Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 730626, 16 pages

http://dx.doi.org/10.1155/2015/730626

## Morphing Wing Structural Optimization Using Opposite-Based Population-Based Incremental Learning and Multigrid Ground Elements

^{1}Department of Mechanical Engineering, Faculty of Engineering, Chiangrai College, Chiangrai 57000, Thailand^{2}Sustainable and Infrastructure Research and Development Center, Department of Mechanical Engineering, Faculty of Engineering, Khon Kaen University, Khon Kaen 40002, Thailand

Received 19 November 2014; Accepted 18 December 2014

Academic Editor: Jinhu Lü

Copyright © 2015 S. Sleesongsom and S. Bureerat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper has twin aims. Firstly, a multigrid design approach for optimization of an unconventional morphing wing is proposed. The structural design problem is assigned to optimize wing mass, lift effectiveness, and buckling factor subject to structural safety requirements. Design variables consist of partial topology, nodal positions, and component sizes of a wing internal structure. Such a design process can be accomplished by using multiple resolutions of ground elements, which is called a multigrid approach. Secondly, an opposite-based multiobjective population-based incremental learning (OMPBIL) is proposed for comparison with the original multiobjective population-based incremental learning (MPBIL). Multiobjective design problems with single-grid and multigrid design variables are then posed and tackled by OMPBIL and MPBIL. The results show that using OMPBIL in combination with a multigrid design approach is the best design strategy. OMPBIL is superior to MPBIL since the former provides better population diversity. Aeroelastic trim for an elastic morphing wing is also presented.

#### 1. Introduction

Design/optimization of a morphing wing aircraft structure is a research field investigated throughout the world. It has been found that the shape and structural flexibility of a morphing aircraft greatly affect its flight performance [1]. This leads to continuous shape change and is called a morphing concept, which is an attempt to avoid using conventional hinged control surfaces. From the literature, external plane transformation of the wing is a recent popular morphing concept [1]. The idea is to change wing aerodynamic performance by deforming the initial wing plane, which can be categorized as wing camber variation [1–5], lateral wing bending [1], and wing twisting [5–9]. The variable camber of an aircraft wing is the most popular method [1], which can be implemented via a conventional hinged mechanism, a smart material, or a compliant mechanism. The compliant mechanism is the best method for such a morphing aircraft wing concept [4]. In the lateral wing bending concept, the wing can bend up and down in a vertical direction so as to produce a wing span camber. Using this concept, wing aerodynamic performance can be varied and it is carried out by using an internal mechanism. The last concept, the wing twisting, can adjust a wing angle of attack by using structural flexibility and applying external torque. Later, there has been some work attempting to exploit adaptive internal structures [6]. The concept can be thought of as a combined lateral wing bending and variable camber concept. From the literature [1–4], it is revealed that the morphing wing aircraft structure is actuated directly by an external force to carry out such aircraft control. Furthermore, the morphing wing structures are elastic rather than perfectly rigid; thus, consideration of their aeroelastic behavior is essential in a design process. This leads to recent work in which the effects of the external force on aeroelastic characteristic of a morphing aircraft wing are studied [10]. The result shows that the actuating force has a significant impact on the aeroelastic and mechanical characteristics and these effects should be taken into account during the design process of a morphing aircraft structure. Also, the aircraft internal structure including the structural layout and sizes significantly affects the aeroelastic characteristics.

In our recent work, it has been proposed to synthesize new internal structural layouts of an aircraft wing [8] and an aircraft morphing wing [9] using multiobjective optimization. The first study leads to an unconventional wing internal layout, which is aeroelastically superior to a traditional rib-spar layout. Design variables include partial topology and sizes of segments on a structure, while design objectives are wing weight, lift effectiveness, and buckling factor. The second study found the unconventional wing structures subject to external actuating torques which are practicable to apply for the morphing aircraft. Design variables include partial topology and sizes of segments on a structure like the first study, but the second study included nodal displacements. Design objectives are wing weight, percentage of change in lift effectiveness, and buckling factor. In both study, the combination of the two types of design variables is carried out by using a ground element approach. The method used to tackle the design problems is multiobjective population-based incremental learning (MPBIL). The optimizer used herein is one of the most powerful evolutionary methods for solving structural topology optimization [11]. The method is category as nongradient optimization methods, which does not require gradient information to converge to a solution. The design strategy proposed in these works can be used as a numerical tool for synthesizing an internal structural layout of a morphing wing, which usually requires high structural and aeroelastic performance for flight operation [8, 9].

