 3 3 3 3

2 2 2 2 2

3 3 3 3 3

3 3 3 3 3

2 2 2 2 2

3 3 3 3 3

3 3 3 3 3

2 2 2 2 2

3 3 3 3 3

3 3 3 3 3

2]s 2]s 2]s 2]s 2]s

Buckling load multiplier kcb

Failure load multiplier kcs

Temperature C

Number of function evaluations (price)

7460.99 7460.99 7460.99 7460.99 7460.99

7565.97 7565.97 7565.97 7565.97 7565.97

635.67 635.67 635.67 635.67 635.67

– 1190 3400 2100 4300

Table 14 Comparative performance of GA and scatter search algorithm for 64 ply laminate S. No.

Algorithm

Stacking sequence

1 2 3 4 5

Enumeration Scatter search GA without elitism GA with elitism GA with multiple elitist strategy

[3 [3 [3 [3 [3

3 3 3 3 3

2 2 2 2 2

3 3 3 3 3

3 3 3 3 3

2 2 2 2 2

3 3 2 3 3

3 3 3 3 3

2 2 3 2 2

3 3 2 3 3

Buckling load Failure load Temperature Number of function multiplier kcb multiplier kcs (C) evaluations (price) 3 3 3 3 3

2 2 3 2 2

3 3 2 3 3

3 3 3 1 1

2 2 2 3 3

3]s 3]s 3]s 2]s 1]s

17443.21 17443.21 16834.40 17372.73 17371.11

9734.40 9734.40 10791.11 7597.62 9168.27

646.52 646.52 549.72 436.11 602.68

– 5510 6300 4850 9100

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Table 15 Material properties of Graphite–Epoxy and Glass–Epoxy Property

Graphite–epoxy(AS/3501)

Glass–epoxy Generic S-glass epoxy 2

6.20 Msi (4.30E10 N/m2) 1.29 Msi (0.907E10 N/m2) 0.65 Msi (0.454E10 N/m2) 0.27 0.072 lb/in.3 (1992.95 kg/m3) 0.005 in. (0.000127 m) 1

20.01 Msi (14.068E10 N/m ) 1.30 Msi (0.913E10 N/m2) 1.03 Msi (0.724E10 N/m2) 0.30 0.058 lb/in.3 (1605.434 kg/m3) 0.005 in. (0.000127 m) 8

Longitudinal modulus, E1 Transverse modulus, E2 In-plane shear modulus, G12 Poisson ratio, m12 Density Thickness, t Cost factor, C

Table 16 Optimal ply sequences of bi-material composite laminate for minimal weight with frequency constraint S. No.

Frequency constraint (Hz)

Sequence

1 2 3 4 5

25 30 35 40 45

[2 [2 [2 [2 [2

2 2 2 2 2

2 2 2 2 2

2 2 2 2 2

2 2 2 2 2

2 2 2 2 2

2]s 2 2]s 2 2 2]s 2 3 2 2]s 2 2 2 3 2]s

Number of plies

Cost

Weight (N)

Frequency (Hz)

Buckling load multiplier kcb

28 32 36 40 44

31.84 36.32 40.88 45.44 50.08

39.04 44.53 50.12 55.72 61.41

28.34 32.80 37.28 41.68 46.23

45.00 67.16 95.63 130.68 174.42

Table 17 Optimal ply sequences of bi-material composite laminate for minimal weight with buckling constraint S. No.

Buckling constraint Sequence

1 2 3 4

90 125 225 275

[2 [2 [2 [2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2]s 2 2]s 2 2 2 2]s 2 2 2 2 2]s

Number of plies Cost

Weight (N) Frequency (Hz) Buckling load multiplier kcb

36 40 48 52

50.13 55.72 66.81 72.50

40.90 45.44 54.53 59.08

37.28 41.77 50.76 55.26

95.63 131.19 226.69 288.22

Table 18 Optimal ply sequences of bi-material composite laminate for minimal cost with frequency constraint S. No.

Frequency Sequence constraint (Hz)

1 2 3 4 5

20 30 35 40 45

[5 [2 [2 [2 [2

5 5 2 2 2

5 5 5 2 5

5 5 5 2 5

5 5 5 5 5

Number of plies Cost Weight (N) Frequency (Hz) Buckling load multiplier kcb 5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

6]s 5 6]s 5 5 6]s 5 5 6]s 5 5 5 5 5 5 6]s

40 44 48 48 64

7.08 69.45 11.60 74.95 16.14 80.44 23.78 77.70 18.98 108.30

23.01 30.67 37.21 42.79 47.78

48.16 91.58 143.67 184.18 309.54

Table 19 Optimal ply sequences of bi-material composite laminate for minimal cost with buckling constraint S. No.

