Global-local Aeroelastic Optimization of Internal ...

AIAA 2017-4321 AIAA AVIATION Forum 5-9 June 2017, Denver, Colorado 18th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference

Global-Local Aeroelastic Optimization of Internal Structure of Transport Aircraft Wing Mohamed Jrad1, Shuvodeep De2, and Rakesh K. Kapania3 Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061-0203

Abstract The SpaRibs (Spars and Ribs) topology has a significant effect on the structural strength and stability as well as the flutter performance of aircraft wings. Although it is desirable to both minimize the structural weight and maximize

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the flutter velocity while satisfying buckling and strength constraints, there is a trade-off between the two objectives. To quantify this tradeoff, a multi-objective multi-disciplinary optimization problem is performed considering both weight and flutter velocity as objectives. The structural optimization of a cantilever transport aircraft wing with stiffeners and curvilinear spars and ribs is conducted. The Global-Local framework EBF3GLWingOpt is employed to optimize the local panels’ thicknesses of each wing design and flutter analysis is performed to compute the flutter velocity. While buckling, von Mises stress, and crippling constraints are satisfied for the optimized panels, the Selective Stiffening algorithm is applied to the optimized structure in order to ensure that the global buckling constraint of the wing is satisfied. A parametric study is first conducted to understand the flutter velocity and wing weight dependency on the different design variables in order to make a proper selection of the most influencing ones for optimization and also to give guidelines for designers on how to place the structure’s components in order to obtain the desired wing weight and flutter velocity. The effect of longitudinal and lateral stiffeners is investigated and a tradeoff is shown for the number of longitudinal stiffeners. The weighted-sum approach is then used to reduce the multiobjective problem into a single-objective problem. The Particle Swarm Optimization (PSO) algorithm approximately determines the Pareto-front of structural weight and flutter velocity while satisfying constraints on buckling and yield strength at flight condition (cruising Mach number of 0.85). Double-level parallel processing is employed in this work in order to reduce the computational cost. The best obtained design has a flutter speed of 556.53 knots and a wing weight of 11152.61 lbs which shows 42% improvement with respect to the weight of the unstiffened baseline model.

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Post-Doctoral Fellow, Kevin T. Crofton Department of Aerospace and Ocean Engineering Graduate Research Assistant, Department of Biomedical Engineering and Mechanics 3 Norris and Wendy Mitchell Professor of Aerospace and Ocean Engineering, Kevin T. Crofton Department of Aerospace and Ocean Engineering, Fellow AIAA 2

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Copyright © 2017 by Mohamed Jrad; Rakesh K. Kapania. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I.

Introduction

Optimizing an aircraft structure is a challenging multi-objective task which involves many disciplines, each characterized with its own constraints and requirements. Flutter is a dynamic instability of elastic structure when put in a fluid in motion. It is characterized by causing oscillations of increasing amplitude which eventually leads to structural failure. The velocity of the structure relative to the fluid at which the onset of flutter occurs is known as flutter velocity and while designing an aircraft it is very important to know the flutter velocity as a function of altitude.


