1. INTRODUCTION

The success of composite materials has been well documented in recent years [1, 2]. Despite their success and insertion in high profile aircraft, significant efficiency and functionality gains may be obtained by undertaking stacking sequence (lay-up) optimization. Studies have shown that significant weight savings and performance gains can be achieved via lay-up optimization [3-5].

In lay-up optimization, the design variables are generally ply thickness and ply orientation. In many practical applications, ply thickness is fixed and ply orientations take a range of discrete values. If ply orientations are used as design variables, the resulting response surface is non-convex. As such, gradient-based methods are not entirely reliable optimizers. Motivated by this deficiency, several heuristic and deterministic methods have been used to determine laminate stacking sequences. With respect to heuristic optimizers, the success of a Genetic Algorithm (GA) has been well documented [4, 5]. Bloomfield et al. [6] compared three popular techniques for stacking sequence optimization, GA, Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO). The research indicated that all three methods were suitable optimizers, yet for different problems, different optimizers may be more appropriate. Furthermore, Bloomfield et al. [6] observed that ACO and PSO could offer significant efficiency savings over a GA. With respect to deterministic optimizers, a branch and bound method has been proposed, chiefly by Todoroki et al. [7, 8]. Todoroki et al. [7, 8] successfully used the branch and bound approach (where the laminate thickness is fixed) for a variety of problems using a single level optimization routine. Despite the apparent success of branch and bound method [9], the algorithm is known to be computationally expensive. This becomes apparent when the set of possible ply orientations or number of plies increases.

It is often desirable to consider the effect of each ply in the lay-up independent of the other plies in the stacking sequence. That is, identification of the i^th ply orientation independent of the other i-1 plies. To achieve this, Kim and Hwang [10] introduced the concept of an idealized laminate. An idealized laminate is artificially enhanced, which cannot be realized with a real stacking sequence. It is noted, that by utilizing idealized laminae, the effect of each ply in the stacking sequence on the objective function can be quantified. Building upon the concept of an idealized laminate and the branch and bound approach, a two-level optimization is proposed. At the first level of the optimization lamination parameters and plate thickness are used to minimize the mass of the composite plate subject to a set of design constraints, i.e. strength, buckling or aeroelastic constraints. The first level of the optimization is not the focus of the current work, but is detailed in [3] as part of a two-level approach for lay-up optimisation. However, it is noted that by representing laminate stiffness in terms of lamination parameters, a convex response surface is created which is ably optimised, for lamination parameters and thicknesses, using a gradient-based method. The first level of the optimization established the potential optimal mass (total thickness) and guarantees the existence of a real stacking sequence which satisfies the set of design constraints, and it takes into account conflicting objective or constraint trade-offs. At the second level, a stacking sequence is determined using a discrete optimizer which seeks to maximize (or minimize) a given function or satisfy design constraints depending upon the nature of the problem. Motivated by the efficiency problems of the standard branch and bound algorithm and the work of Kim and Hwang [10], a Modified Branch and Bound (MBB) is now proposed. The MBB approach expands the formulation by Kim and Hwan [10] to include anisotropic laminates. Furthermore, a modified pruning technique is used to select the ply orientation of each position in the lay-up. This approach yields good designs and offers significant efficiency savings over the conventional branch and bound and other methods such as GA, PSO and ACO. It is shown that by using the MBB with a uni-objective function then global optima are obtained. The novelty of the current work lies in expanding the concept of an idealized laminate and in the introduction of a new and simplified branching and pruning technique compared to the original formulation of Kim and Hwan. This new formulation helps reducing the computational time of more than 10 times when compared to the original algorithm. The proposed modified branch and bound algorithm has been applied to a wide variety of stacking sequence dependent problems in this paper. In addition, a comparison of MBB with GA is provided and discussed to demonstrate the efficacy of the proposed approach.