Buckling Sequence constraint k

1 2 3 4 5

90 150 250 300 425

[2 [2 [2 [2 [2

5 2 5 2 2

5 2 5 5 2

5 5 5 5 2

5 5 5 5 2

Number of plies Cost 5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

6]s 5 6]s 5 5 5 5 5 6]s 5 5 5 5 5 6]s 5 5 5 5 5 6]s

44 48 64 64 64

A close look at the Table 16 indicate that the minimum weight of the laminate sequence is 39.04 N for the frequency constraint of 25 Hz. The minimum weight increases with the increase in the value set as a frequency

Weight (N) Frequency (Hz) Buckling load multiplier kcb

11.60 74.95 19.96 79.06 15.16 109.68 18.98 108.30 30.50 104.18

30.67 40.33 43.30 47.78 57.11

91.58 166.1 257.15 309.54 426.84

constraint. It can be observed that the composite laminate panel consists of only graphite epoxy plies, as they are relatively lighter than the glass–epoxy plies. Similarly, Table 17 shows the weight minimisation of

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Table 20 Trade-off solutions of bi-material laminate composite for multi-objective optimization of both weight and cost using scatter search with frequency and buckling constraints b

Stacking sequence

Number of plies

Weight (N)

Cost

Buckling (k)

Frequency (Hz)

Number of function evaluations (price)

0 0.80 0.9 0.94 0.96 1

[5 [2 [2 [2 [2 [2

48 44 40 36 36 36

83.29 74.95 65.24 54.15 54.15 49.93

8.49 11.61 18.54 29.30 29.30 40.76

83.23 91.59 102.70 93.38 93.38 95.63

27.90 30.68 34.51 35.64 35.64 37.28

27172 27324 27960 27630 28120 27740

5 5 2 2 2 2

5 5 2 2 2 2

5 5 5 2 2 2

5 5 5 2 2 2

5 5 5 2 2 2

5 5 5 5 5 2

5 5 5 5 5 2

5 5 5 6]s 5 5 6]s 5 6]s 6]s 6]s 2]s

Table 21 Trade-off solutions of bi-material laminate composite for multi-objective optimization of both weight and cost using genetic algorithm with frequency and buckling constraints b

Stacking sequence

Number of plies

Weight (N)

Cost

Buckling (k)

Frequency (Hz)

Number of function evaluations (price)

0 0.9 0.94 1

[5 [2 [2 [2

48 40 36 36

83.29 65.24 54.15 49.93

8.49 18.54 29.30 40.76

83.23 95.51 87.07 82.30

27.90 33.28 34.41 34.58

34350 34083 33988 36950

5 2 3 3

5 4 2 3

5 2 2 2

5 5 3 1

5 6 2 3

5 5 6 2

5 6 5 3

5 5 5 6]s 5 6]s 6]s 2]s

the hybrid laminate with buckling constraint. The minimum weight is found to be 50.13 for the buckling load multiplier constraint of 90. In this case also one can observe from Table 17 that the plies of the laminate panel consists of only graphite–epoxy. Next the cost minimisation of the hybrid laminate is attempted with frequency and buckling constraints. Tables 18 and 19 shows the optimal laminate sequences for minimum cost with various frequency and buckling constraints respectively. A close look at the tables indicate that the outer layers of the laminates are of graphite– epoxy whose stiffness is much higher than the glass– epoxy plies and core layers are of glass–epoxy plies. It is quite understandable as the graphite–epoxy plies are much expensive when compared to glass–epoxy plies. Finally, laminate ply sequence optimisation of hybrid composite panels has been attempted for simultaneous optimisation of both weight and cost. The weighted sum method has been employed to solve the multiobjective optimisation problem, where two objective functions are combined into one overall objective function and is given as Objective function ¼ b  Weight þ ð1
bÞCost

ð13Þ

Table 20 summarizes trade-off results obtained for various values of b. The results furnished in the table include the stacking sequences, number of plies and their corresponding weight, cost, buckling load multiplier and fundamental frequencies. While buckling constraint is set for this problem as 80, the frequency constraint is set as 25 Hz. A close look at the results indicate that, the minimum weight of the hybrid laminate is 49.93 N. It is obtained when all plies are made of Graphite–epoxy. The corresponding cost is 40.76. The minimum cost is

found to be 8.49, which is about 20% of the cost of stacking sequence corresponding to the optimal weight. The optimum ply sequence corresponding to the minimum cost is made exclusively of glass–epoxy. ItÕs weight is found to be 83.29 N which is approximately 67% heavier than the optimum weight laminate sequence. Fig. 6 shows the Pareto front obtained by employing the scatter search algorithm with weighted sum approach of multi criteria optimisation. The solid line shows the Pareto trade-off curve generated through the points corresponding to the optimal values obtained for different values of b. This Pareto front is the set of all the non-dominated solutions, which corresponds to lower envelope of all the feasible design points in the cost/weight plane. This confirms the effectiveness of the scatter search algorithm in obtaining optimal solutions for simultaneous cost/weight minimisation. The Pareto trade-off curve can be used by the designer to determine the optimal configurations for his problem. The final choice of the best design will depend on additional information that will enable him to evaluate all the points on the Pareto curve and prioritise these values depending on the application on hand. Finally, the comparisons have been made with the GA implementation [26] and the results are presented in Table 21. It can be observed from Tables 20 and 21 that the trade-off solutions obtained using scatter search and GA are comparable eventhough the stacking sequences are not exactly same. However, the number of function evaluations for scatter search is lesser when compared to GA. From the two case studies, it is evident that the proposed scatter search metaheuristic algorithm is capable of providing practical multiple alternative lay-up se-