A reduction in weight of the wing means less fuel consumption and the increase in flutter velocity ensures safe cruising at a higher Mach number at a given altitude. The objective of this work is to simultaneously minimize the weight and maximize the flutter velocity of an aircraft wing while preventing it to fail in strength and stability. However, a design with minimum weight does not ensure maximum flutter velocity and vice versa i.e. there is a trade-off between the two [1]. Therefore, both weight and flutter velocity are considered as objectives during the optimization in order to quantify this trade-off. The multi-disciplinary optimization was first studied by Schmit in his seminal paper [2] published in 1960. In his work [2, 3, 4], he integrated numerical optimization techniques with finite element analysis. Multi-disciplinary design optimization is increasingly gaining popularity in the aircraft industry, especially due to the exponential increase in computational power. Fulton [5], Haftka [6, 7, 8], and their collaborators performed optimization of aircraft wing considering multiple constraints, such as strength, buckling, and flutter velocity. In recent years, novel additive manufacturing technology, such as electron-beam freeform fabrication (EBF3) [9], has been developed to produce arbitrary curved metallic structures. The concept of curved stiffening members enlarges the design space and leads to the design of a more efficient aircraft structure. This new technology was leveraged by Kapania and his group at Virginia Tech [10, 11, 12] for developing a family of frameworks to optimize the shape of the internal structure of an aircraft wing with curvilinear stiffening members. M4-Engineering Inc. has also developed a set of tools and techniques for modeling and optimizing the structure of a wing with curvilinear SpaRibs [13, 14]. It has already been proved [15, 16, 17] that curvilinear stiffeners can improve the buckling resistance of arbitrary shaped panels. Locatelli et al. [12] optimized an aircraft wing structure using curvilinear spars and ribs (SpaRibs) and showed their potential to minimize the weight of a supersonic civil transport vehicle. While their studies didn’t take into account the aero-elastic performances, Jutte et al. [1] performed extensive parametric study using the NASA Common Research Model varying the number, location and curvature of the stiffening members, i.e. spars,

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ribs and stiffeners, and demonstrated that configurations characterized by weight lower than the baseline aircraft and higher flutter speed exist. The following study has been performed as part of the development of EBF3GLWingOpt framework to demonstrate the usefulness of multi-objective optimization to find out the best configurations in terms of weight and flutter performance. A parametric study is conducted in order to understand the dependency of the wing weight and flutter speed to the different design variables. In order to reduce the computational cost, the most influencing variables are then selected for the next part which consists of Particle Swarm Optimization. A weighted function evaluation that


gathers both weight and flutter speed is considered to transform the multi-objective problem into a single-objective problem. Parallel computing is employed in the Global/Local process and double-level parallel is used at the PSO level in order to reduce the computational time. II.

Global-Local Framework: EBF3GLWingOpt

The framework EBF3GLWingOpt is an integrated global-local optimization tool for aircraft wings. The wing is parameterized by size and shape design variables to create the internal structure as well as stiffened panels. The parametrization of the curvilinear SpaRibs is carried out using the linked shape method developed by Locatelli et al. [12] in a normalized space (Figure 1) and their geometry is constructed using third-order B-splines and projected into the physical space. The objective of the multidisciplinary optimization is to minimize the structural weight of the wing subjected to multiple constraints (buckling, von Mises stress, crippling…). The mathematical formulation of the optimization problem is written as: min W(x) Subjected to: gi(x) ≤ 0 i = 1, 2, …. , p hj(x) = 0 j = 1, 2, …., q

(1)

where x = [x1, x2, …. , xn]T is the vector comprising of the n design variables with ranges: xil ≤ xi ≤ xiu i = 1,2,….n fk: ℝn → ℝ, k = 1, 2, …. , r are the objective functions, gi, hj : ℝn → ℝ, i = 1, 2, …. , p; j = 1, 2, …. , q are the constraint functions

The commercial software MSC.Patran is used to construct the finite element model of the wing structure. The Global-local optimization of a specific wing design consists of finding the local panel thicknesses that minimize the wing weight while satisfying all the constraints, through an iterative global/local process, as described in details

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in [11]. The wing parameterization is divided into two coupled subsystems: the global wing model and the local panel models. During the iterative process, the information is sent from the global model to the local panel models using displacement control by interpolating the boundary condition. This method is called the global/local approach [18]. The values of the optimized local panel thicknesses as well as stiffeners thickness and height are then updated


in the global model.

Figure 1: Linked Shape Parametrization (Locatelli et al. [12]) 

Local Panels Optimization

Local panel edges are determined from the mesh of the global model by finding the intersections of the spars, ribs, and wing skins. Interior panel nodes are then computed using the edge information. The extracted panel mesh is used then to build the panel’s surface, construct the evenly spaced stiffeners on it, and refine the panel’s mesh, as illustrated in Figure 2. This process of construction of the panel’s surface is performed only in the first iteration of the GlobalLocal process. After that, the panel model is updated based on its design variables.

