Hybrid shuffled frog leaping optimisation algorithm for multi-objective optimal design of laminate composites

Computers and Structures 125 (2013) 200–216

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Hybrid shuffled frog leaping optimisation algorithm for multi-objective optimal design of laminate composites K. Lakshmi, A. Rama Mohan Rao ⇑ CSIR-Structural Engineering Research Centre, CSIR Campus, Taramani, Chennai 600 113, India

a r t i c l e

i n f o

Article history: Received 5 October 2012 Accepted 2 May 2013

Keywords: Laminate composites Combinatorial optimisation Shuffled leap frog optimisation Isogrid Pareto optimisation Multi-objective

a b s t r a c t In this paper, a hybrid shuffled frog-leaping algorithm (SFLA) is presented for solving multi-objective optimal design of laminate composite structures. A customized neighborhood search algorithm, an adaptive search factor and a crossover operator are suitably incorporated in the proposed algorithm to improve the convergence characteristics apart from features like Pareto dominance, density estimation, and an external archive to store the non-dominated solutions to handle multiple objectives. The performance of the proposed algorithm is demonstrated by solving laminate composite plate, shell and stiffened shell (isogrid) problems. Further, superiority of the proposed algorithm is demonstrated by comparing with four popular meta-heuristic algorithms. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In many science and engineering problems, it is important to find the minimum or maximum of a function of many variables. For some problems, efficient analytical based algorithms exist, such as linear programming, for obtaining globally optimal solutions. However, for discrete and/or combinatorial optimisation problems, no such efficient algorithms are available for a general problem. It is often necessary to use heuristics, or so-called meta-heuristics, because exact algorithms, such as branch-andbound and dynamic programming, may be limited by computational requirements. Apart from this, most real world optimisation problems involve the optimisation of more than one function, which in turn can require a significant computational time to be evaluated. In this context, deterministic techniques are difficult to apply to obtain the set of Pareto optimal solutions of many multi-objective optimisation problems. In view of this, stochastic methods are being widely used and applied. Among them, the use of evolutionary algorithms for solving multi-objective problems has significantly grown in the last few years, giving raise to a wide variety of algorithms, such as NSGA-II [1], SPEA2 [2], PAES [3], MPSO [4] and several other algorithms [5]. Apart from these popular algorithms, several novel approaches for multi-objective optimisation are proposed for wide variety of applications [6–8].

⇑ Corresponding author. Tel.: +91 44 22549184; fax: +91 44 22541508. E-mail addresses: [email protected] (K. Lakshmi), arm2956@yahoo. com (A. Rama Mohan Rao). 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.05.004

The shuffled frog leaping algorithm (SFLA) [9] is a meta-heuristic optimisation method inspired from the memetic evolution of a group of frogs when seeking for food. The SFLA is formulated by appropriately combining two search techniques, i.e., the local search built into the ‘particle swarm optimisation’ technique; and the competitiveness in mixing of information of the ‘shuffled complex evolution’ technique. This combined strategy enables SFLA to avoid local extremes and to search for a suboptimal solution. The SFLAs have been used to solve discrete as well as continuous optimisation problems. SFLA is employed by Eusuff and Lansey [9] for optimizing the pipe sizes for new pipe networks and for network expansions. An application of SFLA on data clustering is proposed by Amiri et al. [10]. Rahimi-Vahed and Mirzaei [11] have proposed a hybrid version of SFLA and bacteria optimisation algorithms for handling multiple objectives to solve a mixedmodel assembly line sequencing problem. Comparative performance studies have been carried out by Elbeltagi et al. [12] by comparing SFLA with other meta-heuristic algorithms like genetic algorithms (GA), memetic algorithms (MA), particle swarm optimisation (PSO), and ant-colony optimisation (ACO). The studies presented in their paper clearly indicate that the SFLA is a relatively good optimisation technique. The performance of SFLA is similar to PSO and outperforms GA in terms of success rate, solution quality and processing time. The SFLA is thus a promising approach for exploiting their potential for multi-objective combinatorial problems. Motivated by the earlier investigations related to the quality of the original SFLA, this paper presents a modified version of SFLA termed as hybrid SFL algorithm. The proposed hybrid SFL

K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216

algorithm is extended to handle multiple objectives and applied for combinatorial optimisation problems associated with laminate composite structures. The primary aim here is to design a competitive algorithm capable of improving the results produced by stateof-the-art multi-objective evolutionary algorithms, such as MPSO, NSGA-II, PAES and micro-GA. In order to improve the convergence characteristics, we propose a hybrid SFLA algorithm in this paper. Further we have also combined the (external) archive management technique of PAES, and density estimation technique based on crowding distance implemented in NSGA-II in order to deal with multiple objectives. The proposed algorithm is hybridized with a customized neighborhood search algorithm, a search factor and a crossover operator to improve its intensification and diversification capabilities. In the last two decades, the use of laminate composite materials has become widespread not only because of their high strengthto-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, marine and automotive industries. Structures composed of composite materials often contain components, which may be modeled 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 midplane 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 plate’s resonance frequency. The conventional materials are successfully being replaced by composite materials in many structural engineering applications due to their major virtues like higher specific strength, stiffness, better corrosion and wear resistance. In the past, the adoption of composite materials for the aircraft structures is basically driven by the fact that these materials are lighter and exhibit better performance over conventional materials. However, in the current scenario, there is a need for composite structures to match the cost levels of their metal counterparts [13,14]. On the other hand, in order to improve the fuel efficiency apart from benefiting from other features of composite materials, the use of reinforced plastics is gradually increasing from it’s current share of 10% of the total weight of the modern automobiles. Since fuel efficiency is it’s top priority, automotive engineers continue to face challenge in using materials which are light in weight and available at reduced cost [15]. Hence it can be concluded that the ideal choice of the design solution is neither low-cost nor low-weight but rather a combination thereof. It is advantageous sometimes to use two or more fibre composite materials in their construction to obtain improved designs and better tailoring capabilities, as it is possible to combine the desirable properties of the two materials. For instance, it is well known that the cost of composite materials increases very fast with performance. Therefore, it can be advantageous to use a combination of efficient 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 the same high level of performance. For instance, if the designer wants to design a composite panel of size 92  75 cm with a buckling load factor of 50 exclusively with Glass–epoxy plies, the optimum number of plies works out to be 44 with total weight as 76.05 N. Similarly, if designer chooses graphite-epoxy as the material, the optimal number of layers works out to be 32, with a weight of only 34.05 N, but at a cost of approximately 4.7 times the cost of glass–epoxy panel with same performance. Alternatively, the designer can obtain an optimal design with only 32 plies for same performance, using few core layers of Glass–epoxy followed by Graphite-epoxy as outer

201

layers at approximately half the cost of Graphite-epoxy panel and with a marginal 15% increase in weight. So it is clear that the objectives like cost and weight are conflicting in nature i.e. the cost increases when the weight minimisation is our prime objective and vice versa. Similar such several trade-off solutions can be obtained using multi-objective optimisation for the designer to choose the best combination for his application. In this paper we use the proposed hybrid SFL algorithm for multi-objective optimal design of hybrid laminate composite structures like plate, cylinder and stiffened cylinders to obtain set of Pareto optimal (trade-off) solutions with the chosen objectives.

2. Shuffled frog leaping algorithm (SFLA) The SFLA has been designed as a meta-heuristic to perform an informed heuristic search using any mathematical function to seek a solution of a combinatorial optimisation problem. It is based on evolution of memes carried by the interactive individuals, and a global exchange of information among themselves [16]. The benefits of the genetic-based memetic algorithm (MA) and the social behavior-based particle swarm optimisation (PSO) algorithm are used in the SFL algorithm. A memetic algorithm (MA) is a population- based approach for a heuristic search in optimisation problems. The term memetic algorithm comes from ‘meme’. The meme [17] is simply defined as a unit of intellectual or cultural information that survives long enough to be recognized as such and that can pass from mind to mind. As genes propagate themselves in the gene pool via sperm or eggs, memes propagate themselves in the meme pool by leaping from brain to brain via a process that, in the broad sense, can be called imitation. In the other hand, the PSO [18] is an evolutionary algorithm which is inspired by the social behavior of a flock of migrating birds trying to reach an unknown destination. The PSO is initialized with random solutions (named swarm in PSO) and then in each iteration, the individual or potential solution, named particle, flies (moves) with a velocity which is dynamically adjusted according to the flying experiences of its own and its social group. The shuffled frog-leaping algorithm (SFLA) progresses by transforming frogs (solutions) in a memetic evolution. In this algorithm, individual frogs are not so important; rather they are seen as hosts for memes and described as a memetic vector [9]. In the SFLA, the population consists of a set of frogs (solutions) that is partitioned into subsets referred to as memeplexes. The different memeplexes are considered as different cultures of frogs, each performing a local search. Within each memeplex, the individual frogs hold ideas, that can be influenced by the ideas of other frogs, and evolve through a process of memetic evolution. After a defined number of memetic evolution steps, ideas are passed among memeplexes in a shuffling process in order to move towards a global solution [19].The local search and the shuffling processes continue until defined convergence criteria are satisfied [10]. This memetic evolutionary strategy of performing both local and population based search is one of its main advantages [20]. In this paper we have adapted the hybrid shuffled frog leaping optimisation algorithm discussed earlier for multi-objective combinatorial optimisation and applied to stacking sequence optimisation of laminate composites. Unlike single objective optimisation, in multi-objective optimisation, there is no single unique solution which satisfies all the incommensurate objectives of the problem on hand. Hence we use Pareto dominance concept. Further we have also used an external archive to store all the non-dominated solutions obtained during the evolutionary process. In order to limit the number of Pareto optimal solutions and also obtain more meaningful well spread solutions for multi-objective optimisation, we have employed an archive management strategy.