2. MATERIALS AND METHODS

Kim and Hwang [10] defined an ideal lamina to be one with enhanced stiffness, that is practically unachievable, yet relate to the inherent basic ply stiffness. It should be mentioned that the authors restricted their study of ideal laminae to symmetric orthotropic/isotropic laminates. Firstly, the in-plane, coupling and out-of-plane stiffness matrices are defined:

(1)

Where:

n is the number of plies in the lay-up

z is the through thickness co-ordinate, the distance of each layer from the middle plane of the laminate as per classical lamination theory.

Q_ij are the terms of the reduced in-plane stiffness matrix, characterising the planar behaviour of each ply of the laminate. These terms can be expressed in terms of material invariants as follows:

(2)

U_i are the material invariants introduced by Tsai and Pagano [1]. The theoretical maximum values for occur at 0, 45, 90, 45, 60, 30 degrees respectively. Clearly, no single ply can take all these orientations at the same time. However, the maximum or ideal , denoted by, is defined when takes its mathematical maximum values simultaneously. In contrast, Kim and Hwang [10] did not make this definition for . It is observed that for different problems, different idealizations may be required, i.e. certain stiffness terms may be zero or negative. Moreover, since (2θ) , (4θ )are odd functions, and shear being bi-directional, the minimum of may be required for a particular problem. The angles which minimize the corresponding stiffness terms are readily determined. In stacking sequence optimization, the outer plies often drive the design. This is because of the quadratic and cubic volume fraction effect. For a symmetric laminate, i.e. one which does not exhibit any bending-extension coupling, the D matrix is most influenced by the outermost plies. Hence, for flexural, buckling or natural frequency problems, indeed any problem driven by maximising D matrix considerations, it is important to consider the outermost plies first. Alternatively, if a compliant flexural stiffness is sought, then the innermost plies should be considered first. Therefore, the definition of an ideal laminate proposed by Kim and Hwang [10] is extended to include the above formulation and:

1. The fitness (using a defined fitness function) of a potential laminate is maximized for the ideal laminate.
2. When a real layer replaces an ideal layer in the ideal lay-up, a lower value of the fitness function is obtained, simply because a real layer is not as stiff as the ideal one. When each ideal layer is subsequently replaced with a real layer (from the outermost ply inwards), the fitness function decreases monotonically with ply position. It is noted that a monotonically sequence which is bounded below, is necessarily convergent. Therefore, the resulting algorithm converges to stable points.
3. The change in the fitness function should be quantifiable on a ply-by-ply basis. If the ideal stiffness constants are too large. i.e. excessively values are arbitrarily chosen, changing one ply may not significantly alter the fitness of the laminate. Using Eq. (2) prevents this potential ill-conditioning problem.
4. The gradient of the objective function evaluated at the ideal stacking sequence is zero (implying at least local optima).

It is observed that there are potentially different (non-unique) idealizations for different objective functions. This is evident when considering that the optimal angles for a buckling driven problem (typically 45^o) differ from the optimum angles from a strength driven problem (typically 0^o for axial loading). Therefore, an acceptable laminate idealization must satisfy the aforementioned conditions. This implies that the analysis may have to perform different “Trials and Errors” simulations before finding an ideal layer definition for his specific problem. For complex problems, such like multi-direction loading, the ideal layer may be an “Isotropic” layer whose stiffness and strength cannot be achieved by any existing material or orientation of the available material.

Next, the branch and bound algorithm presented by Kim and Hwang [10] is discussed. The first step of their algorithm establishes a random stacking sequence S whose objective function is evaluated. The fitness of S is denoted by F*. Next, the fitness of the ideal stacking sequence, denoted by ID, is evaluated and denoted by F^ID. Following this step, all of the layers in the ideal laminate are sequentially replaced (from the outermost ply inwards) with real layers until the optimal stacking sequence is determined. Starting from the outermost ply, the i^th ideal layer is replaced with the real layer of ply angle θ_i. During this process, an obtained fitness F_i which is worse than F* need not be branched further. In fact, by replacing ideal layers with real layers, the fitness function necessarily decreases. The branch and bound approach adopted by Kim and Hwang [10] is shown in Fig. (1).