Figure 2: Cantilever Wing with Stiffened Panels [11].

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The maximum von Mises stress (σvm)max and the first buckling eigenvalue λp are computed in the static and buckling analyses of local panel using MSC-NASTRAN. During the local panel optimization process, if t0 is the initial panel thickness, then the new panel thickness is computed as follows, topt1 = t0 (σvm)max / σy topt2 = t0 (1/ λp)1/2

(2)

topt = max(topt1 , topt2 )


where (σvm)max is the maximum von Mises stress, σy is the yield stress, and λp is the buckling eigenvalue. The optimal panel thickness is topt the maximum value of the optimal thicknesses calculated by strength and buckling constraints. The Global-Local process is illustrated in Figure 3. In each optimization cycle, the global wing model is updated using the optimized thicknesses of local panels. Then the new displacements are computed and applied on the local panels as boundary conditions for the next iteration.

Figure 3: Global-Local Process 

Parallel Computing

A parallel computing framework has been previously developed within EBF3GLWingOpt. The parallel computational capabilities are implemented in the optimization of the local panels. The license cycle-check method and the memory self-adjustment method have been employed to overcome the limitations of the number of MSC licenses and memory saturation, respectively. More details about these two methods can be found in [11]. The main

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idea of the two methods consists of creating a communication language between the running processing and a set of signs they use to share information about the current status of the simulation in order to be able control themselves and use the available resources efficiently, even if these available resources are not sufficient. The challenge in these two methods is that they all control themselves collectively, unlike the traditional way of thinking where the main process controls the rest of the processes. In the old version of the memory self-adjustment method, the capacity of the machine were determined when the memory saturation is reached. In this work, the method has been updated and the capacity of the machine is computed using a self-learning procedure where the running processes collect


information about the amount of memory they have used for each job in the whole simulation. This dynamically growing database gives, therefore, an accurate estimate of the amount of memory that these new jobs require. Therefore, the simulation runs smoothly and MSC.Nastran only allocates the amount of memory that it really needs. Furthermore, the estimated capacity of the machine is more accurate and the memory saturation is avoided. This Global-Local parallel computational framework is run in Virginia Tech’s Advanced Research Computing System “NewRiver” which has 134 nodes. Each node has 24 core @CPU Speed 2.50 GHz and 128 Gb of RAM. While the simulation time depends heavily on the wing design variables and input parameters (parallel processes, mesh size, etc.), an example of a wing with 4 inner ribs, 25 outer ribs, 18 longitudinal stiffeners and a mesh of 160k elements and 100k nodes converges in about 1 hour and 41 minutes using 60 processes.

Figure 5: Simple Task Running 5 Nastran Jobs Using

Figure 4: Simple Task Running 5 Jobs using License Cycle-

Memory Self-adjustment Method

check Method

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

CRM Wing

The solver MSC.Nastran, which incorporates the static aero-elasticity, buckling, and dynamic flutter modules, is used to perform the finite element analysis. In this work, the analyzed wing is NASA Common Research Model (CRM) studied in [11] (Figure 6). The CRM wing is modeled using aluminum alloy 2024-T3, and its material properties are given in Table 1. During the Global-Local optimization, the aerodynamic loads and the structural deformations are calculated using the doublet-lattice method provided in MSC.Nastran solution 144. The two flight conditions that are considered in this research are -2 deg and 6 deg angle of attack (AOA), at a cruising Mach number


of 0.85 and flight altitude of 35,000 ft. The wing weight optimization history of a typical global/local optimization with EBF3GLWingOpt is shown in Figure 7. Table 1: Aluminum alloy 2014-T3 mechanical properties Property Density Modulus of elasticity Poisson’s ratio Yield stress