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401

Fig. 6. Pareto optimum trade-off solutions obtained using scatter search method for the bi-material composite laminate panel.

quences for laminated composite structures, exploring only a small amount of all possible existing design alternatives.

5. Conclusions In this paper, a scatter search algorithm has been proposed for lay-up sequence optimisation of laminate composite panels. Implementation details of various selection, improvement, combination and reference set update methods proposed while devising the scatter search algorithm for lay-up optimisation is presented. Numerical experiments have been conducted to evaluate the proposed scatter search metaheuristic algorithm for composite laminate optimisation problem. Initially, preliminary studies have been conducted by solving problems given in the literature for buckling load maximisation and have been compared with the best known results available in the literature. These numerical investigations clearly indicated that the proposed scatter search algorithm generates stacking sequences which are either comparable or superior to the best known results and are computationally efficient than GA. Later, two case studies have been considered to further evaluate the proposed scatter search algorithm. The first case study is the stacking sequence optimisation of laminated composite plates with respect to thermal buckling under strain and ply continuity constraints. Scatter search algorithm proposed in this paper is employed to solve the constrained discrete optimisation problem. Numerical studies presented in this paper indicate that, scatter search algorithm with various combination of methods proposed for selection,

improvement, combination and reference update performs successfully and be able to find practical solutions, exploring only a small amount of all possible existing design alternatives. Among the various possible combinations, the combination of Selection method-1 (SM1), Improvement method-1 (IM1) + Solution Combination methods-1 (CM1) and Reference Update Method (RUM1) is found to be performing particularly well both in terms of robustness and computational effectiveness. The comparison of scatter search implementation with genetic algorithm indicate that the results obtained using scatter search algorithm are infact superior to most of the GA implementations both interms of quality of optimal solution and computational efficiency. The second case study considered is the optimisation of a composite laminate made from two materials. Symmetrical laminated plate subjected to bi-axial in-plane loads are considered. First the scatter search algorithm has been employed to solve the problem with a single objective and later extended to solve multiple objectives. The single objective functions considered are cost optimisation and weight optimisation of bi-material composite laminate plates subjected to constraints related to buckling and first natural frequency. Numerical studies presented in this paper indicate that the proposed scatter search is effective in providing practical optimal solutions for both the cases. Finally, the multi objective design study is carried out with the objective being the maximisation of the weighted sum of weight and cost. Scatter search has been employed for the solution of the multi-objective optimisation problem. An attempt has been made to construct pareto-optimal front by optimising a series of composite objective functions

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combining weight and cost by varying the respective weightages of the two objective functions. The constraints are buckling and the first natural frequency. Numerical studies indicate that the trade-off solutions can be obtained for the multi-objective problem employing scatter search metaheuristic algorithm. When cost was a primary consideration the plate was made from glass–epoxy and when weight was a primary consideration, it was made of graphite–epoxy. Compromise designs can easily be selected from the trade-off solutions. Finally, the multi-objective optimisation results obtained using scatter search are compared with the results obtained using a specially tuned genetic algorithm for laminate composite optimisation. Studies indicate that scatter search is quite effective in obtaining practical solutions and the results are comparable and some times even superior to GA both interms of quality and computational efficiency. However, we should mention here that there is ample scope for fine tuning the proposed scatter search algorithm for improved computational performance. For example, the number of function evaluations can be reduced further by the cache-fetch implementation as explained below. In this paper, the evaluation of the objective function (i.e. function evaluation) is not computationally very expensive as we used only analytical solutions for standard problems employing Classical Laminate Theory (CLT). However, for most practical applications, one has to resort to more sophisticated numerical tools like finite element simulations for function evaluations. This makes the evaluation function as the slowest part of the algorithm. In order to improve the performance, cachefetch can be implemented which stores already evaluated patterns or stacking sequences and the corresponding objective values. Since scatter search algorithm involves operators like combination, improvement etc., several repetitive patterns (or stacking sequences) are likely to emerge during evolutionary process. The objective values of these already evaluated patterns can be retrieved from the cache rather than executing the computationally expensive evaluation function.

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Acknowledgment This paper is being published with the permission of the Director, Structural Engineering Research Centre, Chennai.

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