Fig. (1). Scheme of Branch and Bound used by Kim and Hwang [10].

It is observed that whilst the full algorithm, proposed by Kim and Hwang [10], is able to determine good designs, the resulting algorithm is extremely inefficient. The algorithm proposed by Kim and Hwang [10] uses a random initial stacking sequence. In contrast, by using an idealized stacking sequence as a starting point, and changing the ply orientation from the outermost ply inwards, this ensures that the fitness of the laminate monotonically decreases with position. Each ideal layer is replaced with the best real layer with respect to position. Effectively, all branches are pruned except the best and the algorithm continues to the next layer. This pruning mechanism, essentially identifies the largest branch, and is demonstrated in Fig. (2).

Fig. (2). Scheme of modified branch and bound with 3 possible ply orientations [90/45/0].

In other words, the MBB here proposed:

• Does not make any use of the initial random stacking sequence but branches the ideal laminate, as the fitness function decreases in value.
• Only the larger branch is identified and computed (each level in which a layer orientation is selected has only 1 solution that will be branched further). If n is the number of layers and m the number of possible orientations, m x n computations are performed.

Using the new branching technique, it is observed that significant efficiency savings can be achieved. Consider, a laminate of n plies, formed from m different ply orientations, there are possible lay-ups. As such, enumeration quickly becomes unviable with a large number of layers, n, due to the combinatorial explosion of possible lay-ups with increasing thickness. If a heuristic search method (such as a GA) is used, the number of evaluations would be r x s, where r is the population size and s is the number of iterations that have taken place until a solution is determined. Using a heuristic approach, r × s < mⁿ. In contrast, the MBB approach evaluates m x n designs. It is observed that this process represents a significant reduction in the number of evaluations.

If there is more than one objective, different ply orientations may compete for the same position. Therefore, additional criteria must be used, in that case, to select the 'Best' ply at each position. Nonetheless, by selecting the 'Best' ply at each position, in multi-objective problems there may be an abundance of possible combinations of stacking sequences which are missed and may yield better results. Despite this potential shortfall, it will be shown, in the next section, that the method is able to determine good designs (near global optima, with a strong reduction of computational time).

3. RESULTS AND DISCUSSION

In this section, several numerical examples are considered. The first set of numerical examples aims to determine a stacking sequence which maximises or minimizes a given objective function for a laminate of fixed thickness and fixed percentages of ply orientations. Take into account that fixing the percentages is equivalent to fixing the in-plane lamination parameters. The following three objectives are considered:

• Maximize the natural frequency (first mode) of a simply supported rectangular plate. It has been shown [11], that such analysis could be important in the design of laminates which limit resonance due to external excitation.
• Maximize the compression buckling load of a simply supported rectangular plate.
• Minimize the vertical displacements of a clamped laminate subjected to a cantilever tip vertical load.

The second set of numerical examples aims to determine stacking sequences of minimum thickness which satisfy strength and buckling constraints. For each numerical example detailed in this section, the MBB algorithm is used to determine laminate stacking sequences and the results are compared with a permutative GA.

CONCLUSION

The MBB has been shown to give good fidelity and significant computational advantages compared with a GA. Despite the simplicity of the structures in the numerical examples, it is anticipated that the MBB can be used to determine lay-ups in multi-part structures.

The method was used to determine stacking sequences for several problems. The modified branch and bound method was shown to determine good laminate designs and offer significant efficiency savings. It was shown that for a single objective without ply competition, global optima were obtained. However, for multi-objective or multi-criteria problems, different ply orientations may compete for the same ply position. This competition is evident when considering the optimal angles for buckling or the optimum angles for strength driven designs. By selecting only the best ply for a given position there may be an abundance of possible combinations of angles missed which may give improved designs.Obtained designs, which do not have reserve factors greater or equal to one or satisfy some given criteria, could form a part of an initial population of a heuristic method. In addition, it is believed that the method can be applied to determine lay-ups in multi-part structures, a topic of future interest.