Value 0.1 lb/in.3 10,600 ksi 0.33 40 ksi Figure 6: CRM Wing

Figure 7: Weight history 

Selective Stiffening

The Global-Local optimization described in this work ensures that all the local panels satisfy the aforementioned buckling, KS, and crippling constraints are satisfied. However, it does not guarantee that the global buckling factor of the final wing is satisfied. In order to ensure this condition, the Selective Stiffening algorithm (SS), developed by De et al. [19], is applied to find the weakest panels, and increase their thickness in an iterative process using a Thickness Increment Factor (TIF) equal to the square root of the inverse of the buckling factor. To investigate the effect of the SS process on the optimized wing, several cases of the same wing configuration, but with different

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number of longitudinal stiffeners, have been optimized with the Global-Local optimization process (Table 2). The global buckling factor of all these resulted structures is close to 0.7. The Selective Stiffening algorithm is, therefore, applied to each one of them (Table 3) and the global buckling factor became 1. They all converged in few iterations (less than 4 iterations) and took about 10 minutes each. The effect of this SS process on the wing weight and flutter speed is shown to be negligible (Table 4). Therefore, the SS does not affect the decision of the Particle Swarm


Optimization (described below) and will only be applied to the best obtained design. Table 2: Global-Local optimization of several wing configurations Nbr of Weight Flutter Speed Stiffeners (lbs) (knots) 13 12033.99 608.71 14 11838.08 595.72 15 11774.46 579.36 16 11692 567.71 17 11700.3 552.02 18 11748.35 546.08 19 11640.07 526.67 20 11656.08 475.44



Table 3: Global-Local + Selective Stiffening Weight Flutter Speed (lbs) (knots) 12088.13 609.14 11888.67 596.09 11833.59 580.3 11746.59 568.34 11732.77 552.09 11782.28 546.33 11692.91 527.16 11688.63 475.94

Table 4: Difference in weight and flutter speed Diff Weight Diff Flutter (%) Speed (%) 0.45 0.43 0.5 0.47 0.28 0.29 0.45 0.28

0.07 0.06 0.16 0.11 0.013 0.046 0.09 0.1

Flutter Analysis

After implementing the Global/Local optimization process and computing the panels’ thicknesses and stiffeners’ height and thicknesses, the flutter analysis is then performed. The PKNL method is selected in MSC.NASTRAN Sol 145 for flutter analysis of the wing. A flight envelope of altitude 0-35000 ft at a Mach number 0.85 is considered for the analysis. The Frequency and damping of each mode for different equivalent air speeds (EAS) are shown in Figure 8. The first twenty flutter modes are considered for the analysis but the first 10 are plotted. The flutter speed is obtained by finding the lowest speed at which one of the modes’ damping crosses the horizontal axis and then becomes and stays positive.

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(b) Damping


(a) Frequency

Figure 8: Aeroelastic analysis results at Mach 0.85: V-f and V-g plots III.

Parametric Study

During this section, a parametric study is conducted to understand the influence of the different design variables on the wing weight and flutter velocity. For each design, the global/local optimization is performed and the corresponding panels’ thickness distribution as well as stiffeners thickness and height are obtained. The studied CRM wing is composed of two wing-boxes. The spars are parametrized in the inner wing-box using DV6 (starting point at the wing root), DV7 (control point position in the control line), and DV8 (end point at the junction) as seen in Figure 9. These points are then used to build the geometry of the spar using a third-order B-spline. The second part of the spar that lies in the outer wing box needs only DV17 (control point position in the control line) and DV18 (end point at the wing tip) since the C0 continuity is enforced at the wing junction. Based on the linked shape method, the chordwise location of each of these points is obtained using the relationship: 𝐶ℎ𝑜𝑟𝑑𝑤𝑖𝑠𝑒 𝐿𝑜𝑐𝑎𝑡𝑖𝑜𝑛 =

𝐷𝑉𝑖 (1+𝐷𝑉𝑖 )

Figure 10: Number of spars, and number of ribs in the inner and outer wing-boxes

Figure 9: Design variables DV6, DV7, DV8, DV17, and DV18 of spars shape and location

For each of the following curves, a reference point is selected such that it has either the highest or the lowest flutter speed. The case treated in Figure 11 and Figure 12 consists of a wing with 3 spars (including the front and rear