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The proposed multi-objective SFLA combines ideas of two stateof-the-art evolutionary algorithms for solving multi-objective problems. On one hand, an external archive is used to store the non-dominated solutions found during the search, following the scheme applied by PAES, but using the crowding distance of NSGA-II [1] as a niching measure instead of the adaptive grid used by PAES [3]. In this section, we first describe the multi-objective hybrid SFLA and then the archive management of non-dominated solutions will be discussed. 3. Multi-objective hybrid SFLA 3.1. The details of the proposed multi-objective hybrid SFL algorithm are as follows 3.1.1. Initialisation The initial population of N frogs (solutions) is generated randomly. The multiple objectives of each solution (frog) are evaluated. The non-dominated solutions obtained during random generation of initial population will be copied into an external archive maintained to store the historical record of the nondominated individuals found along the search process and also to keep those individuals producing a well-distributed Pareto front. 3.1.2. Fitness evaluation The fitness of each frog is computed using the computed objective values of each frog as follows: k

Fitnessi ¼

1 XNobj objv ali k k¼1 Nobj max objv al

ð1Þ

where Nobj is the number of objectives considered, Fitnessi is the fitk ness associated with ith frog, objv ali is the kth objective value of ith k frog, max objv al is the maximum value of kth objective among all the random solutions generated. Once the fitness of each frog is computed, they are sorted in descending order according to their fitness. 3.1.3. Parameter settings in SFLA We need to set the basic parameters that are essential for running the proposed hybrid SFL algorithm. They include the total number of frogs, N, number of memeplexes, M, Number of submemeplexes, Ns, Number of iterations for shuffling the frogs and maximum number of function evolutions for termination. If the parameter N is too small, SFLA may converge to suboptimal solutions. If N is too large, it may converge to true Pareto solution, but the computational cost will be very high. In view of this, the total number of frogs needs to be chosen carefully through a parametric study. In the present work we have chosen the total size of the frogs as 60 for all the problems solved in this paper based on extensive parametric studies. Similarly, the number of memplexes are set as 5, the number of iterations for shuffling the frogs as 5 and the maximum number of function evaluations as 60,000. Rest of the parameters required for executing multi-objective SFL algorithm can be computed using the above basic parameters. 3.1.4. Formation of memeplexes The sorted frogs are formed into M memeplexes, each holding K frogs such that N = K xM. The division is done in round robin fashion i.e., the first frog going to the first memeplex. Second one going to the second memeplex, the Mth frog to the Mth memeplex and (M + 1)th frog back to the first memeplex. This way of distributing frogs to memeplexes preserves diversity among frogs within each memeplex. In the SFL algorithm, each memeplex is allowed to evolve independently to search locally at different regions of the solution space. In addition, shuffling all the memeplexes and re-dividing

them again into a new set of memeplexes results in a global search through changing the information between memeplexes. As such, the SFL algorithm attempts to balance between a wide search of the solution space and a deep search of promising locations to obtain Pareto optimal solutions. 3.1.5. Evolution steps in each memeplex Various steps involved in each of the memeplex of SFLA is as follows: (i) Construct a submemeplex by giving higher weightage values to the frogs which have better fitness and less weightage to the frogs which have lesser fitness. The weights are assigned to each of the frog in the memeplex with a triangular probability distribution. pj = 2(K + 1  j)/[K(K + 1)], j = 1, . . ., K such that, within a memeplex, the frog with the best performance(objective value) has the higher probability of being selected for the submemeplex, pi = 2/(K + 1) and the frog with the worst fitness value has the lower probability, pK = 2/[K(K + 1)]. ‘S’ number of frogs is then randomly selected to form a submemeplex. The submemeplex is sorted in order of their fitness values i.e. the frog with best objective value will be the first in the order and the frog with least objective will be the last one. Since in multi-objective optimisation, each frog has multiple objectives, this phase is handled slightly differently. We first choose randomly one objective among all the objectives of the problem. We use this randomly chosen objective as a basis for forming the submemeplex. The (guided) search process for finding the optimal solution essentially features in the submemeplex and hence its size, S is crucial in obtaining the optimal results. The size should not be too small to stall the search process involved in SFLA which may obviously leads to a suboptimal solution and at the same time, it should not be too large to increase the computational overheads. Hence parametric studies have been carried out to arrive at an optimal size of the submemeplex size and accordingly set it as 6 for all the problems solved in this paper. (ii) In the submemeplex, the frogs with the best fitness and the worst fitness are identified as Xb and Xw, respectively. Here the fitness refers to the objective value of randomly chosen objective, ‘obj’ of the frog during submemeplex formation. Then the position of the worst frog Xw for the memplex is adjusted as follows:

Di ¼ P  randð0; 1ÞðX ðb;iÞ  X ðw;iÞ Þ xnew ðw;iÞ ¼ xðw;iÞ þ Di ; i ¼ 1; 2; . . .

ð2Þ

number of design variables ð3Þ

ðDmax P Di P Dmin Þ where Dmax ¼ UL  xðw;iÞ Dmin ¼ LL  xðw;iÞ

and ð4Þ

where, P, is the ‘search-acceleration factor’. Assigning a large value to the factor P will accelerate the global search by allowing for a bigger change in the frog’s position and accordingly will widen the global search area. Similarly, a smaller value to P, will focus the process on a deeper local search as it will allow the frogs to change its positions slightly. By adaptively varying the value of P, one can strike a good balance between diversification and intensification. The value of P is taken as RAND(1.5,2.0) (i.e., random value between 1.5 and 2.0) to diversify the search and RAND(0.50, 1.0) (i.e., random value between 0.5 and 1.0) to intensify the search. It may be noted that the multi-objective algorithms will be highly effective, if a search mechanism is devised by maintaining a good balance between intensification and diversification. To facilitate this, initially the value of P is set either RAND(1.5,2.0) or RAND(0.50, 1.0). If the chosen

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objective value of the worst frog is not improved in the last three iterations, we choose the P value based on other alternative. Dmax and Dmin are the maximum and minimum allowed changes in the ith frog’s position (i.e. ith design variable) respectively. UL and LL are the upper and lower limits of the design variables respectively. If the evolution produces a better frog (solution), it replaces the older frog. Otherwise, Xb is replaced by Xg(obj), in Eq. (2) and the process is repeated. Xg(obj), is the best found solution in all submemeplexes with respect to the chosen objective, ‘obj’. The Xg(obj) value at any point of time will be obtained from the archive solutions stored, as the archive solutions are the best non-dominated solutions obtained till that time. If no improvement becomes possible even after this, a two point crossover operator is employed to generate a new solution from the Xw and Xg(obj) solutions. (iii) Although memetic algorithms like SFLA provides promising result, it remains clear that, in many cases, these algorithms cannot compete with customized local (or neighborhood) search algorithms [21]. However, local search methods always suffer from the initialization problem. That is, the performance of a local optimizer is often a function of the initial solution to which it is applied. Therefore hybridization of meta-heuristic algorithm with an effective (application specific) local search algorithm is likely to provide much superior solutions as meta-heuristic algorithms generate good initial solutions for the local search algorithm to explore further to provide a good local optimized solution. Keeping this in view, it is proposed to hybridize SFLA with a customized neighborhood search algorithm called variable depth neighborhood search algorithm. In the variable depth neighborhood search algorithm [22,23], a random cut-off point within the user defined range is used 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 for the chosen variables. 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 to the original solution obtained during the SFLA evolution. Since the influence of extreme fibers in the composite laminate is significant in altering the stiffness of the laminate, the variables to the left of cut-off point in the existing candidate solution are considered for improvement to obtain a local optimal value. In the numerical studies carried out in this paper, the cut-off point is chosen randomly between 1 and 4. After a new solution is found either using Eq. (2) or using cross over operator, we use the variable depth local search algorithm to improve the newly generated solution. The resulting improved solution replaces the worst frog solution Xw. The new solution is also copied into external archive, if it is non-dominated with respect to all non-dominated solutions in the archive. Steps (ii) and (iii) are repeated for specified number of iterations. In the numerical examples explained in this paper, the number of submemeplex iterations is taken as 5. Repeat step (i) to (iii) for a user specified number of evolutions in each memeplex. After specified number of evolutions in each of the memeplex, reshuffle the frogs and sort them again. Repeat steps (3.1.2) to (3.1.5) till the convergence criteria are satisfied. 3.2. External archive

203

be restricted to a prespecified value. The non-dominated trial vectors obtained at each generation are compared one by one with the current archive, which contains the set of non-dominated solutions found so far. There are three cases as illustrated below:  If the trial vector is dominated by a member of the external archive, the trial vector is rejected.  If the trial vector dominates some member(s) of the archive, then the dominated members in the archive are deleted and the trial vector enters the archive.  The trial vector does not dominate any archive members and none of the archive member dominates the solution. This implies that the trial vector belongs to the non-dominated front and it enters the archive. Finally, when the external archived population reaches its maximum allowed capacity, we use crowding distance measure to reduce its size to user defined size. In this way, we preserve the diversity of the archived solutions. The archive maintenance used in multiobjective SFL algorithm is similar to the schemes employed in PAES except in the third case. In PAES, if archive is not full, a new nondominated solution enters archive. Otherwise, the solution replaces a member of the archive residing in the most crowded grid location to maintain the maximum archive size. In this way, every time we have to compare the trial vector with the most crowded archive member. While in the present algorithm, whether or not the archive is full, all nondominated solutions enter the archive and after all nondominated solutions in one submemplex evolution enter the archive, we use crowding distance measure explained below to enforce the maximum archive size. 3.3. Crowding degree estimation To have a good diversity among generated nondominated solutions in the external archive of fixed size, we need to choose a good measure to evaluate the crowding degree around each nondominated solution. The crowding degree estimation method is invoked in two situations. First, when target vector and trial vector do not dominate each other, we evaluate the crowding degree of the target vector and trial vector with respect to the nondominated solutions in the external archive. The less crowded one is chosen as the new target vector of the next generation. Hence, the trial vectors are chosen as the new target vector. Secondly, when the external archive exceeds the prespecified size, the solutions located at the most crowded place should be detected and eliminated. There are several crowding degree estimation methods employed in the Multi-objective Evolutionary algorithms (MOEA) literature. PAES and MPSO use adaptive hypercubes, where we have to choose an appropriate depth parameter (PAES) or the number of divisions (MPSO) to control the hypercube size. Since the size of the hypercubes is adaptive with the bounds of the entire search space, when solutions converge near the Pareto front, the hypercubes are comparatively large. In our approach, we estimate the density of solution with respect to crowding distance in which no user-defined parameter is required. The crowding distance is a simplified version of the 2-nearest neighbor density estimator, where the distance of two points on either side of a particular solution along each of the objectives is used to estimate the crowding degree around this solution. 3.4. Convergence

We use an external archive to keep the best nondominated solutions generated so far by the SFLA. Initially, this archive is empty. As the evolution progresses, good solutions enter the archive. However, the size of the true nondominated set can be huge. The computational complexity of maintaining the archive increases with the archive size. Hence, the size of the archive will

The solution is assumed to have been converged if there is no update of archive for a specified continuous number of iterations or maximum number of function evaluations are reached. The complete process of the hybrid SFL algorithm is explained through a flowchart and is given in Fig. 1.

