Quantum behaved Particle Swarm Optimization (QPSO) for multi-objective design optimization of composite structures

Abstract

We present a new, generic method/model for multi-objective design optimization of laminated composite components using a novel multi-objective optimization algorithm developed on the basis of the Quantum behaved Particle Swarm Optimization (QPSO) paradigm. QPSO is a co-variant of the popular Particle Swarm Optimization (PSO) and has been developed and implemented successfully for the multi-objective design optimization of composites. The problem is formulated with multiple objectives of minimizing weight and the total cost of the composite component to achieve a specified strength. The primary optimization variables are – the number of layers, its stacking sequence (the orientation of the layers) and thickness of each layer. The classical lamination theory is utilized to determine the stresses in the component and the design is evaluated based on three failure criteria; Failure Mechanism based Failure criteria, Maximum stress failure criteria and the Tsai–Wu Failure criteria. The optimization method is validated for a number of different loading configurations – uniaxial, biaxial and bending loads. The design optimization has been carried for both variable stacking sequences as well as fixed standard stacking schemes and a comparative study of the different design configurations evolved has been presented. Also, the performance of QPSO is compared with the conventional PSO.

Introduction

Now-a-days composites are becoming increasingly popular, due to their superior mechanical characteristics, like very high stiffness to weight ratios and amenability to tailoring of these properties. Remarkable variations in the characteristics of composite materials can be achieved by slightly altering their properties. Thus, composite materials offer the possibility to create an unlimited set of different material behaviors that can be tailored to specific structural needs. The use of laminates increases the freedom in design and gives more control to fine-tune the material to meet local design requirements. However, the analysis and design of composite materials is relatively more complex. Composite design optimization typically consists of identifying the optimal configuration that would achieve the required strength with minimum overheads. The possibility to achieve an efficient design that fulfills the global criteria and the difficulty to select the values out of a large set of constrained design variables makes mathematical optimization a natural tool for the design of laminated composite structures (Gürdal, Haftka, & Hajela, 1999). Depending on the nature of application for which the component is being designed, there would be a number of different overheads like weight, cost, etc. which have to be taken into consideration for effective design optimization of composites, thus making this problem multi-objective in nature. There has been considerable amount of work carried out on composites’ design optimization (Adali et al., 1996, Baykasoğlu et al., 2008, Bruyneel, 2006, Griffin et al., 1991, Gürdal and Haftka, 1991, Gürdal et al., 1999, Kumar and Tauchert, 1992, Pelletier and Senthil, 2006). Laminate stacking sequence design optimization has been formulated as a continuous optimization problem and solved using various gradient based methods by Gürdal and Haftka (1991). Bruyneel (2006) has presented a general and effective procedure based on a mathematical programming approach for the optimal design of composite structures subjected to weight, stiffness and strength criteria. Shin, Gürdal, and Griffin (1991) have investigated the minimum-weight design of simply-supported, symmetrically laminated, thin, rectangular, especially orthotropic laminated plates for buckling and post-buckling strengths. Kumar and Tauchert, 1992, Adali et al., 1996 have discussed the multi-objective design of symmetrically laminated plates for different criteria like strength, stiffness and minimal mass. Venkataraman and Haftka (1999) have presented a review of various approaches to the optimization of composite panels.

Composite laminate design problems typically involve multimodal search spaces (Venkataraman & Haftka, 1999) with the design variables capable of taking a wide range of values, making this a combinatorial explosive problem. For such problems, traditional gradient based algorithms are plagued with problem of converging to locally optimal regions of the design space. Multi-objective design of composites warrants the use of modern nonparametric/blind optimization methods.

