MPI-based parallel synchronous vector evaluated particle swarm optimization for multi-objective design optimization of composite structures

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Abstract

This paper presents a decentralized/peer-to-peer architecture-based parallel version of the vector evaluated particle swarm optimization (VEPSO) algorithm for multi-objective design optimization of laminated composite plates using message passing interface (MPI). The design optimization of laminated composite plates being a combinatorially explosive constrained non-linear optimization problem (CNOP), with many design variables and a vast solution space, warrants the use of non-parametric and heuristic optimization algorithms like PSO. Optimization requires minimizing both the weight and cost of these composite plates, simultaneously, which renders the problem multi-objective. Hence VEPSO, a multi-objective variant of the PSO algorithm, is used. Despite the use of such a heuristic, the application problem, being computationally intensive, suffers from long execution times due to sequential computation. Hence, a parallel version of the PSO algorithm for the problem has been developed to run on several nodes of an IBM P720 cluster. The proposed parallel algorithm, using MPI's collective communication directives, establishes a peer-to-peer relationship between the constituent parallel processes, deviating from the more common master-slave approach, in achieving reduction of computation time by factor of up to 10. Finally we show the effectiveness of the proposed parallel algorithm by comparing it with a serial implementation of VEPSO and a parallel implementation of the vector evaluated genetic algorithm (VEGA) for the same design problem.

Introduction

Material history has traversed through the stone, the bronze and the iron ages and at present we are in the “composite age”. Superior mechanical characteristics of composite materials such as high stiffness to weight ratio compared to conventional metals and their inherent amenability towards the tailoring of their properties have made them indispensable. With their low cost and weight benefits, composite laminated plates, a class of composite materials, are extensively used in mechanical, automobile, aerospace, marine, biomedical, civil and other branches of engineering. Composite laminates are a series of lamina or plies of varying thicknesses and various fiber orientations stacked in a certain order to obtain desired directional stiffness and strength properties as required for an acceptable design. Diverse material behaviors, tailored to specific structural needs, can be obtained by slightly altering the properties of these composites. With such characteristics at dispense the choice or the configuration of such materials is left to the manufacturer and is influenced by the application of the composite.

Composite design optimization typically involves identifying the optimal laminate stacking sequence, ply thickness and the optimal number of plies. Such design variables generally have very vast ranges which contribute to the enormity of the number of solutions offered. Hence manufacturers are often burdened with the laborious process of selecting the right combination of elements from the many choices available to them. Hence, optimization of the available solutions becomes necessary.

Composite design optimization involves tackling combinatorial problems determined by their design variables with very vast solution spaces, rendering the problem under the class of constrained non-linear optimization problems (CNOP). Of these solutions those with the least weight and cost are preferred over the rest. Specifically in aerospace applications, weight minimization and reducing manufacturing cost has always been a priority in the manufacturing industry owing to benefits such as reduced fuel consumption and an increased payload. Therefore, the focus is on combined minimization of manufacturing cost and structural weight. This multi-objective nature of the problem and the difficulty in selecting the right values out of a large range of constrained design variables makes mathematical optimization a natural tool for the design of laminated composite structures (Gurdal et al., 1999).

Optimum fiber orientations of laminated composite plates, for the maximum strength taken as the objective function, using state space based methods under multiple in-plane loading conditions with Tsai-Wu failure criterion for several test problems are carried out by Kim et al. (1997)). Adali et al. (1996) have discussed the weighted average method of multi-objective design of symmetrically laminated plates for different criteria like maximum strength, stiffness and minimum mass. Topal and Uzmana (2010) have developed a modified feasible direction (MFD) method for the multi-objective optimization of symmetrical angle-ply, square laminated plates subjected to biaxial compressive and uniform thermal loads in order to maximize the buckling load.

The emergence of heuristics such as Genetic Algorithm, Ant colony Optimization, Tabu Search, etc in solving functional optimization problems has brought in its wake the possibility of solving many problems like Traveling Salesman Problem, Quadratic Assignment and Graph problems, network routing, clustering, data mining, job scheduling and problems of NP-complete nature (Garey and Johnson, 1979) and proves to be a viable alternative to traditional mathematical tools which suffer from drawbacks such as local minimum trapping and single-path searching among others. The multi-objective design optimization of composite laminated structures, also being an NP-complete natured problem, has been solved using various nature-inspired algorithms (Ghasemi and Ehsani., 2007, Pelletier and Senthil, 2006, Deka, 2005, Boyang et al., 2000, Luersen and Holdorf Lopez, 2009). Many researchers have developed different approaches to minimize the weight and cost of laminated composite structures. Of the many heuristic techniques available PSO has become increasingly popular and has found many applications.