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4. Laminate composite structures Combinatorial optimisation problem of stacking sequence design may be stated as follows: For a laminate made up of n plies, where the number m of distinct plies of different orientations is between 1 and n, devise an approach or algorithm, which will gener-

ate 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 fulfil several different incommensurate criteria.

Fig. 1. Flow chart of hybrid SFL algorithm for multiobjective optimisation.

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8 9 > < rxx > =

4.1. Solution representation 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. One string of individuals represents one half of the symmetrically laminated composite plate/shell. In this paper, we concentrate on multi-objective combinatorial optimisation of laminate composite structures, precisely aiming at simultaneously optimizing cost as well as weight of the laminate composite structures. Hence we employ hybrid laminates in our study as effective material utilization by way of hybrid laminates brings down the cost. In this paper we employ glass epoxy and graphite epoxy laminates in our investigations. The solution representation in SFLA is coded accordingly. When a shell structure is considered for layup design, each element in the string is an integer between 0 and 8, where 0 represents an empty ply, 1, 2, 3 and 4 represent 0°, +45°, 90° and 45° plies of graphite epoxy. Similarly 5, 6, 7 and 8 represents 0°, +45°, 90° and 45° plies of glass epoxy. We have to introduce 0 to represent empty plies in the string as the number of individuals in a string are constant where as the number of plies in a laminate is not. For example, the laminate [gr90°, gr45°, gl90°, gl0°]S is encoded as 3 2 7 5. The rightmost 5 corresponds to the layer closest to the laminate plane of symmetry. The leftmost 3 describes the outermost layer. In this paper three laminate composite models are considered as case studies to demonstrate the effectiveness of the proposed multi-objective SFLA for lay-up sequence optimisation of laminate composite structures. The three models considered are hybrid laminate composite plate, isogrid stiffened cylindrical shell and pressure vessel. 4.2. Formulations of laminate composite structures In the present formulation, we use classical laminate theory. Consider a laminate shown in Fig. 2 of total thickness, h composed of n orthotropic layers with the principal natural coordinates X, Y and Z directions with Z axis is taken as positive upward at middle plane. Using the constitutive law, the stress–strain relationship in principal natural coordinate directions (X, Y) can be given as

2

Q 11

Q 12

ryy ¼ 6 4 Q 21 Q 22 > > : rxy ; Q 61 Q 62

9 38 Q 16 > < ex > = 7 Q 26 5 ey > :c > ; Q 66 xy

ð5Þ

where ½Q  ¼ ½T½Q ½TT

2

EL ð1mLT mTL Þ

6 m E ½Q  ¼ 6 4 ð1mTLLT mL TL Þ 0

mLT ET ð1mLT mTL Þ ET ð1mLT mTL Þ

0

0

3

7 0 7 5; GLT

2

cos2 h 6 ½T ¼ 4 sin2 h sin 2h 2

2

sin h

 sin 2h

3

cos2 h

7 sin 2h 5

 sin22h

cos 2h ð6Þ

T is the transformation matrix and h is the fiber orientation with respect to X axis. E and m are Young’s modulus and poison’s ratio respectively of the laminae in the principal material coordinate directions, L and T. Referring to Fig. 2, the displacements at any point in a laminate is given by

@w0 @x @w0 v ðx; y; zÞ ¼ v 0 ðx; yÞ  z @y wðx; y; zÞ ¼ w0 ðx; yÞ uðx; y; zÞ ¼ u0 ðx; yÞ  z

ð7Þ

where u0, v0 and w0 are the displacements along the coordinate lines of a material point on X Y plane, where Z = 0 (mid plane). Combining the strain displacement relationships with Eq. (7)

2 6 4

3

2

3

2

3

ex @u=@x @u0 =@x 6 7 6 7 ey 7 @u=@y @u0 =@y 5¼4 5¼4 5 exy @u=@y þ @ v =@x @u0 =@y þ @ m0 =@x 2 2 3 2 03 2 3 ex jx @ w0 =@x2 6 2 7 6 7 6 7 0  Z 4 @ w0 =@y2 5 ¼ 4 ey 5  Z 4 jy 5 jxy e0xy 2@ 2 w0 =@x@y

or feg ¼ feg0 þ Zfjg 0 x,

0 y,

ð8Þ

ð9Þ

0 xy

Here, e e and e are the mid plane strains, while jx, jy, and jxy are the plate curvatures. The stresses in any layer ‘k’ can be written as

½rk ¼ ½Q k ½e0  þ z½Q k ½j

ð10Þ

Let hk be the thickness of layer k, then the total thickness of the composite laminate with n layers (Fig. 2) can be written as

h¼

n X

hk

ð11Þ

k¼1

Since the stresses in a laminated composite vary from ply to ply, it is convenient to define laminate force (Nx, Ny and Nxy) and moment

Fig. 2. A laminate composite made up of n stacked plies.

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(Mx, My and Mxy) resultants. These resultants of stresses and moments acting on a laminate cross section, provide us with a statically equivalent system of forces and moments acting at the mid plane of the laminated composite. The force and moment resultants per unit width of the laminate are defined in terms of the stresses as follows:

½NT ¼ f Nx ; Ny ; Nxy g ¼

Z

H=2

f rx ;

ry ; sxy gdz

H=2 T

½M ¼ fMx ; M y ; Mxy g ¼

Z

ð12Þ

H=2

f rx z;

ry z; sxy z gdz

H=2

Substitution of Eq. (10) into Eq. (12) yields a relation that relates the force and moment resultants to the midsurface strains and curvatures can be written in compact form as

     A B e0 N ¼ M B D j

ð13Þ

1 2 2 ðQ ij Þk ðhk  hk1 Þ 2 k¼1

and Dij ¼

n 1X 3 3 ðQ ij Þk ðhk  hk1 Þ where i; j ¼ 1; 2; and 6: 3 k¼1

ð14Þ

The matrices A, B and D are known as extensional, bending-extensional coupling and bending stiffness matrices respectively. The six-by-six matrix in Eq. (13) is called the laminate stiffness matrix or ABD matrix. This stiffness matrix depends on Q and Q in turn depends on the orientation angle of each ply. Hence, the force and moment results vary with the stacking sequence. One can precisely optimize the stacking sequence of the laminate to improve the structural performance. Eq. (13) can be inverted to obtain the mid-surface strains and curvatures in terms of the force and moment resultants

a12

a16

b11

b12

a22

a26

b12

b22

a26

a66

b16

b26

b12

b16

d11

d12

b22

b26

d12

d22

b26

b66

d16

d26

9 38 b16 > Nx > > > > > 7 > > b26 7> > Ny > > > > > 7> < Nxy = b66 7 7 d16 7 > Mx > > 7> > > > 7> > > > 5 M d26 > y > > > > : ; M xy d66

ð15Þ

where the abd matrix in (15) is the inverse of the ABD matrix. For simple loading cases where the force and moment resultants are prescribed, the mid-surface strains and curvatures are computed ðnÞ ðnÞ ðnÞ using (15) and the stresses rx and ry and sxy in the global coorðnÞ ðnÞ dinate system are determined using (10). The stresses r1 and r2 ðnÞ and s12 in the principal material coordinate system are obtained by a tensor transformation 8 9ðnÞ > < r1 > = > :

r2 > s12 ;

6 ¼4

k¼1 n X

2

a11 6a 6 12 6 6 a16 ¼6 0 6b > > k > x > 6 11 > > > > 6 > 0 > > > 4 b12 k > > y > > > > > : 0 > ; b16 kxy

2

n X Aij ¼ ðQ ij Þk ðhk  hk1 Þ;

Bij ¼

8 09 ex > > > > > > > > > e0y > > > > > > > > > > < c0xy > =

38 9ðnÞ 2 cos2 /ðnÞ sin /ðnÞ 2sin/ðnÞ cos /ðnÞ > < rx > = 2 ðnÞ ðnÞ ðnÞ 7 2 ðnÞ sin / cos / 2 sin/ cos / 5 ry > : > 2 sxy ; sin /ðnÞ cos /ðnÞ sin/ðnÞ cos /ðnÞ cos2 /ðnÞ  sin /ðnÞ

ð16Þ

When comparing the stiffnesses of different laminates, especially symmetric laminates that are subjected to in-plane loading, it is often convenient to define the effective extensional modulus Ex ; the effective extensional modulus Ey ;and the effective shear modulus Gxy of the laminate, as follows:

Ex ¼

1 ; a11 h

Ey ¼

1 ; a22 h

Gxy ¼

1 a66 h

ð17Þ

4.3. Frequency and Buckling load computation of simply supported laminate plate A Simply supported laminate composite plate of rectangular shape (a x b), with symmetrical stacking sequence is shown in Fig. 3. The laminated composite plate is composed of P layers, each P of thickness hk. The total thickness of the laminate is h = nk¼1 hk .

Fig. 3. Simply supported hybrid laminate composite plate.