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spar) while the tailored design variables are for the middle spar only. The variation of the starting point of the spar at the wing root (DV6) shows that there is a negligible effect on the weight and the flutter speed. However, the variation of the ending point of the spar at the wing tip could make an increase of 17.58% of flutter speed with a weigh compensation of only 1.43%. This could be explained by the fact that changing the spar end-point at the wing tip has a larger effect on the torsional rigidity of the wing and its coupling with the bending stiffness than that of the starting point at the wing root. Furthermore, the location of the elastic axis is directly affected by this design variable, and therefore, it has a significant effect on the flutter speed. Moreover, the third curve in Figure 11 and Figure 12 consists


of varying the values of DV6, DV7, DV8, DV17, and DV18 all together equally in a way that makes a straight spar. It can be seen clearly that moving the straight spar has a significant effect on both objectives and could make 30.4% increase in flutter speed with a weight compensation of 15%. In fact, the position of the straight spar has also a significant effect on the torsional rigidity and the elastic axis of the cantilever wing, and therefore on its flutter speed.

Figure 12: Variation of the flutter speed with the spars shape and location

Figure 11: Variation of the wing weight with the spars shape and location

Figure 13: Variation of the wing weight and flutter speed with the number of spars The variation of the number of spars is also another factor that can be used to increase the flutter speed and reduce the wing weight (Figure 13) due to its effect on the CG location and torsional rigidity. The number of spars in Figure 13 does not include the front and rear spars (Figure 10). However, the number of ribs has a small effect on weight and flutter

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speed since the added or removed ribs will be compensated by a change in the thicknesses of the panels (Figure 14 and Figure 15). 

Effect of Stiffeners

Another parameter that greatly affects our design, is the number of longitudinal stiffeners (Figure 17). For each number of stiffeners in Figure 16, the wing is optimized using the Global-Local framework and the wing weight (orange curve) and flutter speed (blue curve) are computed. It can be seen clearly that attaching more stiffeners to the structure reduces the final weight up to certain extent. For this case, the wing weight is decreasing between 1 and 16 stiffeners. This could be


explained by the fact that longitudinal stiffeners increase the buckling load of the panels and therefore reduces their optimal thickness. However, beyond 16 stiffeners, adding more stiffeners does not help anymore, since the stress constraint becomes the one that affects the optimizer’s decision. In other words, adding more stiffeners in this case does not reduce the optimal panel thickness. Moreover, the additional stiffeners weight causes a further increase in the total wing weight, as shown in right part of the curve. It can be also seen clearly in the figure that adding more longitudinal stiffeners reduces the flutter speed. This can be explained by the fact that longitudinal stiffeners increase the bending rigidity of the wing without affecting much the torsional rigidity. Hence, their ratio is reduced and the flutter speed decreases. Based on this behavior, it is therefore more convenient for this case to use 14 stiffeners.

Figure 14: Variation of the wing weight and flutter speed with the number of inner ribs

Figure 15: Variation of the wing weight and flutter speed with the number of outer ribs

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Figure 17: Longitudinal stiffeners on the CRM wing


Figure 16: Variation of the wing weight and flutter speed with the number of longitudinal stiffeners

In addition to the longitudinal stiffeners (which reduce the flutter speed), lateral stiffeners are added to the structure (Figure 19) in order to increase the torsional rigidity and increase therefore the flutter speed. It could be seen in Figure 18 that 4.64% increase in flutter speed can be obtained with a weight compensation of 11.28%. To sum up, this parametric study gives an idea about the effect of the different design variables on the wing under the Global-Local optimization process, and gives guidelines to designers on how to place the structure’s components in order to get the desired wing weight and flutter speed.

Figure 19: Lateral stiffeners on the CRM wing Figure 18: Variation of the wing weight and flutter speed with the number of lateral stiffeners

IV.