Particle swarm optimization (PSO) developed and introduced by Kennedy and Eberhart (1995), is an effective stochastic, non-gradient based global optimization algorithm derived by simulating social behavior depicted by a flock of birds, where each individual gleans from its own as well as the whole flock's discoveries. It is applicable for a wide range of non-linear function optimization problems. PSO's global search capability and insensitivity to scaling design variables (Schutte et al., 2005) makes it particularly suitable for problems (Kovacs et al., 2004) similar to that considered in this paper. PSO supersedes Genetic algorithm (Hassan et al., 2005) in the context of ease of implementation and proves computationally superior to traditional gradient-based algorithms (Snyman, 2004).

Vector evaluated particle swarm optimization (VEPSO) (Parsopoulos and Vrahatis, 2002), a multi-swarm variant of the PSO, an algorithm that adopts and modifies the main ideas of vector evaluated genetic algorithm (VEGA) (Schaffer, 1985) is particularly suited for multi-objective (MO) optimization problems (Goel et al., 2007, Deb, 2001) where often the objectives are competing, incommensurable and need simultaneous optimization. VEPSO algorithm is based on the principle where each swarm is exclusively evaluated with one of the objective functions, but, information coming from other swarm(s) is used to influence its motion in the search space. The best position attained by each particle separately and the global best positions from a different swarm are the main guidance mechanisms of the swarm. The information exchange among swarms enables optimizing all the objective functions involved.

In the proposed case of the composite design optimization problem, because the minimization of weight does not necessarily ensure the minimization of the corresponding cost, because of the disparate nature of the objective functions, VEPSO has been adopted and fitness evaluations based on the objectives of minimizing the weight and cost are carried out by two separate swarms deployed in a vast solution space where mutual exchange of information between swarms governs them towards a convergence point—indicating movement of the swarms towards a near optimal solution.

Despite its applicability and simplicity, PSO and its variant VEPSO take long durations for the completion of the optimization process due to the application problem’s complexity and the nature of algorithms adopted which look to simulate socio-cognitive behavior by pursuing a sequential approach to phenomena that are innately parallel in nature.

Since the inception of computer science as a discipline, computer scientists and engineers have realized that a possible route to accelerating the execution of a computational task is to exploit the parallelism inherent in its program flow. Recent advancements in scientific computing techniques and the emergence of parallel programming platforms have helped in achieving quicker execution of complex problems through parallelization of serial algorithms. Such related work can be seen in literature (Sung-Kwun et al., 2009, Bin Yua and Chuntian, 2007, Parsopoulos et al., 2004).

Of the many available paradigms, Message-passing on cluster computers is one of the main programming paradigms used for high performance scientific computing these days. Message passing interface (MPI) is a specification standard defined by a broadly based group of parallel computer vendors, library writers, and applications specialists (William Gropp et al., 1996, The MPI Forum, 1995). MPI is a de-facto standard for message-passing used for developing high-performance portable parallel applications (Matsuda et al., 2008, Hempela and Walkerb, 1999). MPI standard defines a library of routines that implement the message-passing model (Gropp et al., 1994). The function of MPI, as the name implies, is to help several concurrently computing processes communicate by passing messages between them (William Gropp et al., 1996). Many researchers have carried out master-slave paradigm based parallel implementations using the message passing model (Coello Coello and Sierra, 2004, Schutte et al., 2004, Dubreuil et al., 2006).

Long execution time owing to the computational complexity of the problem, despite the use of popular heuristics like PSO and its variant VEPSO, and the availability of parallel programming environments, and the need to test the suitability of the relatively unused peer-to-peer paradigm for this line of research, were the driving forces behind the present work leading to parallelization of the VEPSO algorithm for the composite design optimization problem. The novelty of this parallelization lies in the use of MPI’s collective operations, such as bcast and allreduce (Bova and Carey, 2000), which enable the development of a decentralized/peer-to-peer relationship between communicating nodes to solve the composite problem, as opposed to point-to point operations such as send and receive (Houzeaux and Codina, 2003), which support master-slave architecture (Gorlatch, 2002).

To summarize, for this work, a novel parallel VEPSO algorithm is presented, based on decentralized/peer-to-peer paradigm, to optimize the design of composite plates, based on the principles of classical laminate plate theory (CLPT)—to determine the stresses at each layer subjected to Uniformly Distributed Load and Point load. To check for failure two failure criteria are used—Maximum Stress (Narayana Naik et al., 2011) failure criterion and Tsai-Wu (Narayana Naik et al., 2011) failure criterion. This computation problem has been decomposed into parallelizable parts and run concurrently on a Linux-based IBM P720 cluster computer in tandem with MPICH v.1.2.7, a high performance portable implementation of MPI. The decomposition is such that the computation roles of individual particles of the swarms, which solve the optimization problem, are assumed by the different nodes of the cluster computer and to enable communication among these particles/nodes, emulating swarm communication behavior, MPI’s collective communication primitives are made use of. Thus particles configured on different nodes, initialized with stochastically generated design configurations, perform calculations iteratively and in parallel to finally arrive at an optimum or near optimum design configuration. The results show a considerable decrease in the execution time of the parallel algorithm compared to the serial one. To further compare this parallel approach with a more commonly used heuristic for this problem domain, a parallel vector evaluated genetic algorithm has been implemented using MPI and executed on the same platform. Effective execution time comparisons of the algorithms are presented in the results and discussion sections. Execution time of the present parallel approach for VEPSO is found to be comparatively less than its GA counterpart. This approach, at the time of writing the paper, is the first to use MPI’s collective communication to develop a peer-to-peer based parallel VEPSO for composite plate optimization. This work is primarily a follow-up to a previous work (Omkar et al., 2008) in our efforts to find faster ways of solving the composite optimization problem and confirming the efficacy of new methods (such as peer-to-peer parallelization) for the problem domain, starting with a smaller problem used here. The scalability and the speedup that this novel decentralized parallel technique offers shows promise in application to problems of higher complexity in the field and hence our future efforts will consider longer execution problems.