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K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216

Assuming that the plate is loaded, in the x–y plane, by the forces kNx, and kNy, where k is a scalar amplitude parameter, the governing differential equation for the buckling behavior of the plate, under the assumption of the classical plate theory [24,25] is

@4w @4w @4w þ 2 ð D þ 2D Þ þ D 12 66 22 @x4 @x2 @y2 @y4 ! 2 2 2 @ w @ w @ w ¼ k Nx 2 þ Ny 2 þ Nxy @x @y @x@y

D11

ð18Þ

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 [24,25]

kb

p2

¼

D11

m4 a

 2  2  4 þ 2ðD12 þ 2D66 Þ ma nb þ D22 nb m2 n 2 NX a þ Ny b

ð19Þ

influence the stability and strength design of the laminated composite plate and cylindrical shell structures. 4.4. Buckling of a fibre-reinforced isogrid stiffened cylinder The second numerical model considered is a isogrid cylinder shown in Fig. 4 subjected to axial compression loading. An analytical approach based on smeared model to compute equivalent stiffness parameters of the isogrid developed by Wodesenbet et al. [26] is employed here to compute the buckling load. The smeared model is briefly outlined here for the sake of completeness and more details can be found in Wodesenbet et al. [26]. It is first required to determine the equivalent extensional, coupling and bending matrices (A, B and D matrices respectively) of the overall stiffened cylinder in order to calculate the global buckling load of the structure. This involves determining the stiffness contribution of the grid (stiffener) as well as the shell. The stiffness of the stiffeners is given by

derived from the displacement field

2

XX mpx npy wðx; yÞ ¼ Amn sin sin a b m n

6 Ns 6 h 6 s 6 Nxh 6 6 Ms 6 x 6 s 4 Mh Msxh

ð20Þ

that satisfies all imposed boundary conditions of the plate. The smallest value of kb is the critical buckling load kcb and can be evaluated from Eq. (19). Similarly, the first natural frequency, f is given by the following expression

p

f ðm; nÞ ¼ pffiffiffiffiffiffi 2 qh rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m4 m2 n2 n4 þ 2ðD12 þ 2D66 Þ þ D22  D11 a a b b ð21Þ where q is the average density and h is the total thickness of the plate. D11, D12 and D66 are the bending stiffness coefficients of the laminate. f(m,n) are various frequencies correspond to different mode shapes (different values of m and n in Eq. (21)). The fundamental frequency is obtained when both m and n are one. The extensional stiffness matrix, A and bending stiffness matrix, D, which are functions of design variables, are the major factors that

Nsx

2

3

2c3 a

6 2 2sc 7 6 b 7 6 7 6 7 6 0 6 7¼6 7 6 c3 t 7 6 a 7 6 5 6 sc2 t 4 b 0

2s2 c a

0

c3 t a

s2 ct a

ð2s3 þ2Þ b

0

sc2 t b

ð2s3 þ2Þt 2b

0

2sc2 b

0

0

s2 ct a

0

c3 t 2 a

s2 ct 2 a

ð2s3 þ2Þt 2b

0

sc2 t 2 b

ð2s3 þ2Þt 2 2b

0

sc2 t b

0

0

0

3

7 0 7 7 7 sc2 t 7 b 7 7 0 7 7 7 0 7 5

sc2 t 2 b

eox 3 6 eo 7 6 h 7 6 o 7 6 ehx 7 7 6 6j 7 6 x7 7 6 4 jh 5 jxh 2

ð22Þ

where a and b are the width and breadth of the unit cell of the isogrid shown in Fig. 5. c = cos(/), s = sin(/), / is the stiffeners orientation angle. t is the thickness of the shell. The stiffness matrix given in Eq. (22) is a function of the mid-plane strains and curvatures of the shell. The elements of the stiffness matrix are derived by analysing the force and moments due to stiffeners. These stiffness parameters are denoted as Asij ; Bsij , and Dsij . Eq. (22) is a symmetrical matrix due to the geometric relation between the parameters ‘a’, ‘b’, and ‘U’. The total force and moment on the panel is the superposition of the force and moment due to the stiffener and the shell and it can be written as:



N M

"

 ¼

V s N s þ V sh Nsh V s M s þ V sh Msh

#

" ¼

V s As þ V sh Ash

V s Bs þ V sh Bsh

V s Bs þ V sh Bsh

V s Ds þ V sh Dsh

#

eo j



ð23Þ where Vs and Vsh stand for volume fraction of stiffener and shell respectively. Nsh and Msh are the force and moment contribution of the shell respectively. These quantities are easily computed by applying the laminate theory on the shell. Ash, Bsh and Dsh represent the extensional, coupling, and bending stiffness coefficients respectively of the shell and are given in Eq. (14). Similarly, As, Bs and Ds represent the extensional, coupling, and bending stiffness coefficients respectively of the stiffeners and are given in Eq. (22). The resultant stiffness parameters given in Eq. (23) are the equivalent stiffness parameters of the whole panel.

Fig. 4. Laminate composite isogrid stiffened cylinder.

4.4.1. Axial buckling load The total potential energy of the cylinder p, is the sum of the strain energy U and the work done by the external force V. The strain energy is a function of the stiffness parameters of the cylinder panel, the radius of the cylinder ‘r’ and the unknown displacement fields in the radial, axial and hoop direction ‘w’, ‘u’, and ‘v’ respectively. Since the stiffened cylinder panel has already been reduced into an equivalent orthotropic laminate, the strain energy expression can be written as follows:

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K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216

Fig. 5. Typical isogrid cell.

(

2

@u @u @ m w þ 2A12 þ @x @x @h r 0 0 

 2  

 @m @m w w @u @m w þ þ 2 A16 þ þ A26 þ þA22 r @h @h @h r @h r



2 2 @u @ m @u @ m @u @ w  þ A66 þ þ  B11 @h @x @h @x @x @x2 " #

2

2 @ m w @ w @u @ w @m w @2w  B 2B12 þ þ þ 22 @h r @x2 @x @x2 @h r @h2 " # "



@ 2 w @u @ m @u @ 2 w @ 2 w @u @ m  2B16 þ2 2B16 þ þ 2 2 @x @h @x @x @x@h @x @h @x # "



2 # 2 2 @u @ w @ w @u @ m @m w @ w þ2 þ þ  2B26 þ2 @x @x@h @h r @x@h @h2 @h @x !2

@ 2 w @u @ m @2w @2w @2w þ D11 4B66 þ þ 2D12 2 2 @x@h @x @h @x @x @h2 !2 ! @2w @2w @2w @2w þD22 þ 4 D16 2 þ D26 2 2 @x @x@h @h @h !2 9 @2w = þ4D66 dxdh ð24Þ @x@h ;

1 U¼ 2

Z 2pr Z

L

A11

The potential energy due to in-plane load is in turn given by

V¼

1 2

Z 2pr Z 0

L

Nh 0

2 @w dxdh @x

ð25Þ

where Nh is the load per unit length applied on the rim of the cylinder. The strain energy U and the potential energy term V are integrated along the circumference and the height L of the cylinder to obtain the total energy of the cylinder. The displacement field u, v and w can be defined by kinematically admissible functions, i.e., displacement fields satisfying the essential boundary conditions. For a simply supported condition the displacement fields can be assumed as 1   u ¼ R1 m¼1 Rn¼1 Amn cosðmxÞ sinðnsÞ 1 1  cosðn  sÞ m ¼ Rm¼1 Rn¼1 Bmn sinðmxÞ 1   w ¼ R1 m¼1 Rn¼1 C mn sinðmxÞ sinðnsÞ

where m = mp/L, n = n/r, s = rq, and m, n = 1, 2, 3, . . ..

ð26Þ

These displacement fields are substituted into Eqs. (24) and (25). The total energy expression is obtained to be a function of the stiffness matrix elements of the equivalent laminate and the unknown displacement field coefficients Amn, Bmn and Cmn. For the equilibrium to be stable, the total potential energy of the system must be minimum. This can be satisfied by finding the first derivative of the total potential energy with respect to the unknown constants Amn, Bmn, and Cmn and equating to zero. This results in an eigenvalue problem. The resulting equation is then solved for the unknown in-plane load Nh. Numerous loads satisfy the expression for in plane load Nh. The minimum value of these loads corresponds to the buckling load of the structure. 4.5. Laminate composite pressure vessel The third numerical model considered is a laminated fibre-reinforced composite thin walled pressure vessel of radius R (from the center to the mid-surface of the shell), and total thickness h, subjected to internal pressure p shown in Fig. 6. The shell has a symmetric layup consisting of K layers of equal thickness t. Let the midsurface of the shell be the reference surface, and let the origin of the coordinates be located at one end of the pressure vessel. The structure is referenced in an orthogonal coordinate system (x, y, z), where x is the longitudinal, y the circumferential, and z the radial direction. The displacement components u, v and w lie in the x, y and z directions, respectively. The fibre angle is defined as the angle between the fibre direction and the longitudinal (x) axis. The fibre orientations are symmetric with respect to the mid-plane of the shell. The force (Nx, Ny, Nxy) and moment resultants (Mx, My, Mxy) can be written as

pR ; Ny ¼ pR and 2 ¼ Mx ¼ My ¼ M xy ¼ 0

Nx ¼ Nxy

ð27Þ

When the applied loads are greater than or equal to the critical values, buckling or overstressing will occur. The constitutive equation for the fibre-reinforced composite pressure vessel can be rewritten as

fNgxy ¼ ½Aexy ;

fMgxy ¼ ½DfKgxy ;

ð28Þ

where {N}xy is the force resultant tensor, {M}xy the moment resultant tensor, {e}xy the strain tensor, {K}xy the curvature tensor. These matrices will be used in failure analysis of the pressure vessel.

K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216

209

Fig. 6. Laminate composite Pressure vessel.

4.5.1. Tsai-Wu failure criterion The interactive failure criterion proposed by Tsai-Wu [27] is a generic one. It is quadratic failure criterion and is being used most extensively in the design of laminate composites. The theory states that the lamina failure occurs when the value f > 1. Based on the Tsai-Wu failure criterion, the strength level factor n is defined as:

max n

F LL ðrL Þ2 þ F TT ðrT Þ2 þ F SS ðrLT Þ2 þ 2F LT rT o ðkÞ ðkÞ þF L rL þ F T rT

f¼

ðkÞ

ðkÞ

ðkÞ

ðkÞ

k

In Eq. (29), each stress component of the kth layer the material direction can be calculated by

rLðkÞ ð29Þ

ðkÞ ðkÞ rðkÞ L ; rT ; rLT in

ðkÞ

frgLT ¼ ½T½Q ðkÞ ½A1 fNgxy;

ð30Þ

where {N}xy is the force resultants [Nx, Ny, Nxy]T, defined in Eq. (28); [A]1 the inverse of the extension stiffness matrix [A], given in Eq. (14); Q is the reduced stiffness of the kth layer of the laminate; and the strength parameters FLL, FLT, FTT, FSS, FL and FT are given by

F LL ¼

1

rTLU rCLU

FT ¼

1

rTTU

F TT ¼

;



1

rCTU

;

1

rTTU rCTU

F LT ¼ 

;

F SS ¼

1

r2TLU

;

FL ¼

1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rT rC rT rC LU

Lu

TU

1

rTLU



1

rCLU

; ð31Þ

TU

where rTLU ; rCLU ; rTTU , and rCTU are the tensile and compressive strengths of the composite material in the longitudinal and transverse directions, and rTLU is the in-plane shear strength. In order to ensure that first-ply failure does not occur in the cylindrical shell, the following condition must be satisfied

n  1:

ð32Þ

5. Numerical studies The design objective of this study is the selection of the laminate configuration which minimizes the strain energy of the plate (i.e. which maximizes the average stiffness) when subjected to inplane or out-of-plane loads. The layer orientation angles are taken as the design variables, and both the layer thickness and the total

number of layers are assigned, as frequently required in practical design problems. The optimisation problem can be therefore stated as follows: Given a set of m possible orientation choices for each of the N layers, minimize the cost, weight and maximize the stiffness or buckling or failure load factors, while (optionally) satisfying additional constraints (such as, for example, the requirement of special laminate configurations like ply contiguity constraints, ply angle difference, first ply failure etc.,). The application of the proposed multi-objective SFLA to the solution of this design problem is illustrated here by means of numerical examples dealing with rectangular laminated plates made of orthotropic layers, stiffened laminate composite shells and pressure vessel problem with increased number of objectives. Frequently in the design of laminated composite structures, the major objective is to find a laminate lay-up configuration that satisfies the following requirements. (1) The structure does not fail under any of the applied loading conditions and failure margins with respect to the applied loading conditions are always within the limits; failure being defined by an appropriate user defined failure criterion. (2) The structure is as lightweight as possible; the number of layers in a laminate is as small as possible and the performance should be maximum with respect to cost. (3) The structure is easy to manufacture, which requires that the number of different layer orientations is limited and large areas of the structure have the same layer orientations. (4) Effective material utilization by way of hybrid laminates to bring down the cost. For example using plies of higher strength like carbon- epoxy as outer fibre plies which are highly stressed and cheaper and relatively low strength glass plies as core layers to bring down the cost while ensuring a high levels of performance. In the light of the requirements above, we introduce a method for multi-criterion optimisation of composite laminates with the objective of generating a computationally efficient procedure for generating Pareto optimal laminate layup configurations meeting all specified design requirements and specified constraints. In the present work, some basic assumptions concerning the definition of a laminate have been made.

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K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216

where qc is the material density of the Graphite–epoxy layers and qg is the material density of Glass–epoxy layers and h is the total thickness of the laminate and hc is the thickness of the Graphite– epoxy plies in the laminate. Cc and Cg are the cost factors for Carbon-epoxy and Glass–epoxy plies respectively. f is the fundamental frequency and kcb is buckling load factor of the composite plate. Details related to computation of buckling load factor and fundamental frequency using the classical laminate theory is given in the previous chapter. Alternatively, the buckling load factor and fundamental frequency are also considered as design constraints in the numerical studies. As mentioned earlier, the design constraint is treated as an additional objective by formulating the cost function as (f  fc) for frequency and (kcb  kc ) for buckling load factor and is simultaneously minimized along with the design objectives. Apart from these design constraints, combinatorial constraints like ply balancing and ply contiguity [28] are also imposed. The combinatorial constraints are handled in the present work using a correction operator [29,30]. The ply angles are the design variables in the optimisation problem. The ply orientations are considered as 0°, ±45°, and 90°. As mentioned earlier, the solution is represented in the form of an encoded string representing ply angles. 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 0°, ±45°, and 90 2 plies of graphiteepoxy. Similarly 4, 5 and 6 represents 0°, ±45°, and 90 2 plies of Glass–epoxy. We have to introduce 0 to represent empty plies in the string as the number of individuals in a string are constant where as the number of plies in a laminate is not. In the first numerical experiment, we have considered two objectives i.e., cost and weight with buckling load factor of 50.0 as design constraint. The additional objective considered for this problem to treat the design constraint is minimisation of value ðkcb  50Þ; where kcb is the critical buckling load factor of the plate for the given stacking sequence. Therefore the multi-objective design problem for this example can be summarized as minimisation of three objectives, i.e., cost, weight and ðkcb  50Þ. The maximum archive size is taken as 10 for this numerical example. Table 2 shows the trade-off solutions obtained using the proposed multiobjective meta-heuristic algorithm for simultaneous optimisation of cost and weight with constraint on buckling load factor as 50.0. A close look at Table 2 indicates that the optimal stacking sequence with minimum cost is found to be 77.5. The corresponding weight is found to be 76.05 N. It can be observed from the results presented in Table 2 that the stacking sequences corresponding to the minimum cost consists of Glass–epoxy plies which are relatively cheaper when compared to graphite-epoxy plies. The minimum weight is found to be 44.45 N. It can be observed that the stacking sequences corresponding to minimum weight consists of only graphite-epoxy layers which are lighter when compared to Glass–epoxy layers. Further, It is also evident from the stacking

(1) The laminate is assumed to have no free edges, that is, the laminate is assumed to be wide enough and to have no holes. Therefore only in-plane stresses are effective. There are no interlaminar stresses that may cause failure. (2) To improve the ease of laminate lay-up for manufacturing, only a limited number of layer orientations are included in the laminate lay-up and the laminate thickness is limited to multiples of the layer thickness. Therefore, the layer orientations in this paper are limited only to 0, 90, and 45°. (3) To avoid generally undesirable extension-bending coupling effects, the laminate lay-up is limited to a symmetric and balanced laminate structure. We set ply balancing and also ply contiguity as combinatorial constraints. 5.1. Simply supported hybrid laminate composite plate A rectangular hybrid laminate plate of length, a = 92 cm and width, b = 75 cm subjected to in-plane compressive load as shown in Fig. 3, is considered as the first numerical example. The material properties used in this study for Carbon–epoxy and Glass–epoxy are given in Table 1. The stiffness-to-weight ratio of graphite-epoxy is about four times higher than that of Glass–epoxy, with E1/q = 345 against E1/q = 87.5. However, it is also more expensive with a cost per unit weight is 8 times higher than that of Glass–epoxy. In this section different set of numerical studies have been carried out using the laminate composite plate problem by varying the number of design objectives. The design objectives considered for the plate are:

Minimise; Weight ¼ abðqc hc þ qg ðh  hc ÞÞ Minimise; Cost ¼ abðqc hc C c þ qg ðh  hc ÞC g Þ maximize : maximize Buckling load factor kcb ¼ minðkb ðm; nÞÞ Maximise fundamental frequency f ¼ minðf ðm; nÞÞ ð33Þ Table 1 Material properties of graphite–epoxy and Glass–epoxy. Property

Graphite–epoxy

Glass–epoxy

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

14.068e10 N/m2 0.913e10 N/m2 0.724e10 N/m2 0.3 1605.434 kg/m3 0.000127 m 8 1447e06 N/ mm2 1447e06 N/ mm2 51.7e06 N/mm2

4.42e10 N/m2 0.907e10 N/m2 0.464e10 N/m2 0.27 1992.95 kg/m3 0.000127 m 1 1280e06 N/ mm2 690e06 N/mm2

206e06 N/mm2

158e06 N/mm2

93e06 N/mm2

69e06 N/mm2

rTLU

Longitudinal compressive strength, rCLU Transverse tensile strength,

rTTU

Transverse compressive strength, In-plane shear strength, rTLU

rCTU

49e06 N/mm2

Table 2 Trade-off solutions for laminate composite plate using the proposed algorithm for optimisation of cost and weight with buckling constraint as 50 using hybrid SFLA. S.no

Cost

Weight (N)

Number of Layers

Stacking sequence

1

77.5

76.05

44

½902 ; 902 ; 02 ; 902 ; 02 ; 45ðglÞ ; 45ðglÞ ; 02 ; 902 ; 02 ; 902 ; 02 s

2

101.7

60.87

36

½ 45ðgÞ ; 902 ; 02 ; 45ðglÞ; 45ðgÞ ; 45ðgÞ ; 45ðgÞ ; 02 ; 45ðgÞ s

ðglÞ

50.00

3

139.9

59.51

36

50.03

4

171.2

51.23

32

5

209.4

49.88

32

6

247.7

48.52

32

7

285.9

47.16

32

8

324.2

45.81

32

9

362.5

44.45

32

ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ½02 ; 45ðgÞ ; 45ðgÞ ; 02 ; 902 ; 902 ; 02 ; 02 ; 902 s ðgÞ ðgÞ ðglÞ ðglÞ ðglÞ ðgÞ ðglÞ ðglÞ ½902 ; 45 ; 902 ; 45 ; 902 ; 02 ; 902 ; 45 s ðgÞ ðgÞ ðglÞ ðglÞ ½902 ; 902 ; 45ðgÞ ; 02ðgÞ; 902 ; 02 ; 45ðglÞ ; 45ðglÞ s ðgÞ ðgÞ ðgÞ ðglÞ ðglÞ ðgÞ ðgÞ ½02 ; 45 ; 902 ; 902 ; 45 ; 45ðglÞ ; 02 ; 902 s ðgÞ ðgÞ ðgÞ ðgÞ ðglÞ ðglÞ ðgÞ ðgÞ ½ 45 ; 02 ; 902 ; 02 ; 45 ; 02 ; 02 ; 902 s ðgÞ ðgÞ ðgÞ ½02 ; 45ðgÞ ; 902 ; 45ðgÞ ; 45ðgÞ ; 902 ; 45ðgÞ ; 45ðgÞ s ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ ½02 ; 902 ; 45ðgÞ ; 902 ; 02 ; 902 ; 902 ; 02 s

ðglÞ

ðglÞ

ðglÞ

ðglÞ

ðglÞ

Buckling load factor ðglÞ

ðglÞ

ðglÞ

ðglÞ

ðglÞ

ðglÞ

ðglÞ

50.24

50.01 50.00 50.00 50.62 50.00 50.68

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K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216

sequences given in Table 2, that with the increase in cost (or in other words reduction in weight), the number of graphite–epoxy plies in the outer fibres of the laminate composite plate are gradually increasing. Since the design constraint is also treated as an additional design objective, the problem considered here, in fact attempts to optimize three objectives simultaneously. The buckling load factors corresponding to the stacking sequences given in Table 2 are very close to the specified design constraint value. This study clearly indicates that the proposed constraint handling technique appears to be promising as the difference between constraint value and the buckling load factor is smaller. Fig. 7 shows the Pareto optimal curve of the hybrid laminate composite plate obtained using the proposed multi-objective hybrid SFL algorithm for simultaneous optimisation of cost as well as weight with constraint on buckling load factor as 50. Fig. 7 also shows the comparative performance of traditional SFL algorithm with the same parameter settings. It is clearly evident from Fig. 7 that a Pareto front with well spread Pareto solutions can be obtained using the proposed multi-objective hybrid SFL algorithm and its performance is much superior to traditional SFL algorithm in terms of obtaining Pareto solutions. Further the total number of function evaluations for convergence of the proposed HSFLA is found to ⁄be 21622, while the SFLA took 31036 function evaluations to converge. It is evident from this study that the proposed hybrid SFL

algorithm performs better by providing well spread Pareto solutions when compared to SFLA at a much lower computational cost. 5.2. Laminate composite plate subjected to biaxial moments The second numerical study is concerned with stacking sequence optimisation of a laminate composite plate subjected to biaxial moments. The laminate is subjected to the following biaxial moment resultants:

Mx ¼ M;

My ¼

1 M 2

ð34Þ

The force resultants Nx, Ny and Nxy and twisting moment resultant Mxy are zero. The stresses in the principal material coordinate system r1(z), r2(z), and s12(z) for each lamina are obtained as linear functions of M. The stresses are substituted into the Tsai-Wu failure criterion and the resulting quadratic equation is solved to obtain the positive and negative failure values M+ and M for a specified location. The first-ply failure moment Mf of the laminate is defined as the smallest magnitude of the failure value over the entire laminate thickness. That is,

Mf ¼

min ½minðjM  j; jM þ jÞ z½H=2; H=2

ð35Þ

The three objectives considered here is minimization of cost, weight and maximization of the first ply failure moment. The design constraint considered is fundamental frequency as 20 Hz. Apart from the design constraint, the combinatorial constraints related to ply balancing and ply contiguity are also considered. Table 3 shows the non dominated solutions obtained using the proposed multiobjective SFL algorithm. It can be clearly observed from results furnished in Table 3, that the proposed constraint handling technique appears to be promising as the difference between constraint value and the natural frequency is smaller. 5.3. Laminate composite isogrid stiffened cylinder

Fig. 7. Pareto optimal trade-off curve obtained using the proposed multi-objective HSFLA for simply supported laminate composite plate.

The third numerical example considered is a cylinder stiffened with isogrid form of stiffeners. The buckling failure modes depends on the cylindrical shell geometry, skin thickness, laminate stacking sequence, material properties, stiffener configuration, stiffener angle, stiffener cross sectional properties etc. In the present work, it is proposed to optimize stacking sequence and also the stiffener configuration for simultaneously minimizing cost as well as weight for a specified axial buckling load. Accordingly, the shell geometry, the stiffener cross section properties, bending stiffness of stiffeners are kept constant.

Table 3 Multi-objective optimisation of hybrid composite plate with maximization of biaxial moment, minimization of cost and weight with constraint on frequency as 20 Hz using hybrid SFLA. S.No

Moment (Nm)

Cost

Weight (N)

Number of layers

Stacking sequence

1

1999.45

15.60

3.78

24

½902 ; 45ðgÞ ; 02 ; 45ðglÞ ; 45ðglÞ ; 902 s

2

2463.21

16.29

4.48

28

½02 ; 02 ; 45ðgÞ ; 902 ; 902 ; 45ðglÞ ; 45ðglÞ s

20.8655

3

2496.22

19.40

3.65

24

20.1285

4

2852.72

20.10

4.35

28

5

3981.83

20.80

5.05

32

6

2823.2

23.20

3.51

24

7

4650.63

25.30

5.60

36

8

1963.31

27.01

3.37

24

9

2758.64

27.01

3.37

24

10

3043.63

27.70

4.07

28

ðglÞ ðglÞ ðglÞ ðglÞ ½902 ; 45ðgÞ ; 02 ; 02 ; 902 ; 45ðglÞ s ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ½02 ; 02 ; 902 ; 02 ; 45 ; 902 ; 902 s ðgÞ ðgÞ ðgÞ ðglÞ ðglÞ ðglÞ ½02 ; 02 ; 02 ; 45ðglÞ ; 902 ; 902 ; 02 ; 45ðglÞ s ðgÞ ðgÞ ðglÞ ðglÞ ðglÞ ½902 ; 02 ; 45ðgÞ ; 02 ; 02 ; 902 s ðgÞ ðgÞ ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ½902 ; 02 ; 02 ; 45 ; 45 ; 02 ; 45ðglÞ ; 02 ; 45ðglÞ s ðgÞ ðgÞ ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ðglÞ ½902 ; 02 ; 02 ; 45 ; 45 ; 02 ; 45 ; 02 ; 45ðglÞ s ðgÞ ðgÞ ðgÞ ðglÞ ðglÞ ðgÞ ½902 ; 02 ; 02 ; 45 ; 02 ; 02 s ðgÞ ðglÞ ðglÞ ðglÞ ðglÞ ½02 ; 45ðgÞ ; 45ðgÞ ; 02 ; 02 ; 902 ; 02 s

ðgÞ

ðglÞ

Frequency (Hz)

ðglÞ

ðglÞ

ðglÞ

ðglÞ

ðglÞ

20.3280

20.0420 24.2826 20.1194 25.5272 20.0706 20.0734 23.2133

1402.90 1430.98 1405.33 1499.49 1588.71 1697.32 1455.79 1482.19 1595.48 1495.51 NHS: number of horizontal stiffeners; NCS: number of cross stiffeners; Angle: stiffener orientation angle.

Weight(N)

619.08 616.37 611.96 609.26 604.52 603.71 556.27 553.31 530.38 471.28 619.08 693.75 1091.84 1166.50 1297.09 1386.96 1452.09 1662.02 1858.32 2709.80

Cost Material

Glass–epoxy Graphite–epoxy Glass–epoxy Graphite–epoxy Glass–epoxy Graphite–epoxy Glass–epoxy Glass–epoxy Graphite–epoxy Graphite–epoxy 9 9 9 9 9 9 3 6 12 3

NCS NHS

3 3 3 3 3 5 3 5 5 3 30 30 30 30 30 30 60 60 45 30

Angle Stacking sequence

32 32 32 32 32 32 26 26 28 24

[0(gl), 0(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl)]s [45(gl), 0(gl), 0(gl), 45(gl), 0(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 45(gl), 90(gl), 90(gl), 45(gl)]s [90(g), 90(g), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 45(gl), 0(gl), 45(gl), 90(gl)]s [90(g), 90(g), 45(gl), 45(gl), 0(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 45(gl), 0(gl), 45(gl), 90(gl)]s [90(g), 45(g), 90(g), 45(gl), 90(gl), 90(gl), 45(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 45(gl), 0(gl), 45(gl), 90(gl)]s [90(g), 45(g), 45(g), 90(gl), 45(gl), 45(gl), 90(gl), 45(gl), 90(gl), 45(gl), 90(gl), 90(gl), 45(gl), 0(gl), 45(gl), 90(gl)]s [45(g), 90(g), 45(g), 0(g), 45(gl), 90(gl), 90(gl), 45(gl), 0(gl), 0(gl), 45(gl), 0(gl), 45(gl)]s [45(g), 45(g), 90(g), 45(g), 45(g), 0(gl), 45(gl), 45(gl), 0(gl), 45(gl), 90(gl), 45(gl), 0(gl)]s [45(g), 90(g), 45(g), 0(g), 45(g), 0(gl), 45(gl), 0(gl), 45(gl), 45(gl), 90(gl), 45(gl), 45(gl), 0(gl)]s [45(g), 0(g), 45(g), 90(g), 45(g), 0(g), 45(g), 90(g), 45(gl), 0(gl), 45(gl), 90(gl)]s

No. of variables

In the fourth optimisation problem, we increase the dimension of the performance space by increasing the number of objectives to

1 2 3 4 5 6 7 8 9 10

5.4. Multi-objective optimal design of laminate composite pressure vessel

S.no

The fiber-reinforced isogrid stiffened composite cylinder considered for evaluation has a outer radius R = 0.45 m and a length L = 2.75 m. The cross sectional areas of both the stiffeners i.e., stringers and rings are taken as same, with width, bs as 0.00525 m. and depth, ds as 0.0041 m. The external stiffener configuration is used for this study. We use hybrid composite laminates for shell skin and also stiffeners. However, the entire stiffener configuration is assumed to be of same material to meet the practical requirements. The plies and the stiffeners are made of graphite-epoxy of grade IM7/977-2 and glass–epoxy. The laminates are considered to be symmetric and balanced with ply orientations as 0°, ±45°, 90°. Only half of the plies of a laminate are considered as design variables because of symmetry. All the three combinatorial constraints i.e., ply balancing, ply contiguity and ply angle difference are considered. In order to handle the combinatorial constraint related to ply angle difference, each ply need to be represented with an integer. Each element in the string is an integer between 1 and 4. The integers 1, 2, 3 and 4 represent 0°, +45°, 90° and 45° plies of graphite–epoxy. The optimisation problem considered here is to optimize the stacking sequence of the isogrid stiffened cylindrical shell and stiffener configuration for simultaneous optimisation of cost and weight with constraint on axial buckling load. Here the design variables are ply orientations, orientation of cross stiffeners, number of vertical stiffeners, number of cross stiffeners and the material of the stiffener. The combinatorial constraints are handled using the correction operator [29,22]. The maximum archive size is taken as 10 for this numerical example. The optimal stacking sequences and the stiffener configuration are presented in Table 4. The details furnished in Table 4 include the stacking sequences of stiffened laminate composite cylindrical shell, number of horizontal stiffeners (NHS), cross stiffeners (NCS), stiffener orientation and material of stiffeners. Following are the observations based on the results presented in Table 4. The axial buckling load is considered as design constraint and the value is set as 1400 kN. It can be observed from the Pareto optimal stacking sequences given in Table 4 that the axial buckling load value varies from 1402 kN to 1697 kN. It may be recalled that we treat the design constraint as an additional objective and simultaneously optimize with other prescribed objectives so that the optimal configuration closely satisfies the design constrain value. It has been demonstrated in the earlier two numerical examples given Tables 2 and 3 that the design constraint is satisfied very closely, unlike in the current example, where the computed axial buckling load values of the Pareto optimal solutions vary rather quite substantially. This can be explained as follows: The axial buckling load of the isogrid stiffened composite cylinder depends not only on the stacking sequences, but also on the grid configuration. The choices of the number of horizontal stiffeners (NHS), cross stiffeners (NCS), stiffener orientation are discrete values and the choices are very limited. The number of choices for angles is three (30°, 45°, 60°), cross stiffeners is four (3, 6, 9, and 12) and horizontal stiffeners is three (3, 4, and 5). In view of this controlled selection of grid configuration, the stacking sequence together with grid configurations cannot match with the axial buckling constraint specified. It can be observed from Table 4 that plies of all orientations are present in the optimal stacking sequences. Majority of plies in the stacking sequences are either with 0° or 90° ply angles, followed by 45° ply angles. The optimal angle of cross stiffeners is found to be either 30° or 60°.