Particle Swarm Optimization

The internal structure of the wing is described by a set of size and shape design variables. Minimizing the weight and maximizing the flutter velocity of the aircraft is a multi-objective multi-disciplinary design optimization problem with contradictory objectives and with constraints on stress and buckling. Mathematically, a multi-objective problem can be stated as:

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min F(x) = [f1(x), f2(x), …. , fk(x)] Subjected to: gi(x) ≤ 0 i = 1, 2, …. , p hj(x) = 0 j = 1, 2, …., q

(3)

where x = [x1, x2, …. , xn]T is the vector comprising the n design variables with ranges: xil ≤ xi ≤ xiu i = 1,2,….n fk: ℝn → ℝ, k = 1, 2, …. , r are the objective functions, gi, hj : ℝn → ℝ, i = 1, 2, …. , p; j = 1, 2, …. , q are the constraint functions


However, in order to reduce the complexity of the problem, a weighted function evaluation that gathers both weight and flutter speed is considered to transform the multi-objective problem into a single-objective problem:

𝑓=

𝑊 𝑊0

+√

𝑉0 𝑉𝑓

(4)

Where 𝑊 is the wing weight (lbs), 𝑉𝑓 is the flutter speed (knots), 𝑊0 is the effective weight (lbs), and 𝑉0 is the effective flutter velocity (knots). The desired solution to the multi-objective function is a Pareto optimal set of x satisfying the ranges of each of the design variables [20]. The Particle Swarm Optimization will be employed into a double-level parallel computing framework. The upper level corresponds to the PSO particles, while the lower level corresponds to the local panel optimization during the Global-Local process (Figure 20). Additional details about the framework can be found in [11].

Figure 20: Double-Level parallel computing framework

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The values used in this problem for 𝑊0 and 𝑉0 are 𝑊0 = 10𝑙𝑏𝑠 and 𝑉0 = 158.11knots in order to let the wing weight and flutter speed have comparable effects on the weighted evaluation function and determine the level of importance of these quantities with respect to each other while taking decisions during the optimization. Figure 21 shows the isolines of the weighted evaluation function for different values of 𝑊0 and 𝑉0 . Our selected case corresponds to Figure 21-a). During PSO, minimizing the function 𝑓 will direct the optimizer towards the top-left


corner of the frame, which yields smaller weights (x-axis) and higher flutter speeds (y-axis).

(a) 𝑊0 =10 lbs, 𝑉0 =158.11 knots

(b) 𝑊0 =100 lbs, 𝑉0 =158.11 knots

(c) 𝑊0 =10 lbs, 𝑉0 =223.61 knots

Figure 21: Weighted evaluation function isolines: Flutter speed (y-axis) vs wing weight (x-axis) The 13 design variables considered in this Particle Swarm Optimization control the shape of spars as well as the number and shape of ribs in the inner and outer wing boxes. Their description and upper and lower bounds are shown in Table 5. The number of spars is fixed to 3 spars. The average spacing between two adjacent longitudinal stiffeners is 16 inches. The PSO is performed using 21 particles in each population. Double level parallel computing is employed using 21 processes at the PSO level and 24 processes for each particle. The optimization is performed using 7 nodes (24 cores and 128 Gb of RAM each) in the cluster “NewRiver” mentioned above. The analysis converged within 5 iterations in 38 hours where 83 designs were analyzed (Figure 23). The best obtained design (Table 5 and Figure 22) has 3 Spars, 6 inner ribs, and 30 outer ribs. The corresponding value of the objective function is 𝑓 = 2174.99. The wing weight is 𝑊 = 11152.61 𝑙𝑏𝑠 and the flutter velocity is 𝑉𝑓 = 556.53𝑘𝑛𝑜𝑡𝑠. Compared to the unstiffened baseline model used in [11], which has a weight of 19269 𝑙𝑏𝑠, an improvement of 42% has been, therefore, made. The wing’s first 8 vibration modes are shown in Figure 24 through Figure 31. The first wing bending mode and first wing torsion mode frequencies are 1.23Hz and 9.45Hz, respectively (Figure 24 and Figure 28). The flutter mode of this wing design is Mode 1 and the flutter frequency is 4.8 Hz. The optimized panels’ thicknesses are

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shown in Figure 32. The distribution of the thicknesses shows that the highest load is applied near the wing root where the thickness is the highest, while the thickness at the wing tip is the lowest.