The main motivation of this research work is to initiate an idea to synthesize unconventional aircraft wing structures that are expected to outperform their conventional counterparts. This paper is intended as an extension of the above literature. The work has two aims. Firstly, the use of several resolutions of ground elements at the same time is proposed instead of using one ground element resolution for a design problem and this is termed a multigrid ground element approach. Design variables are simultaneous topology, shape, and sizing optimization. The second part is performance enhancement of a multiobjective evolutionary optimizer MPBIL. The opposite-based evolutionary optimization is used to improve the search performance of the original MPBIL and it is named opposite-based multiobjective PBIL (abbreviated as OMPBIL). MPBIL and OMPBIL are employed to solve a morphing wing synthesizing design problem with the use of single-grid and multigrid ground elements. Optimum results reveal that OMPBIL is superior to MPBIL. The multigrid approach leads to better design results than the single-grid one, while the set of design variables that include structural layout, nodal positions, and sizes gives the best structural layout.

The rest of this paper is organized as follows. Section 2 details single- and multiground element design approaches for synthesizing a wing’s internal structure. The opposite-based multiobjective population-based incremental learning is proposed in Section 3. A design problem and its conditions as well as a numerical experiment are given in Section 4, while the design results are in Section 5. The conclusions and discussion of the study are finally drawn in Section 6.

#### 2. Partial Topology, Shape, and Sizing Optimization

Shape and sizing optimization [12] and simultaneous partial topology integrate three different types of structural design variables in the same design problem. This is somewhat complicated but found to be an efficient design strategy [8, 9, 13]. Such a design strategy can be achieved by using the ground finite element technique. The ground element technique is commonly used in a topology optimization process. The basic idea is to generate ground finite elements throughout a design domain being considered. Design variables determine pseudo-densities (such as element thickness and modulus) of those ground elements. Having obtained an optimum solution (e.g., from compliance minimization or dynamic stiffness maximization), ground elements with low pseudo-densities are assigned as holes on a structure whereas elements with high pseudo-density represent a structure. This concept has been proven effective for many design cases [11, 14, 15].

For simultaneous topology, shape, and sizing optimization of an aircraft wing, ground elements include all possible combinations of wing internal segments and wing skins. Figures 1(a) and 1(b) display the ground elements or ground segments for the wing internal structure used for design demonstration in this work. The details of this simple wing-box structure can be found in our previous work [8, 9]. Note that diagonal segments of a structure are used because it has been found that such ground segments result in aeroelastically superior structures compared to using only chordwise and spanwise segments as with a conventional wing structure [8]. The upper and lower wing skins shown in Figure 2 are also set as design variables. It has been found that the variable-thickness of element panels can enhance its aeroelastic performance [5, 16]. The combination of topology and sizing variables of the wing is carried out by assigning values to the thicknesses of those segments in meters (m). For example, let m be design variables for segment thickness where the lower and upper bounds are based on manufacturing tolerance. The th ground segment will be removed from the main structure if the value of is close to its lower bound at the optimum point. Other thickness values higher than the lower bounds are used for structural sizing. It should be noted that the ideal lower bounds for topological design variables are zero segment thickness. However, in order to prevent singularity in a global stiffness matrix of a wing, we have to use a small positive value as lower bounds of the topological design variables. Figure 1 illustrates how to transform from particular design variables (Figure 1(a)) to be a wing structure (Figure 1(b)). Shape design variables, on the other hand, can be added to the partial topology and sizing optimization by assigning positions of internal nodes of those ground segments as design variables as illustrated in Figure 3. The shape design variables determine the positions of the external nodes in - and -directions. Figure 4 shows the segments on lower and upper wing skins as the internal nodes positions are varied. The skin segments are also set as design variables.