Buckling load (kN)

K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216

Table 4 Multi-objective optimisation of hybrid isogrid stiffened laminate composite cylinder-minimization of cost and weight with constraint on axial buckling load as 1400 KN using hybrid SFLA.

212

213

K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216 Table 5 Multi-objective optimisation of pressure vessel with the proposed hybrid SFLA with four objectives and constraint on failure pressure of 1.0 MN/mm2. S.No

Stacking sequence

Plies

Axial rigidity (N/m)

Hoop rigidity (N/m)

Cost

Weight (N)

Pressure (MN/mm2)

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

[45(gl), 90(gl), 45(gl), 45(gl), 90(gl), 90(gl), 45(gl), 90(gl)]s [45(gl), 0(gl), 45(gl), 0(gl), 45(gl), 0(gl), 45(gl), 90(gl), 45(gl)]s [45(gl), 90(gl), 45(gl), 0(gl), 45(gl), 45(gl), 90(gl), 90(gl), 45(gl), 90(gl)]s [90(g), 45(gl), 0(gl), 45(gl), 90(gl), 45(gl), 45(gl), 90(gl)]s [90(g), 90(gl), 45(gl), 0(gl), 45(gl), 90(gl), 45(gl), 45(gl), 90(gl)]s [45(g), 90(g), 45(gl), 90(gl), 45(gl), 45(gl), 90(gl)]s [45(g), 90(g), 90(gl), 45(gl), 90(gl), 90(gl), 45(gl), 0(gl)]s [0(g), 0(g), 90(gl), 45(gl), 0(gl), 45(gl), 0(gl), 45(gl), 0(gl)]s [0(g), 0(g), 45(gl), 90(gl), 45(gl), 0(gl), 0(gl), 45(gl), 0(gl), 45(gl)]s [90(g), 90(g), 45(g), 45(gl), 45(gl), 90(gl), 45(gl)]s [90(g), 45(g), 0(g), 45(gl), 90(gl), 90(gl), 45(gl), 0(gl)]s [0(g), 45(g), 90(g), 45(gl), 45(gl), 0(gl), 45(gl), 45(gl), 90(gl)]s [0(g), 45(g), 0(g), 0(gl), 45(gl), 0(gl), 45(gl), 45(gl), 0(gl), 45(gl)]s [90(g), 45(g), 90(g), 90(g), 90(gl), 45(gl)]s [45(g), 0(g), 45(g), 45(g), 0(gl), 45(gl), 90(gl)]s [45(g), 45(g), 0(g), 0(g), 45(gl), 90(gl), 90(gl), 45(gl)]s [0(g), 45(g), 90(g), 45(g), 90(gl), 90(gl), 45(gl), 0(gl), 45(gl)]s [90(g), 45(g), 45(g), 0(g), 0(gl), 0(gl), 45(gl), 90(gl), 90(gl), 45(gl)]s [45(g), 90(g), 45(g), 0(g), 45(g), 45(gl)]s [90(g), 45(g), 0(g), 45(g), 45(g), 90(gl), 45(gl)]s [0(g), 45(g), 0(g), 45(g), 45(g), 45(gl), 45(gl), 90(gl)]s [0(g), 45(g), 90(g), 45(g), 0(g), 45(gl), 90(gl), 90(gl), 45(gl)]s [45(g), 90(g), 45(g), 45(g), 0(g), 90(gl), 45(gl), 90(gl), 45(gl), 90(gl)]s [0(g), 45(g), 90(g), 45(g), 45(g), 90(g)]s [45(g), 90(g), 90(g), 45(g), 0(g), 45(g), 45(gl)]s [90(g), 45(g), 45(g), 0(g), 0(g), 45(g), 45(gl), 0(gl)]s [0(g), 45(g), 0(g), 45(g), 0(g), 45(g), 45(gl), 45(gl), 90(gl)]s [90(g), 90(g), 45(g), 90(g), 90(g), 45(g), 90(gl), 90(gl), 45(gl), 90(gl)]s

16 18 20 16 18 14 16 18 20 14 16 18 20 12 14 16 18 20 12 14 16 18 20 12 14 16 18 20

3.9E+07 6.8E+07 4.8E+07 4.9E+07 4.9E+07 1E+08 7.9E+07 1.3E+08 1.4E+08 4.3E+07 7.5E+07 8.7E+07 1.4E+08 3.7E+07 7.4E+07 1E+08 1.2E+08 9.2E+07 1.1E+08 8.5E+07 1.1E+08 1.5E+08 1.4E+08 6.4E+07 1.1E+08 1.6E+08 1.5E+08 4.5E+07

1E+08 8.2E+07 1.4E+08 8.8E+07 1.3E+08 8.5E+07 9.6E+07 8.2E+07 8.6E+07 1.4E+08 1.2E+08 8.7E+07 8.4E+07 1.7E+08 1.1E+08 1.2E+08 9.7E+07 1.2E+08 8.2E+07 1.2E+08 8.7E+07 1E+08 1E+08 1.3E+08 1.1E+08 8.6E+07 7.2E+07 2.3E+08

665.2 697.2 729.1 838.6 870.5 980.0 1012.0 1043.9 1075.9 1153.4 1185.3 12173 1249.4 1294.7 1326.7 1358.7 1390.8 1422.8 1468.1 1500.1 1532.1 1564.2 1596.2 1641.4 1673.4 1705.5 1737.6 1769.6

306.08 337.96 369.86 299.88 331.77 261.81 293.69 325.58 357.47 255.62 287.50 319.38 351.28 217.56 249.43 281.31 313.19 345.09 211.37 243.24 275.11 307.00 338.89 205.18 237.04 268.92 300.81 332.70

1.165232 1.106343 1.274815 1.023610 1.279180 1.064914 1.075100 1.023634 1.106780 1.463745 1.029835 1.000586 1.130738 1.170862 1.012538 1.362521 1.087969 1.109045 1.058435 1.947995 1.010736 1.278005 1.001402 1.033684 1.113079 1.022951 1.001051 1.001828

four. We have considered the problem of a thin walled pressure vessel. The pressure vessel considered is symmetric and has radius R = 1 m, composed of laminates made of graphite epoxy and glass epoxy. The pressure vessel is subjected to an internal pressure p, as depicted in Fig. 6. The force resultants are calculated using Eqs. (27) and (28). The Tsai–Wu failure criterion yields a quadratic equation in p, which is used to determine the positive and negative  pressures pþ n and pn respectively, that would cause the nth lamina to fail. We ignore p n ;since the pressure vessel is subjected to only positive internal pressures. The first-ply failure pressure, pf, of the laminated pressure vessel is determined by the smallest positive value of pþ n. A multi-objective optimisation of the pressure vessel is performed for four objective functions with the goal of maximizing the hoop rigidity Ey h, axial rigidity Ex h and minimizing the cost

and weight with a constraint on failure pressure pf. This problem is relevant to the design of stiff, light weight fuel tanks containing compressed gas. The combinatorial constraints considered are ply balancing, ply contiguity and ply angle difference not more than 45°. The maximum archive size is taken as 30 for this numerical example. Table 5 shows the non-dominated solutions obtained using the proposed multi-objective SFL algorithm. It is evident from this study that the proposed algorithm is scalable and larger number of objectives can be considered. Fig. 8 shows the Pareto optimal curve of the hybrid laminate composite pressure vessel obtained using the proposed multi-objective meta-heuristic algorithm for simultaneous optimisation of cost as well as weight with constraint on ply failure (n). Comparisons have also been made with the Pareto solutions obtained using traditional SFLA and the results are shown in Fig. 8. It is clearly evident from Fig. 8 that a Pareto front with well spread Pareto solutions can be obtained using the proposed Hybrid SFL algorithm and the solutions obtained are much superior to traditional SFLA. Further the number of function evaluations for convergence to the Pareto optimal solution is found to be 32,639 while SFLA takes about 44,232. It can be concluded form Figs. 7 and 8 that the proposed hybrid SFLA performance is much superior to traditional SFLA both interms of computational performance and also in generating superior Pareto optimal solutions. 5.5. Performance metrics

Fig. 8. Pareto optimal trade-off curve obtained using the proposed multi-objective HSFLA for pressure vessel.