Table 5: PSO design variables and optimal case Xmin

Xmax

Xopt

DV6

IW spars: Start point spacing

0.2

10

5.52

DV7

IW spars: Control point spacing

0.2

10

3.19

DV8

IW spars: End point spacing

0.2

10

5.75

DV9

IW number of ribs

3

20

6

DV12

IW ribs: Start point spacing

0.2

10

4.33

DV13

IW ribs: Control point spacing

0.2

10

0.69

DV14

IW ribs: End point spacing

0.2

10

4.13

DV17

OW spars: Control point spacing

0.2

10

3.16

DV18

OW spars: End point spacing

0.2

10

6.0

DV19

OW number or ribs

5

40

30

DV22

OW spars: Start point spacing

0.2

10

1.18

DV23

OW spars: Control point spacing

0.2

10

1.56

DV24

OW spars: End point spacing

0.2

10

1.5

Figure 23: PSO analyzed particles

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Figure 22: PSO best obtained design


Figure 24: Mode1 (1.23 Hz)

Figure 25: Mode 2 (3.93 Hz)







Figure 32: Panels thickness distribution of PSO best obtained design V.

Conclusion

This study is a step towards the development of the EBF3GLWingOpt software by Kapania et al. at Virginia Tech that exhibits multi-disciplinary multi-objective optimization capability as well as provision for creating Sparibs geometry for wings of different shapes. The creation of Sparibs using limited number of design variables in a wing with two sections along with the use of PSO to optimize weight and flutter velocity simultaneously was demonstrated.

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Parametric study that gives an idea about the effect of multiple design variables on the wing weight and flutter speed has been conducted in order to make a wise selection of the most influential variables in the particle swarm optimization and in order to give guidelines for designers on how to place the structure’s components in order to obtain the desired wing weight and flutter velocity. It has been shown that tailoring the shape of the middle spar could make an increase of 17.58% of flutter speed with a weigh compensation of 1.43% and its location could make 30.4% increase in flutter speed with a weight compensation of 15%. The effect of longitudinal and lateral stiffeners on the wing weight and flutter velocity was also investigated. While a trade-off was proven for the required number of


longitudinal stiffeners, lateral stiffeners showed an increase in both weight and flutter speed. Particle Swarm Optimization with double-level parallel computing process has been employed using a weighted evaluation function in order to convert the multi-objective optimization problem into single-objective problem. An optimal design was obtained, for which the weight is minimized, the flutter speed is maximized, and all the required constraints are satisfied. The best obtained design has a flutter speed of 556.53 knots and a wing weight of 11152.61 lbs which shows 42% improvement with respect to the weight of the unstiffened baseline model. VI.

Acknowledgments

This effort was funded by the NASA SBIR/STTR program, under contracts NNX14CD16P and NNX15CD08C and NASA NRA, “Lightweight Adaptive Aeroelastic Wing for Enhanced Performance across the Flight Envelope," NRA NNX14AL36A. The authors would like to also thank Dr. Chan-Gi Pak from NASA Armstrong Flight Research Center, Dr. Myles Baker from M4 Engineering Inc., and Dr. Robert Canfield from Virginia Tech for their fruitful discussions.

References [1] C. V. Jutte, B. K. Stanford and C. D. Wieseman, "Internal Structural Design of the Common, Research Model Wing Box for Aeroelastic, Tailoring," Hampton, Virginia: NASA, 2015. [2] L. A. Schmit, "Structural Design by Systematic Synthesis," Pittsburg, 1960. [3] L. A. Schmit and R. K. Ramanathan, "Multilevel Approach to Minimum Weight Design Including Buckling Constraints," AIAA Journal, vol. 16, no. 2, p. 97–104, 1978. [4] L. A. Schmit, "Structural Synthesis: Its Genesis and Development," AIAA Journal, vol. 19, no. 10, p. 1249– 1263, 1981.

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