In pursuit of finding solution to these problems many researchers have been drawing ideas from the field of biology. A host of such biologically inspired evolutionary techniques have been developed namely Genetic Algorithm (GA) (Baykasoğlu et al., 2008, Costa et al., 2004, Grosset et al., 2001, Gürdal Soremekun et al., 2001, Le Riche and Haftka, 1993, Park et al., 2001, Rajendran and Vijayarangan, 2001, Walker and Smith, 2003), Artificial Neural Networks (ANN) (Garg, Roy Mahapatra, Suresh, Gopalakrishna, & Omkar, 2007), Artificial Immune System (AIS) (Omkar, Khandelwal, Santhosh Yathindra, Narayana Naik, & Gopalakrishna, 2008) and Particle Swarm Optimization (PSO) (Omkar, Mudigere, Narayana Naik, & Gopalakrishna, 2008; Parsopoulos, Tasoulis, & Vrahatis, 2004) which are widely used for solving such optimization problems. All of these algorithms with their stochastic means are well equipped to handle such problems.

PSO was introduced by Eberhart and Kennedy (1995a), inspired by the social behaviour of animals such as bird flocking, fish schooling, and the swarm theory. Compared with GA and other similar evolutionary techniques, PSO has some attractive characteristics and in many a cases proved to be more effective (Hassan, Cohanim, Weck, & Venter, 2005). Both GA and PSO have been used extensively for a variety of optimization problems and in most of these cases PSO has proven to have superior computational efficiency (Hassan et al., 2005, Sun, 2008; Zhang et al., 2003). Since 1995, many attempts have been made to improve the performance of the PSO (Clerc, 2004; Zheng, Ma, & Zhang, 2003). Sun et al., 2004, Sun and Xu et al., 2004 introduce quantum theory into PSO and propose a Quantum-behaved PSO (QPSO) algorithm, which is guaranteed theoretically to find good optimal solutions in search space. The experiment results on some widely used benchmark functions show that the QPSO works better than standard PSO (Sun et al., 2004, Sun and Xu et al., 2004) and is a promising algorithm. Hence in the current work we propose to employ a multi objective optimization method based on QPSO and compare it to its predecessor, PSO which has been already implemented by Omkar and Mudigere et al. (2008).

The multiple objectives considered here are – minimizing the weight of the composite component and also minimizing the total cost (manufacturing and material costs). The primary design variables are – number of layers, lamina thickness and the stacking sequence. These variables are altered so as to attain an optimum composite design that achieves both the above mentioned objectives while satisfying the specified strength requirements. In the current work, the stacking sequence is not restricted to the popularly used schemes like {0/45/90}, {0} and {0/90}. Instead, the ply orientation angles are also considered as variables of the optimization process, thereby allowing for evolving new non-standard stacking schemes, appropriate for the specified application. This ensures a truly optimal design for the given application as all the possible stacking sequences are explored. The classical lamination theory is utilized to determine the stresses at each layer for thin laminates subjected to force and/or moment resultants and the design is evaluated based the specified failure criteria. The use of appropriate failure criteria is crucial for the optimal design of composite laminates. Since different failure mechanisms are relevant for different loading combinations, in the current work we evaluate the composite design for three different failure criteria; Tsai–Wu (Groenwold & Haftka, 2006), Maximum stress (Gürdal et al., 1999) and the Failure Mechanism based criteria (Narayana Naik, Krishna Murty, & Gopalakrishnan, 2005). This makes the optimization method truly generic and ensures a completely optimum solution/configuration for the given application.

The generic composite design optimization framework being presented in the current work employs vector evaluated technique for multi-objective optimization alike the Vector Evaluated Particle Swarm Optimization (VEPSO) used by Omkar and Mudigere et al. (2008). The vector evaluated QPSO (VEQPSO) allows for separate evaluation of the multiple objectives, which proves to be very appropriate for the current problem. This is a co-evolutionary method which employs separate swarms for each of the objective and information migration between these swarms ensures an optimal solution with respect to all the objectives.

This paper is structured as follows: Basics of multi-objective problems are presented in Sections 2. Section 3 introduces QPSO and VEQPSO. Details of the problem and its formulation are explained in Section 4. The outline of the optimization process employed is given in Section 5. The numerical results and discussions are presented in Section 6 along with comparison with the classical PSO in Section 7. Finally, the conclusions are given in Section 8.