In order to evaluate the performance of multi-objective optimisation algorithms, two different issues are normally taken into account: minimize the distance of the Pareto-front generated by the proposed algorithm to the exact Pareto front, and to maximize the spread of solutions found, so that we can have a distribution of vectors as smooth and uniform as possible. To determine the first issue it is usually necessary to know the exact location of the true Pareto front. Since we do not have an idea of an exact Pareto front, in the present work, we combine the Pareto optimal solutions obtained for the laminate composite problems, using the five multi-

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objective algorithms i.e. the proposed multi-objective SFL algorithm, Multi-objective PSO (MPSO), Non-dominated sorting GA (NAGA-II), Pareto Archived Evolutionary Strategy (PAES) and micro GA by employing same convergence rule for all the algorithms. The non-dominated solutions obtained from these combined Pareto front solutions are taken as true Pareto front solutions for evaluating the performance of the proposed hybrid SFL algorithm. The performance metrics can be classified into three categories depending on whether to evaluate the closeness to the Pareto front, the diversity in the obtained solutions, or both. The metrics used to evaluate the performance of the proposed hybrid SFL algorithm are as follows: Error Ratio, set convergence metric (SCM) and generational distance (GD) are the metrics used to evaluate the closeness to the Pareto-optimal front. Similarly, spacing, spread, maximum spread and chi-square deviation measure are used to evaluate the diversity among Non-dominated solutions. Finally, the parameter called hypervolume is used for evaluating both closeness and diversity to the true Pareto-front. The details of these metrics are very well documented in the literature [5]. The Pareto optimal solutions obtained using the proposed hybrid SFL algorithm are compared with popular multi objective algorithms like

MPSO, NSGA-II, PAES and micro GA and the results are furnished in Tables 6–8. It can be observed that the proposed hybrid SFL algorithm exhibits superior performance when compared to all the other state-of-the-art multi-objective algorithms except in spacing. This clearly demonstrates the overall superior performance of the proposed algorithm. Since all the algorithms considered for evaluation in this paper are stochastic algorithms, there is no assurance that we get the same results in each run. In view of this, we have executed each of these algorithms 100 times (100 independent runs) with the same input data and control parameters. The best Pareto set obtained in majority of executions (say at least 80 times of 100 executions) is considered for evaluation. We have also recorded the practical reliability [28] of the Pareto solutions obtained for each algorithm in order to evaluate the consistency of these algorithms. The practical reliability of an algorithm is given by the percentage of best Pareto set of solutions obtained, after several (100 in this paper) independent runs of the same algorithm. The practical reliability of the algorithm is recorded for each of the algorithm in Tables 6–8 for the three numerical examples considered. It can be observed that the practical reliability of the proposed HSFL algo-

Table 6 Performance metrics of Hybrid SFL algorithm with other multi-objective algorithms for laminate composite plate. Parameter

Hybrid SFLA

MPSO

MGA

NSGA

PAES

Preferred value

Error ratio Gen dist Spacing Spread Max spread Chi sqr Hyper vol Set coverage Practical reliability Number of dominated solutions Number of function evaluations

0.000000 0.000000 4.390513 0.464370 46.21817 0.000000 1812.236 0.000000 0.95 0 21622

1.000000 0.052203 4.524878 0.592937 35.64376 2.857143 1261.719 1.000000 0.93 2 25634

0.833333 0.058834 2.927007 0.573226 32.36588 1.799471 924.273 0.666667 0.92 3 24148

0.714286 0.023399 2.172740 0.508977 30.77659 1.937809 924.438 0.285714 0.93 3 24264

0.166667 0.031393 6.220860 0.474508 44.93018 1.511858 1774.666 0.166667 0.89 1 22268

SMALLER SMALLER SMALLER SMALLER GREATER SMALLER GREATER SMALLER GREATER SMALLER SMALLER

Note: Bold entries indicate the best values of the respective parameters.

Table 7 Performance metrics of Hybrid SFL algorithm with other multi-objective algorithms for laminate composite isogrid stiffened cylindrical shell. Parameter

Hybrid SFLA

MPSO

MGA

NSGA

PAES

Preferred value

Error Ratio Gen Dist Spacing Spread Max Spread Chi Sqr Hyper Vol Set Coverage Practical Reliability Number of dominated solutions Number of function evaluations

0.090909 0.000734 90.77507 0.369273 4569.65 1.7974 9216.57 0.090909 0.93 0 43897

0.5742 0.02203 64.53 0.4993 3864.37 2.4542 6261.72 0.54000 0.90 1 46336

0.625 0.02965 145.065 0.686052 3623.27 2.7456 5600.75 0.625 0.86 2 44176

0.857143 0.014301 27.19436 0.843593 922.24 3.6374 2462.92 0.857143 0.90 3 47168

0.666667 0.022288 20.11933 0.859521 755.54 3.6689 3625.42 0.666667 0.87 3 44884

SMALLER SMALLER SMALLER SMALLER GREATER SMALLER GREATER SMALLER GREATER SMALLER SMALLER

Note: Bold entries indicate the best values of the respective parameters. Table 8 Performance metrics of Hybrid SFL algorithm with other multi-objective algorithms for laminate composite pressure vessel.

Error ratio Gen dist Spacing Spread Max spread Chi sqr Hyper vol Set coverage Practical reliability Number of dominated solutions Number of function evaluations

Hybrid SFLA

MPSO

MGA

NSGA

PAES

Preferred value

0.000000 0.000000 1.327041 0.181731 22.010796 0.000000 1793.635 0.000000 0.95 0 32639

0.500000 0.078460 2.698080 0.679323 13.200709 2.291288 939.155 0.500000 0.89 2 36366

1.000000 0.058139 2.077170 0.709968 11.006460 1.870829 851.889 1.000000 0.88 2 33176

0.142857 0.016607 1.570138 0.277610 22.010796 0.000000 1759.556 0.142857 0.92 1 36462

0.600000 0.040276 0.000643 0.333374 16.131144 1.527525 1033.209 0.600000 0.88 2 34926

SMALLER SMALLER SMALLER SMALLER GREATER SMALLER GREATER SMALLER GREATER SMALLER SMALLER

Note: Bold entries indicate the best values of the respective parameters.

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K. Lakshmi, A. Rama Mohan Rao / Computers and Structures 125 (2013) 200–216 Table 9 Performance metrics of Hybrid SFL algorithm with SFL algorithm. Parameter

Error ratio Gen dist Spacing Spread Max spread Chi sqr Hyper vol Set coverage Number of function evaluations

Plate

Pressure vessel

Isogrid

Preferred value

SFLA

Hybrid SFLA

SFLA

Hybrid SFLA

SFLA

Hybrid SFLA

0.0 0.0 0.0 0.333 43.85 1.52 971.95 0.0 31036

0.0 0.0 0.0 0.0 46.76 0.0 1893.73 0.0 21622

0.4 0.06 0.78 0.25 16.73 1.54 753.61 0.4 44232

0.0 0.0 0.0 0.08 29.19 0.0 1795.24 0.0 32639

0.33 0.06 0.0 0.43 3829.82 1.63 6962.79 0.33 53232

0.0 0.0 0.0 0.24 4569.71 0.0 9232.17 0.0 43897

Smaller Smaller Smaller Smaller Greater Smaller Greater Smaller Smaller

Note: Bold entries indicate the best values of the respective parameters.

rithm is either superior or comparable to other popular algorithms considered in this paper for evaluation. Apart from the above metrics, we have also included a metric called ‘dominated solutions’, which indicates the number of Pareto solutions of a particular algorithm dominated by the Pareto solutions generated by rest of the algorithms. Hence the performance of an algorithm is considered superior, if the number of dominated solutions is small. The details are recorded in Tables 6–8 for all the three numerical examples considered. It can be clearly concluded that the performance of the proposed algorithm is clearly superior to other multi-objective algorithms considered in this paper for evaluation. Finally, the computational performance of various algorithms is also evaluated for each of the algorithm considered for evaluation. The computational performance is reflected by the number of function evaluations (i.e. evaluation of objective function). From the results recorded in Tables 6–8, it can be concluded that the computational performance of the proposed hybrid SFL algorithm is superior or comparable to rest of the multi-objective algorithms considered in this paper. In order to investigate the role of neighborhood search algorithm in the proposed hybrid SFL algorithm, all the four numerical examples considered in this paper are solved using both SFLA i.e., without local search algorithm and proposed hybrid SFLA i.e. SFLA with variable depth local search algorithm. The details of the performance metrics are shown in Table 9. It is clearly evident from the performance metrics comparing both the algorithms furnished in Table 9, that hybrid SFLA i.e., SFLA with neighborhood search algorithm is far superior to SFLA without local search algorithm. The number of function evaluations for hybrid SFLA is also found to be lower than the SFLA. The Pareto optimal plots given in Figs. 7 and 8 also clearly shows the superiority of the Pareto optimal curves obtained using HSFLA have good spread and also dominates some of the Pareto solutions obtained using SFLA.

6. Conclusions This paper proposes a multi-objective discrete hybrid SFL algorithm for combinatorial optimisation and has been applied to optimal stacking sequence design of laminate composite structures. The SFL algorithm proposed has been built with reconfigurable features, which adaptively selects the scheme based on the feedback on convergence of solutions. An external archive has been maintained for archiving the non-dominated solutions, which will be continuously updated during the evolutionary process. A crowding distance method is employed for resizing the archive. Numerical experiments have been conducted by solving a hybrid laminated composite plate, pressure vessel and stiffened composite cylindrical shells. Studies have been carried out by considering varying number of design objectives and design constraints. The following are some of the observations.

The performance of the meta-heuristic algorithms depends heavily on the control parameters, effectiveness of the neighborhood parameters and also adaptivity built into the algorithm. The success of these algorithms depends on the right balance of diversification and intensification mechanisms provided in the algorithm. While single objective algorithm demands high level of diversification mechanism in search during the initial phase, in the final phase (nearing convergence) high levels of intensification mechanism is needed. In contrast, the multi-objective optimisation algorithms requires right balance of both intensification and diversification mechanisms throughout the search phase so that maximum number of well spread Pareto optimal solutions can be obtained. From the detailed studies presented in this paper, it can easily be verified that hybrid SFLA is built with right balance of intensification mechanism strengthened by a customized neighborhood search and diversification mechanism strengthened by search acceleration factor and the reshuffling mechanism among the memeplexes. Further, the adaptive features built into the algorithm helps in fine tuning according to the state of execution. Numerical studies presented in this paper and performance metrics clearly demonstrate the superiority of the proposed algorithm. The studies carried out in this paper clearly indicate that using plies made of different materials and combining them is in fact an effective way to tailor the structural properties according to the design requirements and thereby offer better designs. Eventhough, population based meta-heuristic algorithms like evolutionary algorithms are being popularly employed for multiobjective optimisation, the studies carried out in this paper clearly indicate that the algorithms like SFL algorithm can be effectively tailored to devise efficient multi-objective optimisation algorithms. The crowding distance algorithm employed for resizing the archive appears to be effective for the problems solved in this paper. This is clearly evident from Pareto optimal curves given in Figs. 7 and 8 for plate and pressure vessel problems respectively. The Pareto curve is represented by well spread non-dominated solutions. In the present work, the design constraints are handled by considering the constraint as a design objective. From the numerical experiments carried out in this paper, it is clear that the present constraint handling technique is very appealing for design optimisation of laminate composite structures. The performance studies carried out using the multi-objective performance metrics clearly show that the proposed algorithm outperforms the state-of-the-art evolutionary algorithms like NSGA-II, PAES, microGA and MPSO.

Acknowledgment This paper is being published with the permission of director, CSIR-Structural Engineering Research Centre, CSIR, Taramani, Chennai. The authors gratefully acknowledge the partial financial

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