The paper is organized as follows. Section 2 includes the brief explanation of basics of the multi-objective problems. Section 3 discusses the emergence of PSO and VEPSO. Section 4 elucidates the details of the problem and its formulation and outlines the optimization process. Section 6 explains the necessity of VEPSO. Section 7 explains MPI's role in the parallelization used for the algorithms and its implementation to the optimization problem. 8 Results and discussion, 9 Conclusions includes the results, discussion, comparisons, and conclusions, respectively.

Section snippets

Multi-objective optimization (MO)

Let SR be an n-dimensional search space andfi(x):SR,i=1,,kbe objective k functions defined over S. Let,gi(x)0,j=1,,mbe m inequality constraints, then MO problem is nothing but finding the vector xT=(x1T,,xnT)Swhich satisfies the m constraints and optimizes the functionF(x)=[f1(x),,fk(x)]:RnRk

In most cases the objective functions may be in conflict, thus, it is not possible to obtain the global minimum at the same point for all the objectives. The goal of such multi-objective

Particle swarm optimization (PSO)

Particle swarm optimization (PSO) is an evolutionary optimization algorithm proposed by Eberhart and Kennedy while attempting to simulate the motion of bird flocks as part of a socio-cognitive study investigating the phenomenon of ‘collective intelligence’ in biological populations. In Particle swarm optimization (PSO) each swarm unit/particle explores a possible solution depending on the point in the search space where it exists. Its trajectory is influenced by its own as well as the entire

Application of VEPSO for design optimization

In the present work, the design of a composite plate, simply supported on all four edges, subjected to uniformly distributed load and point load is considered. The design is governed by the arrangement of constituent plies that make up the composite plate, the stacking sequence, and the stresses and strains developed on each ply of the composite plate are obtained from classical laminate plate theory (CLPT). Using these stresses and strains, failure of a particular design configuration is

The need for VEPSO

The problem at hand looks to obtain a composite laminate which satisfies certain physical constraints such that it bears an applied load without failure. While doing so we deploy a heuristic that looks to minimize the weight and corresponding cost of laminate obtained. These objective functions can in simple terms be summarized as follows:

  • 1.

    Weight function (W=h⁎×abρ where ρ=density h=∑θi*t and t=thickness of each of the plies)

  • a.

    weight ∝ number of plies.

  • b.

    weight ∝ thickness of plies.

  • 2.

    Cost Function (

The optimization process using VEPSO

Vector evaluated particle swarm optimization in the current work has been modified for constrained non-linear optimization problems with discrete design variables unlike previous works carried out for optimizing systems with continuous variables (Gies and Rahmat-Samii., 2004, Vlachogiannis and Lee, 2005). The key point in the constrained optimization process is dealing with the constraints associated with decision variables. In the current work, the constraints are effectively handled to

Message passing interface (MPI) and parallelization

MPI (message passing interface) is a specification for a standard library which is used for message passing between concurrent processes on distributed systems. The MPI standard defines only one API (or three to be more precise, one each for FORTRAN, C, C++ and C#). Every super-computer manufacturer offers its own implementation, optimized for its own hardware. MPI forms the basis of a standard high level communication environment featuring collective communication, point-to-point communication.

Results and discussion

This work is primarily concerned with design of a parallel VEPSO algorithm for the multi-objective design optimization of laminated composite plates problem and to test its efficacy, in terms of execution time and coherence, it has been compared with sequential VEPSO. To further test how our work fares against other popular parallel heuristics we have compared it with parallel vector evaluated genetic algorithm (PVEGA) designed for the same problem. The structural problem for our work is

Conclusions

In our work we have developed a novel parallel approach to VEPSO algorithm, which captures the essence of the peer-to-peer paradigm model of communication and synchronous evaluations, for the design optimization of composite structures using MPI parallel programming platform. MPI being widely available allows its collective communication protocol to be used for a range of problems where peer-to-peer paradigm is intended to be used. The results show reduction in the running time by a

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